The cohomology of the symmetric groups

Mann, Benjamin Michael

doi:10.1090/S0002-9947-1978-0500961-9

The cohomology of the symmetric groups
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Trans. Amer. Math. Soc. 242 (1978), 157-184 Request permission

Abstract:

Let ${{\mathcal {S}}_n}$ be the symmetric group on n letters and SG the limit of the sets of degree +1 homotopy equivalences of the $n - 1$ sphere. Let p be an odd prime. The main results of this paper are the calculations of ${H^{\ast }}({\mathcal {S}_n}, Z/p)$ and ${H^{\ast }}(SG,Z/p)$ as algebras, determination of the action of the Steenrod algebra, $\mathcal {a}(p)$, on ${H^{\ast }}({\mathcal {S}_n}, Z/p)$ and ${H^{\ast }}(SG,Z/p)$ and integral analysis of ${H^{\ast }}({\mathcal {S}_n}, Z, p)$ and ${H^{\ast }}(SG, Z, p)$.

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