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Given a map $f:B^n \to S^n$, where $B^n$ is the unit ball and $S^n$ is the unit sphere, is it true that the degree of $f|_{S^n}$ is always 0, where $f_{S^n}$ is the restriction of $f$ to $S^n$? If so, why?

Thanks!

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 The boundary of $B^n$ is $S^{n-1}$, so you can't restrict $f$ to $S^{n-1}$. I presume you meant something other than what you wrote. If you mean $f:B^{n+1} \rightarrow S^n$, then the map restricted to the sphere is null-homotopic so automatically of degree 0. – Bill Thurston Jan 3 2011 at 16:56
Also asked (simultaneously) at Math.SE: math.stackexchange.com/questions/16247/… – Andres Caicedo Jan 3 2011 at 17:03
Bill - yes, that's what I meant, sorry. Could you please elaborate on why such a map would be null-homotopic (I take it that means it is homotopic to a constant map)? Thanks. – gilad Jan 3 2011 at 17:31

closed as too localized by Andres Caicedo, Deane Yang, Bill Thurston, José Figueroa-O'Farrill, Bill Johnson Jan 3 2011 at 18:08

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