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Singular decompositions of a cap product

Authors: David Chataur, Martintxo Saralegi-Aranguren and Daniel Tanré
Journal: Proc. Amer. Math. Soc. 145 (2017), 3645-3656
MSC (2010): Primary 55N33, 57P10, 57N80
Published electronically: February 22, 2017
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Abstract: In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that this classical cap product is compatible with a cap product in intersection (co)homology that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincaré duality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities.

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David Chataur
Affiliation: LAFMA, Université de Picardie Jules Verne, 33, rue Saint-Leu, 80039 Amiens Cedex 1, France

Martintxo Saralegi-Aranguren
Affiliation: Laboratoire de Mathématiques de Lens, EA 2462, Université d’Artois, SP18, rue Jean Souvraz, 62307 Lens Cedex, France

Daniel Tanré
Affiliation: Département de Mathématiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France

Keywords: Intersection homology, cap product, Poincar\'e duality
Received by editor(s): June 14, 2016
Received by editor(s) in revised form: September 21, 2016
Published electronically: February 22, 2017
Additional Notes: This research was supported through the program “Research in Pairs” at the Mathematisches Forschungsinstitut Oberwolfach in 2016. The authors thank the MFO for its generosity and hospitality.
The third author was also supported by the MINECO grant MTM2016-78647-P and the ANR-11-LABX-0007-01 “CEMPI”
Communicated by: Michael A. Mandell
Article copyright: © Copyright 2017 American Mathematical Society

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