Equivariant chain complexes, twisted homology and relative minimality of arrangements |
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Authors: | Alexandru Dimca |
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Institution: | Laboratoire J.A. Dieudonné, UMR du CNRS 6621, Université de Nice-Sophia-Antipolis, Parc Valrose, 06108 Nice Cedex 02, France; Inst. of Math. “Simion Stoilow”, P.O. Box 1-764, 014700 Bucharest, Romania |
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Abstract: | We show that the π-equivariant chain complex (), , associated to a Morse-theoretic minimal CW-structure X on the complement of an arrangement , is independent of X. The same holds for all scalar extensions, , a field, where X is an arbitrary minimal CW-structure on a space M. When is a section of another arrangement , we show that the divisibility properties of the first Betti number of the Milnor fiber of obstruct the homotopy realization of as a subcomplex of a minimal structure on .If is aspherical and is a sufficiently generic section of , then may be described in terms of π, L and , for an arbitrary local system L; explicit computations may be done, when is fiber-type. In this case, explicit -presentations of arbitrary abelian scalar extensions of the first non-trivial higher homotopy group of , πp(M), may also be obtained. For nonresonant abelian scalar extensions, the -rank of is combinatorially determined. |
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