oantolin, 7 months ago (edited 7 months ago)

@johncarlosbaez @Diffgeometer1 I thought that Thom proved something slightly different. Here's two ways you can use other manifolds to get homology class of M:

1. Take an oriented submanifold N of M, with inclusion j : N → M and consider j_*([N]).

2. Take an arbitrary oriented manifold N with an arbitrary smooth map f : N → M and consider f_*([N]).

I thought Thom's results were about 2, not 1. Do the classes considered in 1 also generate the rational homology?