Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

A degree formula for equivariant cohomology rings

Pages: 345 – 365

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a18

Authors

Mark Blumstein (Poudre Global Academy, Fort Collins, Colorado, U.S.A.)

Jeanne Duflot (Department of Mathematics, Colorado State University, Fort Collins, Colorado, U.S.A.)

Abstract

This paper generalizes a result of Lynn on the “degree” of an equivariant cohomology ring H^\ast_G (X)$. The degree of a graded module is a certain coefficient of its Poincaré series, and is closely related to multiplicity. In the present paper, we study these commutative algebraic invariants for equivariant cohomology rings. The main theorem is an additivity formula for degree:\[\deg (H^\ast_G (X)) =\sum_{[A,c] \in \mathcal{Q}^\prime _{\:\max \:} (G,X)}\dfrac{1}{\lvert W_g (A,c) \rvert}\deg(H^\ast_{C_g (A,c)} (c)) \; \textrm{.}\]We also show how this formula relates to the additivity formula from commutative algebra, demonstrating both the algebraic and geometric character of the degree invariant.

Keywords

homology, homotopy

Received 20 February 2022

Accepted 4 May 2022

Published 26 April 2023