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singular cohomology with local coefficients
Let $\Cal Z$ be a set of all possible nonequivalent systems of local integer coefficients over the classifying space $BO(n_1)\times \dots \times BO(n_m)$. We introduce a cohomology ring $\bigoplus_{\Cal G\in \Cal Z} H^*(BO(n_1)\times \dots \times BO(n_m);\Cal G)$, which has a structure of a $\Bbb Z\oplus (\Bbb Z_2)^m$-graded ring, and describe it in terms of generators and relations. The cohomology ring with integer coefficients is contained as its subring. This result generalizes both the description of the cohomology with the nontrivial system of local integer coefficients of $BO(n)$ in [Č] and the description of the cohomology with integer coefficients of $BO(n_1)\times \dots \times BO(n_m)$ in [M].
[B] Brown E.H. Jr.: The cohomology of $BSO(n)$ and $BO(n)$ with integer coefficients. Proc. Amer. Mat. Soc. 85 (1982), 283-288. MR 0652459
[Č] Čadek M.: The cohomology of $BO(n)$ with twisted integer coefficients. J. Math. Kyoto Univ. 39 2 (1999), 277-286. MR 1709293 | Zbl 0946.55009
[F] Feshbach M.: The integral cohomology rings of the classifying spaces of $O(n)$ and $SO(n)$. Indiana Univ. Math. J. 32 (1983), 511-516. MR 0703281 | Zbl 0507.55014
[M] Markl M.: The integral cohomology rings of real infinite dimensional flag manifolds. Rend. Circ. Mat. Palermo, Suppl. 9 (1985), 157-164. MR 0853138 | Zbl 0591.55007
[MS] Milnor J.W., Stasheff J.D.: Characteristic Classes. Princeton University Press and University of Tokyo Press, Princeton, New Jersey, 1974. MR 0440554 | Zbl 1079.57504
[S] Spanier E.: Algebraic Topology. McGraw-Hill, New York-Toronto, Ont.-London, 1966. MR 0210112 | Zbl 0810.55001
[T] Thomas E.: On the cohomology of the real Grassman complexes and the characteristic classes of the $n$-plane bundle. Trans. Amer. Math. Soc. 96 (1960), 67-89. MR 0121800
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