1

So I have a solid torus, with the boundary of it ($T^2$) identified.

The way I see it is: Since the boundary points are all identified, this manifold then first becomes a "chubby bagel." We can then separate the center of the bagel and deform it to a solid ball. When we identify the boundary $S^{n-1}$ of $D^n$, we get $S^n$, so this is the 3-sphere.

Is it correct?

Bernard
  • 179,256
user134070
  • 377
  • 2
    See here https://math.stackexchange.com/questions/1607076/what-is-the-space-obtained-by-identifying-boundary-mathbb-t2-of-a-solid-toru – Sumanta Sep 15 '20 at 16:32
  • What if $\frac{M}{\partial M}$, a manifold ??? https://math.stackexchange.com/questions/271123/when-does-the-quotient-of-a-manifold-with-boundary-become-a-manifold – Sumanta Sep 15 '20 at 16:42
  • I read the first article and saw that the second reduced homology group is $\mathbb{Z}^g$ for the geneus $g$ surface. Here $g=1$. But the reduced homology group of 3-sphere sould be $\mathbb{Z}$ only for its third reduced homology group. What went wrong in my reasoning? – user134070 Sep 15 '20 at 16:46

0 Answers0