an:07116326
Zbl 1431.55006
Schl??der, Franz Wilhelm; Essig, J. Timo
Multiplicative de Rham theorems for relative and intersection space cohomology
EN
J. Singul. 19, 97-130 (2019).
00426647
2019
j
55N33 55N30 14J17 58A10 58A12 57P10 14J33
singularity; stratified space; pseudomanifold; Poincar?? duality; intersection space cohomology; intersection cohomology; sheaf theory; de Rham theorem; relative de Rham theorem; differential forms; cellular cup products; cup products on cochains
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 [\textit{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 Poincar?? 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 [\textit{M. Banagl}, J. Differ. Geom. 104, No. 1, 1--58 (2016; Zbl 1359.57016)] for pseudomanifolds with isolated singularities and in [\textit{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.
Beatrice Bleile (Armidale)
Zbl 1219.55001; Zbl 1359.57016