Topological invariance of equivariant characteristic classes (Q1355676)

The author proves the topological invariance of certain equivariant characteristic classes with the help of a result of P. May. He gives an alternative proof of Kawakubo's results for Borel cohomology characteristic classes. Kawakubo has considered the equivariant Stiefel-Whitney classes and equivariant rational Pontryagin classes of smooth \(G\)-manifolds. He reduces the problem of equivariant Top-invariance to the non-equivariant one by using a result of Borel-Hirzebruch on the decomposition of tangent bundles. The method proposed in this paper gives a result for any \(G\)-vector bundle. After an introduction, the author recalls Borel cohomology, Borel characteristic classes, and presents the essential step, a result of P. May with the help of which he reduces the equivariant problem to the non-equivariant case. In the third section the main results of the paper are proved. Theorem 1: Equivariant Stiefel-Whitney classes are Top-invariant. Theorem 2: Equivariant rational Pontryagin classes are Top-invariant. Theorem 3: Equivariant integral Pontryagin classes \(\text{mod }p\) are Top-invariant for \(p=3,5\). These results are established with the help of the following theorem which gives a sufficient condition for topological invariance of \(G\)-equivariant characteristic classes. Theorem A: A universal \(G\)-equivariant characteristic class \(x^G\in H_G^*(\text{BSO}(G): \Lambda)\) is Top-invariant if \(x^G\in \text{Image} [\Phi_G^*: H_G^*(\text{BSTOP} (G);\Lambda)\to H_G^*(\text{BSO}(G); \Lambda)]\).