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?
Nontriviality of the Hopf Fibration
differential-geometryfiber-bundlesfibrationhopf-fibration
differential-geometryfiber-bundlesfibrationhopf-fibration
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?
Best Answer
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.