Questions tagged [torus]

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16 votes
2 answers
728 views

What happens to a closed manifold to ensure it is homeomorphic to a torus $T^{n}$?

If $M$ is a smooth connected closed $n$-dimensional manifold, its universal covering space is homeomorphic to Euclidean space $R^{n}$, and its fundamental group is $Z^{n}$, then is it homeomorphic ...
Thom's user avatar
  • 169
14 votes
1 answer
483 views

Can the product of an exotic torus and a circle be the standard torus?

As discussed in this question from last week, if $M$ is a closed manifold such that $M\times S^1$ is homeomorphic to the torus $T^{n+1}$, then $M$ is homeomorphic to $T^n$. Is the corresponding ...
Michael Albanese's user avatar
4 votes
1 answer
414 views

Smallest subgroup of unitary group, containing diagonal matrices and a fixed unitary matrix is the whole group

Suppose that $U_n(\mathbb{C})$ is the group of unitary matrices of dimension $n$ over complex numbers. Fix a unitary matrix $A \in U_n(\mathbb{C})$ and consider the smallest closed subgroup $K \...
Mini's user avatar
  • 85
4 votes
2 answers
375 views

Study of Hex on the Torus

Hex is usually played on a parallelogram shaped board. What if you play it on a Torus? One thing I notice is that the idea of connecting opposite sides doesn't make much sense anymore, since a torus ...
Christopher King's user avatar