On the cohomology of classifying spaces of groups of homeomorphisms (Q2921045)

Let \(M\) be a closed simply connected \(2n\)-manifold. This paper deals with the cohomology of classifying spaces of connected groups of homeomorphism. Let \(G\) be a compact semisimple Lie group. A co-adjoint orbit of \(G\) is generic if there is a non-empty Zariski open subset \(Z\) of the dual of the Lie algebra of \(G\) such that if \(\xi\in Z\), the orbit \(G\xi\) has the property that the action \(G\to\text{Homeo}(G\xi)\) induces a surjective homomorphism of the real cohomology of classifying spaces.NEWLINENEWLINE The first result of this paper asserts that if \(M= G\xi\) is a generic co-adjoint orbit of \(G\) and the action \(G\to\text{Homeo}(M)\) has a finite kernel, then the map \(BG\to B\text{\,Homeo}_0(M)\) induces a surjective homomorphism of real cohomology. The homomorphism is surjective in degree 4 for an arbitrary co-adjoint. The paper offers applications of this result and related concepts. If the product of complex projective spaces \(\mathbb{C} P^m\times\mathbb{C} P^n\) has a product symplectic form such that the symplectic areas of lines in the factors are equal and \(1\leq k< n\leq m\), then rank \((\pi_{2k+1}(\text{Dift}(\mathbb{C} P^m\times\mathbb{C} P^n)))\) is at least \(2n- k+ 2\) if \(m-k\geq n\) and at least \(m+ n- 2k+ 2\) if \(m-k<n\).