Publications

An $A_\infty$-coalgebra structure on a closed compact surface.

Faculty Author(s): Umble, Ronald
Student Author(s): -
Department: MATH
Publication: Georgian Mathematical Journal
Year: 2018
Abstract: Summary: ``Let $P$ be an $n$-gon with $n\geq 3$. There is a formal combinatorial $A_\infty$-coalgebra structure on cellular chains $C_*(P)$ with non-vanishing higher order structure when $n\geq 5$. If $X_g$ is a closed compact surface of genus $g\geq 2$ and $P_g$ is a polygonal decomposition, the quotient map $q : P_g\to X_g$ projects the formal $A_\infty$-coalgebra structure on $C_*(P_g)$ to a quotient structure on $C_*(X_g)$, which persists to homology $H_*(X_g; {\Bbb Z}_2)$, 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 with $g\geq 2$ or unorientable with $g\geq 3$. But whether or not the $A_\infty$-coalgebra structure on homology observed here is topologically invariant is an open question.''
Link: An $A_\infty$-coalgebra structure on a closed compact surface.