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What I am trying to do is to compute $\mathbb{Z}$-graded Bredon homology of $S(1+\sigma)$ over $Q\times\Sigma_2$, where

  • $Q$ is a cyclic group of order 2
  • $\sigma$ is its real sign representation
  • $\Sigma_2$ is acting on $S(1+\sigma)$ by antipodal action.

In particular, different notation for a cyclic group of order 2 is used to emphasize different actions. To make notation more legible, let's put $S:=S(1+\sigma)$. Its Bredon homology with $\underline{\mathbb{F}}_2$ coefficients is given by $H^{Q\times\Sigma_2}_0(S;\underline{\mathbb{F}}_2)=\mathbb{F}_2\oplus\mathbb{F}_2$ and $H^{Q\times\Sigma_2}_1(S;\underline{\mathbb{F}}_2)=\mathbb{F}_2$.

I may compute this homology also by the Universal Coefficient Spectral Sequence, which is given by $$ E^2_{p,q}=Tor^{\mathcal{O}_{Q\times\Sigma_2}}_p\left(H_q(S^\bullet),\underline{\mathbb{F}}_2\right)\Rightarrow H^{Q\times\Sigma_2}_{p+q}(S;\underline{\mathbb{F}}_2) $$

Singular homology here is taken with integral coefficients. (Using this method is a little bit of overkill, but this is supposed to be a toy example for further calculations).

I have inputs of the $E^2$ page computed - they are given by: $$ Tor_i(H_0(S^\bullet),\underline{\mathbb{F}}_2)= \left\{ \begin{array}{cc} \mathbb{F}_2\oplus\mathbb{F}_2 & i=0 \\ \mathbb{F}_2 & i=1 \\ \mathbb{F}_2^{i-1} & i>1 \end{array} \right. $$

and

$$ Tor_i(H_1(S^\bullet),\underline{\mathbb{F}}_2)= \mathbb{F}_2^{i+1} $$ for $i\geq 0$. Everything else vanish.

So I have my second page of the spectral sequence, but I am struggling with identifying the differentials. From the calculations "by hand" I see that they are supposed to be isomorphisms. But I have got no clue how to show that.

Could anybody give me a hand on that?

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    $\begingroup$ The best I can suggest is that, if you give $S$ an equivariant cell structure, it only has nonzero cells in dimensions $0$ and $1$, so there is an exact sequence $$0 \to H_1(S) \to C_1(S) \to C_0(S) \to H_0(S) \to 0.$$ You can deduce (using long exact sequences or spectral sequences) that there is a map $Tor_{i+2}(H_0(S),M) \to Tor_i(H_1(S),M)$, induced by cap product with this Yoneda extension. The middle two terms are projective Mackey functors, so this cap is an isomorphism for large enough $i$. But perhaps this is just shuttling around information you already know... $\endgroup$ – Tyler Lawson Jan 22 at 18:50
  • $\begingroup$ Well, it's exactly opposite :) How you can deduce existence of these maps? $\endgroup$ – Igor Sikora Jan 23 at 16:21
  • $\begingroup$ Which maps? The ones in the exact sequence above, or the ones on Tor? (Or both?) $\endgroup$ – Tyler Lawson Jan 23 at 16:32
  • $\begingroup$ The ones on Tor, I can see the ones in the exact sequence. $\endgroup$ – Igor Sikora Jan 23 at 16:34
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    $\begingroup$ So far as the spectral sequence goes, you need a specific construction of the universal coefficient spectral sequence to give a precise argument of how the differentials happen. One construction is exactly that you filter $C_\bullet(S)$ by "connective covers" and then put together the long exact sequences on $Tor$ precisely so that the differentials are these iterative connecting homomorphisms, which makes the above description easier -- but a different construction of the spectral sequence might make it harder. Do you have a particular construction in mind? $\endgroup$ – Tyler Lawson Jan 23 at 17:20

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