5
$\begingroup$

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)?

$\endgroup$
4
$\begingroup$

(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.

$\endgroup$
2
$\begingroup$

(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.))

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.