On the equivariant cohomology of isotropy actions
Let G be a compact connected Lie group and K \subseteq G a closed subgroup. We show that the isotropy action of K on G/K is equivariantly formal and that the space G/K is formal in the sense of rational homotopy theory whenever K is the identity component of the intersection of the fixed point sets...
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Format:  Dissertation 
Language:  English 
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PhilippsUniversität Marburg
2018
Reine und Angewandte Mathematik 
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Summary:  Let G be a compact connected Lie group and K \subseteq G a closed subgroup. We show that the isotropy action of K on G/K is equivariantly formal and that the space G/K is formal in the sense of rational homotopy theory whenever K is the identity component of the intersection of the fixed point sets of two distinct involutions on G, so that G/K is a \mathbb{Z}_2\times\mathbb{Z}_2symmetric space. If K is the identity component of the fixed point set of a single involution and H \subseteq G is a closed connected subgroup containing K, then we show that the action of K on G/H by leftmultiplication is equivariantly formal. The latter statement follows from the wellknown special case K = H, but is proved by different means, namely by providing an algebraic model for the equivariant cohomology of certain actions. 

Physical Description:  68 Pages 
DOI:  https://doi.org/10.17192/z2018.0496 