Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Guide to Equivariant Parametrized Cohomology

Agnès Beaudry, Chloe Lewis, Clover May, Sabrina Pauli, Elizabeth Tatum

Abstract

This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group $G$. The theory extends the $RO(G)$-graded cohomology of a $G$-space $B$ to a cohomology graded by $RO(ΠB)$, the representations of the equivariant fundamental groupoid of $B$. This paper is meant to serve as a guide to this theory and contains some new computations. We explain the key ingredients for defining parametrized cellular cohomology when $G$ is a finite group, with particular attention to the case of the cyclic group $G=C_2$. We compute some examples and observe that $RO(ΠB)$ is not always free. When $G$ is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with $G = C_2$ and $G=C_4$.

A Guide to Equivariant Parametrized Cohomology

Abstract

This article investigates equivariant parametrized cellular cohomology, a cohomology theory introduced by Costenoble-Waner for spaces with an action by a compact Lie group . The theory extends the -graded cohomology of a -space to a cohomology graded by , the representations of the equivariant fundamental groupoid of . This paper is meant to serve as a guide to this theory and contains some new computations. We explain the key ingredients for defining parametrized cellular cohomology when is a finite group, with particular attention to the case of the cyclic group . We compute some examples and observe that is not always free. When is the trivial group, we explain how to identify equivariant parametrized cellular cohomology with cellular cohomology in local coefficients. Finally, we illustrate the theory with some new computations of parametrized cellular cohomology for several spaces with and .

Paper Structure

This paper contains 34 sections, 19 theorems, 304 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Literature
  3. Organization
  4. Acknowledgements
  5. The Fundamental groupoid and its representations
  6. The equivariant fundamental groupoid
  7. Examples of $\Pi B$
  8. Representations of the fundamental groupoid
  9. Some computations of $RO(\Pi B)$
  10. Parametrized Homotopy Theory
  11. Parametrized Spaces
  12. Base change functors
  13. Change of Groups
  14. Parametrized orbits and spheres
  15. Spectra and Stable Maps
  16. ...and 19 more sections

Key Result

Lemma 1.1

Let $\mathbb{R}^{3,1}$ be the direct sum of a $2$-dimensional trivial representation and the $1$-dimensional sign representation. Consider $B = \mathbb{P}(\mathbb{R}^{3,1})$, the projective space of $\mathbb{R}^{3,1}$ (often called "$\mathbb{R} P^2$-twist"). There is an isomorphism

Figures (15)

  • Figure 1: Some objects and morphisms in $\Pi S^{1,1}$
  • Figure 2: Some morphisms in $\Pi \mathbb{P}(\mathbb{R}^{3,1})$
  • Figure 3: $G_+\wedge_H S^{V,b}$ and its whiskered replacement
  • Figure 4: The restriction
  • Figure 5: Left: A $\text{CW}(\gamma)$ structure on $S^{1,1}$ for $\gamma= (0,0,1)$. Right: A $\text{CW}(\gamma)$ structure on $S^{1,1}$ for $\gamma= (0,1,1)$
  • ...and 10 more figures

Theorems & Definitions (117)

  • Lemma 1.1: c.f. \ref{['lem:ROpiRP2']}
  • Theorem 1.2: c.f. \ref{['thm:localispofinal']}
  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Example 2.6
  • Remark 2.7
  • Definition 2.8: Change-of-Groups
  • ...and 107 more