I've seen a couple of similar questions asking to verify computations of Bredon cohomology here and here, so I will ask one such question myself.

Let $\mathbb{Z}/2$ act on $S^3\subset \mathbb{C}^2$ by restriction of a permutation action on $\mathbb{C}^2.$ I wanted to compute Bredon cohomology $\mathcal{H}^*_{\mathbb{Z}/2}(S^3;\underline{\mathbb{Z}}).$

I have a cell decomposition based on a decomposition of complex $1$-dimensional disk into $3$ cells: $\mathbb{D}=D\sqcup T\sqcup *.$ Here $T\sqcup *=S^1=\partial \mathbb{D}$ and $D$ is the interior of $\mathbb{D}.$ Then we have a decomposition of $S^3=\mathbb{D}\times S^1 \cup S^1\times \mathbb{D}$ into cells compatible with the $\mathbb{Z}/2$ action.

The fixed point set of an action is a circle given by $\{z_1=z_2\}\cap S^3\subset \mathbb{C}^2.$ Since the orbit category of $\mathbb{Z}/2$ consists of $*$ and $\mathbb{Z}/2$ there are the following equivariant chains: \begin{array}{|c|c|c|c|} \hline \operatorname{dim} &*& \mathbb{Z}/2 & \operatorname{cells corresponding to} \underline{C}_n(S^3)(\mathbb{Z}/2)\\ \hline 0 & \mathbb{Z} & \mathbb{Z} & * \times *\\ 1 & 0 & \mathbb{Z}\oplus\mathbb{Z},\quad \begin{pmatrix} 1 \\ 0\end{pmatrix}\xrightarrow{\overline{1}} \begin{pmatrix} 0 \\ 1\end{pmatrix} & T\times *,*\times T\\ 2 & 0 & \mathbb{Z}\oplus \mathbb{Z} \oplus \mathbb{Z},\quad \begin{pmatrix} 1 \\ 0\\0\end{pmatrix}\xrightarrow{\overline{1}}\begin{pmatrix} 0 \\ 1\\0\end{pmatrix};\;\begin{pmatrix} 0 \\ 0\\1\end{pmatrix}\xrightarrow{\overline{1}}\begin{pmatrix} 0 \\ 0\\-1\end{pmatrix} & D\times *, *\times D, T\times T\\ 3 & 0 & \mathbb{Z}\oplus \mathbb{Z},\quad \begin{pmatrix} 1 \\ 0\end{pmatrix}\xrightarrow{\overline{1}} \begin{pmatrix} 0 \\ 1\end{pmatrix} & D\times T, T\times D\\ \hline \end{array}

So it seems that the cochains valued in $\underline{\mathbb{Z}}$ are:

\begin{array}{|c|c|} \hline \operatorname{dim} & \\ \hline 0 & \mathbb{Z}\\ 1 & \mathbb{Z}\\ 2 & \mathbb{Z}\\ 3 & \mathbb{Z}\\ \hline \end{array} Since $(T\times T)^*=0$ in cochains, we have $\mathcal{H}^3_{\mathbb{Z}/2}(S^3;\underline{\mathbb{Z}})=\mathbb{Z}.$ Differential $d_1$ is an isomorphism since $\partial(D\times *)=T\times *.$ It seems that $\mathcal{H}^*_{\mathbb{Z}/2}(S^3;\underline{\mathbb{Z}})=H^*(S^3;\mathbb{Z}).$

It is a bit odd to me that the quotient is a homological sphere. Sure, the group $\mathcal{H}^3_{\mathbb{Z}/2}(S^3;\underline{\mathbb{Z}})=\mathbb{Z}$ since orientation is preserved, but maybe I've missed some $2$-torsion in lower degrees?

  • 2
    $\begingroup$ I think you're correct. $\underline{\Bbb Z}$ coefficients have the property that they compute the integral cohomology of the orbit space $S^3 / (\Bbb Z/2)$, and by a slightly different cell decomposition I think this space is homotopy equivalent to $S^3$. $\endgroup$ Commented Aug 16, 2020 at 15:26

1 Answer 1


Your final answer is correct, but the cell structure you're using isn't a $G$-CW structure: $T\times T$ can't be used as a cell in this way.

I would approach it like this: The action of $G = {\mathbb Z}/2$ on $\mathbb{C}\times\mathbb{C}$ can be written as the representation $\mathbb{C}\oplus\mathbb{C}^\sigma$, where $G$ acts trivially on $\mathbb{C}$ and by negation on $\mathbb{C}^\sigma$. The sphere $S(\mathbb{C}\oplus\mathbb{C}^\sigma)$ is also the one-point compactification $S^{1+2\lambda}$, where $\lambda$ denotes the real line with $G$ acting by negation. This has a $G$-CW structure with

  1. one $G$-fixed 0-cell,
  2. one $G$-fixed 1-cell,
  3. one $G$-free 2-cell, and
  4. one $G$-free 3-cell,

so that the skeleta are $*$, $S^1$, $S^{1+\lambda}$, and $S^{1+2\lambda}$. From here you can work out that the $\underline{\mathbb{Z}}$-cochain complex is $$ \mathbb{Z} \xrightarrow{0} \mathbb{Z} \xrightarrow{1} \mathbb{Z} \xrightarrow{0} \mathbb{Z}. $$

A way to check that the answer is correct is to write $$ H_G^n(S^{1+2\lambda}) \cong \tilde H_G^n(S^0) \oplus \tilde H_G^n(S^{1+2\lambda}) \cong \tilde H_G^n(S^0)\oplus \tilde H_G^{n-1-2\lambda}(S^0) $$ and then use the known calculation of the $RO(G)$-graded cohomology of a point (originally due to Stong (unpublished), since published in various places).

  • $\begingroup$ It is not a G-CW structure because fixed points are not a G-CW subspace? $\endgroup$ Commented Aug 17, 2020 at 8:15
  • 1
    $\begingroup$ All the cells in a $G$-CW complex have to have the form $G/H\times D^n$ where $G$ acts trivially on $D^n$. $T\times T$ doesn't have that form, it looks like the disc of a nontrivial representation of $G$. $\endgroup$ Commented Aug 17, 2020 at 10:23
  • $\begingroup$ Is there a way to work with "cells" with nontrivial action of the stabilizer? I'm now aware that any cell with nontrivial action can be subdivided into "good" G-cells, but is there a way to avoid this step? $\endgroup$ Commented Aug 17, 2020 at 12:28
  • 1
    $\begingroup$ Not easily, at least not if you want to calculate the integer-graded part of the cohomology. Any filtration gives rise to a spectral sequence that could, theoretically, be used for computation, but that's not going to be straightforward. There is a notion of $G$-CW($V$) complexes, using cells of the form $G/H\times D(V)$, but that calculates the cohomology in grading $V$. I think Stefan Waner first noticed this, Gaunce Lewis published an exposition, and then Stefan and I published a book using a very generalized version. $\endgroup$ Commented Aug 17, 2020 at 14:47
  • $\begingroup$ One other situation where you can use arbitrary cells: At least for $G = \mathbb{Z}/p$, if you have a space built out of cells of the form $G/H\times D(V)$ where $V$ is even-dimensional, but can vary, then, with some additional assumptions, you can conclude that the $RO(G)$-graded cohomology is a free module over the cohomology of a point, though the generators may not be where you expect them to be. I think this was first proved by Ferland and Lewis. $\endgroup$ Commented Aug 17, 2020 at 17:21

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.