Shadow forms of Brasselet-Goresky-MacPherson

Author:
Belkacem Bendiffalah

Journal:
Trans. Amer. Math. Soc. **347** (1995), 4747-4763

MSC:
Primary 55N33; Secondary 14F32

MathSciNet review:
1316844

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Abstract: Brasselet, Goresky and MacPherson constructed an explicit morphism, providing a De Rham isomorphism between the intersection homology of a singular variety and the cohomology of some complex of differential forms, called "shadow forms" and generalizing Whitney forms, on the smooth part of . The coefficients of shadow forms are integrals of Dirichlet type. We find an explicit formula for them; from that follows an alternative proof of Brasselet, Goresky and MacPherson's theorem. Next, we give a *duality formula* and a *product formula* for shadow forms and construct the correct algebra structure, for which shadow forms yield a morphism.

**[B]**B. Bendiffalah,*Contributions à l'étude locale des singularité*, Thesis, University of Provence, Marseille, 1994.**[BGM]**J.-P. Brasselet, M. Goresky, and R. MacPherson,*Simplicial differential forms with poles*, Amer. J. Math.**113**(1991), no. 6, 1019–1052. MR**1137533**, 10.2307/2374899**[Ch]**Kuo-Tsai Chen,*Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula*, Ann. of Math. (2)**65**(1957), 163–178. MR**0085251****[GM]**Mark Goresky and Robert MacPherson,*Intersection homology theory*, Topology**19**(1980), no. 2, 135–162. MR**572580**, 10.1016/0040-9383(80)90003-8**[Re]**Rimhak Ree,*Lie elements and an algebra associated with shuffles*, Ann. of Math. (2)**68**(1958), 210–220. MR**0100011****[Ze]**A. Zelevinsky, Letter to J. P. Brasselet, 20 Juin 1990.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1995-1316844-0

Article copyright:
© Copyright 1995
American Mathematical Society