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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
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
The first-named author was partially supported by FCT/Portugal through CAMGSD, IST-ID, and by projects UIDB/04459/2020 and UIDP/04459/2020.
Received 4 February 2022
Received revised 27 June 2022
Accepted 23 July 2022
Published 30 January 2024