An $A_{\infty}$-coalgebra Structure on a Closed Compact Surface

TL;DR

This paper constructs a formal $A_ abla$-coalgebra structure on cellular chains of polygons and surfaces, revealing non-trivial higher order structures that depend on the surface's genus and orientability, with implications for topological invariance.

Contribution

It introduces a combinatorial $A_ abla$-coalgebra structure on polygonal and surface chains, extending to homology and highlighting conditions for non-trivial higher operations.

Findings

Non-vanishing higher order structure for polygons with n≥5

Persistence of structure under quotient maps to surfaces

Higher order structure is non-trivial for orientable and certain unorientable surfaces

Abstract

Let $P$ be an $n$-gon with $n\geq3.$ There is a formal combinatorial $A_\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing higher order structure when $n\geq5$. If $X_g$ is a closed compact surface of genus $g\geq2$ and $P_g$ is a polygonal decomposition, the quotient map $q:P_g\to X_g$ projects the formal $A_\infty$-coalgebra structure on $C_*(P_g)$ to a quotient structure on $C_*(X_g)$, which persists to homology $H_{\ast}\left( X_g;\mathbb{Z}_{2}\right) $, whose operations are determined by the quotient map $q$, and whose higher order structure is non-trivial if and only if $X_g$ is orientable or unorientable with $g\geq3$. But whether or not the $A_{\infty}$-coalgebra structure on homology observed here is topologically invariant is an open question.