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Thomas Schick

Publications and source records attributed to Thomas Schick.

105 records · Page 6Linked to original sources
arXiv2004

A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture

Doing surgery on the 5-torus, we construct a 5-dimensional closed spin-manifold M with $π_1(M) = Z^4times Z/3$, so that the index invariant in the KO-theory of the reduced $C^*$-algebra of $π_1(M)$ is zero. Then we use the theory of minimal surfaces of Schoen/Yau to show that this manifolds cannot carry a metric of positive scalar curvature. The existence of such a metric is predicted by the (unstable) Gromov-Lawson-Rosenberg conjecture.

math.GT↗
arXiv2003

The Ore condition, affiliated operators, and the lamplighter group

Let G be the wreath product of Z and Z/2, the so called lamplighter group and k a commutative ring. We show that kG does not have a classical ring of quotients (i.e. does not satisfy the Ore condition). This answers a Kourovka notebook problem. Assume that kG is contained in a ring R in which the element 1-x is invertible, with x a generator of Z considered as subset of G. Then R is not flat over kG. If k is the field of complex numbers, this applies in particular to the algebra UG of unbounded operators affiliated to the group von Neumann algebra of G. We present two proofs of these results. The second one is due to Warren Dicks, who, having seen our argument, found a much simpler and more elementary proof, which at the same time yielded a more general result than we had originally proved. Nevertheless, we present both proofs here, in the hope that the original arguments might be of use in some other context not yet known to us.

math.RA↗
arXiv2002

Operator algebras and topology

These notes cover the contents of three survey lectures held at the ICTP Trieste Summer school on High dimensional manifold theory 2001. They introduce techniques coming from the theory of operator algebras. We will focus on the basic definitions and properties, and on their relevance to the geometry and topology of manifolds. An central pillar of work in the theory of C*-algebras is the Baum-Connes conjecture. It implies the Novikov conjecture. In the first talk, the Baum-Connes conjecture will be explained and put into our context. One application of the Baum-Connes conjecture is to the positive scalar curvature question. It implies the so called ``stable Gromov-Lawson-Rosenberg conjecture''. The unstable version of this conjecture said that, given a closed spin manifolds M, a certain obstruction, living in a certain (topological) K-theory group, vanishes if and only M admits a Riemannian metric with positive scalar curvature. It turns out that this is wrong, and counterexamples will be presented in the second talk. The third talk introduces $L^2$-cohomology, $L^2$-Betti numbers and other $L^2$-invariants. These invariants, their basic properties, and the central questions about them are introduced.

math.GT↗
arXiv2002

Remarks on a conjecture of Gromov and Lawson

Gromov and Lawson conjectured that a closed spin manifold M of dimension n with fundamental group pi admits a metric with positive scalar curvature if and only if an associated element in KO_n(B pi) vanishes. In this note we present counter examples to the `if' part of this conjecture for groups pi which are torsion free and whose classifying space is a manifold with negative curvature (in the Alexandrov sense).

math.GT↗
arXiv2002

Approximating Spectral invariants of Harper operators on graphs II

We study Harper operators and the closely related discrete magnetic Laplacians (DML) on a graph with a free action of a discrete group, as defined by Sunada. The spectral density function of the DML is defined using the von Neumann trace associated with the free action of a discrete group on a graph. The main result in this paper states that when the group is amenable, the spectral density function is equal to the integrated density of states of the DML that is defined using either Dirichlet or Neumann boundary conditions. This establishes the main conjecture in a paper by Mathai and Yates. The result is generalized to other self adjoint operators with finite propagation.

math.SP↗
arXiv2001

Approximating L^2-signatures by their compact analogues

:Let G be a group together with an descending nested sequence of normal subgroups G=G_0, G_1, G_2 G_3, ... of finite index [G:G_k] such the intersection of the G_k-s is the trivial group. Let (X,Y) be a compact 4n-dimensional Poincare' pair and p: (\bar{X},\bar{Y}) \to (X,Y) be a G-covering, i.e. normal covering with G as deck transformation group. We get associated $G/_k$-coverings (X_k,Y_k) \to (X,Y). We prove that sign^{(2)}(\bar{X},\bar{Y}) = lim_{k\to\infty} \frac{sign(X_k,Y_k)}{[G : G_k]}, where sign or sign^{(2)} is the signature or L^2-signature, respectively, and the convergence of the right side for any such sequence (G_k)_k is part of the statement.

math.GT↗
arXiv2001

The spectral measure of certain elements of the complex group ring of a wreath product

We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk-Zuk. The latter result was used by Grigorchuk-Linnell-Schick-Zuk to produce a counterexample to a strong version of the Atiyah conjecture about the range of L2-Betti numbers. We use our results to construct manifolds with certain L2-Betti numbers (given as convergent infinite sums of rational numbers) which are not obviously rational, but we have been unable to determine whether any of them are irrational.

math.GT↗
arXiv2000

The trace on the K-theory of group C*-algebras

The canonical trace on the reduced C*-algebra of a discrete group gives rise to a homomorphism from the K-theory of this C^*-algebra to the real numbers. This paper addresses the range of this homomorphism. For torsion free groups, the Baum-Connes conjecture and Atiyah's L2-index theorem implies that the range consists of the integers. If the group is not torsion free, Baum and Connes conjecture that the trace takes values in the rational numbers. We give a direct and elementary proof that if G acts on a tree and admits a homomorphism αto another group H whose restriction to every stabilizer group of a vertex is injective, then the range of the trace for G, tr_G(K(C_r^*G)) is contained in the range of the trace for H, tr_H(K(C_r^*H)). This follows from a general relative Fredholm module technique. Examples are in particular HNN-extensions of H where the stable letter acts by conjugation with an element of H, or amalgamated free products G=H*_U H of two copies of the same groups along a subgroup U.

math.KT↗
arXiv2000

Manifolds with boundary and of bounded geometry

For non-compact manifolds with boundary we prove that bounded geometry defined by coordinate-free curvature bounds is equivalent to bounded geometry defined using bounds on the metric tensor in geodesic coordinates. We produce a nice atlas with subordinate partition of unity on manifolds with boundary of bounded geometry, and we study the change of geodesic coordinate maps.

math.DG↗
arXiv1999

Positive and negative results concerning the Gromov-Lawson-Rosenberg conjecture

The Gromov-Lawson-Rosenberg-conjecture for a group G states that a closed spin manifold M^n (n>4) with fundamental group G admits a metric with positive scalar curvature if and only if its C^*-index A(M) in KO_n(C^*_r(G)) vanishes. We prove this for groups G with low-dimensional classifying space, provided the assembly map for G is injective. On the other hand, we construct a spin manifold with no metric with scal>0 but so that already its KO-orientation in KO_*(B pi_1(M)) vanishes. Therefore a corresponding weakened version or the GLR-conjecture is wrong. Last we address non-orientable manifolds. We give a reformulation of the minimal surface method of Schoen and Yau (extended to dimension 8) and introduce a non-orientable version of it. We then construct a manifold M whose orientation cover admits a metric of positive scalar curvature but where M itself does not. M is also a counterexample to a twisted analog of the GLR-conjecture because its twisted index vanishes.

math.DG↗
arXiv1998

L2-index theorem for manifolds with boundary

Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are finite, and their difference, by definition the von Neumann index, equals the index of (A,T). In this way, we extend the classical L^2-index theorem of Atiyah to manifolds with boundary.

math.GT↗
arXiv1998

Analysis and Geometry of Boundary-Manifolds of Bounded Geometry

In this paper, we investigate analytical and geometric properties of certain non-compact boundary-manifolds, namely manifolds of bounded geometry. One result are strong Bochner type vanishing results for the L^2-cohomology of these manifolds: if e.g. a manifold admits a metric of bounded geometry which outside a compact set has nonnegative Ricci curvature and nonnegative mean curvature (of the boundary) then its first relative L^2-cohomology vanishes (this in particular answers a question of Roe). We prove the Hodge-de Rham-theorem for L^2-cohomology of oriented boundary-manifolds of bounded geometry. The technical basis is the study of (uniformly elliptic) boundary value problems on these manifolds, applied to the Laplacian.

math.GT↗
arXiv1998

L2-torsion of hyperbolic manifolds

The L^2-torsion is an invariant defined for compact L^2-acyclic manifolds of determinant class, for example odd dimensional hyperbolic manifolds. It was introduced by John Lott and Varghese Mathai and computed for hyperbolic manifolds in low dimensions. In this paper we show that the L^2-torsion of hyperbolic manifolds of arbitrary odd dimension does not vanish. This was conjectured by J. Lott and W. Lueck. Some concrete values are computed and an estimate of their growth with the dimension is given.

math.DG↗
arXiv1997

Torsion and fibrations

We study the behaviour of analytic torsion under smooth fibrations. Namely, let F \to E \to^{f} B be a smooth fiber bundle of connected closed oriented smooth manifolds and let $V$ be a flat vector bundle over $E$. Assume that $E$ and $B$ come with Riemannian metrics and $V$ comes with a unimodular (not necessarily flat) Riemannian metric. Let $ρ_{an}(E;V)$ be the analytic torsion of $E$ with coefficients in $V$ and let $\Pf_B$ be the Pfaffian $\dim(B)$-form. Let $H^q_{dR}(F;V)$ be the flat vector bundle over $B$ whose fiber over $b \in B$ is $H^q_{dR}(F_b;V)$ with the Riemannian metric which comes from the Hilbert space structure on the space of harmonic forms induced by the Riemannian metrics. Let $ρ_{an}(B;H^q_{dR}(F;V))$ be the analytic torsion of $B$ with coefficients in this bundle. The Leray-Serre spectral sequence for deRham cohomology determines a certain correction term $ρ^{Serre}_{dR}(f)$. We prove $ρ_{an}(E;V) = \int_B ρ_{an}(F_b;V) \cdot \Pf_B + \sum_{q} (-1)^q \cdot ρ_{an}(B;H^q_{dR}(F;V)) + ρ^{Serre}_{dR}(f)$.

dg-ga↗