Smooth and topological bordism and homology

Question

I have some questions about homology, manifolds and bordism. First of all, if X is a smooth manifold, in general an integral homology class in X cannot be represented by a smooth embedded submanifold, as Thom proved.

1) If X is a topological manifold, does the same result hold? Are there in general singular homology classes, which are not representable by a topological embedded submanifold?

Then, let us consider the oriented bordism groups of a topological space X. Its elements are represented by couples (M, f), for M a smooth oriented manifold and f: M -> X continuous. There is a natural map to singular homology, defined as $[(M, f)] \rightarrow f_{*}([M])$, which, in general, is not surjective.

2) If we define the "topological bordism", requiring that M is a topological manifold (not necessarily smooth), is the corresponding map to singular homology surjective?

3) If X is a smooth manifold, and we define bordism requiring that f is a smooth map (not only continuous), do we obtain the same bordism groups (up to isomorphism)?

Answer 1

(3) Yes, this is smooth approximation theory. See Hirsch's "Differential Topology" textbook.

I believe (1) and (2) were effectively answered by Larry Siebenmann in his ICM paper "Topological Manifolds". You can find it on Ranicki's webpage.

In particular, the topological bordism groups are a direct sum of the smooth bordism groups with a complementary factor which is entirely 2-torsion. So I suspect you have similar obstructions to realizability, like odd torsion homology classes in high dimensions. I don't know this material very well but that's where I'd start.

Answer 2

(3) Yes, this is exactly the differential bordism groups $D_k(Y)$ of Conner (see §1.9 in P.E. Conner, Differentiable Periodic Maps, second edition, Springer Lecture Notes in Mathematics 738, Springer-Verlag, Berlin, 1979.)

He proves in Theorem I.9.1 that the natural projection $D_k(Y) → MSO_k(Y)$ is an isomorphism. Hence the so defined bordism groups are isomorphic to the usual definition of bordism groups $MSO_k(Y)$. (As in Conner, or Atiyah (M.F. Atiyah, Bordism and cobordism, Proc. Camb. Phil. Soc. 57 (1961), 200–208.))