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Question: suppose that to every map h squot squot we...

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Suppose that to every map h : S-» S we have assigned an integer, denoted by deg h and called the degree of h, such that: φ Homotopic maps have the same degree. (ii) deg(h ok) - (deg h) (degk). (iii) The identity map has degree 1, any constant map has degree 0, and the reflection map ρ (xi , . . . , xn+1)-(xi, . . . , Xn,-Xn+1) has degree-l [One can construct such a function, using the tools of algebraic topology. Intu- itively, deg h measures how many times h wraps S about itself, the sign tells you whether h preserves orientation or not.] Prove the following: (a) There is no retraction 1- S (b) Ifh : S - S has degree different from (-1)+1, then h has a fixed point. [Hint: Show that if h has no fixed point, then h is homotopic to the antipodal map a (x) =-x antipode-x. exists, show the identity map is homotopic to the antipodal map.] (c) If h : Sn → Sn has degree different from 1, then h maps some point x to its (d) If S has a nonvanishing tangent vector field v, then n is odd. [Hint: If v

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