When do two submanifolds define the same homology class?

Question

Let $X_1$ and $X_2$ be two compact oriented $n$-dimensional submanifolds of an ambient manifold $X$. Are there any checkable conditions which guarantee that $[X_1]=[X_2]$ in $H_n(X)$? This question came to my mind when I was thinking about the axioms of chern classes. One has the normalization axiom which concerns the homology class of some hyperplane in $\mathbf{CP}^n$. In order for this to make sense one has to prove that two hyperplanes are in the same homology class.

Answer 1

A sufficient condition which works in the case of $\mathbb{CP}^n$ is that $X_1$ can be sent to $X_2$ by a path of homeomorphisms, since any such family necessarily acts trivially on homology. For $\mathbb{CP}^n$ and hyperplanes this path can be taken to be a path in $PGL_{n+1}(\mathbb{C})$.

Incidentally, there is no need to require that $X$ is a manifold or that $X_1$ and $X_2$ are submanifolds (arbitrary maps from $X_1, X_2$ into an arbitrary topological space $X$ already produce fundamental classes in the homology of $X$).

Answer 2

Qiaochu's answer already suffices to see why lines in $\Bbb{CP}^n$ are homologous but here's a weaker characterization in greater generality:

The inclusion maps of $X_1, X_2$ in $X$ being homotopic is a sufficient condition for $X_1, X_2$ to be homologous, but a considerably weaker thing to verify is that $X_1$ and $X_2$ are "cobordant inside $X$".

That is to say, there is a $(n+1)$-dimensional manifold $M$ with $\partial M = X_1 \sqcup X_2$ and a map $f : M \to X$ such that $f|_{\partial M}$ is the inclusion of $X_1$ and $X_2$ in each of the components. Sometimes $f$ will be an embedding (eg, the boundary of the pair of pants inside the 2-torus). When $X_1$ and $X_2$ are homeomorphic and $M = X_1 \times [0, 1]$ this is the homotopy condition Qiaochu mentioned.