Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 6

Special Issue in honor of Professor Blaine Lawson’s 80th birthday

Guest Editors: Shiu-Yuen Cheng, Paulo Lima-Filho, and Stephen Shing-Toung Yau

$RO(C_2)$-graded equivariant cohomology and classical Steenrod squares

Pages: 2787 – 2826

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n6.a7

Authors

Pedro F. dos Santos (Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Portugal)

Paulo Lima-Filho (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Abstract

We investigate the restriction to fixed-points and change of coefficient functors in $RO(C_2)$-graded equivariant cohomology, with applications to the equivariant cohomology of spaces with a trivial $C_2$-action for $\underline{\mathbb{Z}}$ and $\underline{\mathbb{F}_2}$ coefficients. To this end, we study the nonequivariant spectra representing these theories and the corresponding functors. In particular, we show that the $RO(C2)$-graded homology class determined by a Real submanifold $Y$ (in the sense of Atiyah) of a Real compact manifold $X$ encodes the total Steenrod square of the dual to $Y^{C_2}$ in $X^{C_2}$.

Keywords

$C_2$-equivariant cohomology, Steenrod squares, real spaces

2010 Mathematics Subject Classification

55N91

Received 4 February 2022

Received revised 27 June 2022

Accepted 23 July 2022

Published 30 January 2024