When is a homology class a fundamental class?

Question

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?

Answer 1

Thom's quoted article in the link to 1. certainly does not contain the "result" that you cite, which is false. The accepted answer you mention is at least ambiguously formulated.

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 $1\cdot [M]$.