What is the difference between $(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})$ and $ S^{1}\times S^{1}\times S^{1}$.

Question

What is the difference between $(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})$ and $S^{1}\times S^{1}\times S^{1}$.

Where their homology groups are: If $\;\;\;\;(S^{2}\times S^{1})\# (S^{2}\times S^{1})\# (S^{2}\times S^{1})=3-(S^{2}\times S^{1}):$

$H_0(3-(S^{2}\times S^{1}))=H_3(3-(S^{2}\times S^{1}))=\mathbb{Z}$

$H_1(3-(S^{2}\times S^{1}))=H_2(3-(S^{2}\times S^{1}))=\mathbb{Z}^3$

and for $S^{1}\times S^{1}\times S^{1}=T^3$:

$H_0(T^3)=H_3(T^3)=\mathbb{Z}$

$H_1(T^3)=H_2(T^3)=\mathbb{Z}^3$

Thus, $\;\;\;\;\; H_0(T^3)=H_3(T^3)=H_0(3-(S^{2}\times S^{1}))=H_3(3-(S^{2}\times S^{1}))=\mathbb{Z}$, $H_1(T^3)=H_2(T^3)=H_1(3-(S^{2}\times S^{1}))=H_2(3-(S^{2}\times S^{1}))=\mathbb{Z}^3$

and $\;\;\;\;\; H_n(T^3)=H_n(3-(S^{2}\times S^{1}))=\mathbb{Z}=0,\;\;\;\forall n>3$

I have a doubt, these two 3 varieties are homeomorphic? If the answer is not, what is the difference?

thanks for the answer.

Answer 1

For $M,N$ with dimension $n>2$ the connected sum $M\# N$ has fundamental group $\pi_1(M\# N) \approx \pi_1(M) * \pi_1(N)$ as the boundary of the $n$-ball used for identification is simply connected. Thus $\pi_1(3-(S^2 \times S^1)) \approx \mathbb Z * \mathbb Z * \mathbb Z$ yet $\pi_1(T^3) \approx \mathbb Z \times \mathbb Z \times \mathbb Z$. So we conclude the two spaces are not homotopic.