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The question may seem silly, but I do not find the answer.

For me, the $n-$torus is "an sphere with $n$ holes".

Topologically, this is called the connected sum of $n$ tori, is it right?

What about the $T^n=\overbrace{S^1\times \cdots \times S^1}^{n \,times}$, what is its name and what are its properties?

Do the two surfaces have some relation?

Thank you so much.

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2 Answers

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What you've described is the n-Torus:

$$ T^n = S^1 \times \dots \times S^1 $$

See the wiki entry:

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The $n$-torus is a torus of dimension $n$, it is defined to be the cartesian product of $n$ one-dimensional circle. This is roughly a $n$-dimensional sphere with a $n$-dimensional hole.

What you describe as a sphere with $m$ holes is indeed the connected sum of $m$ different copies of a $2$-dimensional torus.

Those two surfaces are not homeomorphic whenever $n\neq 2$ or $n=2$ and $m\neq 1$.

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