Steenrod problem

Revision as of 12:18, 31 March 2011

1 Introduction

Given a space X, there is a homomorphism $<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> {{Stub}} == Introduction == <wikitex>; 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? </wikitex> == References == {{#RefList:}} [[Category:Theory]]\Phi : \Omega^{SO}_{∗}(X) \to H_{*}(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 ^[1];: Given a simplicial complex $X$, is every (integral) homology class representable?

2 References