TITLE: When is a Homology Class Represented by a Submanifold?
QUESTION [41 upvotes]: Possible Duplicate:
Cohomology and fundamental classes 

Given an oriented manifold     $M$ and an oriented submanifold     $\phi:N\to M$ we can obtain a homology class     $\phi_*[N]\in H_*(M)$ where     $[N]$ is the fundamental class of     $N$.  In general, it is not true that every homology class of     $M$ can be represented by a submanifold in this manner, however for some special cases it is.
For example, for     $M$ an oriented (and closed maybe?) 4-manifold every homology class can be represented by a submanifold.  Another example is when     $M$ an Euclidean configuration space.
My questions are:
1) Under what circumstances can every homology class of     $M$ be represented by a submanifold and
2) What are some examples of manifolds who have homology classes not representable in this manner?

REPLY [30 votes]: Here are a few simple answers to the question you asked:

Every class in $H_{n-1}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to S^1$ representing the Poincare dual in $H^1(M;Z)=[M,S^1]$ and take the preimage of a point. In dimensions>2 it can be taken connected.
Similarly, every class in $H_{n-2}(M;Z)$ for $M$ orientable is represented by a submanifold: choose a smooth map $f:M\to CP^\infty$ representing the Poincare dual in $H^2(M;Z)=[M,CP^\infty]$, homotop $f$ into a finite skeleton, say $CP^N$, and take the preimage of $CP^{N-1}$.
Transversality says that if you can represent $x\in H_k(M)$ by a map of a smooth manifold (e.g.  elements in the image of the Hurewicz map, or by Thom) , and $2k < n$, then you can represent it by an embedded submanifold (as Andy mentions above). For example, any class in $H_1(M)$ for $dim(M)\ge 3$.  With care you can also make this work for $2k=n$, and there are techniques available in the "metastable" range (no triple points) involving generalizations of Whitney's trick and other ways to replace double points.