Classifying space for U(n): Difference between revisions

Revision as of 08:40, 15 March 2024

Revision as of 22:34, 31 October 2024

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=\mathbb{Z}[c_1,\ldots,c_n].</math>

'''Proof:''' Let us first consider the case ''n'' = 1. In this case, U(1) is the circle '''S'''¹ and the universal bundle is '''S'''^∞ → '''CP'''^∞. It is well known<ref>R. Bott, L. W. Tu-- ''Differential Forms in Algebraic Topology'', Graduate Texts in Mathematics 82, Springer</ref> that the cohomology of '''CP'''^''k'' is isomorphic to <math>\mathbf{RZ}\lbrack c_1\rbrack/c_1^{k+1}</math>, where ''c''₁ is the [[Euler class]] of the U(1)-bundle '''S'''^2''k''+1 → '''CP'''^''k'', and that the injections '''CP'''^''k'' → '''CP'''^''k''+1, for ''k'' ∈ '''N'''*, are compatible with these presentations of the cohomology of the projective spaces. This proves the Proposition for ''n'' = 1.

There are homotopy fiber sequences