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The universality of equivariant complex bordism. (English) Zbl 1008.55015

The authors show that for any abelian compact Lie group \(A\), every \(A\)-equivariant complex vector bundle is \(E\)-orientable, where \(E\) is a complex orientable equivariant cohomology theory (Theorem 1.1). To obtain the result, they calculate the complex orientable homology and cohomology of all complex Grassmannians. Moreover, they show that for a complex oriented cohomology theory \(E^*_A (\cdot)\) with orientation in cohomological degree 2, there is a unique ring map \(MU\to E\) of \(A\)-spectra under which the orientation of \(E\) is the image of the canonical orientation, and conversely, a map \(MU\to E\) of ring \(A\)-spectra gives \(E\) the structure of a complex oriented cohomology theory with orientation in cohomological degree 2 (Theorem 1.2).

MSC:

55R25 Sphere bundles and vector bundles in algebraic topology
55N25 Homology with local coefficients, equivariant cohomology
55R91 Equivariant fiber spaces and bundles in algebraic topology
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