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An A ∞-coalgebra structure on a closed compact surface

An A ∞-coalgebra structure on a closed compact surface AbstractLet P be an n-gon with n≥3{n\geq 3}. There is a formal combinatorial A∞{A_{\infty}}-coalgebra structure on cellular chains C*⁢(P){C_{*}(P)}with non-vanishing higher order structure when n≥5{n\geq 5}.If Xg{X_{g}}is a closed compact surface of genus g≥2{g\geq 2}and Pg{P_{g}}is a polygonal decomposition, the quotient map q:Pg→Xg{q\colon P_{g}\to X_{g}}projects the formal A∞{A_{\infty}}-coalgebra structure on C*⁢(Pg){C_{*}(P_{g})}to a quotient structure on C*⁢(Xg){C_{*}(X_{g})}, which persists to homology H∗⁢(Xg;ℤ2){H_{\ast}(X_{g};\mathbb{Z}_{2})}, whose operations are determined by the quotient map q, and whose higher order structure is non-trivial if and only if Xg{X_{g}}is orientable with g≥2{g\geq 2}or unorientable with g≥3{g\geq 3}.But whether or not the A∞{A_{\infty}}-coalgebra structure on homology observed here is topologically invariant is an open question. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Georgian Mathematical Journal de Gruyter

An A ∞-coalgebra structure on a closed compact surface

Minnich, Quinn; Umble, Ronald
Georgian Mathematical Journal , Volume 25 (4): 10 – Dec 1, 2018
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References (8)

Publisher
de Gruyter
Copyright
© 2018 Walter de Gruyter GmbH, Berlin/Boston
ISSN
1572-9176
eISSN
1572-9176
DOI
10.1515/gmj-2018-0052
Publisher site
See Article on Publisher Site

Abstract

AbstractLet P be an n-gon with n≥3{n\geq 3}. There is a formal combinatorial A∞{A_{\infty}}-coalgebra structure on cellular chains C*⁢(P){C_{*}(P)}with non-vanishing higher order structure when n≥5{n\geq 5}.If Xg{X_{g}}is a closed compact surface of genus g≥2{g\geq 2}and Pg{P_{g}}is a polygonal decomposition, the quotient map q:Pg→Xg{q\colon P_{g}\to X_{g}}projects the formal A∞{A_{\infty}}-coalgebra structure on C*⁢(Pg){C_{*}(P_{g})}to a quotient structure on C*⁢(Xg){C_{*}(X_{g})}, which persists to homology H∗⁢(Xg;ℤ2){H_{\ast}(X_{g};\mathbb{Z}_{2})}, whose operations are determined by the quotient map q, and whose higher order structure is non-trivial if and only if Xg{X_{g}}is orientable with g≥2{g\geq 2}or unorientable with g≥3{g\geq 3}.But whether or not the A∞{A_{\infty}}-coalgebra structure on homology observed here is topologically invariant is an open question.

Journal

Georgian Mathematical Journal – de Gruyter

Published: Dec 1, 2018

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