## New answers tagged manifolds

6
votes

### Is there a flat manifold with trivial first homology?

Here is an idea for making examples.
Let $F$ be a free group and let $N$ be a normal subgroup of $F$. Then $F/[N,N]$ is torsion-free. To see this, suppose $w\in F$ has finite order modulo $[N,N]$, and ...

- 945

12
votes

### Is there a flat manifold with trivial first homology?

Andrzej Szczepański pointed me to Proposition 2.3.13 in the book [Perfect Groups, Derek F. Holt and Wilhelm Plesken, 1989], which gives an answer to my question.
Namely, in a slightly different ...

- 27.3k

13
votes

Accepted

### Stable torus that is not a torus

Suppose $M\times S^1$ is homeomorphic to $T^{n+1}$. Then $\pi_1(M\times S^1) \cong \pi_1(T^{n+1})$, so $\pi_1(M)\oplus\mathbb{Z} \cong \mathbb{Z}^{n+1}$, and hence $\pi_1(M) \cong \mathbb{Z}^n$. ...

- 17.7k

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