# Nontriviality of the Hopf 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?

• Do you mean $S^2 \times S^1$? – Rolf Hoyer Dec 8 '18 at 13:48
• 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! – 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.