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 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.