Homotopy groups of $n$-torus with a point removed.

Question

Is there a simple way how to compute and present homotopy groups of $T^n=S^1\times \ldots\times S^1$ with a point (or several points) removed?

Answer 1

• For $n=2$ $\mathbb{T}^{2}\setminus\{p\}$ is homotopically equivalent to $S^1\vee S^1$ so the fundamental group is isomorphic to $\mathbb{Z}*\mathbb{Z}$ and higher homotopy groups are trivial.

• For $n>2$ Van Kampen theorem assures you that $$\pi_1(\mathbb{T}^{n})\cong\pi_{1}(\mathbb{T}^{n}\setminus\{p\})*_{\pi_1(S^{n-1})} \pi_1(D^n)\cong\pi_{1}(\mathbb{T}^{n}\setminus\{p\})$$ thus $$\pi_{1}(\mathbb{T}^{n}\setminus\{p\})\cong \pi_1(\mathbb{T}^{n})\cong \bigoplus_{i=1}^{n}\mathbb{Z}\ .$$ The universal cover is $\mathbb{R}^{n}\setminus\mathbb{Z}^{n}$ which is homotopically equivalent to an infinite wedge of $S^{n-1}$, so for $k>1$ $$\pi_{k}(\mathbb{T}^{n}\setminus\{p\})\cong\pi_{k}(\bigvee_{i=1}^\infty S^{n-1})\ .$$ In particular $\pi_{k}(\mathbb{T}^{n}\setminus\{p\})\cong0$ for $1<k<n-1$ and $\displaystyle\pi_{n-1}(\mathbb{T}^{n}\setminus\{p\})\cong\bigoplus_{i=1}^{\infty}\mathbb{Z}$.

If you take out more than one point, say $r$ points: for $n=2$ you have a fundamental group that is isomorphic to a free product of $r+1$ copies of $\mathbb{Z}$ and again trivial higher homotopy groups. For $n>2$ it should be the same as the case of a single point removed.