DSpace Collection:
http://hdl.handle.net/2440/1087
2016-07-31T00:35:32ZIndex type invariants for twisted signature complexes and homotopy invariance
http://hdl.handle.net/2440/100258
Title: Index type invariants for twisted signature complexes and homotopy invariance
Author: Benameur, M.T.; Mathai, V.
Abstract: For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,Ɛ,H) for the twisted odd signature operator valued in a flat hermitian vector bundle , where H = ∑ i j+1 H 2j+1 is an odd-degree closed differential form on X and H 2j+1 is a real-valued differential form of degree 2j+1. We show that ρ(X,Ɛ,H) is independent of the choice of metrics on X and Ɛ of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah–Patodi–Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant ρ(X,Ɛ,H) more delicate to establish, and is settled under further hypotheses on the fundamental group of X.2014-01-01T00:00:00ZLocalized index and L2-lefschetz fixed-point formula for orbifolds
http://hdl.handle.net/2440/99827
Title: Localized index and L2-lefschetz fixed-point formula for orbifolds
Author: Wang, B.L.; Wang, H.
Abstract: We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly, and isometrically. These localized indices, generalizing the L²-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group, subject to some trace assumption. Applying the local index technique, we also obtain an L²-version of the Lefschetz fixed-point formulas for orbifolds. These cohomological formulas for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.2016-01-01T00:00:00ZAn Oka principle for equivariant isomorphisms
http://hdl.handle.net/2440/99737
Title: An Oka principle for equivariant isomorphisms
Author: Kutzschebauch, F.; Larusson, F.; Schwarz, G.W.
Abstract: Let G be a reductive complex Lie group acting holomorphically on normal Stein spaces X and Y, which are locally G-biholomorphic over a common categorical quotient Q. When is there a global G-biholomorphism X → Y? If the actions of G on X and Y are what we, with justification, call generic, we prove that the obstruction to solving this local-to-global problem is topological and provide sufficient conditions for it to vanish. Our main tool is the equivariant version of Grauert's Oka principle due to Heinzner and Kutzschebauch. We prove that X and Y are G-biholomorphic if X is K-contractible, where K is a maximal compact subgroup of G, or if X and Y are smooth and there is a G-diffeomorphism Ψ : X → Y over Q, which is holomorphic when restricted to each fibre of the quotient map X → Q. We prove a similar theorem when psi is only a G-homeomorphism, but with an assumption about its action on G-finite functions. When G is abelian, we obtain stronger theorems. Our results can be interpreted as instances of the Oka principle for sections of the sheaf of G-biholomorphisms from X to Y over Q. This sheaf can be badly singular, even for a low-dimensional representation of SL₂(ℂ). Our work is in part motivated by the linearisation problem for actions on C n. It follows from one of our main results that a holomorphic G-action on ℂⁿ, which is locally G-biholomorphic over a common quotient to a generic linear action, is linearisable.2015-01-01T00:00:00ZSpin-structures and proper group actions
http://hdl.handle.net/2440/99736
Title: Spin-structures and proper group actions
Author: Hochs, P.; Mathai, V.
Abstract: Abstract not available2016-01-01T00:00:00Z