# Secondary cohomology of a moment angle complex

# Secondary cohomology of a moment angle complex

***Please note the time change* 10:00AM EST**

**Zoom link: https://princeton.zoom.us/j/92116764865**

**Passcode: 114700**

Given a simplicial complex $K$, one can define a topological space $\mathcal{Z}_K$ called the \emph{moment-angle complex}. The cohomology of $\mathcal{Z}_K$ is captured by a Tor-algebra corresponding to the face ring of $K$, which can be decomposed into the direct sum of the cohomology of full subcomplexes of $K$ by Hochster’s theorem. In this talk, we introduce a certain differential on this Hochster-decomposition of the cohomology $H^\ast(\mathcal{Z}_K)$ to make it a chain complex. This leads us to define a secondary cohomology $HH*(\mathcal{Z}_K)$ of a moment angle complex, which is a new combinatorial invariant of $K$. We will discuss topological and algebraic definitions of $HH*(\mathcal{Z}_K)$, then study several techniques to compute $HH*(\mathcal{Z}_K)$ together with examples.

This is a joint work (in progress) with I. Limonchenko, T. Panov and D. Stanley.