If (M) is a closed oriented manifold, I've heard that the (k)-th homology groups of (M) with coefficients in (R) (denoted ( H_k(M;\mathbb{R}))) is generated by closed oriented submanifolds of (M). So a (k)-cycle is expressed[c=\sum_{i}a_iZ_i ]where (Z_i) is a closed oriented (k)-submanifold of (M) and (a_i\in \mathbb{R}). From here one has a nondegenerate pairing between de Rham cohomology and (H_k(M;\mathbb{R}))
[H^k_{dR}(M)\times H_k(M;\mathbb{R})\rightarrow \mathbb{R}] given by[\langle[\alpha],[c]\rangle:=\int_c \alpha = \sum_ia_i\int_{Z_i}\alpha.]This is the way physicists do it (for example in the string theory book by Schwartz and Becker). In math, we would define maps (\sigma:\Delta^k\rightarrow M) and then define boundary maps, chain complex, and ultimately the definition of the homology groups (in whatever coefficients (\mathbb{Z}, \mathbb{Q}, \mathbb{R}) ect). These two apporaches have to be equivalent although the physics one is nice and intuitive. Can anyone explain why they're equivalent or give a reference where for why homology classes of (H_k(M;\mathbb{R})) can be expressed as a linear combination of closed oriented (k)-dimensional submanifolds?
@Diffgeometer1 - the fact that the homology with real or rational coefficients of a compact oriented manifold M is generated by compact oriented submanifolds is, to the best of my knowledge, a pretty hard result! I don't know the proof, but it was proved by Renee Thom in his 1954 paper "Quelques propriete globales des varietes differentiables". It's not true for homology with integer coefficients, though Thom proved it's true up to and including dimension 8.
@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:
Take an oriented submanifold N of M, with inclusion j : N → M and consider j_*([N]).
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?
@oantolin - The paper by Sullivan that I linked to above seems to claim that the cohomology classes of embedded oriented submanifolds are sufficient to generate the rational homology. Sullivan writes:
"It was known classically to Poincare that codimension one integral homology classes in a manifold are represented by embedded submanifolds. In codimension two this is also true and known to Thom's generation. In higher codimensions there are obstructions which can be described by Thom's duality picture - a homology class in M is represented by a submanifold of M of codimension k iff the Poincare
dual cohomology class in M is induced from the Thom class by a map of M to the Thom space of the universal k-bundle.
The free structure of the rational cohomology (all the powers of the universal Thom class are nonzero for k even) implies all of Thom's realization obstructions are finite. So Thom gets the result that any ray in the rational homology of any dimension in any manifold is represented by an embedded submanifold. One can add the information that the normal bundle of the submanifold may be taken to be trivial in odd codimension always and in even codimension iff the intersection of the homology class with itself is zero."
I believe this "ray" business means that while a rational cohomology class may not be represented by a submanifold, some rational multiple of it will be.
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