# A note on the geometry and topology of almost even-Clifford Hermitian manifolds

A note on the geometry and topology of almost even-Clifford Hermitian manifolds Abstract We compute the structure groups of almost even-Clifford Hermitian manifolds and determine when such groups lead to Spin structures. 1. Introduction Almost even-Clifford Hermitian structures on oriented Riemannian manifolds were introduced recently in [10] under the simpler name of even Clifford structures, which are generalizations of almost Hermitian and almost quaternion-Hermitian structures. They are a subject of current interest [1, 3, 5, 9, 12], although similar types of structures have been studied in the past [4, 7, 11]. The existence of such a structure on a manifold implies the reduction of its structure group to the normalizer of the homomorphic image of a Spin group. In this paper, we identify such structure group (cf. Theorem 3.2) by using the results about their Lie algebras given in [2]. In the case of 4m-dimensional almost quaternion-Hermitian manifolds, we know that such manifolds are Spin when m is even. This is due to the (topological) reduction of the structure group from SO(4m) to the Lie group Sp(m)Sp(1) which, in turn, embeds into Spin(4m) when m is even. Thus, by analogy, we have been led to study when almost even-Clifford Hermitian manifolds admit Spin structures. Recall that an oriented N-dimensional Riemannian manifold is Spin if its orthonormal frame bundle PSO admits a double cover by a principal Spin(N) bundle PSpin  Λ:PSpin⟶PSO,which is Spin(N) equivariant, that is Λ(pg)=Λ(p)λN(g) for all g∈Spin(N). A Riemannian manifold will automatically be Spin if its structure group reduces to a proper subgroup G⊂SO(N) such that there exists a lifting map which makes the following diagram commute:   Spin(N)↗↓G↪SO(N).Indeed, such a lift exists if and only if π1(G) maps trivially into π1(SO(N)). In this paper, we determine when there exists a lifting map which makes the following diagram commute (cf. Theorem 4.1):   Spin(N)↓NSO(N)(S)↪SO(N),where N stands for the dimension of an almost even-Clifford Hermitian manifold, S denotes the homomorphic image of the aforementioned Spin group determined by the even-Clifford structure, and NSO(N)(S) denotes its normalizer in SO(N). In fact, we will verify that there is a lift for the connected component of the identity NSO(N)0(S). Furthermore, note that an almost even-Clifford Hermitian manifold might still be Spin even if there is no such lifting map, as in the case of quaternionic projective spaces HPm of odd quaternionic dimension m. It would be interesting, at least for the authors, to find and study non-Spin almost even-Clifford manifolds of ranks 4, 6 and 8. The note is organized as follows. In Section 2, we recall some preliminaries on Clifford algebras, the Spin group and representations, almost even-Clifford manifolds, etc. In Section 3, we determine the complexifications of real representations of even Clifford algebras containing no trivial summands (cf. Theorem 3.1), identify the subgroups NSO(N)0(S) as quotients of products of classical groups (or real lines in some cases) and Spin groups (cf. Theorem 3.2), and calculate their fundamental groups giving explicit generators (cf. Theorem 3.3). In Section 4, we determine when the aforementioned lifts exist (cf. Theorem 4.1). 2. Preliminaries 2.1. Clifford algebra, spin group and representation The material presented in this subsection can be consulted in [6]. Let Clr denote the 2r-dimensional real Clifford algebra generated by the orthonormal vectors e1,…,er∈Rr subject to the relations   eiej+ejei=−2δij,and Clr=Clr⊗RC its complexification. The even Clifford subalgebra Clr0 is defined as the invariant (+1)-subspace of the involution of Clr induced by the map −IdRr. For any vector Y=y1e1+⋯+yrer, the product   eiY ei=y1e1+⋯+yi−1ei−1−yiei+yi+1ei+1+⋯+yrergives the reflection of the ith coordinate, and the conjugation with the volume element volr=e1⋯er gives the reflection on the origin of Rr, that is   (e1⋯er)Y(er⋯e1)=−Y. There exist algebra isomorphisms   Clr≅{End(C2k)ifr=2k,End(C2k)⊕End(C2k)ifr=2k+1,and the space of (complex) spinors is defined to be   Δr≔C2k=C2⊗…⊗C2︸ktimes.The map   κ:Clr⟶End(C2k)is defined to be either the aforementioned isomorphism for r even, or the isomorphism followed by the projection onto the first summand for r odd. In order to make κ explicit, consider the following matrices:   Id=(1001),g1=(i00−i),g2=(0ii0),T=(0−ii0).In terms of the generators e1,…,er of the Clifford algebra, κ can be described explicitly as follows:   e1↦Id⊗Id⊗…⊗Id⊗Id⊗g1,e2↦Id⊗Id⊗…⊗Id⊗Id⊗g2,e3↦Id⊗Id⊗…⊗Id⊗g1⊗T,e4↦Id⊗Id⊗…⊗Id⊗g2⊗T,⋮…e2k−1↦g1⊗T⊗…⊗T⊗T⊗T,e2k↦g2⊗T⊗…⊗T⊗T⊗T,and, if r=2k+1,   e2k+1↦iT⊗T⊗…⊗T⊗T⊗T.The vectors   u+1=12(1,−i)andu−1=12(1,i),form a unitary basis of C2 with respect to the standard Hermitian product. Thus,   B={uε1,…,εk=uε1⊗…⊗uεk∣εj=±1,j=1,…,k},is a unitary basis of Δr=C2k with respect to the naturally induced Hermitian product. We will denote inner and Hermitian products (as well as Riemannian and Hermitian metrics) by the same symbol ⟨·,·⟩ trusting that the context will make clear which product is being used. A quaternionic structure α on C2 is given by   α(z1z2)=(−z¯2z¯1),and a real structure β on C2 is given by   β(z1z2)=(z¯1z¯2).Following [6, p. 31], the real and quaternionic structures γr on Δr=(C2)⊗[r/2] are built as follows:   γr=(α⊗β)⊗2kifr=8k,8k+1(real),γr=α⊗(β⊗α)⊗2kifr=8k+2,8k+3(quaternionic),γr=(α⊗β)⊗2k+1ifr=8k+4,8k+5(quaternionic),γr=α⊗(β⊗α)⊗2k+1ifr=8k+6,8k+7(real). The Spin group Spin(r)⊂Clr is the subset   Spin(r)={x1x2⋯x2l−1x2l∣xj∈Rr,∣xj∣=1,l∈N},endowed with the product of the Clifford algebra. It is a Lie group and its Lie algebra is   spin(r)=span{eiej∣1≤i<j≤r}.The restriction of κ to Spin(r) defines the Lie group representation   κr≔κ∣Spin(r):Spin(r)⟶GL(Δr),which is, in fact, special unitary. We have the corresponding Lie algebra representation   κr*:spin(r)⟶gl(Δr).Recall that the Spin group Spin(r) is the universal double cover of SO(r), r≥3. For r=2, we consider Spin(2) to be the connected double cover of SO(2). The covering map will be denoted by   λr:Spin(r)→SO(r)⊂GL(Rr).Its differential is given by λr*(eiej)=2Eij, where Eij=ei*⊗ej−ej*⊗ei is the standard basis of the skew-symmetric matrices, and e* denotes the metric dual of the vector e. Furthermore, we will abuse the notation and also denote by λr the induced representation on the exterior algebra ⋀*Rr. By means of κ, we have the Clifford multiplication   μr:Rr⊗Δr⟶Δrx⊗ϕ↦μr(x⊗ϕ)=x·ϕ≔κ(x)(ϕ).The Clifford multiplication μr is skew-symmetric with respect to the Hermitian product   ⟨x·ϕ1,ϕ2⟩=⟨μr(x⊗ϕ1),ϕ2⟩=−⟨ϕ1,μr(x⊗ϕ2)⟩=−⟨ϕ1,x·ϕ2⟩,is Spin(r)-equivariant and can be extended to a Spin(r)-equivariant map   μr:⋀*(Rr)⊗Δr⟶Δrω⊗ψ↦ω·ψ. When r is even, we define the following involution:   Δr⟶Δrψ↦(−i)r2volr·ψ.The ±1 eigenspace of this involution is denoted Δr±. These spaces have equal dimension and are irreducible representations of Spin(r). Note that our definition differs from the one given in [6] by a (−1)r2. The reason for this difference is that we want the spinor u1,…,1 to be always positive. In this case, we will denote the two representations by   κr±:Spin(r)⟶GL(Δr±).Note that while these representations are irreducible, they are not faithful, with kernels isomorphic to Z2 if r≠4. Now, we summarize some results about real representations of Clr0 in the next Table 1 (cf. [8]). Here dr denotes the dimension of an irreducible representation of Clr0 and vr the number of distinct non-trivial irreducible representations. Let Δ˜r denote the irreducible representation of Clr0 for r≢0(mod4) and Δ˜r± denote the irreducible representations for r≡0(mod4). Table 1. Real Clr0-representations. r(mod8)  dr  Clr0  Δ˜r/Δ˜r±≅Rdr  vr  1  2⌊r2⌋  R(dr)  Rdr  1  2  2r2  C(dr/2)  Cdr/2  1  3  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  4  2r2  H(dr/4)⊕H(dr/4)  Hdr/4  2  5  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  6  2r2  C(dr/2)  Cdr/2  1  7  2⌊r2⌋  R(dr)  Rdr  1  8  2r2−1  R(dr)⊕R(dr)  Rdr  2  r(mod8)  dr  Clr0  Δ˜r/Δ˜r±≅Rdr  vr  1  2⌊r2⌋  R(dr)  Rdr  1  2  2r2  C(dr/2)  Cdr/2  1  3  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  4  2r2  H(dr/4)⊕H(dr/4)  Hdr/4  2  5  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  6  2r2  C(dr/2)  Cdr/2  1  7  2⌊r2⌋  R(dr)  Rdr  1  8  2r2−1  R(dr)⊕R(dr)  Rdr  2  View Large 2.2. Maximal torus of Spin(r) In this subsection, we recall explicit expressions for elements of the maximal torus of the Spin group since it will be useful to consider paths within such torus. The rotation   (cos(θ1)−sin(θ1)sin(θ1)cos(θ1)1⋱1)r×rcan be achieved by using the element   e1(−cos(θ1/2)e1+sin(θ1/2)e2)=cos(θ1/2)+sin(θ1/2)e1e2∈Spin(r)as follows:   (cos(θ1/2)+sin(θ1/2)e1e2)y(cos(θ1/2)−sin(θ1/2)e1e2)=(y1cos(θ1)−y2sin(θ1))e1+(y1sin(θ1)+y2cos(θ1))e2+y3e3+⋯+yrer,for y=y1e1+⋯+yrer∈Rr. Thus, we see that the corresponding preimages in Spin(r) are exactly   ±(cos(θ1/2)+sin(θ1/2)e1e2).Furthermore, we can see that a maximal torus of Spin(r) consists of elements of the form   t(θ1,…,θ[r2])=∏j=1[r2](cos(θj/2)+sin(θj/2)e2j−1e2j),noting that the parameters θj must now run between 0 and 4π. Furthermore, using the explicit description of the isomorphisms given above, we see that   (cos(θ1/2)+sin(θ1/2)e1e2)·uε1,…,εk=cos(θ1/2)uε1,…,εk+sin(θ1/2)e1e2·uε1,…,εk=cos(θ1/2)uε1,…,εk+iεksin(θ1/2)uε1,…,εk=(cos(θ1/2)+iεksin(θ1/2))uε1,…,εk=eiεkθ12uε1,…,εk,and similarly,   t(θ1,…,θ[r2])·uε1,…,ε[r2]=ei2∑j=1[r2]εk+1−jθj·uε1,…,εk.Thus, the basis vectors uε1,…,εk are weight vectors of the standard Spin representation with weights   12∑j=1[r2]ε[r2]+1−jθj,which in coordinate vectors are the well-known expressions   (±12,±12,…,±12).Moreover, in terms of the (appropriately ordered) basis B, the matrix associated to an element t(θ1,…,θ[r2]) is   (ei2(θ1+θ2+⋯+θ[r2])ei2(−θ1+θ2+⋯+θ[r2])ei2(θ1−θ2+⋯+θ[r2])⋱ei2(−θ1−θ2+⋯+θ[r2])⋱ei2(−θ1−θ2−⋯−θ[r2])).Note that, when r is even, Δr+ is generated by the basis vectors uε1,…,εr2 with an even number of εj equal to −1, and Δr− is generated by the basis vectors uε1,…,εr2 with an odd number of εj equal to −1. Therefore, after reordering the basis, the matrix above can be split into two blocks of equal size: one block in which the exponents contain an even number of negative signs   (ei2(θ1+θ2+⋯+θr2)ei2(−θ1−θ2+⋯+θr2)ei2(−θ1+θ2−⋯+θr2)⋱)and another block in which the exponents contain an odd number of negative signs   (ei2(−θ1+θ2+⋯+θr2)ei2(θ1−θ2+⋯+θr2)ei2(θ1+θ2−θ3+⋯+θr2)⋱). 2.3. Even-Clifford structures Linear even-Clifford Hermitian structures Definition 2.1 Let N∈N and (e1,…,er) an orthonormal frame of Rr. A linear even-Clifford structure of rank r on RN is an algebra representation   Φ:Clr0⟶End(RN). A linear even-Clifford Hermitian structure of rank r on RN (endowed with a positive definite inner product) is a linear even-Clifford structure of rank r such that each bivector eiej, 1≤i<j≤r, is mapped to a skew-symmetric endomorphism Φ(eiej)=Jij. Remarks Note that   Jij2=−IdRN. (2.1) Given a linear even-Clifford structure of rank r on RN, we can average the standard inner product ⟨,⟩ on RN as follows:   (X,Y)=∑k=1[r/2][∑1≤i1<⋯<i2k<r⟨Φ(ei1…i2k)(X),Φ(ei1…i2k)(Y)⟩],where (e1,…,er) is an orthonormal frame of Rr, so that the linear even-Clifford structure is Hermitian with respect to the averaged inner product. Given a linear even-Clifford Hermitian structure of rank r, the subalgebra spin(r) is mapped injectively into the skew-symmetric endomorphisms End−(RN). Branching of RN From now on, we will denote by Idn the identity endomorphism of a real/complex n-dimensional vector space. First, let us assume r≢0(mod4), r>1. In this case, RN decomposes into a sum of irreducible representations of Clr0. Since Clr0 is simple, its irreducible representations are either trivial or the standard representation Δ˜r of Clr0 (cf. [8]). Due to (2.1), there are no trivial summands RN, that is   RN=Rm⊗Δ˜rfor some m∈N. Thus, we see that spin(r) has an isomorphic image   spin(r)^≔Idm⊗κr*(spin(r))⊂so(drm). Secondly, let us assume r≡0(mod4). Recall that if Δˆr is the irreducible representation of Clr, then by restricting this representation to Clr0 it splits as the sum of two inequivalent irreducible representations   Δˆr=Δ˜r+⊕Δ˜r−.Since RN is a representation of Clr0 satisfying (2.1), there are no trivial summands in RN and   RN=Rm1⊗Δ˜r+⊕Rm2⊗Δ˜r−for some m1,m2∈N. By restricting this representation to spin(r)⊂Clr0, consider the isomorphic image   spin(r)^≔{Idm1⊗ξ+⊕Idm2⊗ξ−∣ξ∈spin(r),ξ+=κr*+(ξ),ξ−=κr*−(ξ)}. Almost even-Clifford Hermitian manifolds Definition 2.2 Let r≥2. A rank r almost even-Clifford structure on a smooth manifold M is a smoothly varying choice of a rank r linear even-Clifford structure on each tangent space of M. A smooth manifold carrying an almost even-Clifford structure will be called an almost even-Clifford manifold. A rank r almost even-Clifford Hermitian structure on a Riemannian manifold M is a smoothly varying choice of a linear even-Clifford Hermitian structure on each tangent space of M. A Riemannian manifold carrying an almost even-Clifford Hermitian structure will be called an almost even-Clifford Hermitian manifold. Remark Our definition of almost even-Clifford Hermitian structure implies that in [10]. 3. Complexifications, structure groups and fundamental groups In this section, we will study the complexification RN⊗C and its decomposition as a representation of the normalizers NSO(N)0(S) (cf. 3.1), where S is one of the homomorphic images of the group Spin(r) corresponding to the Lie subalgebras described in Section 2.3.2. Once we have described such complexifications and decompositions, we will compute the (connected components of the identity of the) structure groups determined by linear even-Clifford structures. They must be closed Lie subgroups of SO(N) whose Lie algebra is the aforementioned normalizer. They are actually isomorphic to quotients of products of classical compact Lie groups (or real lines) with Spin(r) (cf. 3.2). We will also compute their fundamental groups to be used in the last section. All of this will be done in a case by case analysis. Along the way, we will introduce notation that will enable us to state Theorems 3.1, 3.2 and 3.3. The main steps in each case are the following: Complexification: Identify RN⊗C as a representation   G×Spin(r)→ρSO(N)⊂Aut(RN×C),where G denotes a (semi-)simple compact Lie group or SO (2). Structure group: Compute the image of ρ, Im(ρ). Compute ker(ρ) to get   Im(ρ)≅G×Spin(r)ker(ρ). Fundamental group: Identify the universal covering Im(ρ)˜→ρ˜Im(ρ). Compute the preimage of ρ˜−1(IdN) to obtain an explicit description of the fundamental group π1(Im(ρ)). First, let us recall the following Table 2 [2], whose entries’ precise description will be recalled in each case. Table 2. Centralizers and Normalizers of spin(r) and spin±(r) in so(N). r(mod8)  N  Cso(N)(spin(r)^)  Nso(N)0(spin(r)^)  0  dr(m1+m2)  so(m1)^⊕so(m2)^  so(m1)^⊕so(m2)^⊕spin(r)^  1,7  drm  so(m)^  so(m)^⊕spin(r)^  2,6  drm  u(m)^  u(m)^⊕spin(r)^  3,5  drm  sp(m)^  sp(m)^⊕spin(r)^  4  dr(m1+m2)  sp(m1)^⊕sp(m2)^  sp(m1)^⊕sp(m2)^⊕spin(r)^  r(mod8)  N  Cso(N)(spin(r)^)  Nso(N)0(spin(r)^)  0  dr(m1+m2)  so(m1)^⊕so(m2)^  so(m1)^⊕so(m2)^⊕spin(r)^  1,7  drm  so(m)^  so(m)^⊕spin(r)^  2,6  drm  u(m)^  u(m)^⊕spin(r)^  3,5  drm  sp(m)^  sp(m)^⊕spin(r)^  4  dr(m1+m2)  sp(m1)^⊕sp(m2)^  sp(m1)^⊕sp(m2)^⊕spin(r)^  View Large 3.1. r≡1,7(mod8) Complexification. In this case, Δ˜r is the subspace of Δr fixed by the corresponding real structure γr, that is   Δ˜r⊗C=Δr.The centralizer subalgebra of spin(r)^ in so(N) is   Cso(N)(spin(r)^)=so(m)⊗IdΔ˜r≕so(m)^,where N=drm. If z∈C,v⊗ψ∈Rm⊗Δ˜r and A∈so(m),   (A⊗IdΔ˜r)(zv⊗ψ)=zAv⊗ψ,which means   (Rm⊗Δ˜r)⊗C=Cm⊗Δr,where Cm denotes the standard complex representation of SO (m). Thus, we have a representation   SO(m)×Spin(r)→ρSO(N)⊂Aut(Cm⊗Δr). Structure group. Since so(m)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). The exponential of A⊗IdΔ˜r∈so(m)^ gives   eA⊗IdΔ˜r∈SO(m)^≔SO(m)⊗IdΔ˜r≅SO(m).On the other hand, if Idm⊗ξ∈spin(r)^, its exponential is   Idm⊗eξ∈Idm⊗κ(Spin(r))≕Spin(r)^≅Spin(r),since Spin(r) is represented faithfully on Δr. The image of SO(m)×Spin(r) in SO(N) under the aforementioned representation is   NSO(N)0(Spin(r)^)=SO(m)^Spin(r)^,the subgroup of all possible products of elements of the two subgroups, that is we have   SO(m)×Spin(r)→ρSO(m)^Spin(r)^⊂SO(N).Now we need to find ker(ρ) and identify SO(m)^Spin(r)^ as a quotient   SO(m)^Spin(r)^≅SO(m)×Spin(r)ker(ρ).If there are elements g∈SO(m) and h∈Spin(r) such that   ρ(g,h)=IdN,then   Spin(r)^∋ρ(Idm,h)=ρ(g,1)−1∈SO(m)^.Since ρ(Idm,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))=Z2={±1}.Note that −1∈Spin(r) maps to −IdΔr under the Spin(r) representation on Δr, and (Idm,−1) maps to −Idm⊗IdΔr∈SO(N) under ρ. Moreover, −Idm⊗IdΔr belongs to SO(m)^ only if m is even. Thus, ker(ρ)={±(Idm,1)}=Z2 and   SO(m)^Spin(r)^≅SO(m)×Spin(r)Z2if m is even, and ker(ρ)={(Idm,1)},   SO(m)^Spin(r)^≅SO(m)×Spin(r)if m is odd. Fundamental group. Clearly, we only need to deal with the case when m is even. If m≥4, let ρ˜ denote the following composition:   Spin(m)×Spin(r)↓SO(m)×Spin(r)↓SO(m)×Z2Spin(r).We need to find all the elements of Spin(m)×Spin(r) that map to   ±(Idm,1).The elements of Spin(m)×Spin(r) that map to (Idm,1)∈SO(m)×Spin(r) are   (±1,1),and the elements of Spin(m)×Spin(r) that map to (−Idm,−1)∈SO(m)×Spin(r) are   (±volm,−1),that is   ker(ρ˜)={(1,1),(−1,1),(volm,−1),(−volm,−1)}.π1(SO(m)^Spin(r)^)≅{Z2⊕Z2ifm≡0(mod4),Z4ifm≡2(mod4). If m = 2, let ρ˜ denote the following composition:   R×Spin(r)↓SO(2)×Spin(r)↓SO(2)×Z2Spin(r).Similarly,   ker(ρ˜)={(2kπ,1)∣k∈Z}∪{((2k+1)π,−1)∣k∈Z},π1(SO(2)^Spin(r)^)≅Z. 3.2. r≡0(mod8) Complexification. In this case, Δ˜r+ and Δ˜r− are the subspaces of Δr+ and Δr− fixed by the corresponding real structure γr mentioned in Section 2, and   Δ˜r+⊗C=Δr+,Δ˜r−⊗C=Δr−.The centralizer subalgebra of spin(r)^ is   Cso(N)(spin(r)^)=so(m1)^⊕so(m2)^=(so(m1)^so(m2)^)≔(so(m1)⊗IdΔ˜r+so(m2)⊗IdΔ˜r−),where N=dr(m1+m2). If A1∈so(m1), A2∈so(m2), z1,z2∈C, v1⊗ψ1+v2⊗ψ2∈Rm1⊗Δ˜r+⊕Rm1⊗Δ˜r−,  (A1⊗IdΔ˜r+A2⊗IdΔ˜r−)(z1v1⊗ψ1z2v2⊗ψ2)=(z1A1v1⊗ψ1z2A2v2⊗ψ2),which means   (Rm1⊗Δ˜r+⊕Rm2⊗Δ˜r−)⊗C=Cm1⊗Δr+⊕Cm2⊗Δr−,where Cm1 and Cm2 denote the standard complex representation of so(m1) and so(m2), respectively. Thus, we have a representation   SO(m1)×SO(m2)×Spin(r)⟶SO(N)⊂Aut(Cm1⊗Δr+⊕Cm2⊗Δr−). Structure group Since so(m1)^⊕so(m2)^ and spin(r)^ commute, we can take the exponentials of their elements separately within C(N). The exponential of   (A1⊗IdΔr+A2⊗IdΔr−)∈so(m1)^⊕so(m2)^ is   (eA1⊗IdΔr+eA2⊗IdΔr−)∈(SO(m1)^SO(m2)^)≔(SO(m1)⊗IdΔr+SO(m2)⊗IdΔr−),=SO(m1)^×SO(m2)^. On the other hand, if Idm1⊗ξ+⊕Idm2⊗ξ−∈spin(r)^, its exponential is   (Idm1⊗eξ+Idm2⊗eξ−)∈(Idm1⊗κ+(Spin(r))Idm2⊗κ−(Spin(r)))={Spin(r)^≅Spin(r)ifm1>0andm2>0,Spin(r)^+≅κ+(Spin(r))ifm1>0andm2=0,Spin(r)^−≅κ−(Spin(r))ifm1=0andm2>0, where the first representation of Spin(r) is faithful and the last two are not, with   Spin(r)^±≅κ±(Spin(r))≅Spin(r){1,±volr}. The image of SO(m1)×SO(m2)×Spin(r) in SO(N) under the aforementioned representations are   NSO(N)0(Spin(r)^)=(SO(m1)^×SO(m2)^)Spin(r)^,NSO(N)0(Spin(r)^+)=SO(m1)^Spin(r)^+,NSO(N)0(Spin(r)^−)=SO(m2)^Spin(r)^−, respectively, that is in each case we have a map   SO(m1)×SO(m2)×Spin(r)→ρ(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N),SO(m1)×Spin(r)→ρSO(m1)^Spin(r)^−⊂SO(N),SO(m2)×Spin(r)→ρSO(m2)^Spin(r)^+⊂SO(N). Now we need to find ker(ρ) in each case to identify the relevant group as a quotient. Case m1,m2>0. If there are elements gi∈SO(mi) and h∈Spin(r) such that   ρ(g1,g2,h)=IdN,then   Spin(r)^∋ρ(Idm1,Idm2,h)=ρ(g1,g2,1)−1∈SO(m1)^×SO(m2)^.Since ρ(Idm1,Idm2,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))={1,−1,volr,−volr}≅Z2⊕Z2. Note that the element −1 is mapped to −IdΔr± in the Spin(r) representations on Δr±, and (Idm1,Idm2,−1) maps to −(Idm1⊗IdΔr+⊕Idm2⊗IdΔr−)∈SO(N); the element −(Idm1⊗IdΔr+⊕Idm2⊗IdΔr−) belongs to SO(m1)^×SO(m2)^ if m1≡m2≡0(mod2); the element volr is mapped to ±IdΔr±, and (Idm1,Idm2,volr) maps to (Idm1⊗IdΔr+⊕(−1)Idm2⊗IdΔr−)∈SO(N); the element (Idm1⊗IdΔr+⊕(−1)Idm2⊗IdΔr−) belongs to SO(m1)^×SO(m2)^ if m2≡0(mod2); the element −volr is mapped to ∓IdΔr±, and (Idm1,Idm2,−volr) maps to ((−1)Idm1⊗IdΔr+⊕Idm2⊗IdΔr−)∈SO(N);the element ((−1)Idm1⊗IdΔr+⊕Idm2⊗IdΔr−) belongs to SO(m1)^×SO(m2)^ if m1≡0(mod2). Thus, if m1≡m2≡1(mod2),   ker(ρ)={(Idm1,Idm2,1)},(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r); if m1≡0(mod2), m2≡1(mod2),   ker(ρ)={(Idm1,Idm2,1),(−Idm1,Idm2,−volr)}≅Z2,(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r)Z2; if m1≡1(mod2), m2≡0(mod2),   ker(ρ)={(Idm1,Idm2,1),(Idm1,−Idm2,vol)}≅Z2,(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r)Z2; if m1≡m2≡0(mod2),   ker(ρ)={(Idm1,Idm2,1),(−Idm1,−Idm2,−1),(Idm1,−Idm2,vol),(−Idm1,Idm2,−vol)}≅Z2⊕Z2,(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r)Z2⊕Z2. Case m1>0, m2=0. If there are elements g1∈SO(m1) and h∈Spin(r) such that   ρ(g1,h)=IdN, then   Spin(r)^+∋ρ(Idm1,h)=ρ(g1,1)−1∈SO(m1)^. Since ρ(Idm1,h) commutes with every element of Spin(r)^+, it belongs to its center Z(Spin(r)^+)≅Z(Spin(r)/{1,volr})={1,−1,volr,−volr}/{1,volr}≅Z2.Note that the element −1 is mapped to −IdΔr+ in the Spin(r) representation on Δr+, and (Idm1,−1) maps to −Idm1⊗IdΔr+∈SO(N); the element −Idm1⊗IdΔr+ belongs to SO(m1)^ if m1≡0(mod2). Thus, (5)if m1≡1(mod2),   ker(ρ)={(Idm1,1),(Idm1,volr)},SO(m1)^Spin(r)^+≅SO(m1)×Spin(r)Z2; (6)if m1≡0(mod2),   ker(ρ)={(Idm1,1),(−Idm1,−1),(Idm1,volr),(−Idm1,−volr)},SO(m1)^Spin(r)^+≅SO(m1)×Spin(r)Z2⊕Z2. Case m1=0, m2>0. If there are elements g2∈SO(m2) and h∈Spin(r) such that   ρ(g2,h)=IdN,then   Spin(r)^−∋ρ(Idm2,h)=ρ(g2,1)−1∈SO(m2)^.Since ρ(Idm2,h) commutes with every element of Spin(r)^−, it belongs to its center Z(Spin(r)^−)≅Z(Spin(r)/{1,−volr})={1,−1,volr,−volr}/{1,−volr}≅Z2. Note that the element −1 is mapped to −IdΔr− in the Spin(r) representation on Δr−, and (Idm2,−1) maps to −Idm2⊗IdΔr−∈SO(N); the element −Idm2⊗IdΔr− belongs to SO(m2)^ if m2≡0(mod2); Thus, (7)if m2≡1(mod2),   ker(ρ)={(Idm2,1),(Idm2,−volr)},SO(m2)^Spin(r)^−≅SO(m2)×Spin(r)Z2; (8)if m2≡0(mod2),   ker(ρ)={(Idm2,1),(−Idm2,−1),(−Idm2,volr),(Idm2,−volr)},SO(m2)^Spin(r)^−≅SO(m2)×Spin(r)Z2⊕Z2. Fundamental group. We will now analyze each of the previous eight cases: Recall that m1,m2>0 and m1≡m2≡1 (mod 2). If m1,m2≥3, let ρ˜ denote the following map:   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r).Thus,   ker(ρ˜)={(1,1,1),(−1,1,1),(1,−1,1),(−1,−1,1)},π1((SO(m1)^×SO(m2)^)Spin(r)^)≅Z2⊕Z2. If m1=1,m2≥3, SO(m1)={Id1} and let ρ˜ denote the following map:   {Id1}×Spin(m2)×Spin(r)↓{Id1}×SO(m2)×Spin(r).Thus   ker(ρ˜)={(Id1,1,1),(Id1,−1,1)},π1((SO(1)^×SO(m2)^)Spin(r)^)≅Z2. If m1≥3,m2=1, let ρ˜ denote the following map:   Spin(m1)×{Id1}×Spin(r)↓SO(m1)×{Id1}×Spin(r).Thus,   ker(ρ˜)={(1,Id1,1),(−1,Id1,1)},π1((SO(m1)^×SO(1)^)Spin(r)^)≅Z2. If m1=1,m2=1, let ρ˜ be   {Id1}×{Id1}×Spin(r)↓{Id1}×{Id1}×Spin(r).Thus,   ker(ρ˜)={(Id1,Id1,1)},π1((SO(1)^×SO(1)^)Spin(r)^)≅{1}. Recall that m1,m2>0, m1≡0(mod2), m2≡1(mod2). If m1≥4,m2≥3, let ρ˜ denote the composition   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(1,−1,1),(−1,1,1),(volm1,1,−volr)⟩ifm1≡0(mod4),⟨(1,−1,1),(volm1,1,−volr)⟩ifm1≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)≅{Z2⊕Z2⊕Z2ifm1≡0(mod4),Z2⊕Z4ifm1≡2(mod4). If m1=2,m2≥3, SO(m1)=SO(2) and let ρ˜ denote the following composition:   R×Spin(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(0,−1,1),(π,1,−volr)⟩,π1((SO(2)^×SO(m2)^)Spin(r)^)=Z2⊕Z. If m1≥4,m2=1, let ρ˜ denote the composition   Spin(m1)×{Id1}×Spin(r)↓SO(m1)×{Id1}×Spin(r)↓SO(m1)×{Id1}×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(−1,Id1,1),(volm1,Id1,−volr)⟩ifm1≡0(mod4),⟨(volm1,Id1,−volr)⟩ifm1≡2(mod4),π1((SO(m1)^×SO(1)^)Spin(r)^)={Z2⊕Z2ifm1≡0(mod4),Z4ifm1≡2(mod4). If m1=2,m2=1, let ρ˜ denote the composition   R×{Id1}×Spin(r)↓SO(2)×{Id1}×Spin(r)↓SO(2)×{Id1}×Spin(r)Z2.Thus   ker(ρ˜)=⟨(π,Id1,−volr)⟩,π1((SO(2)^×SO(1)^)Spin(r)^)≅Z. Recall that m1,m2>0, m1≡1(mod2), m2≡0(mod2). If m1≥3,m2≥4, let ρ˜ denote the composition   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(−1,1,1),(1,−1,1),(1,volm2,volr)⟩ifm2≡0(mod4),⟨(−1,1,1),(1,volm2,volr)⟩ifm2≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)={Z2⊕Z2⊕Z2ifm2≡0(mod4),Z2⊕Z4ifm2≡2(mod4). If m1≥3,m2=2, let ρ˜ denote the composition   Spin(m1)×R×Spin(r)↓SO(m1)×SO(2)×Spin(r)↓SO(m1)×SO(2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(−1,0,1),(1,π,volr)⟩,π1((SO(m1)^×SO(2)^)Spin(r)^)≅Z2⊕Z. If m1=1,m2≥4, let ρ˜ denote the composition   {Id1}×Spin(m2)×Spin(r)↓{Id1}×SO(m2)×Spin(r)↓{Id1}×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(Id1,−1,1),(Id1,volm2,volr)⟩ifm2≡0(mod4),⟨(Id1,volm2,volr)⟩ifm2≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)={Z2⊕Z2ifm2≡0(mod4),Z4ifm2≡2(mod4). If m1=1,m2=2, let ρ˜ denote the composition   {Id1}×R×Spin(r)↓{Id1}×SO(2)×Spin(r)↓{Id1}×SO(2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(Id1,π,volr)⟩,π1((SO(1)^×SO(2)^)Spin(r)^)≅Z. Recall that m1,m2>0, m1≡m2≡0(mod2). If m1,m2≥4, let ρ˜ denote the composition   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,1,1),(1,−1,1),(1,volm2,volr),(volm1,1,−volr)⟩ifm1≡m2≡0(mod4),⟨(−1,1,1),(volm1,1,−volr),(1,volm2,volr)⟩ifm1≡0(mod4)andm2≡2(mod4),⟨(1,−1,1),(volm1,1,−volr),(1,volm2,volr)⟩ifm2≡2(mod4)andm2≡0(mod4),⟨(volm1,1,−volr),(1,volm2,volr)⟩ifm1≡m2≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)≅{Z2⊕Z2⊕Z2⊕Z2ifm1≡m2≡0(mod4),Z2⊕Z2⊕Z4ifm1≡0(mod4)andm2≡2(mod4),Z2⊕Z4⊕Z2ifm2≡2(mod4)andm2≡0(mod4),Z4⊕Z4ifm1≡m2≡2(mod4). If m1≥4,m2=2, let ρ˜ denote the composition   Spin(m1)×R×Spin(r)↓SO(m1)×SO(2)×Spin(r)↓SO(m1)×SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,0,1),(volm1,0,−volr),(1,π,volr)⟩ifm1≡0(mod4),⟨(volm1,0,−volr),(1,π,volr)⟩ifm1≡2(mod4),π1((SO(m1)^×SO(2)^)Spin(r)^)≅{Z2⊕Z2⊕Zifm1≡0(mod4),Z4⊕Zifm1≡2(mod4). If m1=2,m2≥4, let ρ˜ denote the composition   R×Spin(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(0,−1,1),(0,volm2,volr),(π,1,−volr)⟩ifm2≡0(mod4),⟨(0,volm2,volr),(π,1,−volr)⟩ifm2≡2(mod4),π1((SO(2)^×SO(m2)^)Spin(r)^)≅{Z2⊕Z2⊕Zifm2≡0(mod4),Z4⊕Zifm2≡2(mod4). If m1=m2=2, let ρ˜ denote the composition   R×R×Spin(r)↓SO(2)×SO(2)×Spin(r)↓SO(2)×SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)=⟨(0,π,volr),(π,0,−volr)⟩,π1((SO(2)^×SO(2)^)Spin(r)^)≅Z⊕Z. Recall that m1>0,m2=0, m1≡1(mod2). If m1≥3, let ρ˜ denote the composition   Spin(m1)×Spin(r)↓SO(m1)×Spin(r)↓SO(m1)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(−1,1),(1,volr)⟩,π1(SO(m1)^Spin(r)^+)≅Z2⊕Z2. If m1=1, let ρ˜ denote   {Id1}×Spin(r)↓{Id1}×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(Id1,volr)⟩,π1(SO(1)^Spin(r)^+)≅Z2. Recall that m1>0,m2=0, m1≡0(mod2). If m1≥4, let ρ˜ denote the composition   Spin(m1)×Spin(r)↓SO(m1)×Spin(r)↓SO(m1)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,1),(volm1,−1),(1,volr)⟩ifm1≡0(mod4),⟨(volm1,−1),(1,volr)⟩ifm1≡2(mod4),π1(SO(m1)^Spin(r)^+)≅{Z2⊕Z2⊕Z2ifm1≡0(mod4),Z2⊕Z4ifm1≡2(mod4). If m1=2, let ρ˜ denote the composition   R×Spin(r)↓SO(2)×Spin(r)↓SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)=⟨(π,−1),(0,volr)⟩,π1(SO(2)^Spin(r)^+)≅Z⊕Z2. Recall that m1=0,m2>0, m2≡1 (mod 2). If m2≥3, let ρ˜ denote the composition   Spin(m2)×Spin(r)↓SO(m2)×Spin(r)↓SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(1,−volr)⟩,π1(SO(1)^Spin(r)^−)≅Z2. Recall that m1=0,m2>0, m2≡0 (mod 2). If m2≥4, let ρ˜ denote the composition   Spin(m2)×Spin(r)↓SO(m2)×Spin(r)↓SO(m2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,1),(volm2,−1),(1,−volr)⟩ifm2≡0(mod4),⟨(volm2,−1),(1,−volr)⟩ifm2≡2(mod4),π1(SO(m2)^Spin(r)^−)≅{Z2⊕Z2⊕Z2ifm2≡0(mod4),Z2⊕Z4ifm2≡2(mod4). If m2=2, let ρ˜ denote the composition   R×Spin(r)↓SO(2)×Spin(r)↓SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)=⟨(π,−1),(0,−volr)⟩,π1(SO(2)^Spin(r)^−)≅Z⊕Z2. 3.3. r≡2,6(mod8) Complexification. The volume form volr=e1…er acts as a complex structure J on Δ˜r (cf. [2]). Therefore,   Δ˜r⊗C={ψ−iJψ∣ψ∈Δ˜r}⊕{ψ+iJψ∣ψ∈Δ˜r}.Note that the involution (−i)r/2volr· acts on Δ˜r⊗C as follows: if r≡2(mod8),   (−i)r/2volr·(ψ−iJψ)=(ψ−iJψ),(−i)r/2volr·(ψ+iJψ)=−(ψ+iJψ),that is   Δ˜r⊗C={ψ−iJψ∣ψ∈Δ˜r}⊕{ψ+iJψ∣ψ∈Δ˜r}=Δr+⊕Δr−, if r≡6(mod8),   (−i)r/2volr·(ψ−iJψ)=−(ψ−iJψ),(−i)r/2volr·(ψ+iJψ)=(ψ+iJψ),that is   Δ˜r⊗C={ψ−iJψ∣ψ∈Δ˜r}⊕{ψ+iJψ∣ψ∈Δ˜r}=Δr−⊕Δr+.In any case,   Δ˜r⊗C=Δr.The centralizer subalgebra of spin(r)^ in so(N) is   Cso(N)(spin(r)^)=u(m)^=so(m)⊗IdΔ˜r⊕S2Rm⊗J,where N=drm, so(m) and S2Rm act on Rm as skew-symmetric and symmetric endomorphisms, respectively. Let   v⊗(ψ−iJψ)∈(Rm⊗Δ˜r)⊗Cand   A⊗IdΔ˜r+B⊗J∈so(m)⊗IdΔ˜r⊕S2Rm⊗J.Now,   (A⊗IdΔ˜r+B⊗J)(v⊗(ψ−iJψ))=Av⊗ψ−iAv⊗Jψ+Bv⊗Jψ−iBv⊗JJψ=Av⊗ψ+iBv⊗ψ+(−i)(i)Bv⊗Jψ−iAv⊗Jψ=(A+iB)v⊗(ψ−iJψ),where A+iB∈u(m). Similarly, for v⊗(ψ+iJψ),   (A⊗IdΔ˜r+B⊗J)(v⊗(ψ+iJψ))=(A−iB)v⊗(ψ+iJψ).Thus,   (Rm⊗Δ˜r)⊗C={Cm⊗Δr+⊕Cm¯⊗Δr−ifr≡2(mod8),Cm⊗Δr−⊕Cm¯⊗Δr+ifr≡6(mod8), (3.1)where Cm is the standard representation of U(m). Therefore, we have a representation   U(m)×Spin(r)⟶SO(N)⊂Aut((Rm⊗Δ˜r)⊗C). Structure group. Since u(m)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). With respect to (3.1), an element A⊗IdΔ˜r+B⊗J∈u(m) looks as follows:   ((A+iB)⊗IdΔr±(A−iB)⊗IdΔr∓),so that the exponentials form   {(eA+iB⊗IdΔr±eA−iB⊗IdΔr∓):A∈so(m),B∈S2Rm}≕U(m)^.With respect to (3.1), an element Idm⊗ξ∈spin(r)^ looks as follows:   (Idm⊗κr*+(ξ)Idm⊗κr*−(ξ))=Idm⊗ξ,and its exponential is   Idm⊗eξ∈Idm⊗κ(Spin(r))≕Spin(r)^≅Spin(r),since Spin(r) is represented faithfully on Δr. The image of U(m)×Spin(r) in SO(N)⊂Aut((Rm⊗Δ˜r)⊗C) under the aforementioned representation is   NSO(N)0(Spin(r)^)=U(m)^Spin(r)^,the subgroup of all possible products of elements of the two subgroups, that is we have a map   U(m)×Spin(r)→ρU(m)^Spin(r)^⊂SO(N).Now we need to find ker(ρ) and identify U(m)^Spin(r)^ as a quotient   U(m)^Spin(r)^≅U(m)×Spin(r)ker(ρ).If there are elements g∈U(m) and h∈Spin(r) such that   ρ(g,h)=IdN,then   Spin(r)^∋ρ(Idm,h)=ρ(g,1)−1∈U(m)^.Since ρ(Idm,h) commutes with every element of Spin(r)^, it belongs to the center Z(Spin(r)^)≅Z(Spin(r))={1,−1,volr,−volr}=⟨volr⟩≅Z4.Recall that volr=e1…er acts as ∓i on Δr± if r≡2(mod8), and as ±i on Δr± if r≡6(mod8), so that it maps to   ∓(iIdΔr+⊕(−i)IdΔr−)in the complex Spin(r) representation. Note that (Idm,volr) maps to   (−i)Idm⊗IdΔr+⊕(i)Idm⊗IdΔr−ifr≡2(mod8),(i)Idm⊗IdΔr+⊕(−i)Idm⊗IdΔr−ifr≡6(mod8),in SO(N), and that ((−i)IdCm,1)∈U(m)×Spin(r) maps to such transformations in both cases. Thus, the elements of U(m)×Spin(r) mapping to IdN are   ±(Idm,1),±(iIdm,−volr),which form a copy of Z4 and   U(m)^Spin(r)^≅U(m)×Spin(r)Z4. Fundamental group. Let   R×SU(m)×Spin(r)→ρ˜U(m)^Spin(r)^(t,A,g)↦(eitA⊗κr±(g)e−itA¯⊗κr∓(g)).Thus   ker(ρ˜)=⟨(2πm,e−2πimIdm,1),(π2,Idm,−vol)⟩,π1(U(m)^Spin(r)^)≅{Z,if(m,4)=1,Z⊕Z2,if(m,4)=2,Z⊕Z4,if(m,4)=4.Indeed, let   a≔(2πm,e−2πimIdm,1),b≔(π2,Idm,−vol),and note that (in multiplicative notation)   am=b4.Moreover, If (m,4)=1, there exist t,m∈Z coprime such that   tm+s4=1.The element   btasis such that   (btas)m=bmt(b4)s=b,(btas)4=(am)ta4s=a. If (m,4)=2, m=4k+2 and there exist two generators   c=a−(2k+1)b2,d=ba−k,such that   c2=1,a=d2c,b=d2k+1ck. If (m,4)=4, m=4k and we have two generators   aandc=a−kb,such that   c4=1. 3.4. r≡3,5(mod8) Complexification. In this case, Δ˜r admits three complex structures I,J and K, which behave like quaternions and commute with spin(r) as described in [2]. Indeed, for r≡3(mod8), consider Rr⊂Rr+3 so that the complex structures are induced by Clifford multiplication with the elements   12(1+e1…er)er+1er+2,12(1+e1…er)er+1er+3,12(1+e1…er)er+2er+3.For r≡5(mod8), consider Rr⊂Rr+2 so that the complex structures are induced by Clifford multiplication with the elements   er+1er+2,e1⋯er+1,e1⋯erer+2.Let us consider the complexification of Δ˜r and decompose it as follows:   Δ˜r⊗C={ψ−iIψ∣ψ∈Δ˜r}⊕{ψ+iIψ∣ψ∈Δ˜r},where the first and second subspaces are the +i and −i eigenspaces of I, respectively. Notice that   J(ψ∓iIψ)=Jψ∓iJIψ=Jψ±iIJψ,that is J interchanges the two subspaces and squares to −Iddr. For any ξ∈spin(r)  ξ(ψ±iIψ)=ξψ±iξIψ=ξψ±iIξψ,which means that the subspaces {ψ−iIψ∣ψ∈Δ˜r} and {ψ+iIψ∣ψ∈Δ˜r} are irreducible complex representations of spin(r) of dimension dr/2. Thus, they are isomorphic to Δr as spin(r) representations and   Δ˜r⊗C≅Δr⊕Δr.Now recall that the centralizer subalgebra of spin(r)^ is   Cso(N)(spin(r)^)=so(m)⊗IdΔ˜r⊕S2Rm⊗I⊕S2Rm⊗J⊕S2Rm⊗K≅so(m)⊗IdΔ˜r⊕S2Rm⊗sp(1)≅sp(m),where N=drm. Let us consider the complexification of Rm⊗Δ˜r and decompose it   (Rm⊗Δ˜r)⊗C={v⊗(ψ−iIψ)∣v∈Rm,ψ∈Δ˜r}⊕{v⊗(ψ+iIψ)∣v∈Rm,ψ∈Δ˜r},where the first and second subspaces are the +i and −i eigenspaces of Idm⊗I, respectively. Notice that   (Idm⊗J)(v⊗(ψ∓iIdm⊗Iψ))=v⊗Jψ∓iv⊗JIψ=v⊗Jψ±iv⊗IJψ,that is Idm⊗J interchanges the two subspaces and squares to −IdN. For any Idm⊗ξ∈spin(r)^  (Idm⊗ξ)(v⊗(ψ±iIψ))=v⊗(ξψ±iξIψ)=v⊗(ξψ±iIξψ),which means that the subspaces {v⊗(ψ−iIψ)∣v∈Rm,ψ∈Δ˜r} and {v⊗(ψ+iIψ)∣v∈Rm,ψ∈Δ˜r} are isomorphic to Cm⊗Δr as spin(r)^ representations. Now consider   A⊗IdΔ˜r+B⊗I+C⊗J+D⊗K∈so(m)⊗IdΔ˜r⊕S2Rm⊗I⊕S2Rm⊗J⊕S2Rm⊗K=sp(m),and   (A⊗IdΔ˜r+B⊗I+C⊗J+D⊗K)(v⊗(ψ+iIψ))=Av⊗(ψ+iIψ)+Bv⊗(Iψ+iIIψ)+Cv⊗(Jψ+iJIψ)+Dv⊗(Kψ+iKIψ)=((A−iB)⊗IdΔ˜r+(C+iD)⊗J)(v⊗(ψ+iIψ)).Similarly,   (A⊗IdΔ˜r+B⊗I+C⊗J+D⊗K)(v⊗(ψ−iIψ))=((A+iB)⊗IdΔ˜r+(C−iD)⊗J)(v⊗(ψ−iIψ)).If C=D=0, the subalgebra   u(m)I^={A⊗IdΔ˜r+B⊗I∈so(m)⊗IdΔ˜r⊕S2Rm⊗I∣A∈so(m),B∈S2Rm}is represented as follows:   Δ˜r⊗C=CIm⊗Δr⊕CIm¯⊗Δr,=(CIm⊕CIm¯)⊗Δr,where CIm and CIm¯ denote the standard representation of u(m)I^ and its conjugate, respectively. Since Idm⊗J interchanges the two summands, squares to −IdN and commutes with the action of spin(r)^, we have the standard complex representation of sp(m) as a factor   Δ˜r⊗C=C2m⊗Δr. (3.2)Thus, we have a representation   Sp(m)×Spin(r)⟶SO(N)⊂Aut(C2m⊗Δr). Structure group. Since sp(m)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). By considering (3.2), the exponential of an element Ω⊗IdΔr∈sp(m)⊗IdΔr=spin(r)^ is   eΩ⊗IdΔ˜r∈Sp(m)^=Sp(m)⊗IdΔr≅Sp(m).On the other hand, if Id2m⊗ξ∈spin(r)^, its exponential is   Id2m⊗eξ∈Id2m⊗κ(Spin(r))=Spin(r)^≅Spin(r),since Spin(r) is represented faithfully on Δr. The image of Sp(m)×Spin(r) in SO(N)⊂Aut(C2m⊗Δr) under the aforementioned representation is   NSO(N)0(Spin(r)^)=Sp(m)^Spin(r)^,the subgroup of all possible products of elements of the two subgroups, that is we have a map   Sp(m)×Spin(r)→ρSp(m)^Spin(r)^⊂SO(N).Now we need to find ker(ρ) and identify Sp(m)^Spin(r)^ as a quotient   Sp(m)^Spin(r)^≅Sp(m)×Spin(r)ker(ρ).If there are elements g∈Sp(m) and h∈Spin(r) such that   ρ(g,h)=IdN,then   Spin(r)^∋ρ(Id2m,h)=ρ(g,1)−1∈Sp(m)^.Since ρ(Id2m,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))=Z2={±1}. Note that −1 is mapped to −IdΔr under the Spin(r) representation, and that (Id2m,−1) maps to −Id2m⊗IdΔr∈SO(N) under ρ. Note that −Id2m⊗IdΔr also belongs to Sp(m)^ being the image of (−Id2m,1)∈Sp(m)×Spin(r). Thus,   ker(ρ)={±(Id2m,1)}≅Z2,  Sp(m)^Spin(r)^≅Sp(m)×Spin(r)Z2. Fundamental group. Clearly,   π1(Sp(m)^Spin(r)^)=Z2. 3.5. r≡4(mod8) Recall from [2] that   Δ˜r±=12(1±e1⋯er)Δ˜r+3.In this case, Δ˜r± admits three complex structures I±,J± and K± induced by Clifford multiplication with the elements 12(1±e1…er)er+1er+2, 12(1±e1…er)er+1er+3 and 12(1±e1…er)er+2er+3, respectively. Just as in the previous case,   Δ˜r+⊗C={ψ−iI+ψ∣ψ∈Δ˜r+}⊕{ψ+iI+ψ∣ψ∈Δ˜r+},and both summands are isomorphic to Δr−. Indeed, if   ψ=12(1+e1…er)·ϕ∈Δ˜r+,then,   (−i)r/2(e1…er)·(ψ±iI+ψ)=−(ψ±iI+ψ),that is ψ±iI+ψ∈Δr−. In other words,   Δ˜r+⊗C=Δr−.Similarly,   Δ˜r−⊗C=Δr+.The rest of the proof proceeds as in the previous case,   (Rm1⊗Δ˜r+⊕Rm2Δ˜r−)⊗C=C2m1⊗Δr−⊕C2m2Δr+,and we have a representation   Sp(m1)×Sp(m2)×Spin(r)⟶SO(N)⊂Aut(CN),where N=dr(m1+m2). Structure group. Since sp(m1)^⊕sp(m2)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). The exponential of Ω1⊗IdΔr−⊕Ω2⊗IdΔr+∈sp(m1)^⊕sp(m2)^ gives   (eΩ1⊗IdΔr−eΩ2⊗IdΔr+)∈(Sp(m1)^Sp(m2)^)=(Sp(m1)⊗IdΔr−Sp(m2)⊗IdΔr+)≅Sp(m1)×Sp(m2).On the other hand, if Id2m1⊗ξ−⊕Id2m2⊗ξ+∈spin(r)^, its exponential is   (Id2m1⊗eξ−Id2m2⊗eξ+)∈(Id2m1⊗κ−(Spin(r))Id2m2⊗κ+(Spin(r)))={Spin(r)^≅Spin(r)ifm1>0andm2>0,Spin(r)^−≅κ−(Spin(r))ifm1>0andm2=0,Spin(r)^+≅κ+(Spin(r))ifm1=0andm2>0,where the first case is faithful and the last two are not, with   Spin(r)^±≅κ±(Spin(r))≅Spin(r){1,∓volr},ifr>4Spin(r)^±≅κ±(Spin(r))≅Spin(3),ifr=4.The images of Sp(m1)×Sp(m2)×Spin(r) in SO(N)⊂Aut(CN) under the aforementioned representations are   NSO(N)0(Spin(r)^)=(Sp(m1)^×Sp(m2)^)Spin(r)^,NSO(N)0(Spin(r)^−)=Sp(m1)^Spin(r)^−,NSO(N)0(Spin(r)^+)=Sp(m2)^Spin(r)^+,respectively, that is we have maps   Sp(m1)×Sp(m2)×Spin(r)→ρ(Sp(m1)^×Sp(m2)^)Spin(r)^⊂SO(N),Sp(m1)×Spin(r)→ρSp(m1)^Spin(r)^−⊂SO(N),Sp(m2)×Spin(r)→ρSp(m2)^Spin(r)^+⊂SO(N).Now we need to find ker(ρ) in each case to identify the relevant group as a quotient. Case m1,m2>0. If there are elements gi∈Sp(mi) and h∈Spin(r) such that   ρ(g1,g2,h)=IdN,then   Spin(r)^∋ρ(Id2m1,Id2m2,h)=ρ(g1,g2,1)−1∈Sp(m1)^×Sp(m2)^.Since ρ(Id2m1,Id2m2,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))={1,−1,volr,−volr}≅Z2⊕Z2.The element −1 is mapped to −IdΔr± in the Spin(r) representations on Δr±, and (Id2m1,Id2m2,−1) is mapped to −(Id2m1⊗IdΔr−⊕Id2m2⊗IdΔr+)∈SO(N).The element volr is mapped to ∓IdΔr± in the Spin(r) representations on Δr±, and (Id2m1,Id2m2,volr) is mapped to (Id2m1⊗IdΔr−⊕(−1)Id2m2⊗IdΔr+)∈SO(dr(m1+m2)).In this case, −(Id2m1⊗IdΔr−⊕Id2m2⊗IdΔr+) and (Id2m1⊗IdΔr−⊕(−1)Id2m2⊗IdΔr+) belong to Sp(m1)^×Sp(m2)^. Thus,   ker(ρ)={(Id2m1,Id2m2,1),(−Id2m1,−Id2m2,−1),(Id2m1,−Id2m2,volr),(−Id2m1,Id2m2,−volr)},(Sp(m1)^×Sp(m2)^)Spin(r)^≅Sp(m1)×Sp(m2)×Spin(r)Z2⊕Z2. Case m1>0, m2=0. If there are elements g1∈Sp(m1) and h∈Spin(r) such that   ρ(g1,h)=IdN,then   ρ(Id2m1,h)=ρ(g1,1)−1∈Sp(m1)^and   ρ(Id2m1,h)∈Spin(r)^−∩Sp(m1)^.Since ρ(Id2m1,h) commutes with every element of Spin(r)^−, it belongs to its center   Z(Spin(r)^−)≅{Z(κ−(Spin(r)))=Z(Spin(r)/{1,volr})={1,−1,volr,−volr}/{1,volr}≅{1,−1}≅Z2ifr>4.Z(κ−(Spin(r)))=Z({1}×Spin(3))={(1,1),(1,−1)}≅Z2ifr=4. If r>4, the element −1 is mapped to −IdΔr− in the Spin(r) representation on Δr−, and (Id2m1,−1) is mapped to −(Id2m1⊗IdΔr−)∈SO(N). In this case, −Id2m1⊗IdΔr− belongs to Sp(m1)^. Thus,   ker(ρ)={(Id2m1,1),(Id2m1,volr),(−Id2m1,−1),(−Id2m1,−volr)},Sp(m1)^Spin(r)^−≅Sp(m1)×Spin(r)Z2⊕Z2. If r=4, the element (1,−1)∈{1}×Spin(3) is mapped to −IdΔr− in the Spin(r) representation on Δr−, and (Id2m1,(1,−1)) is mapped to −(Id2m1⊗IdΔr−)∈SO(N). In this case, −Id2m1⊗IdΔr− belongs to Sp(m1)^. Thus,   ker(ρ)={(Id2m1,(1,1)),(−Id2m1,(1,−1))}×(Spin(3)×{1}),Sp(m1)^Spin(4)^−≅Sp(m1)×(Spin(3)×Spin(3)){(Id2m1,(1,1)),(−Id2m1,(1,−1))}×(Spin(3)×{1})≅Sp(m1)×Spin(3)Z2.Note that (1,−1)∈Spin(3)×Spin(3) corresponds −vol4∈Spin(4). Case m1=0, m2>0. If there are elements g2∈Sp(m2) and h∈Spin(r) such that   ρ(g2,h)=IdN,then   ρ(Id2m2,h)=ρ(g2,1)−1∈Sp(m2)^and   ρ(Id2m2,h)∈Spin(r)^+∩Sp(m2)^.Since ρ(Id2m2,h) commutes with every element of Spin(r)^+, it belongs to its center   Z(Spin(r)^−)≅{Z(κ+(Spin(r)))=Z(Spin(r)/{1,−volr})={1,−1,volr,−volr}/{1,−volr}≅{1,−1}≅Z2ifr>4.Z(κ+(Spin(r)))=Z(Spin(3)×{1})={(1,1),(−1,1)}≅Z2ifr=4. The element −1 is mapped to −IdΔr+ in the Spin(r) representation on Δr+, and (Id2m1,−1) is mapped to −(Id2m2⊗IdΔr+)∈SO(N). In this case, −Id2m2⊗IdΔr− belongs to Sp(m2)^. Thus,   ker(ρ)={(Id2m2,1),(Id2m2,−volr),(−Id2m2,−1),(−Id2m2,volr)},Sp(m2)^Spin(r)^+≅Sp(m2)×Spin(r)Z2⊕Z2. If r = 4, the element (−1,1)∈Spin(3)×{1} is mapped to −IdΔr+ in the Spin(r) representation on Δr+, and (Id2m2,(−1,1)) is mapped to −(Id2m2⊗IdΔr+)∈SO(N). In this case, −Id2m2⊗IdΔr− belongs to Sp(m2)^. Thus,   ker(ρ)={(Id2m2,(1,1)),(−Id2m2,(−1,1))}×(Spin(3)×{1}),Sp(m2)^Spin(4)^+≅Sp(m2)×(Spin(3)×Spin(3)){(Id2m2,(1,1)),(−Id2m2,(−1,1))}×({1}×Spin(3))≅Sp(m2)×Spin(3)Z2.Note that (−1,1)∈Spin(3)×Spin(3) corresponds to vol4∈Spin(4). Fundamental group. Clearly,   π1((Sp(m1)^×Sp(m2)^)Spin(r)^)=Z2⊕Z2,π1(Sp(m1)^Spin(r)^−)≅{Z2⊕Z2,ifm1>0,m2=0,r>4,Z2,ifm1>0,m2=0,r=4,π1(Sp(m2)^Spin(r)^+)≅{Z2⊕Z2,ifm1=0,m2>0,r>4,Z2ifm1=0,m2>0,r=4.□ Thus, we have proved the following three theorems. Theorem 3.1 The complexification of a real representation RNof Clr0 without trivial summands decomposes as follows r(mod8)  RN⊗C  0  Cm1⊗Δr+⊕Cm2⊗Δr−  1,7  Cm⊗Δr  2  Cm⊗Δr+⊕Cm¯⊗Δr−  6  Cm¯⊗Δr+⊕Cm⊗Δr−  3,5  C2m⊗Δr  4  C2m2⊗Δr+⊕C2m1⊗Δr−  r(mod8)  RN⊗C  0  Cm1⊗Δr+⊕Cm2⊗Δr−  1,7  Cm⊗Δr  2  Cm⊗Δr+⊕Cm¯⊗Δr−  6  Cm¯⊗Δr+⊕Cm⊗Δr−  3,5  C2m⊗Δr  4  C2m2⊗Δr+⊕C2m1⊗Δr−  where the different Cs denote the corresponding standard complex representations of the classical Lie algebras so(s),u(s)or sp(s/2).□ Theorem 3.2 The connected components of the identity NSO(N)0(Spin(r)^)of the normalizers NSO(N)(Spin(r)^)are isomorphic to the following groups: If r≡1,7(mod8), N=drmand  NSO(N)0(Spin(r)^)≅{SO(m)×Spin(r)Z2,ifmiseven,SO(m)×Spin(r),ifmisodd. If r≡0(mod8), N=dr(m1+m2)and  NSO(N)0(Spin(r)^)≅{SO(m1)×SO(m2)×Spin(r),ifm1>0,m2>0,m1≡m2≡1(mod2),SO(m1)×SO(m2)×Spin(r)Z2,ifm1>0,m2>0,m1+m2≡1(mod2),SO(m1)×SO(m2)×Spin(r)Z2⊕Z2,ifm1>0,m2>0,m1≡m2≡0(mod2),NSO(N)0(Spin(r)^+)≅{SO(m1)×Spin(r)Z2,ifm1>0,m2=0,m1≡1(mod2),SO(m1)×Spin(r)Z2⊕Z2,ifm1>0,m2=0,m1≡0(mod2),NSO(N)0(Spin(r)^−)≅{SO(m2)×Spin(r)Z2,ifm1=0,m2>0,m2≡1(mod2),SO(m2)×Spin(r)Z2⊕Z2,ifm1=0,m2>0,m2≡0(mod2). If r≡2,6(mod8), N=drmand  NSO(N)0(Spin(r)^)≅U(m)×Spin(r)Z4. If r≡3,5(mod8), N=drmand  NSO(N)0(Spin(r)^)≅Sp(m)×Spin(r)Z2. If r≡4(mod8), N=dr(m1+m2)and  NSO(N)0(Spin(r)^)≅Sp(m1)×Sp(m2)×Spin(r)Z2⊕Z2,ifm1>0,m2>0,NSO(N)0(Spin(r)^−)≅{Sp(m1)×Spin(r)Z2⊕Z2,ifm1>0,m2=0,r>4,Sp(m1)×Spin(3)Z2,ifm1>0,m2=0,r=4,NSO(N)0(Spin(r)^+)≅{Sp(m2)×Spin(r)Z2⊕Z2,ifm1=0,m2>0,r>4,Sp(m2)×Spin(3)Z2,ifm1=0,m2>0,r=4.□ Theorem 3.3 The fundamental group of the connected components of the identity of the normalizers NSO(N)0(Spin(r)^)are the following: If r≡1,7(mod8), N=drmand  π1(NSO(N)0(Spin(r)^))≅{Z2⊕Z2,ifm≥4,m≡0(mod4),Z4,ifm≥4,m≡2(mod4),Z2,ifm>1andodd,{1},ifm=1,Z,ifm=2. If r≡0(mod8), N=dr(m1+m2)and either π1(NSO(N)0(Spin(r)^))or π1(NSO(N)0(Spin(r)^+))or π1(NSO(N)0(Spin(r)^−)) are isomorphic to   m2  m1  0  1  2  1(mod2)  2(mod4)  0(mod4)  0    Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  1  Z2  {1}  Z  Z2  Z4  Z2⊕Z2  2  Z⊕Z2  Z  Z⊕Z  Z⊕Z2  Z⊕Z4  Z⊕Z2⊕Z2  1(mod2)  Z2⊕Z2  Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  2(mod4)  Z2⊕Z4  Z4  Z⊕Z4  Z2⊕Z4  Z4⊕Z4  Z2⊕Z2⊕Z4  0(mod4)  Z2⊕Z2⊕Z2  Z2⊕Z2  Z⊕Z2⊕Z2  Z2⊕Z2⊕Z2  Z2⊕Z2⊕Z4  Z2⊕Z2⊕Z2⊕Z2    m2  m1  0  1  2  1(mod2)  2(mod4)  0(mod4)  0    Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  1  Z2  {1}  Z  Z2  Z4  Z2⊕Z2  2  Z⊕Z2  Z  Z⊕Z  Z⊕Z2  Z⊕Z4  Z⊕Z2⊕Z2  1(mod2)  Z2⊕Z2  Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  2(mod4)  Z2⊕Z4  Z4  Z⊕Z4  Z2⊕Z4  Z4⊕Z4  Z2⊕Z2⊕Z4  0(mod4)  Z2⊕Z2⊕Z2  Z2⊕Z2  Z⊕Z2⊕Z2  Z2⊕Z2⊕Z2  Z2⊕Z2⊕Z4  Z2⊕Z2⊕Z2⊕Z2  depending on whether m1,m2>0or m1=0or m2=0, respectively. If r≡2,6(mod8), N=drmand  π1(NSO(n)0(Spin(r)^))={Z,if(m,4)=1,Z×Z2,if(m,4)=2,Z×Z4,if(m,4)=4. If r≡3,5(mod8), N=drmand  π1(NSO(N)0(Spin(r)^))=Z2. If r≡4(mod8), N=dr(m1+m2)and  π1(NSO(N)0(Spin(r)^))≅Z2⊕Z2,ifm1>0,m2>0,π1(NSO(N)0(Spin(r)−^))≅{Z2⊕Z2,ifm1>0,m2=0,r>4,Z2,ifm1>0,m2=0,r=4,π1(NSO(N)0(Spin(r)+^))≅{Z2⊕Z2,ifm1=0,m2>0,r>4,Z2,ifm1=0,m2>0,r=4.□ 4. Lifting maps to the Spin group In this section, we will check how the generators of the fundamental groups π1(NSO(N)0(S)) map into π1(SO(N)). Theorem 4.1 Let r≥3. There exist lifts  Spin(N)↗↓NSO(N)0(S)⟶SO(N),where S denotes the homomorphic image of Spin(r)in SO(N) (either Spin(r)^or Spin(r)^+), in the following cases: r≡1,7(mod8)̲ For all m∈N. r≡0(mod8)̲ For all m1,m2∈N if r>8. For m1≡m2≡0(mod2)if r=8. r≡2,6(mod8)̲ For all m∈N if r>6. For m even if r=6. r≡3,5(mod8)̲ For all m∈Nif r>3. For m even if r=3. r≡4(mod8)̲ For all m1,m2∈Nif r>4. For m1≡m2≡0(mod2)if r=4. The rest of this section is devoted to prove Theorem 4.1 in a case by case analysis. 4.1. r≡1,7(mod8) Recall   π1(SO(m)^Spin(r)^)={⟨(−1,1)⟩×⟨(volm,−1)⟩⊂Spin(m)×Spin(r),ifm≥4,m≡0(mod4),⟨(volm,−1)⟩⊂Spin(m)×Spin(r),ifm≥4,m≡2(mod4),⟨(−1,1)⟩⊂Spin(m)×Spin(r),ifm≥3,misodd,{1}⊂{1}×Spin(r),ifm=1,⟨(π,−1)⟩⊂R×Spin(r),ifm=2.Thus, we only need to check the loops in SO(drm) which are images of paths joining (1,1) to either (−1,1) or (volm,−1) in Spin(m)×Spin(r) or joining (0,1) to (π,−1) in R×Spin(r). Consider the path   δ1:[0,1]⟶Spin(m)×Spin(r)t↦(cos(πt)+sin(πt)v1v2,1)joining (1,1) to (−1,1) in Spin(m)×Spin(r). It projects to the loop   δˆ1:[0,1]⟶SO(m)^Spin(r)^⊂SO(N)t↦(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)1⋱1)m×m⊗IdΔr,which contains 2[r2] blocks   (cos(2πt)−sin(2πt)sin(2πt)cos(2πt)).Thus, δˆ1 represents 2[r2] times the generator of π1(SO(drm)). Since r≥3 and r≡1,7(mod8), 2[r2] is divisible by 8 and δˆ1 is null-homotopic. When m is even and m≥4, also consider the path   δ2:[0,1]⟶Spin(m)×Spin(r)t↦(∏j=1m2cos(πt/2)+sin(πt/2)v2j−1v2j,cos(πt)+sin(πt)e1e2)joining (1,1) to (volm,−1) in Spin(m)×Spin(r). It projects to the loop   δˆ2:[0,1]⟶SO(m)^Spin(r)^⊂SO(N)t↦(cos(πt)−sin(πt)sin(πt)cos(πt)⋱cos(πt)−sin(πt)sin(πt)cos(πt))m×m⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2]which is similar to   (eπite−πit⋱eπite−πit)m×m⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2].It contains 2[r2]−1 blocks that is there are 2[r2]−1m2=2[r2]−2m copies of the generator of π1(SO(drm)). Since r≥3 and r≡1,7(mod8), 2[r2]−2 is divisible by 2 and δˆ2 is null-homotopic. For m=2, consider the path   δ3:[0,1]⟶R×Spin(r)t↦(πt,cos(πt)+sin(πt)e1e2)joining (0,1) to (π,−1) in R×Spin(r), which maps to   δˆ3:[0,1]⟶SO(2)^Spin(r)^⊂SO(N)t↦(cos(πt)−sin(πt)sin(πt)cos(πt))⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2]∼(e2πite−2πit)⊗Id2[r2]−1⊕Id2[r2]−1.This loop represents 2[r2]−1 times the generator of π1(SO(N)), which is null-homotopic since 2[r2]−1 is divisible by 4. 4.2. r≡0(mod8) Let r=8k, {v1,…,vm1} and {v1′,…,vm2′} oriented orthonormal bases of Rm1 and Rm2, respectively. Recall the fundamental group generators for m1,m2≥3: Cases  π1(SO(m1)^SO(m2)^Spin(r)^)  (−1,1,1)  (1,−1,1)  (volm1,1,−volr)  (1,volm2,volr)  (a)  m1≡1(2),m2≡1(2)  Z2⊕Z2  ✓  ✓      (b)  m1≡0(4),m2≡1(2)  Z2⊕Z2⊕Z2  ✓  ✓  ✓    (c)  m1≡2(4),m2≡1(2)  Z2⊕Z4    ✓  ✓    (d)  m1≡1(2),m2≡0(4)  Z2⊕Z2⊕Z2  ✓  ✓    ✓  (e)  m1≡1(2),m2≡2(4)  Z2⊕Z4  ✓      ✓  (f)  m1≡0(4),m2≡0(4)  Z2⊕Z2⊕Z2⊕Z2  ✓  ✓  ✓  ✓  (g)  m1≡0(4),m2≡2(4)  Z2⊕Z2⊕Z4  ✓    ✓  ✓  (h)  m1≡2(4),m2≡0(4)  Z2⊕Z4⊕Z2    ✓  ✓  ✓  (i)  m1≡2(4),m2≡2(4)  Z4⊕Z4      ✓  ✓  Cases  π1(SO(m1)^SO(m2)^Spin(r)^)  (−1,1,1)  (1,−1,1)  (volm1,1,−volr)  (1,volm2,volr)  (a)  m1≡1(2),m2≡1(2)  Z2⊕Z2  ✓  ✓      (b)  m1≡0(4),m2≡1(2)  Z2⊕Z2⊕Z2  ✓  ✓  ✓    (c)  m1≡2(4),m2≡1(2)  Z2⊕Z4    ✓  ✓    (d)  m1≡1(2),m2≡0(4)  Z2⊕Z2⊕Z2  ✓  ✓    ✓  (e)  m1≡1(2),m2≡2(4)  Z2⊕Z4  ✓      ✓  (f)  m1≡0(4),m2≡0(4)  Z2⊕Z2⊕Z2⊕Z2  ✓  ✓  ✓  ✓  (g)  m1≡0(4),m2≡2(4)  Z2⊕Z2⊕Z4  ✓    ✓  ✓  (h)  m1≡2(4),m2≡0(4)  Z2⊕Z4⊕Z2    ✓  ✓  ✓  (i)  m1≡2(4),m2≡2(4)  Z4⊕Z4      ✓  ✓  For the cases (a), (b), (d), (e), (f) and (g), consider the path   δ1:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(cos(πt)+sin(πt)v1v2,1,1)joining (1,1,1) to (−1,1,1) in Spin(m1)×Spin(m2)×Spin(r) which projects to the loop   δˆ1:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)1⋱1)m1×m1⊗IdΔr+⊕Idm2⊗IdΔr−.It contains 2r2−1 copies of the generator of π1(SO(dr(m1+m2))), which is homotopically trivial since 2r2−1 is divisible by 8. For the cases (a), (b), (c), (d), (f) and (h), consider the path   δ2:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(1,cos(πt)+sin(πt)v1′v2′,1)joining (1,1,1) to (1,−1,1) in Spin(m1)×Spin(m2)×Spin(r), which projects to the loop   δˆ2:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦Idm1⊗IdΔr+⊕(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)1⋱1)m2×m2⊗IdΔr−.It contains 2r2−1 copies of the generator of π1(SO(dr(m1+m2))), and is homotopically trivial since 2r2−1 is divisible by 8. For the cases (d), (e), (f), (g), (h) and (i), consider the path   δ3:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(1,∏j=1m22cos(πt/2)+sin(πt/2)v2j−1′v2j′,∏l=1r2cos(πt/2)+sin(πt/2)e2l−1e2l)joining (1,1,1) to (1,volm2,volr) in Spin(m1)×Spin(m2)×Spin(r). It projects to the loop   δˆ3:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦Idm1⊗P+(t)⊕(cos(πt)−sin(πt)sin(πt)cos(πt)⋱cos(πt)−sin(πt)sin(πt)cos(πt))m2×m2⊗P−(t)∼Idm1⊗P+(t)⊕(eπite−πit⋱eπite−πit)m2×m2⊗P−(t),where   P+(t)=diag(e2π(k)it,e2π(k−1)it,…,e2π(k−1)it︸(4k2)times,e2π(k−2)it,…,e2π(k−2)it︸(4k4)times,…,e2π(−k)it),P−(t)=diag(e(2k−1)πit,…,e(2k−1)πit︸(4k1)times,e(2k−3)πit,…,e(2k−3)πit︸(4k3)times,…,e−(2k−1)πit,…,e−(2k−1)πit︸(4k4k−1)times).Thus, δˆ3 contains   m1(k+(4k2)(k−1)+(4k4)(k−2)+⋯+(4k2k−2))+m22((4k1)k+(4k3)(k−1)+⋯+(4k4k−1)(−(k−1)))=m1k(2k−1)8k−2)(4k2k)+24k−3m2copies of the generator of π1(SO(dr(m1+m2))). This number of copies is always even except when k = 1 and m1 is odd. For the cases (b), (c), (f), (g), (h) and (i), consider the path   δ4:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(∏j=1m12(cos(πt/2)+sin(πt/2)v2j−1v2j,1,(cos(πt/2))−sin(πt/2)e1e2)∏l=2r2(cos(πt/2)+sin(πt/2)e2l−1e2l))joining (1,1,1) to (volm1,1,−volr) in Spin(m1)×Spin(m2)×Spin(r). It projects to the loop   δˆ4:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦(cos(πt)−sin(πt)sin(πt)cos(πt)⋱cos(πt)−sin(πt)sin(πt)cos(πt))m1×m1⊗Q+(t)⊕Idm2⊗Q−(t)∼(eπite−πit⋱eπite−πit)m1×m1⊗Q+(t)⊕Idm2⊗Q−(t),where   Q+(t)=diag(e(2k−1)πit,…,e(2k−1)πit︸(4k1)times,e(2k−3)πit,…,e(2k−3)πit︸(4k3)times,e(2k−5)πit,…,e(2k−5)πit︸(4k5)times,…,e−(2k−1)πit,…,e−(2k−1)πit︸(4k4k−1)times),Q−(t)=diag(e2πkit,e2π(k−1)it,…,e2π(k−1)it︸(4k2)times,e2π(k−2)it,…,e2π(k−2)it︸(4k4)times,…,e2π(−k)it).Thus, we have   m12[k(4k1)+(k−1)(4k3)+⋯+(−(k−1))(4k4k−1)]+m2[k(4k0)+(k−1)(4k2)+⋯+1(4k2k−2)]=24k−3m1+m2k(2k−1)8k−2(4k2k),which is odd only if k = 1 and m2≡1 (mod 2). The cases in which either m1≤2 or m2≤2 are treated similarly. 4.3. r≡2,6(mod8) Recall   π1(U(m)^Spin(r)^)={Z,if(m,4)=1,Z⊕Z2,if(m,4)=2,Z⊕Z4,if(m,4)=4.In every case, the fundamental group has generators   (2πm,e−2πimIdm,1),(π2,Idm,−vol). Consider the path   δ1:[0,1]⟶R×SU(m)×Spin(r)t↦(2πtm,(e−2πitme−2πitm⋱e−2πitme2πi(m−1)tm)m×m,1)joining (0,Idm×m,1) to (2πm,e−2πimIdm,1) in R×SU(m)×Spin(r). This path gets mapped to the loop if r≡2(mod8), and to if r≡6(mod8). Since dim(Δr±)=2r2−1, we have 2r2−1 blocks of the form   (e2πite−2πit)∼(cos(2πt)−sin(2πt)sin(2πt)cos(2πt))that is 2r2−1 times the generator of π1(SO(drm)). Since r≥6, 2r2−1 is divisible by 4. Hence, δˆ1 is null-homotopic. Consider the path   δ2:[0,1]⟶R×SU(m)×Spin(r)t↦(πt2,Idm,∏j=1r2(cos(πt/2)−sin(πt/2)e2j−1e2j))joining (0,Idm×m,1) to (π2,Idm,−vol) in R×SU(m)×Spin(r). This path gets mapped to the loop if r≡2(mod8), and if r≡6(mod8), where   P+(t)=diag(e−r2πit2,e−(r2−4)πit2,…,e−(r2−4)πit2︸(r/22)times,e−(r2−8)πit2,…,e−(r2−8)πit2︸(r/24)times,…),P−(t)=diag(e−(r2−2)πit2,…,e−(r2−2)πit2︸(r/21)times,e−(r2−6)πit2,…,e−(r2−6)πit2︸(r/23)times,e−(r2−10)πit2,…,e−(r2−10)πit2︸(r/25)times,…). If r≡2(mod8), r=8k+2 with k≥1. Then r2=4k+1. We have   m[k(4k+10)+(k−1)(4k+12)+(k−2)(4k+14)+⋯+(−k+1)(4k+14k−2)+(−k)(4k+14k)]=−m24k−2copies of the generator of π1(SO(drm)), which is even and δˆ1 is null-homotopic. If r≡6(mod8), r=8k+6 with k≥1. Then r2=4k+3. Thus, we have   m[(k+1)(4k+30)+(k)(4k+32)+(k−1)(4k+34)+⋯+(−k+1)(4k+34k)+(−k)(4k+34k+2)]=m24kcopies of the generator of π1(SO(drm)). If k≥1, this number is always even and δˆ2 is null-homotopic. On the other hand, if r = 6 (k = 0), then the parity of the number depends on m. 4.4. r≡3,5(mod8) Recall   π1(Sp(m)^Spin(r)^)=Z2=⟨(−Id2m,−1)⟩.Thus, consider the path   δ:[0,1]⟶Sp(m)×Spin(r)t↦((eπite−πit⋱eπite−πit)2m×2m,cos(πt)+sin(πt)e1e2)joining (Id2m,1) to (−Id2m,−1) in Sp(m)×Spin(r). It projects to the loop in Sp(m)^Spin(r)^⊂SO(drm)  (eπite−πit⋱eπite−πit)2m×2m⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2].It has 2[r2]−1m blocks of the form   (e2πite−2πit)∼(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)),that is 2[r2]−1m times the generator of π1(SO(drm))=Z2. Note that 2[r2]−1m is divisible by 2 if r>3. 4.5. r≡4(mod8) Let r=8k+4. Recall   π1((Sp(m1)^×Sp(m2)^)Spin(r)^)=⟨(−Id2m1,−Id2m2,−1)⟩×⟨(Id2m1,−Id2m2,volr)⟩⊂Sp(m1)×Sp(m2)×Spin(r).π1(Sp(m1)^Spin(r)−^)≅{⟨(−Id2m1,−volr),(−Id2m1,−1)⟩⊂Sp(m1)×Spin(r),ifm1>0,m2=0,r>4,⟨(−Id2m1,−vol4)⟩⊂Sp(m1)×Spin(4),ifm1>0,m2=0,r=4,π1(Sp(m2)^Spin(r)+^)≅{⟨(−Id2m2,volr),(−Id2m2,−1)⟩⊂Sp(m2)×Spin(r),ifm1=0,m2>0,r>4,⟨(−Id2m2,vol4)⟩⊂Sp(m2)×Spin(4),ifm1=0,m2>0,r=4. Consider the path   δ1:[0,1]⟶Sp(m1)×Sp(m2)×Spin(r)t↦((eπite−πit⋱eπite−πit)2m1×2m1,(eπite−πit⋱eπite−πit)2m2×2m2,cos(πt)+sin(πt)e1e2)joining (Id2m1,Id2m2,1) and (−Id2m1,−Id2m2,−1) in Sp(m1)×Sp(m2)×Spin(r). It maps to the loop   δˆ1:[0,1]⟶(Sp(m1)^×Sp(m2)^)Spin(r)^⊂SO(N)t↦diag(eπit,e−πit,…,eπit,e−πit)2m1×2m1⊗diag(eπit,e−πit,…,eπit,e−πit)2r2−1×2r2−1⊕diag(eπit,e−πit,…,eπit,e−πit)2m2×2m2⊗diag(eπit,e−πit,…,eπit,e−πit)2r2−1×2r2−1∼2r2−2diag(e2πit,e−2πit,…,e2πit,e−2πit)2m1×2m1⊕2r2−2Id2m1⊕2r2−2diag(e2πit,e−2πit,…,e2πit,e−2πit)2m2×2m2⊕2r2−2Id2m2,which is homotopically equivalent to (m1+m2)2r2−2 times the generator of π1(SO(dr(m1+m2))). Hence, δˆ1 is null-homotopic if either r≠4 or r = 4 and m1+m2≡0 (mod 2). Consider the path   δ2:[0,1]⟶Sp(m1)×Sp(m2)×Spin(r)t↦(Id2m1,(eπite−πit⋱eπite−πit)2m2×2m2,∏j=1r2cos(πt/2)+sin(πt/2)e2j−1e2j)joining (Id2m1,Id2m2,1) to (Id2m1,−Id2m2,volr). It maps to the loop   δˆ2⟶(Sp(m1)^×Sp(m2)^)Spin(r)^⊂SO(N)t↦Id2m1⊗P−(t)⊕diag(eπit,e−πit,…,eπit,e−πit)2m2×2m2⊗P+ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Quarterly Journal of Mathematics Oxford University Press

# A note on the geometry and topology of almost even-Clifford Hermitian manifolds

, Volume 69 (1) – Mar 1, 2018
56 pages

/lp/ou_press/a-note-on-the-geometry-and-topology-of-almost-even-clifford-hermitian-COiX4Zkk3m
Publisher
Oxford University Press
ISSN
0033-5606
eISSN
1464-3847
D.O.I.
10.1093/qmath/hax040
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### Abstract

Abstract We compute the structure groups of almost even-Clifford Hermitian manifolds and determine when such groups lead to Spin structures. 1. Introduction Almost even-Clifford Hermitian structures on oriented Riemannian manifolds were introduced recently in [10] under the simpler name of even Clifford structures, which are generalizations of almost Hermitian and almost quaternion-Hermitian structures. They are a subject of current interest [1, 3, 5, 9, 12], although similar types of structures have been studied in the past [4, 7, 11]. The existence of such a structure on a manifold implies the reduction of its structure group to the normalizer of the homomorphic image of a Spin group. In this paper, we identify such structure group (cf. Theorem 3.2) by using the results about their Lie algebras given in [2]. In the case of 4m-dimensional almost quaternion-Hermitian manifolds, we know that such manifolds are Spin when m is even. This is due to the (topological) reduction of the structure group from SO(4m) to the Lie group Sp(m)Sp(1) which, in turn, embeds into Spin(4m) when m is even. Thus, by analogy, we have been led to study when almost even-Clifford Hermitian manifolds admit Spin structures. Recall that an oriented N-dimensional Riemannian manifold is Spin if its orthonormal frame bundle PSO admits a double cover by a principal Spin(N) bundle PSpin  Λ:PSpin⟶PSO,which is Spin(N) equivariant, that is Λ(pg)=Λ(p)λN(g) for all g∈Spin(N). A Riemannian manifold will automatically be Spin if its structure group reduces to a proper subgroup G⊂SO(N) such that there exists a lifting map which makes the following diagram commute:   Spin(N)↗↓G↪SO(N).Indeed, such a lift exists if and only if π1(G) maps trivially into π1(SO(N)). In this paper, we determine when there exists a lifting map which makes the following diagram commute (cf. Theorem 4.1):   Spin(N)↓NSO(N)(S)↪SO(N),where N stands for the dimension of an almost even-Clifford Hermitian manifold, S denotes the homomorphic image of the aforementioned Spin group determined by the even-Clifford structure, and NSO(N)(S) denotes its normalizer in SO(N). In fact, we will verify that there is a lift for the connected component of the identity NSO(N)0(S). Furthermore, note that an almost even-Clifford Hermitian manifold might still be Spin even if there is no such lifting map, as in the case of quaternionic projective spaces HPm of odd quaternionic dimension m. It would be interesting, at least for the authors, to find and study non-Spin almost even-Clifford manifolds of ranks 4, 6 and 8. The note is organized as follows. In Section 2, we recall some preliminaries on Clifford algebras, the Spin group and representations, almost even-Clifford manifolds, etc. In Section 3, we determine the complexifications of real representations of even Clifford algebras containing no trivial summands (cf. Theorem 3.1), identify the subgroups NSO(N)0(S) as quotients of products of classical groups (or real lines in some cases) and Spin groups (cf. Theorem 3.2), and calculate their fundamental groups giving explicit generators (cf. Theorem 3.3). In Section 4, we determine when the aforementioned lifts exist (cf. Theorem 4.1). 2. Preliminaries 2.1. Clifford algebra, spin group and representation The material presented in this subsection can be consulted in [6]. Let Clr denote the 2r-dimensional real Clifford algebra generated by the orthonormal vectors e1,…,er∈Rr subject to the relations   eiej+ejei=−2δij,and Clr=Clr⊗RC its complexification. The even Clifford subalgebra Clr0 is defined as the invariant (+1)-subspace of the involution of Clr induced by the map −IdRr. For any vector Y=y1e1+⋯+yrer, the product   eiY ei=y1e1+⋯+yi−1ei−1−yiei+yi+1ei+1+⋯+yrergives the reflection of the ith coordinate, and the conjugation with the volume element volr=e1⋯er gives the reflection on the origin of Rr, that is   (e1⋯er)Y(er⋯e1)=−Y. There exist algebra isomorphisms   Clr≅{End(C2k)ifr=2k,End(C2k)⊕End(C2k)ifr=2k+1,and the space of (complex) spinors is defined to be   Δr≔C2k=C2⊗…⊗C2︸ktimes.The map   κ:Clr⟶End(C2k)is defined to be either the aforementioned isomorphism for r even, or the isomorphism followed by the projection onto the first summand for r odd. In order to make κ explicit, consider the following matrices:   Id=(1001),g1=(i00−i),g2=(0ii0),T=(0−ii0).In terms of the generators e1,…,er of the Clifford algebra, κ can be described explicitly as follows:   e1↦Id⊗Id⊗…⊗Id⊗Id⊗g1,e2↦Id⊗Id⊗…⊗Id⊗Id⊗g2,e3↦Id⊗Id⊗…⊗Id⊗g1⊗T,e4↦Id⊗Id⊗…⊗Id⊗g2⊗T,⋮…e2k−1↦g1⊗T⊗…⊗T⊗T⊗T,e2k↦g2⊗T⊗…⊗T⊗T⊗T,and, if r=2k+1,   e2k+1↦iT⊗T⊗…⊗T⊗T⊗T.The vectors   u+1=12(1,−i)andu−1=12(1,i),form a unitary basis of C2 with respect to the standard Hermitian product. Thus,   B={uε1,…,εk=uε1⊗…⊗uεk∣εj=±1,j=1,…,k},is a unitary basis of Δr=C2k with respect to the naturally induced Hermitian product. We will denote inner and Hermitian products (as well as Riemannian and Hermitian metrics) by the same symbol ⟨·,·⟩ trusting that the context will make clear which product is being used. A quaternionic structure α on C2 is given by   α(z1z2)=(−z¯2z¯1),and a real structure β on C2 is given by   β(z1z2)=(z¯1z¯2).Following [6, p. 31], the real and quaternionic structures γr on Δr=(C2)⊗[r/2] are built as follows:   γr=(α⊗β)⊗2kifr=8k,8k+1(real),γr=α⊗(β⊗α)⊗2kifr=8k+2,8k+3(quaternionic),γr=(α⊗β)⊗2k+1ifr=8k+4,8k+5(quaternionic),γr=α⊗(β⊗α)⊗2k+1ifr=8k+6,8k+7(real). The Spin group Spin(r)⊂Clr is the subset   Spin(r)={x1x2⋯x2l−1x2l∣xj∈Rr,∣xj∣=1,l∈N},endowed with the product of the Clifford algebra. It is a Lie group and its Lie algebra is   spin(r)=span{eiej∣1≤i<j≤r}.The restriction of κ to Spin(r) defines the Lie group representation   κr≔κ∣Spin(r):Spin(r)⟶GL(Δr),which is, in fact, special unitary. We have the corresponding Lie algebra representation   κr*:spin(r)⟶gl(Δr).Recall that the Spin group Spin(r) is the universal double cover of SO(r), r≥3. For r=2, we consider Spin(2) to be the connected double cover of SO(2). The covering map will be denoted by   λr:Spin(r)→SO(r)⊂GL(Rr).Its differential is given by λr*(eiej)=2Eij, where Eij=ei*⊗ej−ej*⊗ei is the standard basis of the skew-symmetric matrices, and e* denotes the metric dual of the vector e. Furthermore, we will abuse the notation and also denote by λr the induced representation on the exterior algebra ⋀*Rr. By means of κ, we have the Clifford multiplication   μr:Rr⊗Δr⟶Δrx⊗ϕ↦μr(x⊗ϕ)=x·ϕ≔κ(x)(ϕ).The Clifford multiplication μr is skew-symmetric with respect to the Hermitian product   ⟨x·ϕ1,ϕ2⟩=⟨μr(x⊗ϕ1),ϕ2⟩=−⟨ϕ1,μr(x⊗ϕ2)⟩=−⟨ϕ1,x·ϕ2⟩,is Spin(r)-equivariant and can be extended to a Spin(r)-equivariant map   μr:⋀*(Rr)⊗Δr⟶Δrω⊗ψ↦ω·ψ. When r is even, we define the following involution:   Δr⟶Δrψ↦(−i)r2volr·ψ.The ±1 eigenspace of this involution is denoted Δr±. These spaces have equal dimension and are irreducible representations of Spin(r). Note that our definition differs from the one given in [6] by a (−1)r2. The reason for this difference is that we want the spinor u1,…,1 to be always positive. In this case, we will denote the two representations by   κr±:Spin(r)⟶GL(Δr±).Note that while these representations are irreducible, they are not faithful, with kernels isomorphic to Z2 if r≠4. Now, we summarize some results about real representations of Clr0 in the next Table 1 (cf. [8]). Here dr denotes the dimension of an irreducible representation of Clr0 and vr the number of distinct non-trivial irreducible representations. Let Δ˜r denote the irreducible representation of Clr0 for r≢0(mod4) and Δ˜r± denote the irreducible representations for r≡0(mod4). Table 1. Real Clr0-representations. r(mod8)  dr  Clr0  Δ˜r/Δ˜r±≅Rdr  vr  1  2⌊r2⌋  R(dr)  Rdr  1  2  2r2  C(dr/2)  Cdr/2  1  3  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  4  2r2  H(dr/4)⊕H(dr/4)  Hdr/4  2  5  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  6  2r2  C(dr/2)  Cdr/2  1  7  2⌊r2⌋  R(dr)  Rdr  1  8  2r2−1  R(dr)⊕R(dr)  Rdr  2  r(mod8)  dr  Clr0  Δ˜r/Δ˜r±≅Rdr  vr  1  2⌊r2⌋  R(dr)  Rdr  1  2  2r2  C(dr/2)  Cdr/2  1  3  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  4  2r2  H(dr/4)⊕H(dr/4)  Hdr/4  2  5  2⌊r2⌋+1  H(dr/4)  Hdr/4  1  6  2r2  C(dr/2)  Cdr/2  1  7  2⌊r2⌋  R(dr)  Rdr  1  8  2r2−1  R(dr)⊕R(dr)  Rdr  2  View Large 2.2. Maximal torus of Spin(r) In this subsection, we recall explicit expressions for elements of the maximal torus of the Spin group since it will be useful to consider paths within such torus. The rotation   (cos(θ1)−sin(θ1)sin(θ1)cos(θ1)1⋱1)r×rcan be achieved by using the element   e1(−cos(θ1/2)e1+sin(θ1/2)e2)=cos(θ1/2)+sin(θ1/2)e1e2∈Spin(r)as follows:   (cos(θ1/2)+sin(θ1/2)e1e2)y(cos(θ1/2)−sin(θ1/2)e1e2)=(y1cos(θ1)−y2sin(θ1))e1+(y1sin(θ1)+y2cos(θ1))e2+y3e3+⋯+yrer,for y=y1e1+⋯+yrer∈Rr. Thus, we see that the corresponding preimages in Spin(r) are exactly   ±(cos(θ1/2)+sin(θ1/2)e1e2).Furthermore, we can see that a maximal torus of Spin(r) consists of elements of the form   t(θ1,…,θ[r2])=∏j=1[r2](cos(θj/2)+sin(θj/2)e2j−1e2j),noting that the parameters θj must now run between 0 and 4π. Furthermore, using the explicit description of the isomorphisms given above, we see that   (cos(θ1/2)+sin(θ1/2)e1e2)·uε1,…,εk=cos(θ1/2)uε1,…,εk+sin(θ1/2)e1e2·uε1,…,εk=cos(θ1/2)uε1,…,εk+iεksin(θ1/2)uε1,…,εk=(cos(θ1/2)+iεksin(θ1/2))uε1,…,εk=eiεkθ12uε1,…,εk,and similarly,   t(θ1,…,θ[r2])·uε1,…,ε[r2]=ei2∑j=1[r2]εk+1−jθj·uε1,…,εk.Thus, the basis vectors uε1,…,εk are weight vectors of the standard Spin representation with weights   12∑j=1[r2]ε[r2]+1−jθj,which in coordinate vectors are the well-known expressions   (±12,±12,…,±12).Moreover, in terms of the (appropriately ordered) basis B, the matrix associated to an element t(θ1,…,θ[r2]) is   (ei2(θ1+θ2+⋯+θ[r2])ei2(−θ1+θ2+⋯+θ[r2])ei2(θ1−θ2+⋯+θ[r2])⋱ei2(−θ1−θ2+⋯+θ[r2])⋱ei2(−θ1−θ2−⋯−θ[r2])).Note that, when r is even, Δr+ is generated by the basis vectors uε1,…,εr2 with an even number of εj equal to −1, and Δr− is generated by the basis vectors uε1,…,εr2 with an odd number of εj equal to −1. Therefore, after reordering the basis, the matrix above can be split into two blocks of equal size: one block in which the exponents contain an even number of negative signs   (ei2(θ1+θ2+⋯+θr2)ei2(−θ1−θ2+⋯+θr2)ei2(−θ1+θ2−⋯+θr2)⋱)and another block in which the exponents contain an odd number of negative signs   (ei2(−θ1+θ2+⋯+θr2)ei2(θ1−θ2+⋯+θr2)ei2(θ1+θ2−θ3+⋯+θr2)⋱). 2.3. Even-Clifford structures Linear even-Clifford Hermitian structures Definition 2.1 Let N∈N and (e1,…,er) an orthonormal frame of Rr. A linear even-Clifford structure of rank r on RN is an algebra representation   Φ:Clr0⟶End(RN). A linear even-Clifford Hermitian structure of rank r on RN (endowed with a positive definite inner product) is a linear even-Clifford structure of rank r such that each bivector eiej, 1≤i<j≤r, is mapped to a skew-symmetric endomorphism Φ(eiej)=Jij. Remarks Note that   Jij2=−IdRN. (2.1) Given a linear even-Clifford structure of rank r on RN, we can average the standard inner product ⟨,⟩ on RN as follows:   (X,Y)=∑k=1[r/2][∑1≤i1<⋯<i2k<r⟨Φ(ei1…i2k)(X),Φ(ei1…i2k)(Y)⟩],where (e1,…,er) is an orthonormal frame of Rr, so that the linear even-Clifford structure is Hermitian with respect to the averaged inner product. Given a linear even-Clifford Hermitian structure of rank r, the subalgebra spin(r) is mapped injectively into the skew-symmetric endomorphisms End−(RN). Branching of RN From now on, we will denote by Idn the identity endomorphism of a real/complex n-dimensional vector space. First, let us assume r≢0(mod4), r>1. In this case, RN decomposes into a sum of irreducible representations of Clr0. Since Clr0 is simple, its irreducible representations are either trivial or the standard representation Δ˜r of Clr0 (cf. [8]). Due to (2.1), there are no trivial summands RN, that is   RN=Rm⊗Δ˜rfor some m∈N. Thus, we see that spin(r) has an isomorphic image   spin(r)^≔Idm⊗κr*(spin(r))⊂so(drm). Secondly, let us assume r≡0(mod4). Recall that if Δˆr is the irreducible representation of Clr, then by restricting this representation to Clr0 it splits as the sum of two inequivalent irreducible representations   Δˆr=Δ˜r+⊕Δ˜r−.Since RN is a representation of Clr0 satisfying (2.1), there are no trivial summands in RN and   RN=Rm1⊗Δ˜r+⊕Rm2⊗Δ˜r−for some m1,m2∈N. By restricting this representation to spin(r)⊂Clr0, consider the isomorphic image   spin(r)^≔{Idm1⊗ξ+⊕Idm2⊗ξ−∣ξ∈spin(r),ξ+=κr*+(ξ),ξ−=κr*−(ξ)}. Almost even-Clifford Hermitian manifolds Definition 2.2 Let r≥2. A rank r almost even-Clifford structure on a smooth manifold M is a smoothly varying choice of a rank r linear even-Clifford structure on each tangent space of M. A smooth manifold carrying an almost even-Clifford structure will be called an almost even-Clifford manifold. A rank r almost even-Clifford Hermitian structure on a Riemannian manifold M is a smoothly varying choice of a linear even-Clifford Hermitian structure on each tangent space of M. A Riemannian manifold carrying an almost even-Clifford Hermitian structure will be called an almost even-Clifford Hermitian manifold. Remark Our definition of almost even-Clifford Hermitian structure implies that in [10]. 3. Complexifications, structure groups and fundamental groups In this section, we will study the complexification RN⊗C and its decomposition as a representation of the normalizers NSO(N)0(S) (cf. 3.1), where S is one of the homomorphic images of the group Spin(r) corresponding to the Lie subalgebras described in Section 2.3.2. Once we have described such complexifications and decompositions, we will compute the (connected components of the identity of the) structure groups determined by linear even-Clifford structures. They must be closed Lie subgroups of SO(N) whose Lie algebra is the aforementioned normalizer. They are actually isomorphic to quotients of products of classical compact Lie groups (or real lines) with Spin(r) (cf. 3.2). We will also compute their fundamental groups to be used in the last section. All of this will be done in a case by case analysis. Along the way, we will introduce notation that will enable us to state Theorems 3.1, 3.2 and 3.3. The main steps in each case are the following: Complexification: Identify RN⊗C as a representation   G×Spin(r)→ρSO(N)⊂Aut(RN×C),where G denotes a (semi-)simple compact Lie group or SO (2). Structure group: Compute the image of ρ, Im(ρ). Compute ker(ρ) to get   Im(ρ)≅G×Spin(r)ker(ρ). Fundamental group: Identify the universal covering Im(ρ)˜→ρ˜Im(ρ). Compute the preimage of ρ˜−1(IdN) to obtain an explicit description of the fundamental group π1(Im(ρ)). First, let us recall the following Table 2 [2], whose entries’ precise description will be recalled in each case. Table 2. Centralizers and Normalizers of spin(r) and spin±(r) in so(N). r(mod8)  N  Cso(N)(spin(r)^)  Nso(N)0(spin(r)^)  0  dr(m1+m2)  so(m1)^⊕so(m2)^  so(m1)^⊕so(m2)^⊕spin(r)^  1,7  drm  so(m)^  so(m)^⊕spin(r)^  2,6  drm  u(m)^  u(m)^⊕spin(r)^  3,5  drm  sp(m)^  sp(m)^⊕spin(r)^  4  dr(m1+m2)  sp(m1)^⊕sp(m2)^  sp(m1)^⊕sp(m2)^⊕spin(r)^  r(mod8)  N  Cso(N)(spin(r)^)  Nso(N)0(spin(r)^)  0  dr(m1+m2)  so(m1)^⊕so(m2)^  so(m1)^⊕so(m2)^⊕spin(r)^  1,7  drm  so(m)^  so(m)^⊕spin(r)^  2,6  drm  u(m)^  u(m)^⊕spin(r)^  3,5  drm  sp(m)^  sp(m)^⊕spin(r)^  4  dr(m1+m2)  sp(m1)^⊕sp(m2)^  sp(m1)^⊕sp(m2)^⊕spin(r)^  View Large 3.1. r≡1,7(mod8) Complexification. In this case, Δ˜r is the subspace of Δr fixed by the corresponding real structure γr, that is   Δ˜r⊗C=Δr.The centralizer subalgebra of spin(r)^ in so(N) is   Cso(N)(spin(r)^)=so(m)⊗IdΔ˜r≕so(m)^,where N=drm. If z∈C,v⊗ψ∈Rm⊗Δ˜r and A∈so(m),   (A⊗IdΔ˜r)(zv⊗ψ)=zAv⊗ψ,which means   (Rm⊗Δ˜r)⊗C=Cm⊗Δr,where Cm denotes the standard complex representation of SO (m). Thus, we have a representation   SO(m)×Spin(r)→ρSO(N)⊂Aut(Cm⊗Δr). Structure group. Since so(m)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). The exponential of A⊗IdΔ˜r∈so(m)^ gives   eA⊗IdΔ˜r∈SO(m)^≔SO(m)⊗IdΔ˜r≅SO(m).On the other hand, if Idm⊗ξ∈spin(r)^, its exponential is   Idm⊗eξ∈Idm⊗κ(Spin(r))≕Spin(r)^≅Spin(r),since Spin(r) is represented faithfully on Δr. The image of SO(m)×Spin(r) in SO(N) under the aforementioned representation is   NSO(N)0(Spin(r)^)=SO(m)^Spin(r)^,the subgroup of all possible products of elements of the two subgroups, that is we have   SO(m)×Spin(r)→ρSO(m)^Spin(r)^⊂SO(N).Now we need to find ker(ρ) and identify SO(m)^Spin(r)^ as a quotient   SO(m)^Spin(r)^≅SO(m)×Spin(r)ker(ρ).If there are elements g∈SO(m) and h∈Spin(r) such that   ρ(g,h)=IdN,then   Spin(r)^∋ρ(Idm,h)=ρ(g,1)−1∈SO(m)^.Since ρ(Idm,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))=Z2={±1}.Note that −1∈Spin(r) maps to −IdΔr under the Spin(r) representation on Δr, and (Idm,−1) maps to −Idm⊗IdΔr∈SO(N) under ρ. Moreover, −Idm⊗IdΔr belongs to SO(m)^ only if m is even. Thus, ker(ρ)={±(Idm,1)}=Z2 and   SO(m)^Spin(r)^≅SO(m)×Spin(r)Z2if m is even, and ker(ρ)={(Idm,1)},   SO(m)^Spin(r)^≅SO(m)×Spin(r)if m is odd. Fundamental group. Clearly, we only need to deal with the case when m is even. If m≥4, let ρ˜ denote the following composition:   Spin(m)×Spin(r)↓SO(m)×Spin(r)↓SO(m)×Z2Spin(r).We need to find all the elements of Spin(m)×Spin(r) that map to   ±(Idm,1).The elements of Spin(m)×Spin(r) that map to (Idm,1)∈SO(m)×Spin(r) are   (±1,1),and the elements of Spin(m)×Spin(r) that map to (−Idm,−1)∈SO(m)×Spin(r) are   (±volm,−1),that is   ker(ρ˜)={(1,1),(−1,1),(volm,−1),(−volm,−1)}.π1(SO(m)^Spin(r)^)≅{Z2⊕Z2ifm≡0(mod4),Z4ifm≡2(mod4). If m = 2, let ρ˜ denote the following composition:   R×Spin(r)↓SO(2)×Spin(r)↓SO(2)×Z2Spin(r).Similarly,   ker(ρ˜)={(2kπ,1)∣k∈Z}∪{((2k+1)π,−1)∣k∈Z},π1(SO(2)^Spin(r)^)≅Z. 3.2. r≡0(mod8) Complexification. In this case, Δ˜r+ and Δ˜r− are the subspaces of Δr+ and Δr− fixed by the corresponding real structure γr mentioned in Section 2, and   Δ˜r+⊗C=Δr+,Δ˜r−⊗C=Δr−.The centralizer subalgebra of spin(r)^ is   Cso(N)(spin(r)^)=so(m1)^⊕so(m2)^=(so(m1)^so(m2)^)≔(so(m1)⊗IdΔ˜r+so(m2)⊗IdΔ˜r−),where N=dr(m1+m2). If A1∈so(m1), A2∈so(m2), z1,z2∈C, v1⊗ψ1+v2⊗ψ2∈Rm1⊗Δ˜r+⊕Rm1⊗Δ˜r−,  (A1⊗IdΔ˜r+A2⊗IdΔ˜r−)(z1v1⊗ψ1z2v2⊗ψ2)=(z1A1v1⊗ψ1z2A2v2⊗ψ2),which means   (Rm1⊗Δ˜r+⊕Rm2⊗Δ˜r−)⊗C=Cm1⊗Δr+⊕Cm2⊗Δr−,where Cm1 and Cm2 denote the standard complex representation of so(m1) and so(m2), respectively. Thus, we have a representation   SO(m1)×SO(m2)×Spin(r)⟶SO(N)⊂Aut(Cm1⊗Δr+⊕Cm2⊗Δr−). Structure group Since so(m1)^⊕so(m2)^ and spin(r)^ commute, we can take the exponentials of their elements separately within C(N). The exponential of   (A1⊗IdΔr+A2⊗IdΔr−)∈so(m1)^⊕so(m2)^ is   (eA1⊗IdΔr+eA2⊗IdΔr−)∈(SO(m1)^SO(m2)^)≔(SO(m1)⊗IdΔr+SO(m2)⊗IdΔr−),=SO(m1)^×SO(m2)^. On the other hand, if Idm1⊗ξ+⊕Idm2⊗ξ−∈spin(r)^, its exponential is   (Idm1⊗eξ+Idm2⊗eξ−)∈(Idm1⊗κ+(Spin(r))Idm2⊗κ−(Spin(r)))={Spin(r)^≅Spin(r)ifm1>0andm2>0,Spin(r)^+≅κ+(Spin(r))ifm1>0andm2=0,Spin(r)^−≅κ−(Spin(r))ifm1=0andm2>0, where the first representation of Spin(r) is faithful and the last two are not, with   Spin(r)^±≅κ±(Spin(r))≅Spin(r){1,±volr}. The image of SO(m1)×SO(m2)×Spin(r) in SO(N) under the aforementioned representations are   NSO(N)0(Spin(r)^)=(SO(m1)^×SO(m2)^)Spin(r)^,NSO(N)0(Spin(r)^+)=SO(m1)^Spin(r)^+,NSO(N)0(Spin(r)^−)=SO(m2)^Spin(r)^−, respectively, that is in each case we have a map   SO(m1)×SO(m2)×Spin(r)→ρ(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N),SO(m1)×Spin(r)→ρSO(m1)^Spin(r)^−⊂SO(N),SO(m2)×Spin(r)→ρSO(m2)^Spin(r)^+⊂SO(N). Now we need to find ker(ρ) in each case to identify the relevant group as a quotient. Case m1,m2>0. If there are elements gi∈SO(mi) and h∈Spin(r) such that   ρ(g1,g2,h)=IdN,then   Spin(r)^∋ρ(Idm1,Idm2,h)=ρ(g1,g2,1)−1∈SO(m1)^×SO(m2)^.Since ρ(Idm1,Idm2,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))={1,−1,volr,−volr}≅Z2⊕Z2. Note that the element −1 is mapped to −IdΔr± in the Spin(r) representations on Δr±, and (Idm1,Idm2,−1) maps to −(Idm1⊗IdΔr+⊕Idm2⊗IdΔr−)∈SO(N); the element −(Idm1⊗IdΔr+⊕Idm2⊗IdΔr−) belongs to SO(m1)^×SO(m2)^ if m1≡m2≡0(mod2); the element volr is mapped to ±IdΔr±, and (Idm1,Idm2,volr) maps to (Idm1⊗IdΔr+⊕(−1)Idm2⊗IdΔr−)∈SO(N); the element (Idm1⊗IdΔr+⊕(−1)Idm2⊗IdΔr−) belongs to SO(m1)^×SO(m2)^ if m2≡0(mod2); the element −volr is mapped to ∓IdΔr±, and (Idm1,Idm2,−volr) maps to ((−1)Idm1⊗IdΔr+⊕Idm2⊗IdΔr−)∈SO(N);the element ((−1)Idm1⊗IdΔr+⊕Idm2⊗IdΔr−) belongs to SO(m1)^×SO(m2)^ if m1≡0(mod2). Thus, if m1≡m2≡1(mod2),   ker(ρ)={(Idm1,Idm2,1)},(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r); if m1≡0(mod2), m2≡1(mod2),   ker(ρ)={(Idm1,Idm2,1),(−Idm1,Idm2,−volr)}≅Z2,(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r)Z2; if m1≡1(mod2), m2≡0(mod2),   ker(ρ)={(Idm1,Idm2,1),(Idm1,−Idm2,vol)}≅Z2,(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r)Z2; if m1≡m2≡0(mod2),   ker(ρ)={(Idm1,Idm2,1),(−Idm1,−Idm2,−1),(Idm1,−Idm2,vol),(−Idm1,Idm2,−vol)}≅Z2⊕Z2,(SO(m1)^×SO(m2)^)Spin(r)^≅SO(m1)×SO(m2)×Spin(r)Z2⊕Z2. Case m1>0, m2=0. If there are elements g1∈SO(m1) and h∈Spin(r) such that   ρ(g1,h)=IdN, then   Spin(r)^+∋ρ(Idm1,h)=ρ(g1,1)−1∈SO(m1)^. Since ρ(Idm1,h) commutes with every element of Spin(r)^+, it belongs to its center Z(Spin(r)^+)≅Z(Spin(r)/{1,volr})={1,−1,volr,−volr}/{1,volr}≅Z2.Note that the element −1 is mapped to −IdΔr+ in the Spin(r) representation on Δr+, and (Idm1,−1) maps to −Idm1⊗IdΔr+∈SO(N); the element −Idm1⊗IdΔr+ belongs to SO(m1)^ if m1≡0(mod2). Thus, (5)if m1≡1(mod2),   ker(ρ)={(Idm1,1),(Idm1,volr)},SO(m1)^Spin(r)^+≅SO(m1)×Spin(r)Z2; (6)if m1≡0(mod2),   ker(ρ)={(Idm1,1),(−Idm1,−1),(Idm1,volr),(−Idm1,−volr)},SO(m1)^Spin(r)^+≅SO(m1)×Spin(r)Z2⊕Z2. Case m1=0, m2>0. If there are elements g2∈SO(m2) and h∈Spin(r) such that   ρ(g2,h)=IdN,then   Spin(r)^−∋ρ(Idm2,h)=ρ(g2,1)−1∈SO(m2)^.Since ρ(Idm2,h) commutes with every element of Spin(r)^−, it belongs to its center Z(Spin(r)^−)≅Z(Spin(r)/{1,−volr})={1,−1,volr,−volr}/{1,−volr}≅Z2. Note that the element −1 is mapped to −IdΔr− in the Spin(r) representation on Δr−, and (Idm2,−1) maps to −Idm2⊗IdΔr−∈SO(N); the element −Idm2⊗IdΔr− belongs to SO(m2)^ if m2≡0(mod2); Thus, (7)if m2≡1(mod2),   ker(ρ)={(Idm2,1),(Idm2,−volr)},SO(m2)^Spin(r)^−≅SO(m2)×Spin(r)Z2; (8)if m2≡0(mod2),   ker(ρ)={(Idm2,1),(−Idm2,−1),(−Idm2,volr),(Idm2,−volr)},SO(m2)^Spin(r)^−≅SO(m2)×Spin(r)Z2⊕Z2. Fundamental group. We will now analyze each of the previous eight cases: Recall that m1,m2>0 and m1≡m2≡1 (mod 2). If m1,m2≥3, let ρ˜ denote the following map:   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r).Thus,   ker(ρ˜)={(1,1,1),(−1,1,1),(1,−1,1),(−1,−1,1)},π1((SO(m1)^×SO(m2)^)Spin(r)^)≅Z2⊕Z2. If m1=1,m2≥3, SO(m1)={Id1} and let ρ˜ denote the following map:   {Id1}×Spin(m2)×Spin(r)↓{Id1}×SO(m2)×Spin(r).Thus   ker(ρ˜)={(Id1,1,1),(Id1,−1,1)},π1((SO(1)^×SO(m2)^)Spin(r)^)≅Z2. If m1≥3,m2=1, let ρ˜ denote the following map:   Spin(m1)×{Id1}×Spin(r)↓SO(m1)×{Id1}×Spin(r).Thus,   ker(ρ˜)={(1,Id1,1),(−1,Id1,1)},π1((SO(m1)^×SO(1)^)Spin(r)^)≅Z2. If m1=1,m2=1, let ρ˜ be   {Id1}×{Id1}×Spin(r)↓{Id1}×{Id1}×Spin(r).Thus,   ker(ρ˜)={(Id1,Id1,1)},π1((SO(1)^×SO(1)^)Spin(r)^)≅{1}. Recall that m1,m2>0, m1≡0(mod2), m2≡1(mod2). If m1≥4,m2≥3, let ρ˜ denote the composition   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(1,−1,1),(−1,1,1),(volm1,1,−volr)⟩ifm1≡0(mod4),⟨(1,−1,1),(volm1,1,−volr)⟩ifm1≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)≅{Z2⊕Z2⊕Z2ifm1≡0(mod4),Z2⊕Z4ifm1≡2(mod4). If m1=2,m2≥3, SO(m1)=SO(2) and let ρ˜ denote the following composition:   R×Spin(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(0,−1,1),(π,1,−volr)⟩,π1((SO(2)^×SO(m2)^)Spin(r)^)=Z2⊕Z. If m1≥4,m2=1, let ρ˜ denote the composition   Spin(m1)×{Id1}×Spin(r)↓SO(m1)×{Id1}×Spin(r)↓SO(m1)×{Id1}×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(−1,Id1,1),(volm1,Id1,−volr)⟩ifm1≡0(mod4),⟨(volm1,Id1,−volr)⟩ifm1≡2(mod4),π1((SO(m1)^×SO(1)^)Spin(r)^)={Z2⊕Z2ifm1≡0(mod4),Z4ifm1≡2(mod4). If m1=2,m2=1, let ρ˜ denote the composition   R×{Id1}×Spin(r)↓SO(2)×{Id1}×Spin(r)↓SO(2)×{Id1}×Spin(r)Z2.Thus   ker(ρ˜)=⟨(π,Id1,−volr)⟩,π1((SO(2)^×SO(1)^)Spin(r)^)≅Z. Recall that m1,m2>0, m1≡1(mod2), m2≡0(mod2). If m1≥3,m2≥4, let ρ˜ denote the composition   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(−1,1,1),(1,−1,1),(1,volm2,volr)⟩ifm2≡0(mod4),⟨(−1,1,1),(1,volm2,volr)⟩ifm2≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)={Z2⊕Z2⊕Z2ifm2≡0(mod4),Z2⊕Z4ifm2≡2(mod4). If m1≥3,m2=2, let ρ˜ denote the composition   Spin(m1)×R×Spin(r)↓SO(m1)×SO(2)×Spin(r)↓SO(m1)×SO(2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(−1,0,1),(1,π,volr)⟩,π1((SO(m1)^×SO(2)^)Spin(r)^)≅Z2⊕Z. If m1=1,m2≥4, let ρ˜ denote the composition   {Id1}×Spin(m2)×Spin(r)↓{Id1}×SO(m2)×Spin(r)↓{Id1}×SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)={⟨(Id1,−1,1),(Id1,volm2,volr)⟩ifm2≡0(mod4),⟨(Id1,volm2,volr)⟩ifm2≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)={Z2⊕Z2ifm2≡0(mod4),Z4ifm2≡2(mod4). If m1=1,m2=2, let ρ˜ denote the composition   {Id1}×R×Spin(r)↓{Id1}×SO(2)×Spin(r)↓{Id1}×SO(2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(Id1,π,volr)⟩,π1((SO(1)^×SO(2)^)Spin(r)^)≅Z. Recall that m1,m2>0, m1≡m2≡0(mod2). If m1,m2≥4, let ρ˜ denote the composition   Spin(m1)×Spin(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)↓SO(m1)×SO(m2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,1,1),(1,−1,1),(1,volm2,volr),(volm1,1,−volr)⟩ifm1≡m2≡0(mod4),⟨(−1,1,1),(volm1,1,−volr),(1,volm2,volr)⟩ifm1≡0(mod4)andm2≡2(mod4),⟨(1,−1,1),(volm1,1,−volr),(1,volm2,volr)⟩ifm2≡2(mod4)andm2≡0(mod4),⟨(volm1,1,−volr),(1,volm2,volr)⟩ifm1≡m2≡2(mod4),π1((SO(m1)^×SO(m2)^)Spin(r)^)≅{Z2⊕Z2⊕Z2⊕Z2ifm1≡m2≡0(mod4),Z2⊕Z2⊕Z4ifm1≡0(mod4)andm2≡2(mod4),Z2⊕Z4⊕Z2ifm2≡2(mod4)andm2≡0(mod4),Z4⊕Z4ifm1≡m2≡2(mod4). If m1≥4,m2=2, let ρ˜ denote the composition   Spin(m1)×R×Spin(r)↓SO(m1)×SO(2)×Spin(r)↓SO(m1)×SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,0,1),(volm1,0,−volr),(1,π,volr)⟩ifm1≡0(mod4),⟨(volm1,0,−volr),(1,π,volr)⟩ifm1≡2(mod4),π1((SO(m1)^×SO(2)^)Spin(r)^)≅{Z2⊕Z2⊕Zifm1≡0(mod4),Z4⊕Zifm1≡2(mod4). If m1=2,m2≥4, let ρ˜ denote the composition   R×Spin(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)↓SO(2)×SO(m2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(0,−1,1),(0,volm2,volr),(π,1,−volr)⟩ifm2≡0(mod4),⟨(0,volm2,volr),(π,1,−volr)⟩ifm2≡2(mod4),π1((SO(2)^×SO(m2)^)Spin(r)^)≅{Z2⊕Z2⊕Zifm2≡0(mod4),Z4⊕Zifm2≡2(mod4). If m1=m2=2, let ρ˜ denote the composition   R×R×Spin(r)↓SO(2)×SO(2)×Spin(r)↓SO(2)×SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)=⟨(0,π,volr),(π,0,−volr)⟩,π1((SO(2)^×SO(2)^)Spin(r)^)≅Z⊕Z. Recall that m1>0,m2=0, m1≡1(mod2). If m1≥3, let ρ˜ denote the composition   Spin(m1)×Spin(r)↓SO(m1)×Spin(r)↓SO(m1)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(−1,1),(1,volr)⟩,π1(SO(m1)^Spin(r)^+)≅Z2⊕Z2. If m1=1, let ρ˜ denote   {Id1}×Spin(r)↓{Id1}×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(Id1,volr)⟩,π1(SO(1)^Spin(r)^+)≅Z2. Recall that m1>0,m2=0, m1≡0(mod2). If m1≥4, let ρ˜ denote the composition   Spin(m1)×Spin(r)↓SO(m1)×Spin(r)↓SO(m1)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,1),(volm1,−1),(1,volr)⟩ifm1≡0(mod4),⟨(volm1,−1),(1,volr)⟩ifm1≡2(mod4),π1(SO(m1)^Spin(r)^+)≅{Z2⊕Z2⊕Z2ifm1≡0(mod4),Z2⊕Z4ifm1≡2(mod4). If m1=2, let ρ˜ denote the composition   R×Spin(r)↓SO(2)×Spin(r)↓SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)=⟨(π,−1),(0,volr)⟩,π1(SO(2)^Spin(r)^+)≅Z⊕Z2. Recall that m1=0,m2>0, m2≡1 (mod 2). If m2≥3, let ρ˜ denote the composition   Spin(m2)×Spin(r)↓SO(m2)×Spin(r)↓SO(m2)×Spin(r)Z2.Thus,   ker(ρ˜)=⟨(1,−volr)⟩,π1(SO(1)^Spin(r)^−)≅Z2. Recall that m1=0,m2>0, m2≡0 (mod 2). If m2≥4, let ρ˜ denote the composition   Spin(m2)×Spin(r)↓SO(m2)×Spin(r)↓SO(m2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)={⟨(−1,1),(volm2,−1),(1,−volr)⟩ifm2≡0(mod4),⟨(volm2,−1),(1,−volr)⟩ifm2≡2(mod4),π1(SO(m2)^Spin(r)^−)≅{Z2⊕Z2⊕Z2ifm2≡0(mod4),Z2⊕Z4ifm2≡2(mod4). If m2=2, let ρ˜ denote the composition   R×Spin(r)↓SO(2)×Spin(r)↓SO(2)×Spin(r)Z2⊕Z2.Thus,   ker(ρ˜)=⟨(π,−1),(0,−volr)⟩,π1(SO(2)^Spin(r)^−)≅Z⊕Z2. 3.3. r≡2,6(mod8) Complexification. The volume form volr=e1…er acts as a complex structure J on Δ˜r (cf. [2]). Therefore,   Δ˜r⊗C={ψ−iJψ∣ψ∈Δ˜r}⊕{ψ+iJψ∣ψ∈Δ˜r}.Note that the involution (−i)r/2volr· acts on Δ˜r⊗C as follows: if r≡2(mod8),   (−i)r/2volr·(ψ−iJψ)=(ψ−iJψ),(−i)r/2volr·(ψ+iJψ)=−(ψ+iJψ),that is   Δ˜r⊗C={ψ−iJψ∣ψ∈Δ˜r}⊕{ψ+iJψ∣ψ∈Δ˜r}=Δr+⊕Δr−, if r≡6(mod8),   (−i)r/2volr·(ψ−iJψ)=−(ψ−iJψ),(−i)r/2volr·(ψ+iJψ)=(ψ+iJψ),that is   Δ˜r⊗C={ψ−iJψ∣ψ∈Δ˜r}⊕{ψ+iJψ∣ψ∈Δ˜r}=Δr−⊕Δr+.In any case,   Δ˜r⊗C=Δr.The centralizer subalgebra of spin(r)^ in so(N) is   Cso(N)(spin(r)^)=u(m)^=so(m)⊗IdΔ˜r⊕S2Rm⊗J,where N=drm, so(m) and S2Rm act on Rm as skew-symmetric and symmetric endomorphisms, respectively. Let   v⊗(ψ−iJψ)∈(Rm⊗Δ˜r)⊗Cand   A⊗IdΔ˜r+B⊗J∈so(m)⊗IdΔ˜r⊕S2Rm⊗J.Now,   (A⊗IdΔ˜r+B⊗J)(v⊗(ψ−iJψ))=Av⊗ψ−iAv⊗Jψ+Bv⊗Jψ−iBv⊗JJψ=Av⊗ψ+iBv⊗ψ+(−i)(i)Bv⊗Jψ−iAv⊗Jψ=(A+iB)v⊗(ψ−iJψ),where A+iB∈u(m). Similarly, for v⊗(ψ+iJψ),   (A⊗IdΔ˜r+B⊗J)(v⊗(ψ+iJψ))=(A−iB)v⊗(ψ+iJψ).Thus,   (Rm⊗Δ˜r)⊗C={Cm⊗Δr+⊕Cm¯⊗Δr−ifr≡2(mod8),Cm⊗Δr−⊕Cm¯⊗Δr+ifr≡6(mod8), (3.1)where Cm is the standard representation of U(m). Therefore, we have a representation   U(m)×Spin(r)⟶SO(N)⊂Aut((Rm⊗Δ˜r)⊗C). Structure group. Since u(m)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). With respect to (3.1), an element A⊗IdΔ˜r+B⊗J∈u(m) looks as follows:   ((A+iB)⊗IdΔr±(A−iB)⊗IdΔr∓),so that the exponentials form   {(eA+iB⊗IdΔr±eA−iB⊗IdΔr∓):A∈so(m),B∈S2Rm}≕U(m)^.With respect to (3.1), an element Idm⊗ξ∈spin(r)^ looks as follows:   (Idm⊗κr*+(ξ)Idm⊗κr*−(ξ))=Idm⊗ξ,and its exponential is   Idm⊗eξ∈Idm⊗κ(Spin(r))≕Spin(r)^≅Spin(r),since Spin(r) is represented faithfully on Δr. The image of U(m)×Spin(r) in SO(N)⊂Aut((Rm⊗Δ˜r)⊗C) under the aforementioned representation is   NSO(N)0(Spin(r)^)=U(m)^Spin(r)^,the subgroup of all possible products of elements of the two subgroups, that is we have a map   U(m)×Spin(r)→ρU(m)^Spin(r)^⊂SO(N).Now we need to find ker(ρ) and identify U(m)^Spin(r)^ as a quotient   U(m)^Spin(r)^≅U(m)×Spin(r)ker(ρ).If there are elements g∈U(m) and h∈Spin(r) such that   ρ(g,h)=IdN,then   Spin(r)^∋ρ(Idm,h)=ρ(g,1)−1∈U(m)^.Since ρ(Idm,h) commutes with every element of Spin(r)^, it belongs to the center Z(Spin(r)^)≅Z(Spin(r))={1,−1,volr,−volr}=⟨volr⟩≅Z4.Recall that volr=e1…er acts as ∓i on Δr± if r≡2(mod8), and as ±i on Δr± if r≡6(mod8), so that it maps to   ∓(iIdΔr+⊕(−i)IdΔr−)in the complex Spin(r) representation. Note that (Idm,volr) maps to   (−i)Idm⊗IdΔr+⊕(i)Idm⊗IdΔr−ifr≡2(mod8),(i)Idm⊗IdΔr+⊕(−i)Idm⊗IdΔr−ifr≡6(mod8),in SO(N), and that ((−i)IdCm,1)∈U(m)×Spin(r) maps to such transformations in both cases. Thus, the elements of U(m)×Spin(r) mapping to IdN are   ±(Idm,1),±(iIdm,−volr),which form a copy of Z4 and   U(m)^Spin(r)^≅U(m)×Spin(r)Z4. Fundamental group. Let   R×SU(m)×Spin(r)→ρ˜U(m)^Spin(r)^(t,A,g)↦(eitA⊗κr±(g)e−itA¯⊗κr∓(g)).Thus   ker(ρ˜)=⟨(2πm,e−2πimIdm,1),(π2,Idm,−vol)⟩,π1(U(m)^Spin(r)^)≅{Z,if(m,4)=1,Z⊕Z2,if(m,4)=2,Z⊕Z4,if(m,4)=4.Indeed, let   a≔(2πm,e−2πimIdm,1),b≔(π2,Idm,−vol),and note that (in multiplicative notation)   am=b4.Moreover, If (m,4)=1, there exist t,m∈Z coprime such that   tm+s4=1.The element   btasis such that   (btas)m=bmt(b4)s=b,(btas)4=(am)ta4s=a. If (m,4)=2, m=4k+2 and there exist two generators   c=a−(2k+1)b2,d=ba−k,such that   c2=1,a=d2c,b=d2k+1ck. If (m,4)=4, m=4k and we have two generators   aandc=a−kb,such that   c4=1. 3.4. r≡3,5(mod8) Complexification. In this case, Δ˜r admits three complex structures I,J and K, which behave like quaternions and commute with spin(r) as described in [2]. Indeed, for r≡3(mod8), consider Rr⊂Rr+3 so that the complex structures are induced by Clifford multiplication with the elements   12(1+e1…er)er+1er+2,12(1+e1…er)er+1er+3,12(1+e1…er)er+2er+3.For r≡5(mod8), consider Rr⊂Rr+2 so that the complex structures are induced by Clifford multiplication with the elements   er+1er+2,e1⋯er+1,e1⋯erer+2.Let us consider the complexification of Δ˜r and decompose it as follows:   Δ˜r⊗C={ψ−iIψ∣ψ∈Δ˜r}⊕{ψ+iIψ∣ψ∈Δ˜r},where the first and second subspaces are the +i and −i eigenspaces of I, respectively. Notice that   J(ψ∓iIψ)=Jψ∓iJIψ=Jψ±iIJψ,that is J interchanges the two subspaces and squares to −Iddr. For any ξ∈spin(r)  ξ(ψ±iIψ)=ξψ±iξIψ=ξψ±iIξψ,which means that the subspaces {ψ−iIψ∣ψ∈Δ˜r} and {ψ+iIψ∣ψ∈Δ˜r} are irreducible complex representations of spin(r) of dimension dr/2. Thus, they are isomorphic to Δr as spin(r) representations and   Δ˜r⊗C≅Δr⊕Δr.Now recall that the centralizer subalgebra of spin(r)^ is   Cso(N)(spin(r)^)=so(m)⊗IdΔ˜r⊕S2Rm⊗I⊕S2Rm⊗J⊕S2Rm⊗K≅so(m)⊗IdΔ˜r⊕S2Rm⊗sp(1)≅sp(m),where N=drm. Let us consider the complexification of Rm⊗Δ˜r and decompose it   (Rm⊗Δ˜r)⊗C={v⊗(ψ−iIψ)∣v∈Rm,ψ∈Δ˜r}⊕{v⊗(ψ+iIψ)∣v∈Rm,ψ∈Δ˜r},where the first and second subspaces are the +i and −i eigenspaces of Idm⊗I, respectively. Notice that   (Idm⊗J)(v⊗(ψ∓iIdm⊗Iψ))=v⊗Jψ∓iv⊗JIψ=v⊗Jψ±iv⊗IJψ,that is Idm⊗J interchanges the two subspaces and squares to −IdN. For any Idm⊗ξ∈spin(r)^  (Idm⊗ξ)(v⊗(ψ±iIψ))=v⊗(ξψ±iξIψ)=v⊗(ξψ±iIξψ),which means that the subspaces {v⊗(ψ−iIψ)∣v∈Rm,ψ∈Δ˜r} and {v⊗(ψ+iIψ)∣v∈Rm,ψ∈Δ˜r} are isomorphic to Cm⊗Δr as spin(r)^ representations. Now consider   A⊗IdΔ˜r+B⊗I+C⊗J+D⊗K∈so(m)⊗IdΔ˜r⊕S2Rm⊗I⊕S2Rm⊗J⊕S2Rm⊗K=sp(m),and   (A⊗IdΔ˜r+B⊗I+C⊗J+D⊗K)(v⊗(ψ+iIψ))=Av⊗(ψ+iIψ)+Bv⊗(Iψ+iIIψ)+Cv⊗(Jψ+iJIψ)+Dv⊗(Kψ+iKIψ)=((A−iB)⊗IdΔ˜r+(C+iD)⊗J)(v⊗(ψ+iIψ)).Similarly,   (A⊗IdΔ˜r+B⊗I+C⊗J+D⊗K)(v⊗(ψ−iIψ))=((A+iB)⊗IdΔ˜r+(C−iD)⊗J)(v⊗(ψ−iIψ)).If C=D=0, the subalgebra   u(m)I^={A⊗IdΔ˜r+B⊗I∈so(m)⊗IdΔ˜r⊕S2Rm⊗I∣A∈so(m),B∈S2Rm}is represented as follows:   Δ˜r⊗C=CIm⊗Δr⊕CIm¯⊗Δr,=(CIm⊕CIm¯)⊗Δr,where CIm and CIm¯ denote the standard representation of u(m)I^ and its conjugate, respectively. Since Idm⊗J interchanges the two summands, squares to −IdN and commutes with the action of spin(r)^, we have the standard complex representation of sp(m) as a factor   Δ˜r⊗C=C2m⊗Δr. (3.2)Thus, we have a representation   Sp(m)×Spin(r)⟶SO(N)⊂Aut(C2m⊗Δr). Structure group. Since sp(m)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). By considering (3.2), the exponential of an element Ω⊗IdΔr∈sp(m)⊗IdΔr=spin(r)^ is   eΩ⊗IdΔ˜r∈Sp(m)^=Sp(m)⊗IdΔr≅Sp(m).On the other hand, if Id2m⊗ξ∈spin(r)^, its exponential is   Id2m⊗eξ∈Id2m⊗κ(Spin(r))=Spin(r)^≅Spin(r),since Spin(r) is represented faithfully on Δr. The image of Sp(m)×Spin(r) in SO(N)⊂Aut(C2m⊗Δr) under the aforementioned representation is   NSO(N)0(Spin(r)^)=Sp(m)^Spin(r)^,the subgroup of all possible products of elements of the two subgroups, that is we have a map   Sp(m)×Spin(r)→ρSp(m)^Spin(r)^⊂SO(N).Now we need to find ker(ρ) and identify Sp(m)^Spin(r)^ as a quotient   Sp(m)^Spin(r)^≅Sp(m)×Spin(r)ker(ρ).If there are elements g∈Sp(m) and h∈Spin(r) such that   ρ(g,h)=IdN,then   Spin(r)^∋ρ(Id2m,h)=ρ(g,1)−1∈Sp(m)^.Since ρ(Id2m,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))=Z2={±1}. Note that −1 is mapped to −IdΔr under the Spin(r) representation, and that (Id2m,−1) maps to −Id2m⊗IdΔr∈SO(N) under ρ. Note that −Id2m⊗IdΔr also belongs to Sp(m)^ being the image of (−Id2m,1)∈Sp(m)×Spin(r). Thus,   ker(ρ)={±(Id2m,1)}≅Z2,  Sp(m)^Spin(r)^≅Sp(m)×Spin(r)Z2. Fundamental group. Clearly,   π1(Sp(m)^Spin(r)^)=Z2. 3.5. r≡4(mod8) Recall from [2] that   Δ˜r±=12(1±e1⋯er)Δ˜r+3.In this case, Δ˜r± admits three complex structures I±,J± and K± induced by Clifford multiplication with the elements 12(1±e1…er)er+1er+2, 12(1±e1…er)er+1er+3 and 12(1±e1…er)er+2er+3, respectively. Just as in the previous case,   Δ˜r+⊗C={ψ−iI+ψ∣ψ∈Δ˜r+}⊕{ψ+iI+ψ∣ψ∈Δ˜r+},and both summands are isomorphic to Δr−. Indeed, if   ψ=12(1+e1…er)·ϕ∈Δ˜r+,then,   (−i)r/2(e1…er)·(ψ±iI+ψ)=−(ψ±iI+ψ),that is ψ±iI+ψ∈Δr−. In other words,   Δ˜r+⊗C=Δr−.Similarly,   Δ˜r−⊗C=Δr+.The rest of the proof proceeds as in the previous case,   (Rm1⊗Δ˜r+⊕Rm2Δ˜r−)⊗C=C2m1⊗Δr−⊕C2m2Δr+,and we have a representation   Sp(m1)×Sp(m2)×Spin(r)⟶SO(N)⊂Aut(CN),where N=dr(m1+m2). Structure group. Since sp(m1)^⊕sp(m2)^ and spin(r)^ commute with each other, we can take separately the exponentials of their elements within C(N). The exponential of Ω1⊗IdΔr−⊕Ω2⊗IdΔr+∈sp(m1)^⊕sp(m2)^ gives   (eΩ1⊗IdΔr−eΩ2⊗IdΔr+)∈(Sp(m1)^Sp(m2)^)=(Sp(m1)⊗IdΔr−Sp(m2)⊗IdΔr+)≅Sp(m1)×Sp(m2).On the other hand, if Id2m1⊗ξ−⊕Id2m2⊗ξ+∈spin(r)^, its exponential is   (Id2m1⊗eξ−Id2m2⊗eξ+)∈(Id2m1⊗κ−(Spin(r))Id2m2⊗κ+(Spin(r)))={Spin(r)^≅Spin(r)ifm1>0andm2>0,Spin(r)^−≅κ−(Spin(r))ifm1>0andm2=0,Spin(r)^+≅κ+(Spin(r))ifm1=0andm2>0,where the first case is faithful and the last two are not, with   Spin(r)^±≅κ±(Spin(r))≅Spin(r){1,∓volr},ifr>4Spin(r)^±≅κ±(Spin(r))≅Spin(3),ifr=4.The images of Sp(m1)×Sp(m2)×Spin(r) in SO(N)⊂Aut(CN) under the aforementioned representations are   NSO(N)0(Spin(r)^)=(Sp(m1)^×Sp(m2)^)Spin(r)^,NSO(N)0(Spin(r)^−)=Sp(m1)^Spin(r)^−,NSO(N)0(Spin(r)^+)=Sp(m2)^Spin(r)^+,respectively, that is we have maps   Sp(m1)×Sp(m2)×Spin(r)→ρ(Sp(m1)^×Sp(m2)^)Spin(r)^⊂SO(N),Sp(m1)×Spin(r)→ρSp(m1)^Spin(r)^−⊂SO(N),Sp(m2)×Spin(r)→ρSp(m2)^Spin(r)^+⊂SO(N).Now we need to find ker(ρ) in each case to identify the relevant group as a quotient. Case m1,m2>0. If there are elements gi∈Sp(mi) and h∈Spin(r) such that   ρ(g1,g2,h)=IdN,then   Spin(r)^∋ρ(Id2m1,Id2m2,h)=ρ(g1,g2,1)−1∈Sp(m1)^×Sp(m2)^.Since ρ(Id2m1,Id2m2,h) commutes with every element of Spin(r)^, it belongs to its center Z(Spin(r)^)≅Z(Spin(r))={1,−1,volr,−volr}≅Z2⊕Z2.The element −1 is mapped to −IdΔr± in the Spin(r) representations on Δr±, and (Id2m1,Id2m2,−1) is mapped to −(Id2m1⊗IdΔr−⊕Id2m2⊗IdΔr+)∈SO(N).The element volr is mapped to ∓IdΔr± in the Spin(r) representations on Δr±, and (Id2m1,Id2m2,volr) is mapped to (Id2m1⊗IdΔr−⊕(−1)Id2m2⊗IdΔr+)∈SO(dr(m1+m2)).In this case, −(Id2m1⊗IdΔr−⊕Id2m2⊗IdΔr+) and (Id2m1⊗IdΔr−⊕(−1)Id2m2⊗IdΔr+) belong to Sp(m1)^×Sp(m2)^. Thus,   ker(ρ)={(Id2m1,Id2m2,1),(−Id2m1,−Id2m2,−1),(Id2m1,−Id2m2,volr),(−Id2m1,Id2m2,−volr)},(Sp(m1)^×Sp(m2)^)Spin(r)^≅Sp(m1)×Sp(m2)×Spin(r)Z2⊕Z2. Case m1>0, m2=0. If there are elements g1∈Sp(m1) and h∈Spin(r) such that   ρ(g1,h)=IdN,then   ρ(Id2m1,h)=ρ(g1,1)−1∈Sp(m1)^and   ρ(Id2m1,h)∈Spin(r)^−∩Sp(m1)^.Since ρ(Id2m1,h) commutes with every element of Spin(r)^−, it belongs to its center   Z(Spin(r)^−)≅{Z(κ−(Spin(r)))=Z(Spin(r)/{1,volr})={1,−1,volr,−volr}/{1,volr}≅{1,−1}≅Z2ifr>4.Z(κ−(Spin(r)))=Z({1}×Spin(3))={(1,1),(1,−1)}≅Z2ifr=4. If r>4, the element −1 is mapped to −IdΔr− in the Spin(r) representation on Δr−, and (Id2m1,−1) is mapped to −(Id2m1⊗IdΔr−)∈SO(N). In this case, −Id2m1⊗IdΔr− belongs to Sp(m1)^. Thus,   ker(ρ)={(Id2m1,1),(Id2m1,volr),(−Id2m1,−1),(−Id2m1,−volr)},Sp(m1)^Spin(r)^−≅Sp(m1)×Spin(r)Z2⊕Z2. If r=4, the element (1,−1)∈{1}×Spin(3) is mapped to −IdΔr− in the Spin(r) representation on Δr−, and (Id2m1,(1,−1)) is mapped to −(Id2m1⊗IdΔr−)∈SO(N). In this case, −Id2m1⊗IdΔr− belongs to Sp(m1)^. Thus,   ker(ρ)={(Id2m1,(1,1)),(−Id2m1,(1,−1))}×(Spin(3)×{1}),Sp(m1)^Spin(4)^−≅Sp(m1)×(Spin(3)×Spin(3)){(Id2m1,(1,1)),(−Id2m1,(1,−1))}×(Spin(3)×{1})≅Sp(m1)×Spin(3)Z2.Note that (1,−1)∈Spin(3)×Spin(3) corresponds −vol4∈Spin(4). Case m1=0, m2>0. If there are elements g2∈Sp(m2) and h∈Spin(r) such that   ρ(g2,h)=IdN,then   ρ(Id2m2,h)=ρ(g2,1)−1∈Sp(m2)^and   ρ(Id2m2,h)∈Spin(r)^+∩Sp(m2)^.Since ρ(Id2m2,h) commutes with every element of Spin(r)^+, it belongs to its center   Z(Spin(r)^−)≅{Z(κ+(Spin(r)))=Z(Spin(r)/{1,−volr})={1,−1,volr,−volr}/{1,−volr}≅{1,−1}≅Z2ifr>4.Z(κ+(Spin(r)))=Z(Spin(3)×{1})={(1,1),(−1,1)}≅Z2ifr=4. The element −1 is mapped to −IdΔr+ in the Spin(r) representation on Δr+, and (Id2m1,−1) is mapped to −(Id2m2⊗IdΔr+)∈SO(N). In this case, −Id2m2⊗IdΔr− belongs to Sp(m2)^. Thus,   ker(ρ)={(Id2m2,1),(Id2m2,−volr),(−Id2m2,−1),(−Id2m2,volr)},Sp(m2)^Spin(r)^+≅Sp(m2)×Spin(r)Z2⊕Z2. If r = 4, the element (−1,1)∈Spin(3)×{1} is mapped to −IdΔr+ in the Spin(r) representation on Δr+, and (Id2m2,(−1,1)) is mapped to −(Id2m2⊗IdΔr+)∈SO(N). In this case, −Id2m2⊗IdΔr− belongs to Sp(m2)^. Thus,   ker(ρ)={(Id2m2,(1,1)),(−Id2m2,(−1,1))}×(Spin(3)×{1}),Sp(m2)^Spin(4)^+≅Sp(m2)×(Spin(3)×Spin(3)){(Id2m2,(1,1)),(−Id2m2,(−1,1))}×({1}×Spin(3))≅Sp(m2)×Spin(3)Z2.Note that (−1,1)∈Spin(3)×Spin(3) corresponds to vol4∈Spin(4). Fundamental group. Clearly,   π1((Sp(m1)^×Sp(m2)^)Spin(r)^)=Z2⊕Z2,π1(Sp(m1)^Spin(r)^−)≅{Z2⊕Z2,ifm1>0,m2=0,r>4,Z2,ifm1>0,m2=0,r=4,π1(Sp(m2)^Spin(r)^+)≅{Z2⊕Z2,ifm1=0,m2>0,r>4,Z2ifm1=0,m2>0,r=4.□ Thus, we have proved the following three theorems. Theorem 3.1 The complexification of a real representation RNof Clr0 without trivial summands decomposes as follows r(mod8)  RN⊗C  0  Cm1⊗Δr+⊕Cm2⊗Δr−  1,7  Cm⊗Δr  2  Cm⊗Δr+⊕Cm¯⊗Δr−  6  Cm¯⊗Δr+⊕Cm⊗Δr−  3,5  C2m⊗Δr  4  C2m2⊗Δr+⊕C2m1⊗Δr−  r(mod8)  RN⊗C  0  Cm1⊗Δr+⊕Cm2⊗Δr−  1,7  Cm⊗Δr  2  Cm⊗Δr+⊕Cm¯⊗Δr−  6  Cm¯⊗Δr+⊕Cm⊗Δr−  3,5  C2m⊗Δr  4  C2m2⊗Δr+⊕C2m1⊗Δr−  where the different Cs denote the corresponding standard complex representations of the classical Lie algebras so(s),u(s)or sp(s/2).□ Theorem 3.2 The connected components of the identity NSO(N)0(Spin(r)^)of the normalizers NSO(N)(Spin(r)^)are isomorphic to the following groups: If r≡1,7(mod8), N=drmand  NSO(N)0(Spin(r)^)≅{SO(m)×Spin(r)Z2,ifmiseven,SO(m)×Spin(r),ifmisodd. If r≡0(mod8), N=dr(m1+m2)and  NSO(N)0(Spin(r)^)≅{SO(m1)×SO(m2)×Spin(r),ifm1>0,m2>0,m1≡m2≡1(mod2),SO(m1)×SO(m2)×Spin(r)Z2,ifm1>0,m2>0,m1+m2≡1(mod2),SO(m1)×SO(m2)×Spin(r)Z2⊕Z2,ifm1>0,m2>0,m1≡m2≡0(mod2),NSO(N)0(Spin(r)^+)≅{SO(m1)×Spin(r)Z2,ifm1>0,m2=0,m1≡1(mod2),SO(m1)×Spin(r)Z2⊕Z2,ifm1>0,m2=0,m1≡0(mod2),NSO(N)0(Spin(r)^−)≅{SO(m2)×Spin(r)Z2,ifm1=0,m2>0,m2≡1(mod2),SO(m2)×Spin(r)Z2⊕Z2,ifm1=0,m2>0,m2≡0(mod2). If r≡2,6(mod8), N=drmand  NSO(N)0(Spin(r)^)≅U(m)×Spin(r)Z4. If r≡3,5(mod8), N=drmand  NSO(N)0(Spin(r)^)≅Sp(m)×Spin(r)Z2. If r≡4(mod8), N=dr(m1+m2)and  NSO(N)0(Spin(r)^)≅Sp(m1)×Sp(m2)×Spin(r)Z2⊕Z2,ifm1>0,m2>0,NSO(N)0(Spin(r)^−)≅{Sp(m1)×Spin(r)Z2⊕Z2,ifm1>0,m2=0,r>4,Sp(m1)×Spin(3)Z2,ifm1>0,m2=0,r=4,NSO(N)0(Spin(r)^+)≅{Sp(m2)×Spin(r)Z2⊕Z2,ifm1=0,m2>0,r>4,Sp(m2)×Spin(3)Z2,ifm1=0,m2>0,r=4.□ Theorem 3.3 The fundamental group of the connected components of the identity of the normalizers NSO(N)0(Spin(r)^)are the following: If r≡1,7(mod8), N=drmand  π1(NSO(N)0(Spin(r)^))≅{Z2⊕Z2,ifm≥4,m≡0(mod4),Z4,ifm≥4,m≡2(mod4),Z2,ifm>1andodd,{1},ifm=1,Z,ifm=2. If r≡0(mod8), N=dr(m1+m2)and either π1(NSO(N)0(Spin(r)^))or π1(NSO(N)0(Spin(r)^+))or π1(NSO(N)0(Spin(r)^−)) are isomorphic to   m2  m1  0  1  2  1(mod2)  2(mod4)  0(mod4)  0    Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  1  Z2  {1}  Z  Z2  Z4  Z2⊕Z2  2  Z⊕Z2  Z  Z⊕Z  Z⊕Z2  Z⊕Z4  Z⊕Z2⊕Z2  1(mod2)  Z2⊕Z2  Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  2(mod4)  Z2⊕Z4  Z4  Z⊕Z4  Z2⊕Z4  Z4⊕Z4  Z2⊕Z2⊕Z4  0(mod4)  Z2⊕Z2⊕Z2  Z2⊕Z2  Z⊕Z2⊕Z2  Z2⊕Z2⊕Z2  Z2⊕Z2⊕Z4  Z2⊕Z2⊕Z2⊕Z2    m2  m1  0  1  2  1(mod2)  2(mod4)  0(mod4)  0    Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  1  Z2  {1}  Z  Z2  Z4  Z2⊕Z2  2  Z⊕Z2  Z  Z⊕Z  Z⊕Z2  Z⊕Z4  Z⊕Z2⊕Z2  1(mod2)  Z2⊕Z2  Z2  Z⊕Z2  Z2⊕Z2  Z2⊕Z4  Z2⊕Z2⊕Z2  2(mod4)  Z2⊕Z4  Z4  Z⊕Z4  Z2⊕Z4  Z4⊕Z4  Z2⊕Z2⊕Z4  0(mod4)  Z2⊕Z2⊕Z2  Z2⊕Z2  Z⊕Z2⊕Z2  Z2⊕Z2⊕Z2  Z2⊕Z2⊕Z4  Z2⊕Z2⊕Z2⊕Z2  depending on whether m1,m2>0or m1=0or m2=0, respectively. If r≡2,6(mod8), N=drmand  π1(NSO(n)0(Spin(r)^))={Z,if(m,4)=1,Z×Z2,if(m,4)=2,Z×Z4,if(m,4)=4. If r≡3,5(mod8), N=drmand  π1(NSO(N)0(Spin(r)^))=Z2. If r≡4(mod8), N=dr(m1+m2)and  π1(NSO(N)0(Spin(r)^))≅Z2⊕Z2,ifm1>0,m2>0,π1(NSO(N)0(Spin(r)−^))≅{Z2⊕Z2,ifm1>0,m2=0,r>4,Z2,ifm1>0,m2=0,r=4,π1(NSO(N)0(Spin(r)+^))≅{Z2⊕Z2,ifm1=0,m2>0,r>4,Z2,ifm1=0,m2>0,r=4.□ 4. Lifting maps to the Spin group In this section, we will check how the generators of the fundamental groups π1(NSO(N)0(S)) map into π1(SO(N)). Theorem 4.1 Let r≥3. There exist lifts  Spin(N)↗↓NSO(N)0(S)⟶SO(N),where S denotes the homomorphic image of Spin(r)in SO(N) (either Spin(r)^or Spin(r)^+), in the following cases: r≡1,7(mod8)̲ For all m∈N. r≡0(mod8)̲ For all m1,m2∈N if r>8. For m1≡m2≡0(mod2)if r=8. r≡2,6(mod8)̲ For all m∈N if r>6. For m even if r=6. r≡3,5(mod8)̲ For all m∈Nif r>3. For m even if r=3. r≡4(mod8)̲ For all m1,m2∈Nif r>4. For m1≡m2≡0(mod2)if r=4. The rest of this section is devoted to prove Theorem 4.1 in a case by case analysis. 4.1. r≡1,7(mod8) Recall   π1(SO(m)^Spin(r)^)={⟨(−1,1)⟩×⟨(volm,−1)⟩⊂Spin(m)×Spin(r),ifm≥4,m≡0(mod4),⟨(volm,−1)⟩⊂Spin(m)×Spin(r),ifm≥4,m≡2(mod4),⟨(−1,1)⟩⊂Spin(m)×Spin(r),ifm≥3,misodd,{1}⊂{1}×Spin(r),ifm=1,⟨(π,−1)⟩⊂R×Spin(r),ifm=2.Thus, we only need to check the loops in SO(drm) which are images of paths joining (1,1) to either (−1,1) or (volm,−1) in Spin(m)×Spin(r) or joining (0,1) to (π,−1) in R×Spin(r). Consider the path   δ1:[0,1]⟶Spin(m)×Spin(r)t↦(cos(πt)+sin(πt)v1v2,1)joining (1,1) to (−1,1) in Spin(m)×Spin(r). It projects to the loop   δˆ1:[0,1]⟶SO(m)^Spin(r)^⊂SO(N)t↦(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)1⋱1)m×m⊗IdΔr,which contains 2[r2] blocks   (cos(2πt)−sin(2πt)sin(2πt)cos(2πt)).Thus, δˆ1 represents 2[r2] times the generator of π1(SO(drm)). Since r≥3 and r≡1,7(mod8), 2[r2] is divisible by 8 and δˆ1 is null-homotopic. When m is even and m≥4, also consider the path   δ2:[0,1]⟶Spin(m)×Spin(r)t↦(∏j=1m2cos(πt/2)+sin(πt/2)v2j−1v2j,cos(πt)+sin(πt)e1e2)joining (1,1) to (volm,−1) in Spin(m)×Spin(r). It projects to the loop   δˆ2:[0,1]⟶SO(m)^Spin(r)^⊂SO(N)t↦(cos(πt)−sin(πt)sin(πt)cos(πt)⋱cos(πt)−sin(πt)sin(πt)cos(πt))m×m⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2]which is similar to   (eπite−πit⋱eπite−πit)m×m⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2].It contains 2[r2]−1 blocks that is there are 2[r2]−1m2=2[r2]−2m copies of the generator of π1(SO(drm)). Since r≥3 and r≡1,7(mod8), 2[r2]−2 is divisible by 2 and δˆ2 is null-homotopic. For m=2, consider the path   δ3:[0,1]⟶R×Spin(r)t↦(πt,cos(πt)+sin(πt)e1e2)joining (0,1) to (π,−1) in R×Spin(r), which maps to   δˆ3:[0,1]⟶SO(2)^Spin(r)^⊂SO(N)t↦(cos(πt)−sin(πt)sin(πt)cos(πt))⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2]∼(e2πite−2πit)⊗Id2[r2]−1⊕Id2[r2]−1.This loop represents 2[r2]−1 times the generator of π1(SO(N)), which is null-homotopic since 2[r2]−1 is divisible by 4. 4.2. r≡0(mod8) Let r=8k, {v1,…,vm1} and {v1′,…,vm2′} oriented orthonormal bases of Rm1 and Rm2, respectively. Recall the fundamental group generators for m1,m2≥3: Cases  π1(SO(m1)^SO(m2)^Spin(r)^)  (−1,1,1)  (1,−1,1)  (volm1,1,−volr)  (1,volm2,volr)  (a)  m1≡1(2),m2≡1(2)  Z2⊕Z2  ✓  ✓      (b)  m1≡0(4),m2≡1(2)  Z2⊕Z2⊕Z2  ✓  ✓  ✓    (c)  m1≡2(4),m2≡1(2)  Z2⊕Z4    ✓  ✓    (d)  m1≡1(2),m2≡0(4)  Z2⊕Z2⊕Z2  ✓  ✓    ✓  (e)  m1≡1(2),m2≡2(4)  Z2⊕Z4  ✓      ✓  (f)  m1≡0(4),m2≡0(4)  Z2⊕Z2⊕Z2⊕Z2  ✓  ✓  ✓  ✓  (g)  m1≡0(4),m2≡2(4)  Z2⊕Z2⊕Z4  ✓    ✓  ✓  (h)  m1≡2(4),m2≡0(4)  Z2⊕Z4⊕Z2    ✓  ✓  ✓  (i)  m1≡2(4),m2≡2(4)  Z4⊕Z4      ✓  ✓  Cases  π1(SO(m1)^SO(m2)^Spin(r)^)  (−1,1,1)  (1,−1,1)  (volm1,1,−volr)  (1,volm2,volr)  (a)  m1≡1(2),m2≡1(2)  Z2⊕Z2  ✓  ✓      (b)  m1≡0(4),m2≡1(2)  Z2⊕Z2⊕Z2  ✓  ✓  ✓    (c)  m1≡2(4),m2≡1(2)  Z2⊕Z4    ✓  ✓    (d)  m1≡1(2),m2≡0(4)  Z2⊕Z2⊕Z2  ✓  ✓    ✓  (e)  m1≡1(2),m2≡2(4)  Z2⊕Z4  ✓      ✓  (f)  m1≡0(4),m2≡0(4)  Z2⊕Z2⊕Z2⊕Z2  ✓  ✓  ✓  ✓  (g)  m1≡0(4),m2≡2(4)  Z2⊕Z2⊕Z4  ✓    ✓  ✓  (h)  m1≡2(4),m2≡0(4)  Z2⊕Z4⊕Z2    ✓  ✓  ✓  (i)  m1≡2(4),m2≡2(4)  Z4⊕Z4      ✓  ✓  For the cases (a), (b), (d), (e), (f) and (g), consider the path   δ1:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(cos(πt)+sin(πt)v1v2,1,1)joining (1,1,1) to (−1,1,1) in Spin(m1)×Spin(m2)×Spin(r) which projects to the loop   δˆ1:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)1⋱1)m1×m1⊗IdΔr+⊕Idm2⊗IdΔr−.It contains 2r2−1 copies of the generator of π1(SO(dr(m1+m2))), which is homotopically trivial since 2r2−1 is divisible by 8. For the cases (a), (b), (c), (d), (f) and (h), consider the path   δ2:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(1,cos(πt)+sin(πt)v1′v2′,1)joining (1,1,1) to (1,−1,1) in Spin(m1)×Spin(m2)×Spin(r), which projects to the loop   δˆ2:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦Idm1⊗IdΔr+⊕(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)1⋱1)m2×m2⊗IdΔr−.It contains 2r2−1 copies of the generator of π1(SO(dr(m1+m2))), and is homotopically trivial since 2r2−1 is divisible by 8. For the cases (d), (e), (f), (g), (h) and (i), consider the path   δ3:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(1,∏j=1m22cos(πt/2)+sin(πt/2)v2j−1′v2j′,∏l=1r2cos(πt/2)+sin(πt/2)e2l−1e2l)joining (1,1,1) to (1,volm2,volr) in Spin(m1)×Spin(m2)×Spin(r). It projects to the loop   δˆ3:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦Idm1⊗P+(t)⊕(cos(πt)−sin(πt)sin(πt)cos(πt)⋱cos(πt)−sin(πt)sin(πt)cos(πt))m2×m2⊗P−(t)∼Idm1⊗P+(t)⊕(eπite−πit⋱eπite−πit)m2×m2⊗P−(t),where   P+(t)=diag(e2π(k)it,e2π(k−1)it,…,e2π(k−1)it︸(4k2)times,e2π(k−2)it,…,e2π(k−2)it︸(4k4)times,…,e2π(−k)it),P−(t)=diag(e(2k−1)πit,…,e(2k−1)πit︸(4k1)times,e(2k−3)πit,…,e(2k−3)πit︸(4k3)times,…,e−(2k−1)πit,…,e−(2k−1)πit︸(4k4k−1)times).Thus, δˆ3 contains   m1(k+(4k2)(k−1)+(4k4)(k−2)+⋯+(4k2k−2))+m22((4k1)k+(4k3)(k−1)+⋯+(4k4k−1)(−(k−1)))=m1k(2k−1)8k−2)(4k2k)+24k−3m2copies of the generator of π1(SO(dr(m1+m2))). This number of copies is always even except when k = 1 and m1 is odd. For the cases (b), (c), (f), (g), (h) and (i), consider the path   δ4:[0,1]⟶Spin(m1)×Spin(m2)×Spin(r)t↦(∏j=1m12(cos(πt/2)+sin(πt/2)v2j−1v2j,1,(cos(πt/2))−sin(πt/2)e1e2)∏l=2r2(cos(πt/2)+sin(πt/2)e2l−1e2l))joining (1,1,1) to (volm1,1,−volr) in Spin(m1)×Spin(m2)×Spin(r). It projects to the loop   δˆ4:[0,1]⟶(SO(m1)^×SO(m2)^)Spin(r)^⊂SO(N)t↦(cos(πt)−sin(πt)sin(πt)cos(πt)⋱cos(πt)−sin(πt)sin(πt)cos(πt))m1×m1⊗Q+(t)⊕Idm2⊗Q−(t)∼(eπite−πit⋱eπite−πit)m1×m1⊗Q+(t)⊕Idm2⊗Q−(t),where   Q+(t)=diag(e(2k−1)πit,…,e(2k−1)πit︸(4k1)times,e(2k−3)πit,…,e(2k−3)πit︸(4k3)times,e(2k−5)πit,…,e(2k−5)πit︸(4k5)times,…,e−(2k−1)πit,…,e−(2k−1)πit︸(4k4k−1)times),Q−(t)=diag(e2πkit,e2π(k−1)it,…,e2π(k−1)it︸(4k2)times,e2π(k−2)it,…,e2π(k−2)it︸(4k4)times,…,e2π(−k)it).Thus, we have   m12[k(4k1)+(k−1)(4k3)+⋯+(−(k−1))(4k4k−1)]+m2[k(4k0)+(k−1)(4k2)+⋯+1(4k2k−2)]=24k−3m1+m2k(2k−1)8k−2(4k2k),which is odd only if k = 1 and m2≡1 (mod 2). The cases in which either m1≤2 or m2≤2 are treated similarly. 4.3. r≡2,6(mod8) Recall   π1(U(m)^Spin(r)^)={Z,if(m,4)=1,Z⊕Z2,if(m,4)=2,Z⊕Z4,if(m,4)=4.In every case, the fundamental group has generators   (2πm,e−2πimIdm,1),(π2,Idm,−vol). Consider the path   δ1:[0,1]⟶R×SU(m)×Spin(r)t↦(2πtm,(e−2πitme−2πitm⋱e−2πitme2πi(m−1)tm)m×m,1)joining (0,Idm×m,1) to (2πm,e−2πimIdm,1) in R×SU(m)×Spin(r). This path gets mapped to the loop if r≡2(mod8), and to if r≡6(mod8). Since dim(Δr±)=2r2−1, we have 2r2−1 blocks of the form   (e2πite−2πit)∼(cos(2πt)−sin(2πt)sin(2πt)cos(2πt))that is 2r2−1 times the generator of π1(SO(drm)). Since r≥6, 2r2−1 is divisible by 4. Hence, δˆ1 is null-homotopic. Consider the path   δ2:[0,1]⟶R×SU(m)×Spin(r)t↦(πt2,Idm,∏j=1r2(cos(πt/2)−sin(πt/2)e2j−1e2j))joining (0,Idm×m,1) to (π2,Idm,−vol) in R×SU(m)×Spin(r). This path gets mapped to the loop if r≡2(mod8), and if r≡6(mod8), where   P+(t)=diag(e−r2πit2,e−(r2−4)πit2,…,e−(r2−4)πit2︸(r/22)times,e−(r2−8)πit2,…,e−(r2−8)πit2︸(r/24)times,…),P−(t)=diag(e−(r2−2)πit2,…,e−(r2−2)πit2︸(r/21)times,e−(r2−6)πit2,…,e−(r2−6)πit2︸(r/23)times,e−(r2−10)πit2,…,e−(r2−10)πit2︸(r/25)times,…). If r≡2(mod8), r=8k+2 with k≥1. Then r2=4k+1. We have   m[k(4k+10)+(k−1)(4k+12)+(k−2)(4k+14)+⋯+(−k+1)(4k+14k−2)+(−k)(4k+14k)]=−m24k−2copies of the generator of π1(SO(drm)), which is even and δˆ1 is null-homotopic. If r≡6(mod8), r=8k+6 with k≥1. Then r2=4k+3. Thus, we have   m[(k+1)(4k+30)+(k)(4k+32)+(k−1)(4k+34)+⋯+(−k+1)(4k+34k)+(−k)(4k+34k+2)]=m24kcopies of the generator of π1(SO(drm)). If k≥1, this number is always even and δˆ2 is null-homotopic. On the other hand, if r = 6 (k = 0), then the parity of the number depends on m. 4.4. r≡3,5(mod8) Recall   π1(Sp(m)^Spin(r)^)=Z2=⟨(−Id2m,−1)⟩.Thus, consider the path   δ:[0,1]⟶Sp(m)×Spin(r)t↦((eπite−πit⋱eπite−πit)2m×2m,cos(πt)+sin(πt)e1e2)joining (Id2m,1) to (−Id2m,−1) in Sp(m)×Spin(r). It projects to the loop in Sp(m)^Spin(r)^⊂SO(drm)  (eπite−πit⋱eπite−πit)2m×2m⊗(eπite−πit⋱eπite−πit)2[r2]×2[r2].It has 2[r2]−1m blocks of the form   (e2πite−2πit)∼(cos(2πt)−sin(2πt)sin(2πt)cos(2πt)),that is 2[r2]−1m times the generator of π1(SO(drm))=Z2. Note that 2[r2]−1m is divisible by 2 if r>3. 4.5. r≡4(mod8) Let r=8k+4. Recall   π1((Sp(m1)^×Sp(m2)^)Spin(r)^)=⟨(−Id2m1,−Id2m2,−1)⟩×⟨(Id2m1,−Id2m2,volr)⟩⊂Sp(m1)×Sp(m2)×Spin(r).π1(Sp(m1)^Spin(r)−^)≅{⟨(−Id2m1,−volr),(−Id2m1,−1)⟩⊂Sp(m1)×Spin(r),ifm1>0,m2=0,r>4,⟨(−Id2m1,−vol4)⟩⊂Sp(m1)×Spin(4),ifm1>0,m2=0,r=4,π1(Sp(m2)^Spin(r)+^)≅{⟨(−Id2m2,volr),(−Id2m2,−1)⟩⊂Sp(m2)×Spin(r),ifm1=0,m2>0,r>4,⟨(−Id2m2,vol4)⟩⊂Sp(m2)×Spin(4),ifm1=0,m2>0,r=4. Consider the path   δ1:[0,1]⟶Sp(m1)×Sp(m2)×Spin(r)t↦((eπite−πit⋱eπite−πit)2m1×2m1,(eπite−πit⋱eπite−πit)2m2×2m2,cos(πt)+sin(πt)e1e2)joining (Id2m1,Id2m2,1) and (−Id2m1,−Id2m2,−1) in Sp(m1)×Sp(m2)×Spin(r). It maps to the loop   δˆ1:[0,1]⟶(Sp(m1)^×Sp(m2)^)Spin(r)^⊂SO(N)t↦diag(eπit,e−πit,…,eπit,e−πit)2m1×2m1⊗diag(eπit,e−πit,…,eπit,e−πit)2r2−1×2r2−1⊕diag(eπit,e−πit,…,eπit,e−πit)2m2×2m2⊗diag(eπit,e−πit,…,eπit,e−πit)2r2−1×2r2−1∼2r2−2diag(e2πit,e−2πit,…,e2πit,e−2πit)2m1×2m1⊕2r2−2Id2m1⊕2r2−2diag(e2πit,e−2πit,…,e2πit,e−2πit)2m2×2m2⊕2r2−2Id2m2,which is homotopically equivalent to (m1+m2)2r2−2 times the generator of π1(SO(dr(m1+m2))). Hence, δˆ1 is null-homotopic if either r≠4 or r = 4 and m1+m2≡0 (mod 2). Consider the path   δ2:[0,1]⟶Sp(m1)×Sp(m2)×Spin(r)t↦(Id2m1,(eπite−πit⋱eπite−πit)2m2×2m2,∏j=1r2cos(πt/2)+sin(πt/2)e2j−1e2j)joining (Id2m1,Id2m2,1) to (Id2m1,−Id2m2,volr). It maps to the loop   δˆ2⟶(Sp(m1)^×Sp(m2)^)Spin(r)^⊂SO(N)t↦Id2m1⊗P−(t)⊕diag(eπit,e−πit,…,eπit,e−πit)2m2×2m2⊗P+

### Journal

The Quarterly Journal of MathematicsOxford University Press

Published: Mar 1, 2018

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