Equivariant cohomology of projective spaces

Authors

• Samik Basu, Indian Statistical Institute, Kolkata
• Pinka Dey, Indian Statistical Institute, Kolkata
• Aparajita Karmakar, Indian Statistical Institute, Kolkata

Article Type

Research Article

Publication Title

Algebraic and Geometric Topology

Abstract

We compute the equivariant homology and cohomology of projective spaces with integer coefficients. More precisely, in the case of cyclic groups, we show that the cellular filtration of the projective space P (k ρ) of lines inside copies of the regular representation yields a splitting of H ℤ ∧ P (k ρ) + as a wedge of suspensions of H ℤ. This is carried out both in the complex case, and also in the quaternionic case, and further, for the C2 -action on ℂ Pⁿ by complex conjugation. We also observe that these decompositions imply a degeneration of the slice tower in these cases. Finally, we describe the cohomology of the projective spaces when G is of prime power order, with explicit formulas for (formula presenetd) -coefficients. Letting k = ∞, this also describes the equivariant homology and cohomology of the classifying spaces of S¹ and S³.

First Page

3049

Last Page

3088

DOI

10.2140/agt.2025.25.3049

Publication Date

1-1-2025

Recommended Citation

Basu, Samik; Dey, Pinka; and Karmakar, Aparajita, "Equivariant cohomology of projective spaces" (2025). Journal Articles. 5341.
https://digitalcommons.isical.ac.in/journal-articles/5341