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samiraahansaz
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Homework Statement
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:
w1 (ξ1 ⊗ η1) = w1(ξ1) + w1(η1)
More generally there is a universal formula of the form:
w(ξm ⊗ ηn) = Pm,n(w1(ξm),…,wm(ξm), w1(ηn),…,wn(ηn))
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,n (σ1,…, σm , σʹ1,…, σʹn) = ∏mi=1∏nj=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).