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I have some questions and would be infinitely grateful to you for your answers:

1- $f^{*}$ being the dual of $f_{*}$ so the degree (between top dimensional (co)homology groups) is the same for both maps. Is that true? (yes/no)

2- if f is a polynomial (defined on a p-manifold say $M^{p}$) of degree n what can I say about the degree of f? Is it n?

3-f real homogeneous polynomial. what can I say about the dimension of the submanifold f(X)=0?

I am not familiar with differential geometry however I need to know about it for me to feel easy with some notions that appear in algebraic topology. So those questions might be obvious however I shall be really grateful to the person who would help know their answers.

Many thanks

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The degree is defined for maps between compact oriented manifolds (so $\mathbb R^n$ doesn't work). Regarding the third point, if the zeros of $f$ are a submanifold, then its dimension is $n-1$ (this should follow from some version of the implicit function theorem). –  Giacomo d'Antonio May 31 '11 at 13:55
    
thanks Giacomo I have edited my second question now it's a manifold instead of $R^{n}$ –  El Moro May 31 '11 at 13:59
    
What is a polynomial defined on a p-manifold? –  Rasmus May 31 '11 at 14:06
1  
$f^*$ is not the dual of $f_*$, as $H^*$ is not, in general, the dual of $H_*$. –  Rasmus May 31 '11 at 14:06
    
Not very sure about my answers: I think a polynomial of degree $n$ should generally have degree $n$ as a map, but there should be exceptional cases. For 3 you can have easy problems like $x^{3}-y^{3}=0$ only have solution $x=y$, so you need some condition like the Jacobian is not degenerate, etc. –  Kerry May 31 '11 at 14:31
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