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

Abstract

Let $P$ be an $n$-gon with $n\ge 3.$ There is a formal combinatorial $A_{\infty }$-coalgebra structure on cellular chains $C_{\ast }(P)$ with non-vanishing higher order structure when $n\ge 5$. If $X_g$ is a closed compact surface of genus $g\ge 2$ and $P_g$ is a polygonal decomposition, the quotient map $q:P_g\to X_g$ projects the formal $A_{\infty }$-coalgebra structure on $C_{\ast }(P_g)$ to a quotient structure on $C_{\ast }(X_g)$, 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 $g\ge 3$. But whether or not the $A_{\infty }$-coalgebra structure on homology observed here is topologically invariant is an open question.

13 pages, 6 figures