Steenrod problem
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1 Introduction
Given a space X there is a homomorphism Φ : Ω∗(X) → H∗(X,Z), called the Thom homomorphism, given by [M,f] → f∗([M]) where [M] is the fundamental class of M. The elements in the image of Φ are called representable. In certain situations it is convenient to assume that a homology class is repre- sentable. In dimensions 0 and 1 it is clear that Φ 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 [2]: Given a simplicial complex X, is every (integral) homology class representable?
