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Thomas Schick

Publications and source records attributed to Thomas Schick.

At least 55 records · Page 3Linked to original sources
arXiv2015

Large time limit and local L^2-index theorems for families

We compute explicitly, and without any extra regularity assumptions, the large time limit of the fibrewise heat operator for Bismut-Lott type superconnections in the L^2-setting. This is motivated by index theory on certain non-compact spaces (families of manifolds with cocompact group action) where the convergence of the heat operator at large time implies refined L^2-index formulas. As applications, we prove a local L^2-index theorem for families of signature operators and an L^2-Bismut-Lott theorem, expressing the Becker-Gottlieb transfer of flat bundles in terms of Kamber-Tandeur classes. With slightly stronger regularity we obtain the respective refined versions: we construct L^2-eta forms and L^2-torsion forms as transgression forms.

math.DG↗
arXiv2014

Codimension two index obstructions to positive scalar curvature

We derive a general obstruction to the existence of Riemannian metrics of positive scalar curvature on closed spin manifolds in terms of hypersurfaces of codimension two. The proof is based on coarse index theory for Dirac operators that are twisted with Hilbert C*-module bundles. Along the way we give a complete and self-contained proof that the minimal closure of a Dirac type operator twisted with a Hilbert C*-module bundle on a complete Riemannian manifold is a regular and self-adjoint operator on the Hilbert C*-module of L2-sections of this bundle. Moreover, we give a new proof of Roe's vanishing theorem for the coarse index of the Dirac operator on a complete Riemannian manifold whose scalar curvature is uniformly positive outside of a compact subset. This proof immediately generalizes to Dirac operators twisted with Hilbert C*-module bundles.

math.KT↗
arXiv2014

The topology of positive scalar curvature

In this survey article, given a smooth closed manifold M we study the space of Riemannian metrics of positive scalar curvature on M. A long-standing question is: when is this space non-empty (i.e. when does M admit a metric of positive scalar curvature)? More generally: what is the topology of this space? For example, what are its homotopy groups? Higher index theory of the Dirac operator is the basic tool to address these questions. This has seen tremendous development in recent years, and in this survey we will discuss some of the most pertinent examples. In particular, we will show how advancements of large scale index theory (also called coarse index theory) give rise to new types of obstructions, and provide the tools for a systematic study of the existence and classification problem via the K-theory of C*-algebras. This is part of a program "mapping the topology of positive scalar curvature to analysis". In addition, we will show how advanced surgery theory and smoothing theory can be used to construct the first elements of infinite order in the k-th homotopy groups of the space of metrics of positive scalar curvature for arbitrarily large k. Moreover, these examples are the first ones which remain non-trivial in the moduli space of such metrics.

math.GT↗
arXiv2014

The Gromoll filtration, KO-characteristic classes and metrics of positive scalar curvature

Let X be a closed m-dimensional spin manifold which admits a metric of positive scalar curvature and let Pos(X) be the space of all such metrics. For any g in Pos(X), Hitchin used the KO-valued alpha-invariant to define a homomorphism A_{n-1} from π_{n-1}(Pos(X) to KO_{m+n}. He then showed that A_0 is not 0 if m = 8k or 8k+1 and that A_1 is not 0 if m = 8k-1 or 8$. In this paper we use Hitchin's methods and extend these results by proving that A_{8j+1-m} is not 0 whenever m>6 and 8j - m >= 0. The new input are elements with non-trivial alpha-invariant deep down in the Gromoll filtration of the group Γ^{n+1} = π_0(\Diff(D^n, \del)). We show that α(Γ^{8j+2}_{8j-5}) is not 0 for j>0. This information about elements existing deep in the Gromoll filtration is the second main new result of this note.

math.GT↗
arXiv2014

The space of metrics of positive scalar curvature

We study the topology of the space of positive scalar curvature metrics on high dimensional spheres and other spin manifolds. Our main result provides elements of infinite order in higher homotopy and homology groups of these spaces, which, in contrast to previous approaches, are of infinite order and survive in the (observer) moduli space of such metrics. Along the way we construct smooth fiber bundles over spheres whose total spaces have non-vanishing A-hat-genera, thus establishing the non-multiplicativity of the A-hat-genus in fibre bundles with simply connected base.

math.GT↗
arXiv2013

A Counterexample to a Conjecture about Positive Scalar Curvature

Conjecture 1 of Stanley Chang: "Positive scalar curvature of totally nonspin manifolds" asserts that a closed smooth manifold M with non-spin universal covering admits a metric of positive scalar curvature if and only if a certain homological condition is satisfied. We present a counterexample to this conjecture, based on the counterexample to the unstable Gromov-Lawson-Rosenberg conjecture given in Schick: "A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture".

math.GT↗
arXiv2013

Families index for Boutet de Monvel operators

We define the analytical and the topological indices for continuous families of operators in the C*-closure of the Boutet de Monvel algebra. Using techniques of C*-algebra K-theory and the Atiyah-Singer theorem for families of elliptic operators on a closed manifold, we prove that these two indices coincide.

math.KT↗
arXiv2012

C*-Algebra approach to the index theory of boundary value problems

Boutet de Monvel's calculus provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported $K$-theory classes on the cotangent bundle over the interior of the manifold. The Atiyah-Singer topological index map, applied to this class, then furnishes the index of the operator. Based on this result, Fedosov, Rempel-Schulze and Grubb have given index formulas in terms of the symbols. In this paper we survey previous work how C*-algebra K-theory can be used to give a proof of Boutet de Monvel's index theorem for boundary value problems, and how the same techniques yield an index theorem for families of Boutet de Monvel operators. The key ingredient of our approach is a precise description of the K-theory of the kernel and of the image of the boundary symbol.

math.KT↗
arXiv2012

Landweber exact formal group laws and smooth cohomology theories

The main aim of this paper is the construction of a smooth (sometimes called differential) extension \hat{MU} of the cohomology theory complex cobordism MU, using cycles for \hat{MU}(M) which are essentially proper maps W\to M with a fixed U(n)-structure and U(n)-connection on the (stable) normal bundle of W\to M. Crucial is that this model allows the construction of a product structure and of pushdown maps for this smooth extension of MU, which have all the expected properties. Moreover, we show, using the Landweber exact functor principle, that \hat{R}(M):=\hat{MU}(M)\otimes_{MU^*}R defines a multiplicative smooth extension of R(M):=MU(M)\otimes_{MU^*}R whenever R is a Landweber exact MU*-module. An example for this construction is a new way to define a multiplicative smooth K-theory.

math.KT↗
arXiv2011

Differential K-theory. A survey

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.

math.KT↗
arXiv2011

Hodge Theory on Metric Spaces

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional α-scale homology.

math.KT↗
arXiv2011

Differential orbifold K-Theory

We construct differential equivariant K-theory of representable smooth orbifolds as a ring valued functor with the usual properties of a differential extension of a cohomology theory. For proper submersions (with smooth fibres) we construct a push-forward map in differential equivariant K-theory. Finally, we construct a non-degenerate intersection pairing with values in C/Z for the subclass of smooth orbifolds which can be written as global quotients by a finite group action. We construct a real subfunctor of our theory, where the pairing restricts to a non-degenerate R/Z-valued pairing.

math.KT↗
arXiv2011

Approximating L2-invariants, and the Atiyah conjecture

Let G be a torsion free discrete group and let \bar{Q} denote the field of algebraic numbers in C. We prove that \bar{Q}[G] fulfills the Atiyah conjecture if G lies in a certain class of groups D, which contains in particular all groups which are residually torsion free elementary amenable or which are residually free. This result implies that there are no non-trivial zero-divisors in C[G]. The statement relies on new approximation results for L2-Betti numbers over \bar{Q}[G], which are the core of the work done in this paper. Another set of results in the paper is concerned with certain number theoretic properties of eigenvalues for the combinatorial Laplacian on L2-cochains on any normal covering space of a finite CW complex. We establish the absence of eigenvalues that are transcendental numbers, whenever the covering transformation group is either amenable or in the Linnell class \mathcal{C}. We also establish the absence of eigenvalues that are Liouville transcendental numbers whenever the covering transformation group is either residually finite or more generally in a certain large bootstrap class \mathcal{G}. Please take the errata to Schick: "L2-determinant class and approximation of L2-Betti numbers" into account, which are added at the end of the file, rectifying some unproved statements about "amenable extension". As a consequence, throughout, amenable extensions should be extensions with normal subgroups.

math.GT↗
arXiv2011

Integrality of L2-Betti numbers

The Atiyah conjecture predicts that the L2-Betti numbers of a finite CW-complex with torsion-free fundamental group are integers. We show that the Atiyah conjecture holds (with an additional technical condition) for direct and inverse limits of directed systems of groups for which it is true. As a corollary it holds for residually torsion-free solvable groups, e.g. for pure braid groups or for positive 1-relator groups with torsion free abelianization. Putting everything together we establish a new class of groups for which the Atiyah conjecture holds, which contains all free groups and in particular is closed under taking subgroups, direct sums, free products, extensions with elementary amenable quotient, and under direct and inverse limits of directed systems. This is a corrected version of an older paper with the same title. The proof of Proposition 2.1 of the earlier version contains a gap, as was pointed out to me by Pere Ara. This gap could not be fixed. Consequently, in this new version everything based on this result had to be removed. Please take the errata to "L2-determinant class and approximation of L2-Betti numbers" into account, which are added, rectifying some unproved statements about "amenable extension". As a consequence, throughout, amenable extensions should be extensions with normal subgroups.

math.GT↗
arXiv2011

L2-determinant class and approximation of L2-Betti numbers

A standing conjecture in L2-cohomology is that every finite CW-complex X is of L2-determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L2-acyclic, we also prove that the L2-determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L2-Betti numbers. Errata are added, rectifying some unproved statements about "amenable extension": throughout, amenable extensions should be extensions with \emph{normal} subgroups.

math.GT↗
arXiv2011

A parametrized version of the Borsuk Ulam theorem

The main result of this note is a parametrized version of the Borsuk-Ulam theorem. We show that for a continuous family of Borsuk-Ulam situations, parameterized by points of a compact manifold W, its solution set also depends continuously on the parameter space W. Continuity here means that the solution set supports a homology class which maps onto the fundamental class of W. When W is a subset of Euclidean space, we also show how to construct such a continuous family starting from a family depending in the same way continuously on the points of the boundary of W. This solves a problem related to a conjecture which is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information. A new method (of independent interest) used in this context is a canonical symmetric squaring construction in Cech homology with coefficients in Z/2Z.

math.AT↗
arXiv2010

Periodic twisted cohomology and T-duality

The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality and integration. The main result is the construction of a functorial periodization functor associated to a U(1)-gerbe. As applications we verify the $T$-duality isomorphism in periodic twisted cohomology and in periodic twisted orbispace cohomology.

math.AT↗