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S.D.Theriault@soton.ac.uk School of Mathematics, Beben and Wu showed that if M is a (2n − 2)-connected (4n − 1)-dimensional Poincaré University of Southampton, 2n Duality complex such that n ≥ 3and H (M; Z) consists only of odd torsion, then M Southampton SO17 1BJ, UK Full list of author information is can be decomposed up to homotopy as a product of simpler, well-studied spaces. We available at the end of the article use a result from Beben and Theriault (Doc Math 27:183-211, 2022) to greatly simplify Research supported in part by and enhance Beben and Wu’s work and to extend it in various directions. the National Natural Science Foundation of China (Grant Keywords: Poincaré duality space, Loop space decomposition, Whitehead product Nos. 11801544 and 11688101), the National Key R&D Program of Mathematics Subject Classiﬁcation: Primary 55P35, 57N65; Secondary 55Q15 China (No. 2021YFA1002300), the Youth Innovation Promotion Association of Chinese Academy Sciences, and the “Chen Jingrun” Future Star Program of AMSS. 1 Introduction An orientable Poincaré Duality complex is a connected CW -complex whose cohomology satisﬁes Poincaré Duality. An orientable manifold is an example. In [6], Beben and Wu gave a homotopy decomposition of M where M is any (2n − 2)-connected (4n − 1)- 2n dimensional orientable Poincaré Duality complex, provided n ≥ 3and H (M; Z)has no 2-torsion. They used this to show that the homotopy type of M depended only on homological properties of M. This is in contrast to the homotopy type of M, which is known to depend on other properties as well. In particular, their result implies that the homotopy groups of M depend only on its homological properties. In this paper, we revisit Beben and Wu’s result. We give a simpler approach involving much less spectral sequence calculation, instead relying on a result proved in [5]. This allows for the results to be signiﬁcantly extended and advanced in various directions, complementing work in [3,4]. It should also be noted that earlier work of Selick [14] using diﬀerent methods can be used 2n to give a p-local homotopy decomposition of M when p is an odd prime and H (M; Z) Z/p Z. This has the advantage that it avoids calculating the mod-p homology of M entirely but it also cedes a level of precision that we will later require; this is explained more fully in Sect. 3. © The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 0123456789().,–: volV 53 Page 2 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 For any (2n − 2)-connected (4n − 1)-dimensional Poincaré Duality complex M,wehave 2n d k H (M; Z) = Z ⊕ Z/p Z, k=1 where d ≥ 0, each p is prime and each r ≥ 1. The case when d ≥ 1 has been dealt k k with in [5, Examples 4.4 and 5.3] (see [2] for a diﬀerent approach to the homotopy type) 2n so we restrict to the case when d = 0. In this case, the description of H (M; Z)implies that the 2n-skeleton M of M is homotopy equivalent to a wedge of Moore spaces, 2n 2n k M P (p ). If each p is an odd prime, we show the following. Let m be the 2n k k=1 r r r¯ r¯ 1 1 s least common multiple of {p , ... ,p } and let m = p¯ ··· p¯ be its prime decomposition. 1 1 Notice that {p¯ , ... , p¯ } is the set of distinct primes in {p , ... ,p } and each r¯ is the 1 s 1 j r r maximum power of p¯ appearing in the list {p , ... ,p }.By[12], the wedge of Moore r¯ s j 2n 2n 2n spaces P (¯p ) is homotopy equivalent to P (m). Write M P (m) ∨ A 2n j=1 j where A is the wedge of the remaining Moore spaces in M .Let f be the composite 2n of inclusions f : A −→ M −→ M and deﬁne the space V and the map h by the 2n f h homotopy coﬁbration A −→ M −→ V . We show that V is a Poincaré Duality complex 2n with H (V ; Z) = Z/mZ,and h has a right homotopy inverse s : V −→ M and prove the following. In general, for a space X let ev : X −→ X be the canonical evaluation map. Given two maps a: X −→ Z and b: Y −→ Z,let [a, b]: X ∧ Y −→ Z be the Whitehead 2n+1 r r 2n+1 product of a and b.Let S {p } be the homotopy ﬁbre of the degree p map on S . Theorem 1.1 Let M be a (2n − 2)-connected, (4n − 1)-dimensional Poincaré Duality complex such that n ≥ 2. Suppose that 2n H (M; Z) Z/p Z, k=1 where each p is an odd prime. Then with V and A chosen as above: (a.) there is a homotopy ﬁbration [γ ,f ]+f h (V ∧ A) ∨ A −− − −→ M − −− − → V, s ev where γ is the composite γ : V −→ M −→ M; (b.) the homotopy ﬁbration in (a) splits after looping to give a homotopy equivalence M V × ((V ∧ A) ∨ A); (c.) there is a homotopy equivalence r¯ 2n−1 4n−1 V S {p¯ }× S . j=1 As a notable special case, if the primes p for 1 ≤ k ≤ all equal a common prime p, 2n−1 r 4n−1 and r is the maximum of {r , ... ,r } then V S {p }× S . 1 k In [6], the decompositions in parts (b) and (c) of Theorem 1.1 were proved for n ≥ 3. Part (a) is new as is the n = 2 case for 2-connected 7-dimensional Poincaré Duality complexes. Further, while [6] gives no information in 2-torsion cases, in Theorem 5.7 we 2n k r ∼ s prove analogues of parts (a) and (b) when H (M; Z) Z/p Z ⊕ Z/2 Z, k=1 s=1 k R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 3 of 24 53 where each p is an odd prime, each r ≥ 2, and ≥ 1, and in Proposition 6.1 we consider k s special cases when 2-primary analogues of part (c) of Theorem 1.1 hold. An interesting consequence is a rigidity result. In Remark 4.6, we show that the space (V ∧ A) ∨ A is homotopy equivalent to a wedge W of Moore spaces, so part (b) may be written more succinctly as M V × W . As observed in [6], the homotopy 2n types of V and M depend only on information from H (M; Z). Thus, if M and M are both (2n − 2)-connected (4n − 1) dimensional Poincaré Duality complexes satisfying the 2n 2n hypotheses of Theorem 1.1,and H (M; Z) = H (M ; Z), then M M . We also prove an additional statement that was unaddressed in [6]. Let I : M −→ 2n M be the inclusion of the 2n-skeleton. We show that there is a homotopy coﬁbration G I +H 4n−1 4n−1 P (m) −→ M ∨ S −→ M where (I + H) has a right homotopy inverse 2n 4n−1 S : M −→ (M ∨ S ), and prove the following. 2n Theorem 1.2 With the same hypotheses as in Theorem 1.1, there is a homotopy ﬁbration [G,]+G I +H 4n−1 4n−1 4n−1 (P (m) ∧ M) ∨ P (m) − −− − → M ∨ S −− − −→ M, 2n S ev 4n−1 4n−1 where is the composite M −→ (M ∨ S ) −→ M ∨ S ,and this 2n 2n homotopy ﬁbration splits after looping to give a homotopy equivalence 4n−1 4n−1 4n−1 (M ∨ S ) M × ((P (m) ∧ M) ∨ P (m)). 2n Theorem 1.2 is interesting. Since M is homotopy equivalent to a wedge of simply 2n connected Moore spaces, it is a suspension. The theorem therefore shows that M retracts oﬀ a loop suspension, it identiﬁes the complementary factor, and it explicitly describes how the complementary factor maps into the loop suspension. The authors would like to thank the referee for several comments that have improved the paper. 2 Preliminary results This section contains preliminary results that will be referred to frequently in the subse- quent sections. We start with a general result from [5, Proposition 3.5]. Theorem 2.1 Let A −→ Y −→ Z be a homotopy coﬁbration. Suppose that hhas a s ev right homotopy inverse s : Z −→ Y . Let γ be the composite γ : Z −→ Y −→ Y. Then there there is a homotopy ﬁbration [γ ,f ]+f (Z ∧ A) ∨ A − −− −→ Y − −− − → Z which splits after looping to give a homotopy equivalence Y Z × ((Z ∧ A) ∨ A). 53 Page 4 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 Remark 2.2 As pointed out in [15, Remark 2.2], Theorem 2.1 has a naturality property. If there is a homotopy coﬁbration diagram A Y Z A Y Z and both h and h have right homotopy inverses s and s , respectively, such that there is a homotopy commutative diagram Z Y Z Y then the homotopy ﬁbration in Theorem 2.1 is also natural. Next, we prove two general lemmas about the existence of certain right homotopy inverses. Lemma 2.3 Suppose that there is a homotopy equivalence f ×g e : X × Y −→ Z × Z −→ Z for some maps f and g, where μ is the loop multiplication, and suppose that there is a map W −→ Z. If both f and g lift through h, then h has a right homotopy inverse. Proof Let s : X −→ W and t : Y −→ W be lifts of f and g, respectively, through h. Consider the diagram s×t X × Y W × W W h×h h f ×g Z × Z Z. The left triangle homotopy commutes by deﬁnition of s and t and the right square homotopy commutes since h is an H-map. The lower direction around the diagram is the deﬁnition of the homotopy equivalence e, so the upper row is a lift of e through W . Therefore, h has a right homotopy inverse. Lemma 2.4 Suppose that there is a homotopy ﬁbration diagram F F E E B B of path-connected CW -complexes, where b, fand p have right homotopy inverses. Then, e has a right homotopy inverse. R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 5 of 24 53 Proof Let r : B −→ B, s : F −→ F and t : B −→ E be right homotopy inverses for b, f and p, respectively. Let θ be the composite t×q r×s μ θ : B × F − −→ B × F − − → E × E −− → E. Consider the diagram F F F B × F E E π p r b B B B where i is the inclusion of the second factor and π is the projection onto the ﬁrst factor. 2 1 The lower left square homotopy commutes by deﬁnition of θ and p being an H-map. The left column is a ﬁbration, so the homotopy commutativity of the lower left square implies there is an induced map of ﬁbres F −→ F.Since θ ◦ i = q ◦ s, a choice of map of ﬁbres is s. Thus, the left side of the diagram is a map of homotopy ﬁbrations, as is the right by hypothesis. Therefore, the composite from the left to the right column is a self-map of a homotopy ﬁbration in which the top and bottom maps are homotopic to the identity. Therefore, by the Five Lemma, e ◦ θ induces an isomorphism on homotopy groups and so is a homotopy equivalence by Whitehead’s Theorem. 2n r 3 The case when H (M; Z) = Z/p Z Let V be a (2n − 2)-connected (4n − 1)-dimensional Poincaré Duality complex with 2n r 2n r 4n−1 H (V ; Z) Z/p Z where p is aprime.Asa CW -complex, V = P (p ) ∪ e ,and there is a homotopy coﬁbration f i 4n−2 2n r S −→ P (p ) −→ V, where f is the attaching map for the top cell and i is the inclusion of the (4n − 2)-skeleton. In this section, we prove Theorems 1.1 and 1.2 in the special case when M = V , assuming that n ≥ 2and p is odd. Some 2-primary cases of Theorem 1.1 (c) will be deferred to Sect. 6. A decomposition of V was proved by Beben and Wu [6] for n ≥ 3and p odd. Their method was more elaborate as it kept track of homology information in the general case of M where M is any (2n − 2)-connected (4n − 1)-dimensional Poincaré Duality complex with degree 2n cohomology consisting only of odd torsion. We give a much simpler approach to the general case in Sect. 4, and so only need to keep track of homology information for the special case of V . Diﬀerent methods were used by Selick [14]togivea p-local decomposition of V for n ≥ 2and p odd. He used a generalization of methods developed by Dyer-Lashof and Ganea to produce a p-local homotopy ﬁbration ∂ h 2n−1 r 4n−1 S {p } −→ S −→ V, where ∂ is null homotopic, implying that there is a p-local homotopy equivalence V 2n−1 r 4n−1 S {p }× S , analogous to our Proposition 3.7. The advantage of Selick’s method 53 Page 6 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 is that it avoids homology calculations entirely. Ideally, we would like this to be an integral result rather than a p-local one; the method itself cannot be upgraded to do this as it 2n−1 depends on S being an H-space which rarely happens integrally; however, a Sullivan square type argument could potentially be used to rectify this given that V localized at 4n−1 primes not equal to p or rationally is homotopy equivalent to S . The real disadvantage of Selick’s method for our purposes is that it does not describe the homotopy class of the map h with enough precision for later use in Lemma 3.8 and Proposition 3.9.Itwould be interesting to see if his techniques could be enhanced to do this, but in the meantime, we fall back to homology calculations to deal with V . Lemma 3.1 In degrees ≤ 4n, there is an algebra isomorphism H (V ; Z/pZ) Z/pZ[x, y] where |x|= 2n − 2 and |y|= 2n − 1. Proof Throughout, take cohomology and homology with mod-p coeﬃcients. By Poincaré Duality, there is an algebra isomorphism H (V ) = (a, b) where |a|= 2n − 1, |b|= 2n. Dualizing, there is a coalgebra isomorphism H (V ) = (u, v) where u, v are the duals of a, b, respectively. In particular, if is the reduced diagonal, then u and v are primitive and (uv) = u ⊗ v + v ⊗ u. Consider the mod-p homology Serre spectral sequence for the principal homotopy ﬁbration V −→ ∗ −→ V.Wehave E = H (V ) ⊗ H (V ) and the spectral sequence ∗ ∗ 2n−1 converges to H (∗). For degree reasons, the ﬁrst possible nontrivial diﬀerential is d , 2n−1 and for convergence reasons, we must have d (u) = x for some x ∈ H (V ). 2n−2 2n Also, for convergence reasons, we must have d (v) = y for some y ∈ H (V ). 2n−1 Thus, in the E -term, we also have the elements x ⊗ u, x ⊗ v, y ⊗ u, y ⊗ v.Since the 2n−1 2n 2n−1 2 spectral sequence is principal, d and d are diﬀerentials, so d (x ⊗ u) = x , 2n−1 2n 2 2n−1 d (y ⊗u) = xy and d (y ⊗v) = y .Weclaim that d (uv) = t ·(x ⊗v) for some unit t ∈ Z/pZ. The diagonal map gives a morphism of ﬁbrations from V −→ ∗ −→ V to V × V −→ ∗ − → V × V that induces a morphism of mod-p homology Serre spectral sequences. Note that in the product ﬁbration there is a Künneth isomorphism that lets us regard the homology of the product as the tensor product of the homologies of the factors. Since the diagonal map induces the coalgebra structure in homology, this morphism of Serre spectral sequences implies that the diﬀerentials commute with the reduced diagonal. 2n−1 2n−1 2n−1 Therefore (d (uv)) = (d ⊗ d )(u ⊗ v + v ⊗ u) = x ⊗ v + v ⊗ x (noting that 2n−1 2n−1 2n−1 d (v) = 0). In particular, (d (uv)) = 0, so d (uv) = 0. For degree reasons, 2n−1 2n+1 this implies that d (uv) = t · (x ⊗ v) for some unit t ∈ Z/pZ.Thusatthe E -page, all elements of degree ≤ 4n have vanished. Consequently, in degrees ≤ 4n, there is an algebra isomorphism H (V ) Z/pZ[x, y]. 2n r Next consider the eﬀect of the map P (p ) −→ V in mod-p homology. Since 2n r n ≥ 2, P (p ) is a suspension, so by the Bott–Samelson Theorem there is an algebra isomorphism 2n r H (P (p ); Z/pZ) T(x, y), where T( ) is the free tensor algebra functor, |x|= 2n − 2, |y|= 2n − 1and β y = x. Since i is the inclusion of the (4n − 2)-skeleton, it induces a homotopy equivalence in dimensions ≤ 4n − 3, so i induces a homotopy equivalence in dimensions ≤ 4n − 4. In particular, (i) induces an isomorphism in degrees 2n−2and 2n−1 in mod-p homology. As (i) is an algebra map, from Lemma 3.1 we obtain the following. ∗ R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 7 of 24 53 Lemma 3.2 In mod-p homology, the generator of least degree in the kernel of (i) is [x, y]. 4n−3 2n r m Let f : S −→ P (p ) be the adjoint of f .Let ι ∈ H (S ; Z/pZ) be a choice of a m m generator. Lemma 3.3 In mod-p homology, there is a choice of ι such that f (ι ) = [x, y]. 4n−3 ∗ 4n−3 f i 4n−2 2n r Proof Recall the coﬁbration S −→ P (p ) −→ V . Deﬁne the space F by the homo- 2n r topy ﬁbration F −→ P (p ) −→ V and consider the mod-p homology Serre spectral 2n r 2 sequence for the principal ﬁbration V −→ F −→ P (p ). The E -page of the spectral 2n r 2n r sequence is given by H (P (p )) ⊗ H (V ). Let u, v be the generators of H (P (p )) in ∗ ∗ ∗ degrees 2n − 1, 2n, respectively. By Lemma 3.1, H (V ) Z/pZ[x, y] in degrees ≤ 4n, where |x|= 2n − 2and |y|= 2n − 1. Since i is the inclusion of the 2n-skeleton, we 2n−1 2n have d (u) = x and d (v) = y. As the ﬁbration is principal, the diﬀerentials in the 2n−1 2 2n−1 spectral sequence are derivations so we obtain d (u ⊗ x) = x , d (u ⊗ y) = xy 2n 2 2n+1 and d (v ⊗ y) = v . Thus by the E -page of the spectral sequence, there is only one element left in degrees ≤ 4n − 2, and that is the image of the E -page element v ⊗ x. For degree reasons, this element is in the kernel of all higher diﬀerentials and therefore 4n−2 survives the spectral sequence. Thus, the (4n − 2)-skeleton of F is S . f i 4n−2 2n r Returning again to the coﬁbration S −→ P (p ) −→ V , there is clearly a lift 4n−2 2n r F P (p ) for some map λ. By the Blakers–Massey Theorem, λ is a homotopy equivalence in dimensions less than 4n − 2, so up to multiplication by a unit, λ may be regarded as the inclusion of the bottom cell of F. Taking adjoints, f factors as the composite 4n−3 2n r S −→ F −→ P (p ), where λ is the adjoint of λ.Now λ is the inclusion of the bottom cell in F, and Lemma 3.2 implies that the inclusion of this bottom cell has image equal to the generator of least degree in the kernel of (i) , which is [x, y]. Thus, there is 4n−3 a choice of generator ι in H (S ; Z/pZ) such that f (ι ) = [x, y]. 4n−3 4n−3 ∗ 4n−3 The low degree calculations made so far now let us calculate H (V ; Z/pZ)and (i) ∗ ∗ in full. Proposition 3.4 Let V be a (2n − 2)-connected, (4n − 1)-dimensional Poincaré Duality complex with H (V ; Z) = Z/p Z for p a prime and r ≥ 1. Then, there is an algebra 2n−1 isomorphism H (V ; Z/pZ) = Z/pZ[x, y], r r th where |x|= 2n − 2, |y|= 2n − 1 and β y = x, where β is the r -Bockstein. Further, in 2n r mod-p homology, the map P (p ) −→ V induces the algebra epimorphism T(x, y) −→ Z/pZ[x, y]. Proof In general, if X is a simply connected CW -complex and R is a ring, then there is an Adams–Hilton model AH(X) for calculating H (X; R)asanalgebra.The model is ∗ 53 Page 8 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 a diﬀerential graded algebra of the form T(a , ... ,a ; d) where T() is the free tensor 1 k algebra functor, there is a generator a for each cell of X, the degree of a is one less i i than the dimension of the corresponding cell, and d is a diﬀerential. There is an algebra isomorphism H(AH(X)) = H (X; R). To describe d,let X be the t-skeleton of X and let S −→ X attach a (t + 1)-cell t t corresponding to a .Let AH(X ) be the Adams–Hilton model obtained from AH(X)by i t restriction to the generators corresponding to cells in X .Then, d(a ) is determined by t i t−1 the image of the adjoint S −→ X in the Adams–Hilton model AH(X ). t t In our case, as V has three cells there is an Adams–Hilton model AH(V ) = T(x, y, z; d) with |x|= 2n − 2, |y|= 2n − 1and |z|= 4n − 2, and an algebra isomorphism H(AH(V )) = H (V ; Z/pZ). The inclusion of the (4n − 2)-skeleton of V is the map 2n r 2n r P (p ) −→ V,so AH(P (p )) = T(x, y; d ) is an Adams–Hilton model whose homol- 2n r ogy is isomorphic as an algebra to H (P (p ); Z/pZ). By the Bott–Samelson theorem, the latter is known to be T(x, y), so d must be identically zero. Thus, in this case, 2n r 2n r AH(P (p )) H (P (p ); Z/pZ), so to determine the diﬀerential dz, which corre- 4n−2 2n r sponds to the attaching map S −→ P (p ) for the top cell of V , we need to determine 4n−3 2n r the image in mod-p homology of the adjoint S −→ P (p ). By Lemma 3.3, this image is [x, y]. Thus, dz = [x, y], so we obtain algebra isomorphisms ∼ ∼ ∼ H (V ; Z/pZ) H(AH(V )) H(T(x, y, z; dz = [x, y])) Z/pZ[x, y]. = = = 2n r Further, the skeletal inclusion P (p ) −→ V induces the map of Adams–Hilton models (i) T(x, y; d ) −→ T(x, y, z; d), which upon taking homology gives the projection T(x, y) −→ Z/pZ[x, y]. Now specialize to p being an odd prime; we will return to p = 2 in Sect. 6.By[1], for m ≤ 2n r r 2n r (2n − 2)p, the homotopy groups π (P (p )) have the property that p · π (P (p )) 0. m m 4n−2 2n r Notice that 4n − 2 ≤ (2n − 2)p for all n ≥ 2and p ≥ 3. Thus, S −→ P (p ) extends 4n−1 r 2n r to a map g : P (p ) −→ P (p ), and there is a homotopy coﬁbration diagram 4n−2 4n−1 r 4n−1 S P (p ) S g (1) 4n−2 2n r S P (p ) V where q is the pinch map to the top cell and h is an induced map of coﬁbres. Let 4n−2 h: S −→ V be the adjoint of h. Lemma 3.5 Let p be an odd prime and take mod-p homology. If n ≥ 3,then h (ι ) = y . ∗ 4n−2 2 3 If n = 2,then h (ι ) = y + t · x for some t ∈ Z/pZ. ∗ 4n−2 4n−2 r 2n r Proof Let g : P (p ) −→ P (p ) be the adjoint of g and ﬁrst consider g .By Lemma 3.3, in mod-p homology, we have f (ι ) = [x, y]. So if u and v are the gen- ∗ 4n−3 4n−2 r eratorsindimensions4n −3and 4n −2of H (P (p ); Z/pZ), respectively, then the left square in (1)implies that g (u) = [x, y]. The naturality of the Bockstein therefore implies r r 2n r that [x, y] = g (u) = g (β (v)) = β ( g (v)). The only generator of H (P (p ); Z/pZ) ∗ ∗ ∗ 4n−2 th r 2 2 with a nonzero r -Bockstein is β (y ) = xy − yx = [x, y]. Thus, g (v) = y + z where ∗ R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 9 of 24 53 r 2n r β (z) = 0. If n ≥ 3 then, for degree reasons, the only generator of H (P (p ); Z/pZ) 4n−2 2 2 2n r is y .Thus, g (v) = y .If n = 2, then H (P (p ); Z/pZ) has one other generator, ∗ 4n−2 3 2 3 that being x ,so g (v) = y + t · x for some t ∈ Z/pZ. q¯ 4n−2 r 4n−2 Next, consider h . Note that q is the suspension of the pinch map P (p ) −→ S . Taking adjoints for the right square in (1)thenimplies that i ◦ g h ◦ q¯.Since q¯ (v) isachoiceof ι and (i) is an epimorphism by Proposition 3.4, from the description 4n−2 ∗ 2 2 3 of g (v)weobtain h (ι ) = y if n ≥ 3and h (ι ) = y + t · x for some t ∈ Z/pZ if ∗ ∗ 4n−2 ∗ 4n−2 n = 2. Remark 3.6 Observe that (1) implies that in mod-q homology for q a prime diﬀerent from p, or in rational homology, the map h induces an isomorphism. 2n−1 r r 2n−1 For any prime p,let S {p } be the homotopy ﬁbre of the degree p map on S . 2n−1 2n−1 r 2n−1 r The principal ﬁbration S −→ S {p }−→ S is induced by the degree p map so the mod-p homology Serre spectral sequence collapses at the E -term, giving an isomorphism of Z/pZ-modules 2n−1 r 2n−1 2n−1 H (S {p }; Z/pZ) H (S ; Z/pZ) ⊗ H (S ; Z/pZ) ∗ ∗ ∗ (a) ⊗ Z/pZ[b], r 2n r where |a|= 2n − 1, |b|= 2n − 2and β (a) = b.Since P (p ) is the homotopy coﬁbre of r 2n−1 the degree p map on S , there is a homotopy ﬁbration diagram 2n−1 2n−1 r 2n−1 2n−1 S S {p } S S (2) j s j 2n r 2n r 2n r P (p ) P (p ) ∗ P (p ) where j is the inclusion of the bottom cell and s is an induced map of ﬁbres. Observe that j 2n−2 2n−1 r is the suspension of the map j : S −→ P (p ) that includes the bottom cell, and this inclusion induces an isomorphism in degree 2n − 2 in mod-p homology. The naturality of the Bott–Samelson Theorem therefore implies that (j) = (j) is an algebra map ∗ ∗ sending Z/pZ[b] isomorphically onto the subalgebra Z/pZ[x] ⊆ T(x, y). The left square 2n−1 r in (2)thenimplies that s sends Z/pZ[b] ⊆ H (S {p }; Z/pZ) isomorphically onto the ∗ ∗ th subalgebra Z/pZ[x] ⊆ T(x, y). The r -Bockstein is a diﬀerential, implying that s sends (a) ⊗ Z/pZ[b] isomorphically onto the sub-module (y) ⊗ Z/pZ[x] ⊆ T(x, y). Let t be the composite s i 2n−1 r 2n r t : S {p } −→ P (p ) −→ V. Then, the description of s implies that t injects onto the submodule (y) ⊗ Z/pZ[x] ⊆ ∗ ∗ Z/pZ[x, y]. Let e be the composite t×h 2n−1 r 4n−1 e : S {p }× S − −− −→ V × V − −− −→ V, where μ is the loop space multiplication. Again, we focus on odd primes, leaving p = 2to Sect. 6. 2n−1 r 4n−1 Proposition 3.7 Let p be an odd prime. If n ≥ 2, then the map S {p }× S −→ V is a homotopy equivalence. 53 Page 10 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 Proof We will show that after localizing at each prime and rationally, e is a homotopy equivalence. This would imply that e is an integral homotopy equivalence. 4n−1 First consider the case when n ≥ 3. Localizing at p, H (S ; Z/pZ) Z/pZ[c] for 4n−1 |c|= 4n − 2. The restriction of h to the bottom cell of S is h, so by Lemma 3.5, (h) (c) = y .As(h) is an algebra map, it sends Z/pZ[c] isomorphically onto the ∗ ∗ subalgebra Z/pZ[y ] ⊆ Z/pZ[x, y]. The description of t then implies that e induces an ∗ ∗ isomorphism in mod-p homology, implying that e is a p-local homotopy equivalence by 2n−1 r Whitehead’s Theorem. Localized at a prime q = p or rationally, S {p } is contractible, 4n−1 V is equivalent to S ,and Remark 3.6 implies that h is a q-local or rational homotopy equivalence. Thus, in these cases, e is also a q-local or rational homotopy equivalence. Next, consider the case when n = 2. Localize at p. Going back to the description of V as a f q 6 4 r 4 CW -complex, observe that the composite S −→ P (p ) −→ S is null homotopic, where q is the pinch map to the top cell. This is because the generator of π (S ) = Z/2Z cannot factor through an odd primary Moore space. Thus, q extends to a map q : V −→ S . Since q extends q, in mod-p homology we have q inducing the projection Z/pZ[x, y] −→ h q 7 4 Z/pZ[y]. Now consider the composite S −→ V −→ S . The restriction of h to the bottom cell is h, so Lemma 3.5 implies that (q ◦ h) is an injection onto the subalgebra 2 4 3 7 Z/pZ[y ] ⊆ Z/pZ[y]. Since S S × S because of the existence of an element 4 7 of Hopf invariant one, there is a projection π : S −→ S which in mod-p homology h π 2 7 4 7 projects Z/pZ[y] onto Z/pZ[y ]. Thus, the composition S −→ V −→ S −→ S induces an isomorphism in homology and so is a homotopy equivalence. Consequently, there is a homotopy equivalence V F × S where F is the homotopy ﬁbre of π ◦ q. Notice that as q extends q, from the deﬁnition of s there is a homotopy commutative diagram s i 3 r 4 r S {p } P (p ) V q q 3 4 4 S S S where E is the inclusion of the bottom cell. The top row is the deﬁnition of t. Consequently, 3 r π ◦ q ◦ t is null homotopic, so t lifts to a map t : S {p }−→ F.Since t is an injection in mod-p homology, so is t . The decomposition V F × S implies that F has the 3 r same Euler–Poincaré series as S {p }; therefore, t is an isomorphism. Hence, the map e induces an isomorphism in mod-p homology and so is a p-local homotopy equivalence by Whitehead’s Theorem. Localizing at a prime q = p or rationally, arguing exactly as in the n ≥ 3 case shows that e is also a q-local or rational homotopy equivalence. We can go further. In general, suppose that there is a homotopy pushout A B C D of simply connected spaces where A is a suspension. The suspension hypothesis implies that the set of homotopy classes of maps [A, Z] is a group for any space Z. A Mayer– Vietoris style argument then shows that there is a homotopy coﬁbration a−b c+d A −− → B ∨ C − − → D. R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 11 of 24 53 4n−1 r 4n−2 r Since P (p ) is the suspension of P (p ), applying this to the right square in (1)we obtain a homotopy coﬁbration g −q i+h 4n−1 r 2n r 4n−1 P (p ) − − → P (p ) ∨ S − −→ V. (3) (i+h) 2n r 4n−1 Lemma 3.8 The map (P (p ) ∨ S ) −− − −→ V has a right homotopy inverse. t×h 2n−1 r 4n−1 Proof By Proposition 3.7, there is a homotopy equivalence S {p }× S −− → V × V − −→ V . By Lemma 2.3, to show that (i + h) has a right homotopy inverse, it suﬃces to show that both t and h lift through (i + h). 2n r 2n r 4n−1 4n−1 2n r 4n−1 Let i : P (p ) −→ P (p ) ∨S and i : S −→ P (p ) ∨S be the inclusions 1 2 of the left and right wedge summands, respectively. Then, (i+h)◦i = i and (i+h)◦i = h. 1 2 By deﬁnition, t = i ◦ s, so the composite i (i+h) 2n−1 r 2n r 2n r 4n−1 S {p } −→ P (p ) −→ (P (p ) ∨ S ) − − −− → V i (i+h) 4n−1 2n r 4n−1 equals t, while S −→ (P (p ) ∨ S ) − − −− → V is h.Thus, both t and h lift through (i + h), as required. 2n r 4n−1 Next, the homotopy ﬁbre of (i + h) is identiﬁed. Let s : V −→ (P (p ) ∨ S ) be a right homotopy inverse for (i + h). Let γ be the composite s ev 2n r 4n−1 2n r 4n−1 γ : V −→ (P (p ) ∨ S ) −→ P (p ) ∨ S . Let g = g − q. Proposition 3.9 There is a homotopy ﬁbration [g,γ ]+g i+h 4n−1 r 4n−1 r 2n r 4n−1 (P (p ) ∧ V ) ∨ P (p ) − −− −→ P (p ) ∨ S − −− −→ V, which splits after looping to give a homotopy equivalence 2n r 4n−1 4n−1 r 4n−1 r (P (p ) ∨ S ) V × ((P (p ) ∧ V ) ∨ P (p )). i+h 4n−1 r 2n r 4n−1 Proof Since there is a homotopy coﬁbration P (p ) −→ P (p ) ∨ S −→ V and, by Lemma 3.8, (i + h) has a right homotopy inverse, the assertions follow immediately from Theorem 2.1. Note that Proposition 3.7 proves Theorem 1.1 in the special case when M = V while Proposition 3.9 proves Theorems 1.2. 2n 4 The general case when H (M; Z) is odd torsion Let M be a (2n − 2)-connected (4n − 1)-dimensional Poincaré Duality complex such that n ≥ 2and 2n k H (M; Z) = Z/p Z, k=1 where each p is an odd prime. Then, the 2n-skeleton M of M is homotopy equivalent k 2n to a wedge of Moore spaces 2n k M P (p ). 2n k=1 53 Page 12 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 2n−1 2n For 1 ≤ k ≤ ,let a ∈ H (M; Z/p Z)and b ∈ H (M; Z/p Z) be generators k k k k 2n k corresponding to the wedge summand P (p )of M .In[6, Section 6], Beben and Wu 2n used a Poincaré Duality argument to prove the following. Lemma 4.1 Let p ∈{p , ... ,p } be an odd prime. Let {i , ... ,i }⊆{1, ... , } be the subset 1 1 t satisfying p = p and let r = max{r , ... ,r }.Ifp = p ,thena ∪ b is a generator of i i i i i j 1 t j j 4n−1 H (M; Z/pZ). r r As in the Introduction, let m be the least common multiple of {p , ... ,p } and let r¯ r¯ 1 s m = p¯ ··· p¯ be its prime decomposition. Notice that {p¯ , ... , p¯ } is the set of distinct s 1 s primes in {p , ... ,p } and each r¯ is the maximum power of p¯ appearing in the list 1 j j r r {p , ... ,p }. In general, if a and b are coprime, then by [12, proof of Proposition 1.5] t t t there is a homotopy equivalence P (ab) P (a) ∨ P (b). In our case, since {p¯ , ... , p¯ } are 1 s r¯ r¯ 1 s distinct primes and m = p¯ ··· p¯ , there is a homotopy equivalence r¯ 2n 2n P (m) P (¯p ). j=1 Therefore, M can be rewritten as 2n 2n M P (m) ∨ A, (4) 2n where A is the wedge of the remaining Moore spaces in M . 2n Deﬁne j and j by the composites 2n 2n j : P (m) → P (m) ∨ A −→ M −→ M 2n 2n j : A → P (m) ∨ A −→ M −→ M. 2n Deﬁne the space V and the map h by the homotopy coﬁbration j h A −→ M −→ V. 2n 4n−1 Then, V is a three-cell complex, V = P (m) ∪ e , and the inclusion of the (4n − 2)- skeleton is given by the composite 2n i : P (m) −→ M −→ V. Observe that Lemma 4.1 implies that V is a Poincaré Duality complex since the power of r¯ r¯ 1 s each p¯ appearing as a factor of m = p¯ ··· p¯ is maximal. j s 4n−2 Let F : S −→ M be the attaching map for the top cell of M. Deﬁne f by the 2n 4n−2 2n 2n composite f : S −→ M −→ P (m) where q collapses A in M P (m) ∨ A 2n 2n to a point. Observe that there is a homotopy pushout diagram pushout diagram A A 4n−2 S M M (5) 2n 4n−2 2n S P (m) V 2n 2n that deﬁnes the map i . By deﬁnition of q the composite P (m) → P (m) ∨ A −→ 2n M −→ P (m) is the identity map. Therefore, i is homotopic to the composite 2n R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 13 of 24 53 2n 2n P (m) → P (m) ∨ A −→ M −→ M −→ V , which, by deﬁnition of j ,is h ◦ j .But 2n h ◦ j is the deﬁnition of i,sowehave i = i. Therefore, f is the attaching map for the top cell of V,and F is a lift of f through q. We wish to show that h has a right homotopy inverse. Doing so will involve decom- posing V in a manner analogous to that for the special case when m = p in Sect. 3.We ﬁrst aim for the analogue of (1). 4n−2 4n−1 Lemma 4.2 The map S −→ M extends to a map G : P (m) −→ M . 2n 2n Proof It is equivalent to show that the map F has order m, and showing this is equivalent 4n−3 to showing that the adjoint F : S −→ M of F has order m. 2n 2n k Since M P (p ), by the Hilton–Milnor Theorem 2n k=1 ( ) r r r j j 2n k 2n−1 1 2n−1 2 M P (p ) × (P (p ) ∧ P (p )) × N, 2n k j j 1 2 k=1 j=1 where 1 ≤ j ,j ≤ , j = j ,and N is (4n − 2)-connected. Thus, F is a sum of maps of 1 2 1 2 r r r j j 4n−3 2n k 4n−3 2n−1 1 2n−1 2 the form F : S −→ P (p )and F : S −→ (P (p ) ∧ P (p )). As k j j 1 2 r r k k before, by [1] each map F has order at most p .As p is a factor of m,weobtainanull k k homotopy for F ◦ m, for 1 ≤ k ≤ .By[12, Corollary 6.6], if p and q are distinct primes a r b s r then P (p ) ∧ P (q ) is contractible, and if r ≤ s and p = 2, then there is a homotopy a r b s a+b r a+b−1 r equivalence P (p ) ∧ P (p ) P (p ) ∨ P (p ). Thus, if p = p , then F is null j j j 1 2 homotopic, while if p = p and we assume without loss of generality that r ≤ r , j j j j 1 2 1 2 then for dimensional reasons the Hilton-Milnor Theorem implies that F factors through r r j j 4n−3 4n−1 1 4n−2 1 F : S −→ P (p ) × P (p ). For dimension and connectivity reasons, F j j j j 1 1 4n−1 1 is trivial on the P (p ) factor and is a multiple of the inclusion of the bottom cell on r r r j j j 4n−2 1 1 1 the P (p ) factor. This inclusion has order p ,soas p isafactorof m,weobtain j j j 1 1 1 a null homotopy for F ◦ m, and therefore one for F ◦ m. Hence, F ◦ m is null homotopic. Lemma 4.2 implies that there is a homotopy coﬁbration diagram 4n−2 4n−1 4n−1 S P (m) S (6) G H F I 4n−2 S M M 2n where I is the skeletal inclusion, q is the pinch map to the top cell, and H is an induced map of coﬁbres. Combining this with (5) gives an iterated homotopy pushout diagram 4n−2 4n−1 4n−1 S P (m) S G H F I 4n−2 (7) S M M 2n 4n−2 2n S P (m) V. 53 Page 14 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 r¯ 2n 2n We now give a homotopy decomposition of V . By deﬁnition, P (m) P (¯p ). j=1 j For 1 ≤ j ≤ s,deﬁne S by the composite s i r¯ j r¯ j j j 2n−1 2n 2n S : S {p¯ } −→ P (¯p ) −→ P (m), j j th where s is from (2)and i is the inclusion of the j -wedge summand. Deﬁne S by the j j composite s s s j=1 j r¯ μ 2n−1 2n 2n S : S {p¯ } − − −− → P (m) − − −− → P (m) j=1 j=1 and deﬁne T by the composite r¯ S i 2n−1 2n T : S {p¯ } −→ P (m) −→ V. j=1 Finally, deﬁne e by the composite r¯ T ×(h◦H) μ 2n−1 4n−1 e : S {p¯ } × S −− − −− − → V × V − −− − −− → V. j=1 Proposition 4.3 If n ≥ 2, then the map e is a homotopy equivalence. Proof We will show that after localizing at each prime p and rationally, e is a homotopy equivalence. This would imply that e is an integral homotopy equivalence. Localize at a prime p where p = p¯ for some 1 ≤ j ≤ s.Let r = r¯ .If q is a prime j j a s distinct from p, then the Moore space P (q ) is contractible for a ≥ 2. Therefore, as r¯ a a P (m) P (¯p ) and the primes p¯ , ... , p¯ are distinct, there is a p-local homotopy 1 s j=1 equivalence. a a r P (m) P (p ). Applying this to (7), we obtain a p-local homotopy coﬁbration diagram 4n−2 4n−1 r 4n−1 S P (p ) S 4n−2 2n r S P (p ) V where g = q ◦ G and h = h ◦ H.Thisisa p-local version of (1) so we may argue as in Lemma 3.5 and Proposition 3.7 to show that the composite r¯ e 2n−1 r 4n−1 2n−1 4n−1 S {p }× S → S {p¯ } × S −→ V j=1 r¯ 2n−1 is a p-local homotopy equivalence. Notice that the spaces S {p¯ } are contractible if p¯ = p, so in fact we have shown that e is a p-local homotopy equivalence. Next, localize at a prime p ∈{ / p¯ , ... , p¯ }.Then, P (m) for a ≥ 2 and the Moore 1 s space wedge summands of M are all contractible. Therefore, in (7), both M and V are 2n 4n−1 homotopy equivalent to S and the maps H and h are both homotopy equivalences. r¯ 2n−1 On the other hand, the spaces S {p¯ } are also contractible so e reduces to (h ◦ H), which we have just seen is a homotopy equivalence. The same argument shows that e is also a rational homotopy equivalence. R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 15 of 24 53 Proposition 4.3 will be used to show that the map M −→ V has a right homotopy inverse. Thinking ahead, this is drawn from a slightly stronger statement. (j +H) h 2n 4n−1 Lemma 4.4 The composite (P (m) ∨S ) −− −− → M −− −− → V has a right homo- topy inverse. Proof By Proposition 4.3, there is a homotopy equivalence T ×(h◦H) μ r¯ 2n−1 4n−1 S {p¯ } × S −− − −− −→ V × V − −− −− − → V. j=1 By Lemma 2.3, to show that h ◦ (j + H) has a right homotopy inverse, it suﬃces to show that both T and (h ◦ H) lift through h ◦ (j + H). r¯ S i s j 2n−1 2n By deﬁnition, T is the composite S {p¯ } −→ P (m) −→ V and, by j=1 j j h 2n deﬁnition, i is the composite P (q) −→ M −→ V.Thus, T = h ◦ j ◦ S.Thisimplies that T lifts through h◦j and hence through h◦(j +H). Clearly, (h◦H) h◦H lifts through h ◦ (j + H). Corollary 4.5 The map M −→ V has a right homotopy inverse. We can now prove Theorem 1.1. j h Proof of Theorem 1.1 From the homotopy coﬁbration A −→ M −→ V and the right homotopy inverse of h in Corollary 4.5, parts (a) and (b) follow immediately from The- orem 2.1. Part (c) is Proposition 4.3. Remark 4.6 By Theorem 1.1, M V × ((V ∧ A) ∨ A). We claim that the space (V ∧ A) ∨ A is homotopy equivalent to a wedge W of spheres and odd primary Moore spaces. If so, then we may more simply write M V × W . To prove the claim, ﬁst consider r¯ 2n−1 4n−1 V S {p¯ } × S . j=1 In general, if B and C are path-connected spaces, then (B×C) B∨C ∨(B∧C); by t+1 [10], the space S is homotopy equivalent to a wedge of suspended spheres; by [9], the 2n−1 r r space S {p } is homotopy equivalent to a wedge of mod-p Moore spaces; and by [12, a r b s a+b s a+b−1 s Corollary 6.6], there is a homotopy equivalence P (p ) ∧ P (p ) P (p ) ∨ P (p ) a r b s if s ≤ r and p is odd while P (p ) ∧ P (q ) is contractible if p and q are distinct primes. Collectively, these statements imply that V is homotopy equivalent to a wedge of spheres and odd primary Moore spaces. Since A is deﬁned as a wedge of odd primary Moore spaces, we therefore also obtain that (V ∧ A) ∨ A is homotopy equivalent to a wedge of spheres and odd primary Moore spaces. Next, we consider the analogue of Proposition 3.9. This will be done in two steps, ﬁrst i I 2n with respect to P (m) −→ V and then with respect to M −→ M. First, the homotopy 2n 53 Page 16 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 pushout 4n−1 4n−1 P (m) S q◦G h◦H 2n P (m) V in (7) implies that there is a homotopy coﬁbration (q◦G)−q i+(h◦H) 4n−1 2n 4n−1 P (m) − − −− → P (m) ∨ S − − −− → V. (i+(h◦H)) 2n 4n−1 Lemma 4.7 The map (P (m) ∨ S ) − −− − −− → V has a right homotopy inverse. Proof This follows immediately from Lemma 4.4 since i = h ◦ j . Second, the homotopy pushout in (6) implies that there is a homotopy coﬁbration G−q I +H 4n−1 4n−1 P (m) −− − −→ M ∨ S − −− −→ M. 2n (I +H) 4n−1 Lemma 4.8 The map (M ∨ S ) − −− −→ M has a right homotopy inverse. 2n Proof The plan is to use the right homotopy inverse for (i + (h ◦ H)) in Lemma 4.7 and the naturality of Remark 2.2. This will be done in steps. 2n Step 1.By (4), M A ∨ P (m). Let A −→ M be the inclusion of the wedge 2n 2n a I summand and recall that the composite A −→ M −→ M is the deﬁnition of the 2n map j appearing in (5), whose coﬁbre is the map M −→ V . From this and the homotopy G−q I +H 4n−1 4n−1 coﬁbration P (m) −− − −→ M ∨ S −− − −→ M, we obtain a homotopy pushout 2n diagram 4n−1 4n−1 P (m) P (m) i G−q a+(G−q) 4n−1 4n−1 A ∨ P (m) M ∨ S V 2n 1 I +H j h A M V where i is the inclusion of the second wedge summand, p is the pinch map onto the ﬁrst 2 1 wedge summand, and h is deﬁned as h ◦ (I + H). 4n−1 Step 2. By Lemma 4.4, h has a right homotopy inverse s : V −→ (M ∨ S ). 2n s (I +H) 4n−1 Let s be the composite s : V − − −− → (M ∨ S ) − − −− → M.Then, s is a right 2n homotopy inverse for h and there is a homotopy commutative diagram 4n−1 V (M ∨ S ) 2n (8) (I +H) V M. R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 17 of 24 53 j h Step 3. The homotopy coﬁbration A −→ M −→ V and the existence of a right homotopy inverse s for h led to the identiﬁcation of the homotopy ﬁbre of h as (V ∧ A) ∨ A via Theorem 2.1. Similarly, the homotopy coﬁbration a+(G−q) h 4n−1 4n−1 A ∨ P (m) − −− −→ M ∨ S − −− −→ V 2n and the existence of a right homotopy inverse s for h lets us use Theorem 2.1 to identify 4n−2 4n−1 the homotopy ﬁbre of h as (V ∧ (A ∨ P (m))) ∨ (A ∨ P (m)). The compati- bility of the s and s in (8) lets us apply the naturality property in Remark 2.2 to obtain a homotopy ﬁbration diagram 4n−2 4n−1 4n−1 (V ∧ (A ∨ P (m))) ∨ (A ∨ P (m)) M ∨ S V 2n (9) (1∧p )∨p I +H 1 1 (V ∧ A) ∨ A M V. Step 4. Finally, observe that the map (1 ∧ p ) ∨ p has a right homotopy inverse, and 1 1 clearly the identity map on V does as well. Since (i + (h ◦ H)) has a right homotopy inverse by Lemma 4.7 and it factors as 2n 4n−1 4n−1 (i + (h ◦ H)) : (P (m) ∨ S ) −− −− − −→ (M ∨ S ) − −− −− − → V, 2n h also has a right homotopy inverse. Therefore, Lemma 2.4 implies that (I + H)has a right homotopy inverse. Nowwecan proveTheorem 1.2. G I +H 4n−1 4n−1 Proof of Theorem 1.2 From the homotopy coﬁbration P (m) −→ M ∨S −→ M, 2n where G = G−q, and the right homotopy inverse for (I +H) in Lemma 4.8, the assertions follow immediately from Theorem 2.1. 5 An extension to some 2-torsion cases I In this section, we consider a partial extension for parts (a) and (b) of Theorem 1.1 to cases involving 2-torsion. A full extension may not be possible due to issues involving Poincaré Duality as indicated by the lack of a 2-primary analogue of Lemma 4.1.Let M be a (2n − 2)-connected (4n − 1)-dimensional Poincaré Duality complex such that n ≥ 2 and 2n r k s H (M; Z) Z/p Z ⊕ Z/2 Z s=1 k=1 where each p is an odd prime and ≥ 1. Then, the 2n-skeleton M of M is homotopy 2n equivalent to a wedge of Moore spaces 2n k 2n r M P (p ) ∨ P (2 ). 2n k=1 s=1 Note the absence of mod-2 Moore spaces: this has to do with the smash product of two mod-2 Moore spaces as described in Remark 5.2. 53 Page 18 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 As in the Introduction and Sect. 4,let m be the least common multiple of {p , ... ,p } r¯ r¯ 1 s and let m = p¯ ··· p¯ be its prime decomposition. Notice that {p¯ , ... , p¯ } is the set of s 1 s distinct primes in {p , ... ,p } and each r¯ is the maximum power of p¯ appearing in the 1 j j r r list {p , ... ,p }. Therefore, M can be rewritten as 2n 2n 2n r M P (m) ∨ P (2 ) ∨ A, (10) 2n s=1 where A is the wedge of the remaining Moore spaces in M . 2n Deﬁne j and j by the composites t t 2n r 2n 2n r s s j : A ∨ P (2 ) → P (m) ∨ P (2 ) ∨ A −→ M −→ M, 2n s=1 s=1 2n 2n r j : A → P (m) ∨ P (2 ) ∨ A −→ M −→ M. 2n s=1 Deﬁne the spaces V and V , and the maps h and h , by the homotopy pushout diagram A A j h 2n r A ∨ P (2 ) M V (11) s=1 2n r P (2 ) V V. s=1 2n 4n−1 2n 2n r 4n−1 Then, V = P (m) ∪ e and V = P (m) ∨ P (2 ) ∪ e . Observe that the s=1 bottom row implies that there is a p-local homotopy equivalence V V for any odd prime p. We wish to show that h has a right homotopy inverse. That is, the analogue of Theorem 1.1 we aim to prove is based on a decomposition of M involving V as a factor rather than V . To do so we will take a local-to-global approach by applying the fracture theorem of [11, Theorem 8.1.3]. However, ﬁrst we need a functional version of Lemma 4.2 and a modiﬁcation of Proposition 4.3. 4n−2 Let F : S −→ M be the attaching map for the top cell of M. Deﬁne f and f by the 2n composites 4n−2 2n f : S −→ M −→ P (m) 2n F q 4n−2 2n 2n r f : S −→ M −→ P (m) ∨ P (2 ) 2n s=1 2n r where q and q collapse A ∨ P (2 )and A in M to a point, respectively. Then, 2n s=1 f and f are the attaching maps for the top cell of V and V , respectively. In particular, there is a homotopy pushout 4n−2 S M M 2n q (12) 4n−2 2n S P (m) V R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 19 of 24 53 where i is the inclusion of the 2n-skeleton. r r 1 r r 1 t Let m be the least common multiple of {p , ... ,p }∪{2 , ... , 2 }. In particular, m = v v r r 1 t 2 m with 2 = max{2 , ... , 2 }. Anticipating that the upper bound on the exponent for 2n r v π (P (2 )) in [1] is higher than for odd primes, let v = v + 1and let m = 2 m.By[12, proof of Proposition 1.5], there is a canonical morphism of homotopy coﬁbrations 4n−2 4n−1 4n−1 S P (m ) S (13) 4n−2 4n−1 4n−1 S P (m) S , 4n−1 4n−1 v 4n−1 4n−1 where Q collapses P (m ) P (2 ) ∨ P (m)to P (m)and q and q are the pinch maps to the top cell. The following lemma is the analogue of Lemma 4.2. 4n−2 4n−2 2n Lemma 5.1 The maps S −→ M and S −→ P (m) extend to maps 2n 4n−1 4n−1 2n G : P (m ) −→ M and g : P (m) −→ P (m), respectively. Moreover, the exten- 2n sions are compatible, that is, there is the homotopy commutative diagram 4n−1 P (m ) M 2n 4n−1 2n P (m) P (m). Proof The existence of G follows exactly as in the proof of Lemma 4.2, using the fact that r+1 2n r [1]implies that 2 · π (P (2 )) = 0if r ≥ 2. 4n−2 A choice of the map g is given by Lemma 4.2, but we need to make sure that a choice is made that also gives the asserted homotopy commutative diagram. Notice that there is ω Q 4n−1 v 4n−1 4n−1 a homotopy coﬁbration P (2 ) −→ P (m ) −→ P (m) where ω is the inclusion 4n−1 4n−1 4n−1 v into P (m ) P (m) ∨ P (2 ). If the composite ω G 4n−1 v 4n−1 2n P (2 ) −→ P (m ) −→ M −→ P (m) 2n 4n−1 2n is null homotopic, then q ◦ G extends along Q to a map g : P (m) −→ P (m)and we are done. To see that q ◦ G ◦ ω is null homotopic, observe that it represents an element 2n 2n of 2-torsion in π (P (m)). But the space P (m) is 2-locally contractible since m is a 4n−2 product of odd primes. Remark 5.2 It is the use of Lemma 4.2 that prevents us from considering 2-torsion in the a r b r cohomology of M. Its proof uses the property that the smash product P (p ) ∧ P (p )is r r homotopy equivalent to a wedge of two mod-p Moore spaces: this only holds if p = 2. From the extension of F to G in Lemma 5.1, we obtain a homotopy coﬁbration diagram 4n−2 4n−1 4n−1 S P (m ) S (14) G H F I 4n−2 S M M 2n 53 Page 20 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 where I is the skeletal inclusion and H is an induced map of coﬁbres. Lemma 5.3 There is a homotopy commutative diagram 4n−1 S M 4n−1 S V for a map h satisfying h ◦ q i ◦ g. Proof Consider the cube 4n−1 4n−1 P (m ) S 4n−1 4n−1 G P (m) S M M 2n q h 2n P (m) V where the map h will be deﬁned momentarily. The top face is a homotopy pushout by (13), the rear face homotopy commutes by (12), the left face homotopy commutes by Lemma 5.1, and the bottom face homotopy commutes by (12). The homotopy com- mutativity of these four faces implies that i ◦ g ◦ Q h ◦ H ◦ q. Therefore, as the top face 4n−1 is a homotopy pushout, there is a pushout map h: S −→ V such that h ◦ q i ◦ g v v and h ◦ 2 h ◦ H. In particular, the homotopy h ◦ 2 h ◦ H gives the homotopy commutative diagram asserted by the lemma. Next, we modify Proposition 4.3. Similarly to the map e in Sect. 4,deﬁne e by the composite r¯ T ×h μ 2n−1 4n−1 e : S {p¯ } × S − −− − −− → V × V − −− − −− → V. j=1 Notice that e replaces the map h ◦ H in the deﬁnition of e appearing in Sect. 4 by h,but the property from Lemma 5.3 that h ◦ q i ◦ g ensures that the argument for Proposition 4.3 also applies to e . Proposition 5.4 If n ≥ 2, then the map e is a homotopy equivalence. Finally, we show that h has a right homotopy inverse using a local-to-global approach. Let T be the set of odd primes and T ={2}. o e Lemma 5.5 The map M −→ V has: (i) aT -local right homotopy inverse θ : V −→ Mand o o R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 21 of 24 53 (ii) aT -local right homotopy inverse θ : V −→ M, e e 4n−1 both of whose rationalizations are the identity map on S . h h Proof For (i), by (11) the composite M −→ V −→ V is homotopic to M −→ V.As V −→ V is a T -local equivalence, to show that h has a T -local right homotopy inverse o o it suﬃces to prove that h has a T -local right homotopy inverse θ : V −→ M.We then take θ to be the composite V −→ V −→ M. Localize spaces and maps at T . Arguing as for Lemma 4.4 and using Lemma 5.3 gives a homotopy commutative diagram (j +H) 2n 4n−1 (P (m) ∨ S ) M μ◦(S×( )) (15) r¯ T ×h 2n−1 4n−1 S {p¯ } × S V × V V j=1 while Proposition 5.4 implies that the bottom row is the homotopy equivalence e . There- fore, θ = (j + H) ◦ μ ◦ (S × ( )) ◦ e is a (T -local) right homotopy inverse for o v 4n−1 h. Rationally, h is the identity map on S ,asis h since e is an integral homotopy equivalence (technically, h could have degree ±1 but if it is degree −1 we can replace h by its negative). Thus, the homotopy commutativity of (15) implies that, rationally, θ must 4n−1 be the identity map on S . For (ii), the homotopy coﬁbration A −→ M −→ V from (11) implies that h is a T -local homotopy equivalence since A is a wedge of odd primary Moore spaces and so is contractible when localized at 2. Therefore, h has a T -local right homotopy inverse θ . 4n−1 Further, as the rationalization of h is the identity map on S , so is the rationalization of θ .Thus, θ = θ is a T -local right homotopy inverse for h whose rationalization e e e e 4n−1 is the identity map on S . The local right homotopy inverses for h in Lemma 5.5 are now assembled into an integral one. Lemma 5.6 The map M −→ V has a right homotopy inverse θ. Proof By the fracture theorem of [11, Theorem 8.1.3], for any simply connected space X, there is a homotopy pullback X X X × X X × X T T Q Q o e where X , X and X are the T , T and Q-localizations of X, respectively, r is rational- T T Q o e o e ization and is the diagonal map. In our case, consider the diagram V × V V × V V T T Q Q Q o e θ ×θ θ ×θ θ o e Q Q Q M × M M × M M T T Q Q Q o e 53 Page 22 of 24 R. Huang, S. Theriault Res Math Sci (2022) 9:53 where θ and θ , respectively, are the T and T -local right homotopy inverses for h in o e o e 4n−1 Lemma 5.5 and θ is the rationalization of the identity map on S . The left square homotopy commutes by Lemma 5.5 and the right square commutes by the naturality of the diagonal map. By the fracture theorem, the homotopy pullback of the maps in the top row is V and the homotopy pullback of the maps in the bottom row is M.The pullback property for M and the homotopy commutativity of the two squares implies that there is a pullback map θ : V −→ M with the property that its T -localization is θ ,its T -localization is θ and its rationalization is θ .Thus, θ is a right homotopy inverse e o o Q for h because it is when localized at any prime or rationally. From the homotopy coﬁbration A −→ M −→ V and the right homotopy inverse θ of h in Lemma 5.6, the following theorem follows immediately from Theorem 2.1. Theorem 5.7 Let M be a (2n − 2)-connected (4n − 1)-dimensional Poincaré Duality complex such that n ≥ 2 and 2n r k s H (M; Z) Z/p Z ⊕ Z/2 Z s=1 k=1 where each p is an odd prime, each r ≥ 2,and ≥ 1.ThenwithV and A chosen as k s above: (a) there is a homotopy ﬁbration [γ ,j ]+j h (V ∧ A) ∨ A − −− − → M − − −− → V θ ev where γ is the composite γ : V −→ M −→ M; (b) the homotopy ﬁbration in (a) splits after looping to give a homotopy equivalence M V × ((V ∧ A) ∨ A). Note that when t = 0, Theorem 5.7 reduces to part (a) and (b) of Theorem 1.1.Note also that, unlike Theorems 1.1, 5.7 does not decompose V any further. 6 An extension to some 2-torsion cases II Finally, we consider an extension for part (c) of Theorem 1.1 to certain special cases 2n r 4n−1 involving 2-torsion. In general, when V = P (2 ) ∪ e , it is unreasonable to expect a 2n−1 r 4n−1 2n−1 r decomposition V S {2 }× S since this implies that the space S {2 } is 2n−1 an H-space. Often this is not the case, for example, if n = 3or n ≥ 5 then S {2} is not 2n−1 r an H-space [8]. A full classiﬁcation of when S {2 } is an H-space seems not to appear 3 r in the literature. However, by [7, Corollary 21.6], it is known that S {2 } is an H-space if 7 r r ≥ 3and S {2 } is an H-space if r ≥ 4. In these cases, we show that the arguments in Sect. 3 hold, giving a decomposition of V . Lemma 3.3 and Proposition 3.4 were proved for all primes p. The ﬁrst point in Sect. 3 where the restriction p ≥ 3 occurred was in the the existence of the extension g for f r 2n r in (1). In general, it may not be the case that 2 · π (P (2 )) 0. However, Sasao [13] 4n−2 r 4 r r 8 r ∼ ∼ showed that 2 · π (P (2 )) 0if r ≥ 3and 2 · π (P (2 )) 0if r ≥ 4. Thus, in these = = 6 14 cases, we obtain a homotopy coﬁbration diagram as in (1). The argument for Lemma 3.5 now goes through in exactly the same manner. The maps s, t and e following Lemma 3.5 were deﬁned for all primes p, and the restriction to odd primes in Proposition 3.7 was R. Huang, S. Theriault Res Math Sci (2022) 9:53 Page 23 of 24 53 present only to: (i) invoke Lemma 3.5 and (ii) in the n = 2 case, ensure that the composite f q 6 4 r 4 S −→ P (2 ) −→ S is null homotopic so that there is an extension of q to a map V −→ S . Therefore, Proposition 3.7 will hold: (i) for n = 4and r ≥ 4, and (ii) for n = 2 and r ≥ 3 with the extra assumption that there is a map V −→ S inducing a surjection in mod-2 homology. 2n r 4n−1 Proposition 6.1 Let V = P (2 ) ∪ e be a Poincaré Duality complex. (a) If n = 2,r ≥ 3 and there is a map V −→ S inducing a surjection in mod-2 homology, 3 r 7 then there is a homotopy equivalence V S {2 }× S ; 7 r 15 (b) if n = 4 and r ≥ 4, then there is a homotopy equivalence V S {2 }× S . 2n 2n 2n For example, if τ(S ) is the unit tangent bundle of S then, as a CW -complex, τ(S ) = 2n 4n−1 2n−1 2n 2n P (2) ∪ e , and there is a ﬁbration S −→ τ(S ) −→ S .For r ≥ 2, deﬁne the “mod-2 tangent bundle” by the homotopy pullback 2n−1 2n 2n S τ (S ) S r−1 2n−1 2n 2n S τ(S ) S r−1 r−1 2n 2n r 4n−1 where 2 is the map of degree 2 .Asa CW -complex, τ (S ) = P (2 ) ∪ e and ∗ 2n H (τ (S )) satisﬁes Poincaré Duality. Proposition 6.1 implies that there are homotopy 4 3 r 7 8 7 r 15 equivalences τ (S ) S {2 }× S if r ≥ 3and τ (S ) S {2 }× S if r ≥ 4. r r Remark 6.2 The argument for Proposition 6.1 is independent of prior knowledge that 3 r 7 r S {2 } for r ≥ 3or S {2 } for r ≥ 4are H-spaces. So the loop space decompositions of r 3 r the mod-2 tangent bundles is a new proof of this property, since the retractions of S {2 } 7 r for r ≥ 3and S {2 } for r ≥ 4 oﬀ loop spaces imply that they are H-spaces. The previous argument in [7] examined the H-deviation of the degree 2 map. 2n 4n−1 Generalizing to the case V = P (2m) ∪ e where m is divisible by more than one prime seems to be much more diﬃcult. Our argument breaks down with the loss of Lemma 4.1. It would be interesting to know if a diﬀerent argument can be used to make progress. Data Availability Statement Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Author details Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, School of Mathematics, University of Southampton, Southampton SO17 1BJ, UK. Received: 13 July 2021 Accepted: 9 June 2022 Published online: 5 August 2022 References 1. 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Theriault, S.: Homotopy ﬁbrations with a section after looping. Mem. Amer. Math. Soc. (to appear) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.
Research in the Mathematical Sciences – Springer Journals
Published: Sep 1, 2022
Keywords: Poincaré duality space; Loop space decomposition; Whitehead product; Primary 55P35; 57N65; Secondary 55Q15
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