# Steenrod problem

## 1 Introduction


Given a simplicial complex $X$$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$$0 \leq m \leq 6$ this is true but in general this is not the case. He constructed a counter example in dimension $m=7$$m=7$. Thom also showed that the corresponding problem with $\mathbb{Z}/2$$\mathbb{Z}/2$ coefficients is correct, that is the corresponding homomorphism $\Phi : \Omega^{O}_{\ast}(X) \to H_{\ast}(X,\mathbb{Z}/2)$$\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$$p$ of integral homology of a finite polyhedron K, there exists a non zero integer $N$$N$ which depends only on p, such that the product $Nz$$Nz$ is the image of the fundamental class of a closed oriented differentiable manifold.

More about that can be found in [Sullivan2004].