An A∞-coalgebra structure on a closed compact surface

Quinn Minnich; Ronald Umble

doi:10.1515/gmj-2018-0052

Article

An A_∞-coalgebra structure on a closed compact surface

Published/Copyright: September 10, 2018

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From the journal Georgian Mathematical Journal Volume 25 Issue 4

Abstract

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

Keywords: associahedron; non-orientable surface

MSC 2010: 57N05; 57N65; 55P35

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Received: 2018-02-02

Accepted: 2018-06-12

Published Online: 2018-09-10

Published in Print: 2018-12-01

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Articles in the same Issue

https://doi.org/10.1515/gmj-2018-0052

Keywords for this article

associahedron; non-orientable surface