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Hello! My question is about the realization of homology class.

The definition of the realizaion of homology class is: for manifold M and a homology class $z\in H_k(M)$, k is an integer. If we find a k-dimensional manifold N and a map $f:N \rightarrow M$ such that $f_* [N]=z$, $[N]$ is the fundamental class, then we call the homology class $z$ can be realized.

For this problem, Thom has the following theorem:

Thom[1954] For every manifold M, consider a interger coefficient homology class $z\in H_*(M)$, that there exist a interger $l$ and $lz$ can be realized.

My Question is Simple: Why we should add this interger $l$?

Thom's original paper is written by French and I cann't understand it. Recently, I am reading a paper by A.Gaifullin:Combinatorial Realisation of Cycles and Small Covers and the result is related to Thom's paper.

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As far as I can tell, Thom's paper is about realization by EMBEDDED submanifolds. –  Igor Rivin Oct 19 '12 at 16:43
Well, Thom gives examples of compact differentiable manifolds where one cannot take $l=1$. Are you asking for an english explaination of these examples? –  J.C. Ottem Oct 19 '12 at 16:44
Some torsion classes can not be realized as fundamental classes of manifolds, that's why $l>1$ can happen. –  Ryan Budney Oct 19 '12 at 16:45
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1 Answer

up vote 2 down vote accepted

Duplicate of this question, which has a very good answer by Eric Wofsey.

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@Allen: thanks! –  Igor Rivin Oct 19 '12 at 18:26
Oh, that really helps me a lot, thank you! –  Siqi He Oct 20 '12 at 23:50
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