By Werner Hildbert Greub

ISBN-10: 0123027039

ISBN-13: 9780123027030

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

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

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N. Examples: 1. Exterior algebra: Let E", E be dual n-dimensional vector spaces. Then AE" is a PoincarC duality algebra, with PoincarC = AE given by isomorphism D : AE" - D@ = i(@)e, where e is a basis vector of AnE. 2. Tensor products: Let A and B be finite-dimensional graded algebras. Then A @ B is a PoincarC duality algebra if and only if both A and B are. I n fact, write A = AP and B = Cr Bq, where A" # 0 and Bm # 0. Then A @ B = Xttm(A @ B Y , and x: dim(A @ B)"+m= dim A n . dim Bm. Thus dim(A @ B)n+m = 1 if and only if dim A n = 1 = dim Bm.

If the differential spaces are graded differential spaces, and ip and y are homogeneous of degrees k and I, then d is homogeneous and deg d = deg dX - k - 1. I n this case we obtain a long exact sequence (m = deg a) - HP( Y ) a 4 HP+m( W )f + v* Hp+m+k(X) - HP+m+k+l( Y ) * Let ( X , 6,) and ( Y , 6,) be graded differential spaces where 6, and dP are homogeneous of the same odd degree k. Then their tensor product is the graded differential space ( X 0 Y, B X B y ) given by We will often write Consider the inclusion Z ( X ) 0Z ( Y ) + Z ( X 0 Y ) .

Then the mapping is injective (surjective). In particular, if pMis injective (surjective), then p)# is injective (surjective). 14) that pa')is injective (surjective) if pk) is. Now the corollary follows from Proposition VII, applied to the graded filtered spaces H ( M ) and H ( f i ) . D. - Corollary 11: Let v: M i@ be a homomorphism of graded filtered differential spaces whose spectral sequences are convergent and collapse at the Kth term. Then (p#y:H r ( M ) Iir(@) is injective (surjective) whenever &): EF) + EF) is injective (surjective).

### Connections, curvature and cohomology. Vol. III: Cohomology of principal bundles and homogeneous spaces by Werner Hildbert Greub

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