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Could someone tell me what is the cohomology algebra $H^*(\mathbb{R}P^n \# \mathbb{R}P^n; \mathbb{Z}_2)$ and how to compute it. Here $\#$ is the connected sum.

Thanks.

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    First compute the cohomology groups using Mayer-Vietoris. See here: http://math.stackexchange.com/questions/31287/homology-of-punctured-projective-space2012-01-14

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