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Poincaré-Verdier duality in o-minimal structures. (English) Zbl 1235.03070

This paper continues the development of sheaf theory for sets definable in o-minimal structures, as introduced in work by M. J. Edmundo, N. J. Peatfield and the reviewer [J. Math. Log. 6, No. 2, 163–179 (2006; Zbl 1120.03024)] and furthered by A. Berarducci and A. Fornasiero [J. Math. Log. 9, No. 2, 167–182 (2009; Zbl 1236.03030)]. Motivated by the problem of computing the cohomology of definably compact definable groups in arbitrary o-minimal structures, the authors prove a version of the Poincaré-Verdier duality theorem. Although they work in this very general setting, the result is new in the case when the o-minimal structure expands a field (and indeed, even when the o-minimal structure is just a real closed field). The paper concludes with a sheaf-theoretic take on orientation theory and shows that if the o-minimal structure is an expansion of a field then this theory coincides with that developed by A. Berarducci and M. Otero [J. Symb. Log. 68, No. 3, 785–794 (2003; Zbl 1060.03059)].

MSC:

03C64 Model theory of ordered structures; o-minimality
55N30 Sheaf cohomology in algebraic topology
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References:

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