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Homework Help: Stiefel-Whitney Classes and

  1. May 5, 2010 #1
    Stiefel-Whitney Classes and immersions

    1. The problem statement, all variables and given/known data

    I don't know whether this goes here or somewhere in the math section of these forums as I am brand new here.
    My algebraic topology professor is rather cryptic and the other two guys in the class are just as stuck as I am. I have gone to him and he told me to use what i have derived and a few properties of cohomology to get the desired result. So here it goes.
    Given two bundles V and W such that [tex] V \oplus W [/tex] is trivial. Let [tex] w_j(V)=w_j [/tex] and [tex]w_j(W)=w^j[/tex] we can show that if
    [tex] w(V)=1+ w_1 +w_2+w_3+ \dotsc [/tex]
    [tex] w(W)= 1+(w_1+w_2+ \dotsc)+(w_1+w_2+ \dotsc)^2 +(w_1+w_2+ \dotsc)^3 + \dotsc [/tex]

    We have shown that when [tex] X=\mathbb{R}\textsf{P}^n [/tex],
    [tex]w(TX)=(1+x)^{n+1} \in H^*(X)=\mathbb{Z}[x]/x^{n+1} [/tex]

    So the part I am having trouble with is the following:
    If [tex] X=\mathbb{R}\textsf{P}^n [/tex] immerses in [tex]R^{n+c}[/tex] then [tex]\binom{-n-1}{j}[/tex] is even for c<j<=n.
    The hint given is Show [tex]w^j=\binom{-n-1}{j}x^j[/tex] where V=TX

    2. Relevant equations
    [tex] w^k=w_1w^{k-1}+w_2w^{k-2}+w_3w^{k-3} +\dotsc w_{k-1}w^1 + w_k [/tex]
    [tex] w_1=w^1 [/tex]
    [tex] w^2=(w_1)^2 + w_2 [/tex]
    [tex] w^3=(w_1)^3 +w_3 [/tex]

    3. The attempt at a solution
    I have tried many ways but have not found any success. Just now while writing this i tried an induction:
    [tex] w^k=\binom{n+1}{1} \binom{-n-1}{k-1} x^k + \binom{n+1}{2} \binom{-n-1}{k-2} x^k + \dotsc + \binom{n+1}{k-1} \binom{-n-1}{1} x^k + \binom{n+1}{k} \binom{-n-1}{0} x^k [/tex]
    but got stuck when this did not work out to anything nice:
    [tex] \binom{n+1}{i} \binom{-n-1}{k-i} [/tex]

    Anybody know how to do this? Are there binomial identities that I am missing or is this misunderstanding entirely in the structure of the cohomology ring? This is the last homework of my undergrad and I really want to be done.
    Last edited: May 6, 2010
  2. jcsd
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