By Jean-Luc Brylinski
This booklet examines the differential geometry of manifolds, loop areas, line bundles and groupoids, and the relatives of this geometry to mathematical physics. functions provided within the booklet contain anomaly line bundles on loop areas and anomaly functionals, valuable extensions of loop teams, Kähler geometry of the distance of knots, and Cheeger--Chern--Simons secondary features sessions. It additionally covers the Dirac monopole and Dirac’s quantization of charge.
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Additional info for Loop Spaces, Characteristic Classes and Geometric Quantization (Modern Birkhäuser Classics)
Indeed, since a direct limit of exact sequences is still exact, we will then get an exact sequence for the stalks at x. 7, we will then have an exact sequence of sheaves. Take V = (120, for some LI20 which contains x. 2. 4. Proposition. For any sheaf A of abelian groups on a space X and any open covering U of X, there is a canonical group homomorphism fji (U, A) (X, A) from Oech cohomology to sheaf cohomology, induced by a morphism of resolutions from C'(U, A) to some injective resolution of A.
Then f'(A)(U) is the set of af(z) E (A) such continuous sections x E U Both definitions of the sheaf can be found in the literature. a closed subspace, we set F(Z, A) = For A a sheaf on X and for F(Z, i'A). 4. Lemma. Let X be a paracompact space, and let i Z '—+ X be the inclusion of a closed subset. Then for any sheaf A of sets on X we have F(Z, A) = lLm F(U, A), where the direct limit is taken over all open (I neighborhoods of Z. Proof. There is a natural map lim F(U, A) F(Z, A). The crucial point is to show this map is surjective.
6. We consider the double complex of abelian groups F(X, 1'). , the sheaf cohomology HQ(X, KP), which is zero for q > 0. Hence the second spectral sequence degenerates at E2, and the total cohomology of the double complex F(X, identifies with the cohomology of the complex ['(X, K). q = Ker (d: is an injective resolution of = (2) is an injective resolution of (3) is an injective resolution of if'(K). 12. Proposition. For any bounded below complex of sheaves K' on = a space X, there is a convergent spectral sequence 24 urith COMPLEXES OF SHEAVES AND THEIR HYPERCOHOMOLOCY -term the graded quotients of some filtration of H' (X, K').