Download Connections, curvature and cohomology. Vol. III: Cohomology by Werner Hildbert Greub PDF

By Werner Hildbert Greub

Greub W., Halperin S., James S Van Stone. Connections, Curvature and Cohomology (AP Pr, 1975)(ISBN 0123027039)(O)(617s)

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Extra info for Connections, curvature and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces

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1 (B, 61, , ; :(: ; O), Moreover, in view of sec. 11, the bigradation off& is given by Ep BP, @=o, q>o @so H*(B), Bq= 0, q (i=O,l), and > 0 (i 2 2). Next, consider the inclusion e : B -+ M . It induces a linear map e*: H(B) --t H ( M ) . Moreover, e is filtration preserving, and so it de- termines a homomorphism ei: ($ ,di) + ( E d ,di) of spectral sequences. Proposition VI: The maps are isomorphisms. I n particular, mo BP and EPo Hp(B). Proof: We show first that each el"*' is an isomorphism. 0 0 38 I.

Then vD = Dv, and so 9 determines a linear map I $ : H ( M , D ) -+ H ( m , b). On the other hand, v preserves the filtrations and so it induces a homomorphism of spectral sequences 'pi: i 2 0. 'if Ri is the induced isomorphism. where Fi: Passing to cohomology and using Proposition I11 we obtain the relations p7k# 0 tk#= {f Vg 0 and 9: 0 ok = sk 0 P)k+l. I. Spectral Sequences 30 Thus the diagram commutes. 8. The homogeneous case. Suppose now that ( M , 6) is as above but that 6 is homogeneous of degree k.

I t follows that for i 5 k the inclusion map j : Mp + Fp(M) can be composed with qf to yield an isomorphism tf : M p

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