On the cohomology of quotients of moment-angle complexes

TL;DR

This paper characterizes the cohomology of quotients of moment-angle complexes by free torus actions, establishing a ring isomorphism with a Tor-algebra and proving spectral sequence collapse for arbitrary coefficients.

Contribution

It provides a detailed proof of the cohomology ring structure for quotients of moment-angle complexes, extending previous results to non-field coefficients and finite characteristic.

Findings

Cohomology ring of Z_K/H is isomorphic to a Tor-algebra over R[K].

Spectral sequence collapses for arbitrary commutative ring coefficients.

Extension of previous results to nontrivial H and finite characteristic cases.

Abstract

We describe the cohomology of the quotient Z_K/H of a moment-angle complex Z_K by a freely acting subtorus H in T^m by establishing a ring isomorphism of H*(Z_K/H,R) with an appropriate Tor-algebra of the face ring R[K], with coefficients in an arbitrary commutative ring R with unit. This result was stated in [BP02, 7.37] for a field R, but the argument was not sufficiently detailed in the case of nontrivial H and finite characteristic. We prove the collapse of the corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with respect to `strongly homotopy multiplicative' maps in the category DASH, following Munkholm [Mu74]. Our collapse result does not follow from the general results of Gugenheim-May and Munkholm.