1.a)let f,g:X→Sⁿ be smooth maps s.t.ǁf(x)-g(x)ǁ<2 for any x in X.show that f is smoothly homotopic to g.

   b)let m<p & show that any 2 mapping Mᵐ→Sᵖ are homotopic.

   c)let f:Sⁿ→Sⁿ be given with deg(f)=2k+1.show that there exists x in Sⁿ s.t. f(-x)=-f(x).

2.a)define Guass map for a manifold Xᵏ⊆Rᵏ.

   b)state & prove Hopf lemma.

3.give an example of 2 suitable manifolds M,N s.t.:

    a)deg:C∞(M,N) →Z is onto(Z is set of all intigers)

    b)deg:C∞(M,N)→{1,-1} is onto;

    c)compute the degree of f:lR-{0} →lR+ define by f(x)=l x l. Is this map smoothly homotopic to the constant map? Justify your answer.

    d)let n be even.Is it possible to use deg2 to show that the anti podal map Sⁿ→Sⁿ is not homotopic to the identity?

4.state the homogeneity lemma & prove it for Rᵏ.

5.a)let Mᵐ⊆Rᵏbe a compact manifold without boundary & v:M→Rᵏ a nonzero vector field.Is it possible to find a k-dim manifold N s.t. M⊆N &N admits a nonzero vector field   w:N→Rᵏ s.t. w(M)=v?

   b)let M=Sⁿ when n=1 & v:M→Rᵏ ( k=2) be given by v(x,y)=(y,-x). find an example for N,M as part (a).

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