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Multiplicative de Rham theorems for relative and intersection space cohomology. (English) Zbl 1431.55006
The authors “construct an explicit de Rham isomorphism relating the cohomology rings of Banagl’s de Rham and spatial approach to intersection space cohomology for stratified pseudomanifolds with isolated singularities.” Banagl assigned intersection spaces to certain classes of stratified spaces by a process of spatial homology truncation [M. Banagl, Intersection spaces, spatial homology truncation, and string theory. Dordrecht: Springer (2010; Zbl 1219.55001)]. These intersection spaces are cell complexes whose ordinary rational homology satisfies generalized Poincaré duality. Their cohomology is not isomorphic to intersection cohomology and, in contrast to the latter, intersection space cohomology is naturally equipped with perversity-internal cup-products. “De Rham Theorems for intersection space cohomology are given in [M. Banagl, J. Differ. Geom. 104, No. 1, 1–58 (2016; Zbl 1359.57016)] for pseudomanifolds with isolated singularities and in [J. T. Essig, About a de Rham complex describing intersection space cohomology in a non-isolated singularity case. University of Heidelberg (Master’s thesis) (2012)] for pseudomanifolds of depth one with product link bundles.”
However, it remains unclear whether the de Rham isomorphisms constructed in these papers respect the multiplicative structure and this is the question addressed in paper on hand. Its main result establishes an isomorphism of the cohomology rings in the case of isolated singularities. On the way the authors prove the de Rham Theorem for cohomology rings of pairs of smooth manifolds.
55N33 Intersection homology and cohomology in algebraic topology
55N30 Sheaf cohomology in algebraic topology
14J17 Singularities of surfaces or higher-dimensional varieties
58A10 Differential forms in global analysis
58A12 de Rham theory in global analysis
57P10 Poincaré duality spaces
14J33 Mirror symmetry (algebro-geometric aspects)
Full Text: DOI arXiv
[1] Markus Banagl. Intersection Spaces, Spatial Homology Truncation, and String Theory, volume 1997 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010.DOI: 10.1007/978-3-642-12589-8 2 · Zbl 1219.55001
[2] Markus Banagl. Foliated stratified spaces and a De Rham complex describing intersection space cohomology. J. Differential Geom., 104(1):1-58, 2016.DOI: 10.4310/jdg/1473186538 · Zbl 1359.57016
[3] Markus Banagl and Eugenie Hunsicker. Hodge Theory for Intersection Space Cohomology.arχiv: 1502.03960 to appear in Geom. Topol.
[4] J.-P. Brasselet, G. Hector, and M. Saralegi. Th´eor‘eme de de Rham pour les vari´et´es stratifi´ees. Ann. Global Anal. Geom., 9(3):211-243, 1991.DOI: 10.1007/bf00136813 · Zbl 0733.57010
[5] J. P. Brasselet and A. Legrand. Differential forms on singular varieties and cyclic homology. In Singularity theory (Liverpool, 1996), volume 263 of London Math. Soc. Lecture Note Ser., pages xviii, 175-187. Cambridge Univ. Press, Cambridge, 1999. · Zbl 0949.55003
[6] Jean-Paul Brasselet and Andr´e Legrand. Un complexe de formes diff´erentielles ‘a croissance born´ee sur une vari´et´e stratifi´ee. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 21(2):213-234, 1994.
[7] Glen E. Bredon. Sheaf theory, volume 170 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997.
[8] Jean-Luc Brylinski. Equivariant intersection cohomology. In Kazhdan-Lusztig theory and related topics (Chicago, IL, 1989), volume 139 of Contemp. Math., pages 5-32. Amer. Math. Soc., Providence, RI, 1992. DOI: 10.1090/conm/139/1197827 · Zbl 0803.55002
[9] J. Timo Essig. About a de rham complex discribing intersection space cohomology in a non-isolated singularity case. Master’s thesis, University of Heidelberg, 2012.
[10] J. Timo Essig. Intersection Space Cohomology of Three-Strata Pseudomanifolds. to appear in J. Topol. Anal. DOI: 10.1142/s1793525320500120
[11] Rudolf Fritsch and Renzo A. Piccinini. Cellular structures in topology, volume 19 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990. · Zbl 0837.55001
[12] M. Goresky and R.D. MacPherson. Intersection homology theory. Topology,19(2):135-162,1980. DOI: 10.1016/0040-9383(80)90003-8 · Zbl 0448.55004
[13] M. Goresky and R.D. MacPherson. Intersection homology ii. Invent. Math.,72(1):77-129,1983. DOI: 10.1007/bf01389130 · Zbl 0529.55007
[14] John M. Lee. Introduction to Smooth Manifolds, volume 218 of Graduate Texts in Mathematics. Springer, New York, second edition, 2013. · Zbl 1258.53002
[15] William S. Massey. Singular homology theory, volume 70 of Graduate Texts in Mathematics. Springer-Verlag, New York-Berlin, 1980.
[16] J. P. May. A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 1999. · Zbl 0923.55001
[17] Martin Saralegi. Homological properties of stratified spaces. Illinois J. Math.,38(1):47-70,1994. DOI: 10.1215/ijm/1255986886 · Zbl 0792.57009
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