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I've run across a way of combining the integral cohomology of the real projective space $RP^\infty$ with its cohomology with twisted coefficients, that seems very simple and natural, but which I don't recall every seeing before, so my question is: Has this been noticed before and, if so, where is it published?

Let me describe the result in a relatively simple way, then comment on how I actually came to it. We know that $H^n(RP^\infty;\mathbb{Z})$ is $\mathbb Z$ if $n=0$, $\mathbb Z/2$ if $n>0$ is even, and 0 otherwise. If $\mathbb Z_-$ denotes the local coefficient system on $RP^\infty$ on which the nontrivial loop acts as $-1$ on $\mathbb Z$, we know that $H^n(RP^\infty;\mathbb Z_-)$ is $\mathbb Z/2$ if $n>0$ is odd, and 0 otherwise.

Let $R = \mathbb{Z}\times\mathbb{Z}/2$ and write elements of $R$ as $n + \epsilon\gamma$, where $n\in\mathbb Z$ and $\epsilon\in\mathbb Z/2$. Define a group graded on $R$ by $$ H^{n+\epsilon\gamma}(RP^\infty) = \begin{cases} H^n(RP^\infty;\mathbb Z) & \text{if $\epsilon=0$} \\ H^n(RP^\infty;\mathbb Z_-) & \text{if $\epsilon=1$.} \end{cases} $$ This is a graded ring, using the pairing of coefficient systems $\mathbb Z_- \otimes \mathbb Z_- \cong \mathbb Z$, etc. With this definition, $$ H^*(RP^\infty) \cong \mathbb Z[w]/\langle 2w \rangle $$ where $w\in H^{1+\gamma}(RP^\infty) = H^1(RP^\infty;\mathbb Z_-)$. (The class $w$ is the Euler class of the canonical line bundle over $RP^\infty$.) This could also be viewed as a description of the cohomology of the group $\mathbb Z/2$, of course. Similar simple descriptions can be given for the cohomologies of the truncated spaces $RP^k$.

Where this actually came from: Stefan Waner and I recently published a Springer Lecture Notes volume in which we describe ordinary equivariant (co)homology graded on "representations of the fundamental groupoid" of a space. This can be applied in the nonequivariant case as well — the group of representations of the fundamental groupoid of $RP^\infty$ is exactly the $R$ above, and the general theory leads to the calculation of $H^*(RP^\infty)$ above with this grading.

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    $\begingroup$ This looks like the $RO(G)$-grading on the Borel cohomology of the point $\endgroup$ – Denis Nardin Dec 4 '17 at 8:22
  • $\begingroup$ Related, but actually rather different. $RO(\mathbb Z/2) \cong \mathbb Z \times \mathbb Z$ and the $RO(\mathbb Z/2)$-graded cohomology of a point is quite a bit more complicated than this calculation. $\endgroup$ – Steve Costenoble Dec 4 '17 at 13:41
  • $\begingroup$ I said, Borel cohomology, which is just the cohomology of the homotopy orbits (yes, that is also $RO(G)$-graded) $\endgroup$ – Denis Nardin Dec 4 '17 at 13:54
  • $\begingroup$ @DenisNardin: Sorry, I was thinking of Bredon cohomology. Yes, you're right. $\endgroup$ – Steve Costenoble Dec 4 '17 at 13:54
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This description is given in Lemma 1 of

  • M. Cadek. The cohomology of $BO(n)$ with twisted integer coefficients. J. Math. Kyoto Univ. 39-2 (1999), 277-286.

(and Cadek goes on to give similar descriptions of the cohomology of infinite real Grassmannians)

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  • $\begingroup$ Exactly what I was looking for. Thanks! $\endgroup$ – Steve Costenoble Dec 4 '17 at 13:39

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