Use this URL to cite or link to this record in EThOS: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.682535
Title: Cohomology for multicontrolled stratified spaces
Author: Lukiyanov, Vladimir
ISNI:       0000 0004 5924 2741
Awarding Body: Loughborough University
Current Institution: Loughborough University
Date of Award: 2016
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Abstract:
In this thesis an extension of the classical intersection cohomology of Goresky and MacPherson, which we call multiperverse cohomology, is defined for a certain class of depth 1 controlled stratified spaces, which we call multicontrolled stratified spaces. These spaces are spaces with singularities -- this being their controlled structure -- with additional multicontrol data. Multiperverse cohomology is constructed using a cochain complex of tau-multiperverse forms, defined for each case tau of a parameter called a multiperversity. For the spaces that we consider these multiperversities, forming a lattice M, extend the general perversities of intersection cohomology. Multicontrolled stratified spaces generalise the structure of (the compactifications of) Q-rank 1 locally symmetric spaces. In this setting multiperverse cohomology generalises some of the aspects of the weighted cohomology of Harder, Goresky and MacPherson. We define two special cases of multicontrolled stratified spaces: the product-type case, and the flat-type case. In these cases we can calculate the multiperverse cohomology directly for cones and cylinders, this yielding the local calculation at a singular stratum of a multicontrolled space. Further, we obtain extensions of the usual Mayer-Vietoris sequences, as well as a partial Kunneth Theorem. Using the concept a dual multiperversity we are able to obtain a version of Poincare duality for multiperverse cohomology for both the flat-type and the product-type case. For this Poincare duality there exist self-dual multiperversities in certain cases, such as for non-Witt spaces, where there are no self-dual perversities. For certain cusps, called double-product cusps, which are naturally compactified to multicontrolled spaces, the multiperverse cohomology of the compactification of the double-product cusp for a certain multiperversity is equal to the L2-cohomology, analytically defined, for certain doubly-warped metrics.
Supervisor: Not available Sponsor: EPSRC
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID: uk.bl.ethos.682535  DOI: Not available
Keywords: Cohomology ; Stratified ; Controlled ; Intersection
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