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?

  • $\begingroup$ 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

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.


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