Rewriting systems and the multiplicative structure in the cohomology of $π_1(Σ_{g_1}\times\cdots\timesΣ_{g_n})$ with twisted $\mathbb{Z}$-coefficients

Natalia Cadavid-Aguilar, Jesús González

Published 2019-07-24Version 1

Let $\pi_g$ stand for the fundamental group of the closed orientable surface $\Sigma_g$ of genus $g$. We use a finite complete rewriting system for $\pi_g$ in order to produce an explicit contracting homotopy for the standard minimal $\pi_g$-free resolution $M_*^g$ of the trivial $\pi_g$-module $\mathbb{Z}$. This allows us to construct an explicit diagonal approximation for any $\mathbb{Z}$-tensor product $M_*^{g_1}\otimes\cdots\otimes M_*^{g_n}$ which, in turn, yields an efficient method to compute the structure of the cohomology product maps $H^*(\pi_{g_1}\times\cdots\times\pi_{g_n};M)\otimes H^*(\pi_{g_1}\times\cdots\times\pi_{g_n};M')\to H^*(\pi_{g_1}\times\cdots\times\pi_{g_n};M\otimes M').$ Details and explicit examples are spelled out for $n=1$ and $n=2$ when the abelian group structure underlying both $M$ and $M'$ is $\mathbb{Z}$.