0
$\begingroup$

A simple question how to understand why even though locally $S^3$ is homeomorphic to $S^2\times S^1$, how do you see that globally this is not true?

$\endgroup$
  • $\begingroup$ Do you mean $S^2 \times S^1$? $\endgroup$ – Rolf Hoyer Dec 8 '18 at 13:48
  • $\begingroup$ Yes sorry! In my head I was thinking about the wrong theorem showing that the quotient of two lie groups forming a bundle with the quotient being the fibres! I'll amend the question! $\endgroup$ – Bunneh Dec 9 '18 at 12:31
2
$\begingroup$

A standard way to show that $S^3$ is not homeomorphic to $S^2 \times S_1$ is to look at its fundamental group: $S^3$ is simply connected (that is, $\pi_1$ is trivial), but $\pi_1(S^2 \times S^1) \simeq \pi_1(S^1) = \Bbb{Z}$.

More intuitively, try this: in $S^3$, any imbedded 2-sphere will separate the space into two connected components. But in $S^2 \times S^1$, a 2-sphere $S^2 \times \{x\}$ does not separate the space.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.