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If winding number of two curves are same then are they homotopic ?

I need this to prove $\pi(S^1)$ is isomorphic to $\mathbb{Z}$ (I know converse is true)

dragoboy
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  • Yes.. there is a similar post in the Related column, http://math.stackexchange.com/questions/657822/equal-winding-number-implies-two-paths-are-path-homotopic?rq=1 –  Feb 19 '15 at 03:37
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    isn't there any proof without lifting ? i don't know lifting – dragoboy Feb 19 '15 at 03:49
  • The other standard proof is to use the van Kampen theorem for the fundamental groupoid $\pi_1(X,A)$ on a set $A$ of base points, chosen according to the geometry. For the circle, one needs $2$ base points. Have a look at http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one – Ronnie Brown Feb 20 '15 at 16:59

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