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INVARIANT SULLIVAN–DE RHAM FORMS ON CYCLIC SETS

Published online by Cambridge University Press:  01 April 1998

CHRISTOPHER ALLDAY
Affiliation:
Department of Mathematics, University of Hawaii at Manoa, 2565 The Mall, Honolulu, Hawaii 96822
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Abstract

Let Y be a topological space; and suppose that the circle group, S1, which will be denoted by G throughout this paper, is acting on Y. Then the equivariant cohomology H*G(Y) of Y is defined to be H*(YG), where YG=(EG×Y)/G, the Borel construction. Because of the bundle map YGBG, the cohomology H*G(Y) is an H*(BG)-module. With rational coefficients H*(BG; ℚ)=ℚ[u], where deg(u)=2. And, when Y is a reasonably nice finite-dimensional space, for example, a finite-dimensional G-CW-complex, then the localized module H*G(Y; ℚ)[u−1] is isomorphic to H*G (YG; ℚ)[u−1] via the restriction homomorphism, where YGY is the fixed point set. (See, for example, [2].) That this very useful Localization Theorem does not generalize to infinite dimensional spaces can be seen by taking Y=EG, for example.

Type
Notes and Papers
Copyright
The London Mathematical Society 1998

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