Secondary cohomology of a moment angle complex

Secondary cohomology of a moment angle complex

Jongbaek Song, KIAS

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

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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.