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The stifle-Whitney classes of a tensor product

  1. Jan 16, 2014 #1
    1. The problem statement, all variables and given/known data

    Let ξm and ηn be vector bundles over a paracompact base space. Show that the stifle-Whitney classes of the tensor product ξm ⊗ ηn (or of the isomorphic bundle Hom (ξm, ηn) can be computed as follows. If the fiber dimensions m and n are both 1 then:
    w11 ⊗ η1) = w11) + w11)
    More generally there is a universal formula of the form:
    w(ξm ⊗ ηn) = Pm,n(w1m),…,wmm), w1n),…,wnn))
    Where the polynomial Pm,n in m+n variables can be characterized as minutes t1,…, tm and if σʹ1,…, σʹn are the elementary symmetric functions of tʹ1,…., tʹn
    Pm,n1,…, σm , σʹ1,…, σʹn) = ∏mi=1nj=1(1+ti+ tʹj)
    (Help: The chohomology of Gm × Gn can be computed by the künneth theorem. The formula for w (ξm × ηn) can be verified first in the special case when ξm and ηn are Whitney sums of line bundles).


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
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