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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Additional resources for Connections, curvature and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces
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. Spectral Sequences (cf. formula ( l . l l ) , sec. 10). '. Now the commutative diagram implies that e ? * O is an isomorphism. Next note that the differential operator d, is homogeneous of bidegree (1, 0). 12) P it follows that e:: H ( E , , d,) H( c d,) ~ r - 0 , .
Spectral Sequences 40 that ( T * ) ~is n-regular. Now the theorem follows from Proposition VII, below. D. Corollary: If vt:Ed Ed is an isomorphism for a certain i, then -+ and are isomorphisms. -+a Proposition VII: Let v: M be a homomorphism of graded filtered spaces and consider the induced linear maps is Suppose ~ 2is) injective (respectively, surjective). Then vr:Mr -+ injective (respectively, surjective). In particular, if ~ ( a ' )is an isomorphism, so is vr. Proof: (1) Assume that pl(a') is injective, and fix x E ker vr.
In particular, it is nondegenerate if and only if both of them are nondegenerate. 7. Differential spaces. A dzyerential space ( X , 6) is a vector space X together with a linear transformation 6 of X such that 62= 0; 6 is called the dzperential operator. We write ker 6 = Z ( X ) , Im 6 = B ( X ) , Z ( X ) / B ( X )= H ( X , 6 ) (or simply H ( X ) ) and call these spaces, respectively, the cocycle, coboundary, and cohomology spaces of X . 0. Algebraic Preliminaries 11 A homomorphism p : ( X , S ,) + ( Y , 6), of differential spaces is a linear map v: X -,Y such that pdX = 6,~).
Connections, curvature and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces by Werner Hildbert Greub