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.



Geometry Realization of Homology Class
Duplicate of [this question](https://mathoverflow.net/questions/1489/cohomology-and-fundamental-classes), which has a very good answer by Eric Wofsey.


