Splitting principle in equivariant cohomology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T03:41:28Z http://mathoverflow.net/feeds/question/78159 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/78159/splitting-principle-in-equivariant-cohomology Splitting principle in equivariant cohomology Julia Sauter 2011-10-14T18:32:11Z 2011-10-14T18:41:44Z <p>The following is a weaker version of what is called splitting principle in<br> <a href="http://www.ma.huji.ac.il/~karshon/monograph" rel="nofollow"> Appendix C, page 12</a>, see also for a lighter version <a href="http://www-fourier.ujf-grenoble.fr/~mbrion/notes.html" rel="nofollow">Brions Eq cohom and eq intersection theory, page 6</a>:<br> Let \$G\$ be a compact (complex) connected Lie group with torus \$T\subset G\$ and \$N\$ its normalizer, \$W=N/T\$ its Weyl group. Let \$X\$ be any \$G\$-variety. Then, there is an isomorphism </p> <p>\$ H_G^* (X) \cong (H_T^*(X))^W \$<br> of graded algebras. For \$X=pt\$ this is known to be Chevalley's restriction theorem. </p> <p>My question is, can one drop the assumption on \$G\$ and \$T\$ to be compact? I came across an article <a href="http://arxiv.org/abs/0901.3992" rel="nofollow">VarVas, page 12</a>, where that has been claimed (with G=Gl_n, X quasi-projective) to be a standard result and no reference is provided.</p> http://mathoverflow.net/questions/78159/splitting-principle-in-equivariant-cohomology/78161#78161 Answer by Angelo for Splitting principle in equivariant cohomology Angelo 2011-10-14T18:41:44Z 2011-10-14T18:41:44Z <p>The embedding of the unitary group \$U_n\$ into \$GL_n(\mathbb C)\$ is a homotopy equivalence; this is easily seen to imply that \$H^*_{U_n}(X)\$ is isomorphic to \$H^*_{GL_n}(X)\$. So the result for compact groups implies that for \$GL_n\$.</p> <p>The same idea works for any reductive complex algebraic group \$G\$, since the embedding of a maximal compact subgroup into \$G\$ is a homotopy equivalence.</p>