The rational homotopy of Thom spaces and the smoothing of homology classes (Q1075650)

Let G be a closed connected subgroup of the special orthogonal group SO(n) of \({\mathbb{R}}^ n\) and \(i: G\to SO(n)\) an embedding. Let V be a closed oriented manifold of dimension m. A homology class \(z\in H_{m- n}(V; {\mathbb{Z}})\) is said to be G-smoothable if z is represented by an (m- n)-dimensional submanifold W of V which has G as the structure group of its normal bundle. Let \(\gamma^ n\) be the universal oriented n-plane bundle over the classifying space BSO(n) of SO(n), and let \(\gamma^ G\) be the pull-back of \(\gamma^ n\) over the classifying space BG of G. If V is a finite polyhedron, a cohomology class \(u\in H^ n(V; {\mathbb{Z}})\) is said to be G-realizable if \(u=g^*(u_ G)\) for some map \(g: V\to MG\), where MG is the Thom space of \(\gamma^ G\) and \(u_ G\in H^ n(MG; {\mathbb{Z}})\) is the universal Thom class. Thom's results say that \(z\in H_{m-n}(V; {\mathbb{Z}})\) is G-smoothable if and only if its Poincaré dual \(u\in H^ n(V; {\mathbb{Z}})\) is G-realizable, and if \(G=SO(n)\), then some non-zero multiple of u is G-realizable for any u. The main results of this paper are the following. Theorem. Let \(e_ G=e(\gamma^ G)\in H^ n(BG; {\mathbb{Q}})\) be the universal Euler class. (i) If \(e_ G=0\), then some nonzero multiple of u is G-realizable if and only if \(u^ 2\) is a torsion element. (ii) If \(e_ G=0\), then some nonzero multiple of any u is G-realizable if and only if \(e_ G\) is not decomposable in \(H^*BG\). (iii) If V is a finite connected polyhedron such that \(H^ i(V; {\mathbb{Q}})=0\) for \(i>2n+3\), then some nonzero multiple of any u is G-realizable for any \(G\to SO(n)\) with \(e_ G=0\). Corollary. If G is one of the classical groups U(r), Sp(r), \(r\geq 1\), or SU(r), \(r\geq 2\), with standard embeddings, then some nonzero multiple of any u is G-realizable. These results are obtained by studying the rational homotopy of MG.