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

**Example text**

Suppose M and A? are filtered vector spaces. i is called filtration preserving (or a homomorphism. i). In this case 9 induces a 19 I. Spectral Sequences 20 unique linear map vA:A M -+ Aa of graded spaces such that the diagrams FP(M) LFp(rn) A&-A&, -O0 < p < 00, PA commute. If M = C pMp is a graded vector space, then by setting Fp(M) = CpspMp we obtain a filtration of M. This filtration is said to be induced r by the gradation. In this case e p restricts to an isomorphism M p A&, and these isomorphisms define an isomorphism M AM of graded spaces.

6, this implies that eP,o: E P . 0. D. Corollary: The maps eiso: H p ( B )+ El,' are surjective for K 2 2. 14. Homomorphisms of graded filtered differential spaces. Assume that Q : M + h? is a homomorphism of graded filtered differential spaces. Then the induced maps v i : Ei+ (cf. sec. ll) 3. 14) Recall from sec. 9 that a homomorphism p: Cpz0A p -+ Cp20B p of graded spaces is called n-regular, if p p : A p B p is an isomorphism for p 5 n and injective for p = n 1. --f + Theorem I (Comparison theorem): Suppose p : M + I@ is a homomorphism of graded filtered differential spaces whose spectral sequences are convergent.

Y" are dual (v*)~:VX" t V Y* and = 9". are dual as well. We write (v,")~ T h e substitution operator is(x) determined by x derivation in VX" satisfying is(x)x" xi xx E x), E X is the unique X". Its dual is multiplication by x in VX and is denoted by ,us(x). Finally, assume X* = Y" @ Z" and X = Y @ 2. Then (@v Y, a v b) = (@, a)(Y, b) = (@ @ E V Y " , YEVZ", 0Y, a 0b), aEVY, bEVZ. 6. Poincar6 duality algebras. A Poincare' duality algebra is a finitedimensional positively graded associative algebra A = C&oAP subject to the following conditions: (1) dimAn = 1.