Steenrod problem

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[edit] 1 Introduction

Given a space X, there is a homomorphism $\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Phi : \Omega^{SO}_{\ast}(X) \to H_{\ast}(X,\mathbb{Z})$ , 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:

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.