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Cohomology of \(BO(n_1)\times \cdot \cdot \cdot \times BO(n_m)\) with local integer coefficients. (English) Zbl 1121.55012

Summary: Let \(\mathcal Z\) 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 \(\bigoplus _{\mathcal G\in \mathcal Z} H^*(BO(n_1)\times \dots \times BO(n_m); \mathcal G)\), which has a structure of a \(\mathbb Z\oplus (\mathbb Z_2)^m\)-graded ring, and describe it in terms of generators and relations. The cohomology ring with integer coefficients is contained as its subring.

MSC:

55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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