Results 1  10
of
29
Heat kernels on covering spaces and topological invariants
 J. Dierential Geom
, 1992
"... It is well known that there are relationships between the heat flow, acting on differential forms on a closed oriented manifold M, and the topology of M. From Hodge theory, one can recover the Betti numbers of M from the heat flow. Furthermore, Ray and Singer [42] defined an analytic tor ..."
Abstract

Cited by 64 (0 self)
 Add to MetaCart
(Show Context)
It is well known that there are relationships between the heat flow, acting on differential forms on a closed oriented manifold M, and the topology of M. From Hodge theory, one can recover the Betti numbers of M from the heat flow. Furthermore, Ray and Singer [42] defined an analytic tor
Hilbert modules and modules over finite von Neumann algebras and applications to L²invariants
 MATH. ANN. 309, 247285 (1997)
, 1997
"... ..."
L²Invariants of Locally Symmetric Spaces
 DOCUMENTA MATH.
, 2002
"... Let X = G/K be a Riemannian symmetric space of the noncompact type, Γ ⊂ G a discrete, torsionfree, cocompact subgroup, and let Y = Γ\X be the corresponding locally symmetric space. In this paper we explain how the HarishChandra Plancherel Theorem for L²(G) and results on (g, K)cohomology can be u ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
Let X = G/K be a Riemannian symmetric space of the noncompact type, Γ ⊂ G a discrete, torsionfree, cocompact subgroup, and let Y = Γ\X be the corresponding locally symmetric space. In this paper we explain how the HarishChandra Plancherel Theorem for L²(G) and results on (g, K)cohomology can be used in order to compute the L²Betti numbers, the NovikovShubin invariants, and the L²torsion of Y in a uniform way thus completing results previously obtained by Borel, Lott, Mathai, Hess and Schick, Lohoue and Mehdi. It turns out that the behaviour of these invariants is essentially determined by the fundamental rank m = rkCG−rkCK of G. In particular, we show the nonvanishing of the L²torsion of Y whenever m = 1.
L² torsion without the determinant class condition and extended L² cohomology
 COMMUN. CONTEMP. MATH
, 2004
"... We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and the analytic L² torsions which, unlike the work of the previous authors, requires no additional assumptions; in pa ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
We associate determinant lines to objects of the extended abelian category built out of a von Neumann category with a trace. Using this we suggest constructions of the combinatorial and the analytic L² torsions which, unlike the work of the previous authors, requires no additional assumptions; in particular we do not impose the determinant class condition. The resulting torsions are elements of the determinant line of the extended L² cohomology. Under the determinant class assumption the L² torsions of this paper specialize to the invariants studied in our previous work [6]. Applying a recent theorem of D. Burghelea, L. Friedlander and T. Kappeler [3] we obtain a Cheeger Müller type theorem stating the equality between the combinatorial and the analytic L² torsions.
L 2 torsion and 3manifolds
 LowDimensional Topology, Conf. Proc. Lecture Notes Geom. Topology, III
, 1994
"... by ..."
Delocalized L 2 invariants
 J. Funct. Anal
, 1999
"... Abstract. We define extensions of the L 2analytic invariants of closed manifolds, called delocalized L 2invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show th ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We define extensions of the L 2analytic invariants of closed manifolds, called delocalized L 2invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spectrum of an odddimensional hyperbolic manifold can be recovered from its delocalized L 2analytic torsion. There are technical convergence questions. 1.
INDEX TYPE INVARIANTS FOR TWISTED SIGNATURE COMPLEXES AND HOMOTOPY INVARIANCE
, 2013
"... For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X, E, H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H = ∑ ij+1H2j+1 is an odddegree closed differential form on X and H2j+1 is a realvalued differential form of degree ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
For a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X, E, H) for the twisted odd signature operator valued in a flat hermitian vector bundle E, where H = ∑ ij+1H2j+1 is an odddegree closed differential form on X and H2j+1 is a realvalued differential form of degree 2j + 1. We show that ρ(X, E, H) is independent of the choice of metrics on X and E and 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 3dimensional manifolds with a degree three flux form. A core technique used is our analogue of the AtiyahPatodiSinger 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, E, H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.
Diffeomorphisms, analytic torsion and noncommutative geometry
, 1996
"... We prove an index theorem concerning the pushforward of flat Bvector bundles, where B is an appropriate algebra. We construct an associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of π∗(Diff(Z)). ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
We prove an index theorem concerning the pushforward of flat Bvector bundles, where B is an appropriate algebra. We construct an associated analytic torsion form T. If Z is a smooth closed aspherical manifold, we show that T gives invariants of π∗(Diff(Z)).