Steenrod problem

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[edit] 1 Introduction

Given a space X, there is a homomorphism \Phi : \Omega^{SO}_{\ast}(X) \to H_{\ast}(X,\mathbb{Z}), called the Thom homomorphism, given by [M,f] \to f_{*}([M]) where [M] is the fundamental class of M. The elements in the image of \Phi are called representable. In certain situations it is convenient to assume that a homology class is representable. In dimensions 0 and 1 it is clear that \Phi is surjective (even an isomorphism). It is less obvious in dimension 2, but also can be shown geometrically. This made Steenrod raise his famous problem in 1946 [Eilenberg1949]:

Given a simplicial complex X, is every (integral) homology class representable?

The answer was given by Thom in 1954 [Thom1954]. He showed that in dimensions 0 \leq m \leq 6 this is true but in general this is not the case. He constructed a counter example in dimension m=7. Thom also showed that the corresponding problem with \mathbb{Z}/2 coefficients is correct, that is the corresponding homomorphism \Phi : \Omega^{O}_{\ast}(X) \to H_{\ast}(X,\mathbb{Z}/2) is surjective. Thom also proved the following:

Theorem 1.1 [Thom1954, Theorem III.4]. For every class in dimension p of integral homology of a finite polyhedron K, there exists a non zero integer N which depends only on p, such that the product Nz is the image of the fundamental class of a closed oriented differentiable manifold.

More about that can be found in [Sullivan2004].

[edit] 2 References

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