# When is a homology class a fundamental class?

Let $X$ be a real connected orientable closed $n$-dimensional compact differentiable manifold.

A connected oriented closed $d$-dimensional submanifold $i:M\to X$ (i.e. $M$ is a real connected orientable closed compact differentiable manifold and $i$ is a topological embedding) has a fudamental class $[M]\in H_d(M,\mathbb{Z})$. This can be considered as an element $i_*([M])$ of the singular homology $H_d(X,\mathbb{Z})$.

1. The accepted answer to this question states, that a multiple $\lambda x$ of every element $x\in H_i(X,\mathbb{Z})$ for $0\leq i\leq n$ is of the form $i_*([M])$ for some $M$. If $n\leq 8$ then one may chose $\lambda=1$.
2. An answer to this question however states that for example $2[M]$ is for $M=S^1$ not representable in this way. Also the other answers seem to be in conflict with (1.).

Where is my misunderstanding? Is the first answer not about embeddings $i:M\to X$ but about immersions $i$ or arbitrary continuous maps?

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This is a very good question! – Georges Elencwajg Jan 19 '13 at 13:25

Indeed, Don Stanley, who gave an answer to the MO question linked to your 2. is perfectly right: the homology class $n[M]\in H_n(M,\mathbb Z)$ is certainly not represented by a closed submanifold of $M$ as soon as $n\geq 2$.
The reason is that $H_n(M,\mathbb Z)$ is canonically isomorphic to $\mathbb Z$ once an orientation has been fixed and thus has no torsion: we cannot have $n[M]=[M]$, and obviously the only closed manifold of $M$ that can represent a homology class in $H_n(M,\mathbb Z)$ is $M$, which represents only $1\cdot [M]$.
In Thom's theorem the case when $i=n$ is excluded from the theorem, but the rest of the cases remain true, if I understood it correctly. – user17786 Jan 19 '13 at 13:45