Thomas Schick | Georg-August-Universität Göttingen (original) (raw)
Papers by Thomas Schick
Proceedings of the American Mathematical Society, 2015
Annales de l’institut Fourier, 2015
Pacific Journal of Mathematics, 2001
Eprint Arxiv Math 0702826, Feb 1, 2007
We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it.... more We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible.
New York Journal of Mathematics, Sep 13, 2002
Let GGG be a word hyperbolic group in the sense of Gromov and PPP its associated Rips complex. We... more Let GGG be a word hyperbolic group in the sense of Gromov and PPP its associated Rips complex. We prove that the fixed point set PHP^HPH is contractible for every finite subgroups HHH of GGG. This is the main ingredient for proving that PPP is a finite model for the universal space e.g.e.g.e.g. of proper actions. As a corollary we get that a hyperbolic group has only finitely many conjugacy classes of finite subgroups.
Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a ... more Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a certain subset of the Banach bidual module V**. We give another (equivalent) definition of quasi-multipliers for Hilbert C*-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C*-algebras, provided these C*-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule l_2(A) and for bimodules of sections of Hilbert C*-bimodule bundles over locally compact spaces.
Eprint Arxiv Math 0604466, Apr 21, 2006
In this note, we study non-standard models of the rational numbers with countably many elements. ... more In this note, we study non-standard models of the rational numbers with countably many elements. These are ordered fields, and so it makes sense to complete them, using non-standard Cauchy sequences. The main result of this note shows that these completions are real closed, i.e. each positive number is a square, and each polynomial of odd degree has a root. This way, we give a direct proof of a consequence of a theorem of Hauschild. In a previous version of this note, not being aware of these results, we missed to mention this reference. We thank Matthias Aschenbrenner for pointing out this and related work. We also give some information about the set of real parts of the finite elements of such completions -about the more interesting results along this we have been informed by Matthias Aschenbrenner. The main idea to achieve the results relies on a way to describe real zeros of a polynomial in terms of first order logic. This is achieved by carefully using the sign changes of such a polynomial.
Survey article on loop groups and their representations, following a course of three lectures hel... more Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.
J Reine Angew Math, 2006
0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact i... more 0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.
Algebr Geom Topol, 2004
In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-th... more In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-theory of a real C^*-algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum-Connes assembly map is an isomorphism if and only if the real one is, thus reproving a result of Baum and Karoubi. After inverting 2, the same is true for the injectivity or surjectivity part alone.
We compute explicitly, and without any extra regularity assumptions, the large time limit of the ... more 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.
Geom Dedic, Oct 4, 2010
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter grou... more We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
Russian Journal of Mathematical Physics, Sep 12, 2009
It is proved that the asymptotic dimension of any building is finite and equal to the asymptotic ... more It is proved that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.
Let G be a finitely generated discrete group. In this paper we establish vanishing results for rh... more Let G be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite dimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G. We prove that all these rho-invariants vanish if the group G is torsion-free and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C^*-algebra. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof re-establishes this result and also extends it to the delocalized eta-invariant of Lott. Our method also gives some information about the eta-invariant itself (a much more saddle object than the rho-invariant).
v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian ... more v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible. This condition is used in the proof of Lemma 4.62.
High-Dimensional Manifold Topology, 2003
Proceedings of the American Mathematical Society, 2015
Annales de l’institut Fourier, 2015
Pacific Journal of Mathematics, 2001
Eprint Arxiv Math 0702826, Feb 1, 2007
We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it.... more We give a counterexample to a conjecture of D.H. Gottlieb and prove a strengthened version of it. The conjecture says that a map from a finite CW-complex X to an aspherical CW-complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial. As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible.
New York Journal of Mathematics, Sep 13, 2002
Let GGG be a word hyperbolic group in the sense of Gromov and PPP its associated Rips complex. We... more Let GGG be a word hyperbolic group in the sense of Gromov and PPP its associated Rips complex. We prove that the fixed point set PHP^HPH is contractible for every finite subgroups HHH of GGG. This is the main ingredient for proving that PPP is a finite model for the universal space e.g.e.g.e.g. of proper actions. As a corollary we get that a hyperbolic group has only finitely many conjugacy classes of finite subgroups.
Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a ... more Quasi-multipliers for a Hilbert C*-bimodule V were introduced by Brown, Mingo and Shen 1994 as a certain subset of the Banach bidual module V**. We give another (equivalent) definition of quasi-multipliers for Hilbert C*-bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C*-algebras, provided these C*-algebras act non-degenerately. A topological picture of quasi-multipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule l_2(A) and for bimodules of sections of Hilbert C*-bimodule bundles over locally compact spaces.
Eprint Arxiv Math 0604466, Apr 21, 2006
In this note, we study non-standard models of the rational numbers with countably many elements. ... more In this note, we study non-standard models of the rational numbers with countably many elements. These are ordered fields, and so it makes sense to complete them, using non-standard Cauchy sequences. The main result of this note shows that these completions are real closed, i.e. each positive number is a square, and each polynomial of odd degree has a root. This way, we give a direct proof of a consequence of a theorem of Hauschild. In a previous version of this note, not being aware of these results, we missed to mention this reference. We thank Matthias Aschenbrenner for pointing out this and related work. We also give some information about the set of real parts of the finite elements of such completions -about the more interesting results along this we have been informed by Matthias Aschenbrenner. The main idea to achieve the results relies on a way to describe real zeros of a polynomial in terms of first order logic. This is achieved by carefully using the sign changes of such a polynomial.
Survey article on loop groups and their representations, following a course of three lectures hel... more Survey article on loop groups and their representations, following a course of three lectures held at the summer school "algebraic groups" at the Georg-August-Universitaet zu Goettingen, June 27--July 13, 2005. We discuss loop groups, their central extensions, and positive energy representations.
J Reine Angew Math, 2006
0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact i... more 0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.
Algebr Geom Topol, 2004
In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-th... more In this paper, we use the KK-theory of Kasparov to prove exactness of sequences relating the K-theory of a real C^*-algebra and of its complexification (generalizing results of Boersema). We use this to relate the real version of the Baum-Connes conjecture for a discrete group to its complex counterpart. In particular, the complex Baum-Connes assembly map is an isomorphism if and only if the real one is, thus reproving a result of Baum and Karoubi. After inverting 2, the same is true for the injectivity or surjectivity part alone.
We compute explicitly, and without any extra regularity assumptions, the large time limit of the ... more 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.
Geom Dedic, Oct 4, 2010
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter grou... more We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
Russian Journal of Mathematical Physics, Sep 12, 2009
It is proved that the asymptotic dimension of any building is finite and equal to the asymptotic ... more It is proved that the asymptotic dimension of any building is finite and equal to the asymptotic dimension of an apartment in that building.
Let G be a finitely generated discrete group. In this paper we establish vanishing results for rh... more Let G be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental group G. The invariants we consider are more precisely - the Atiyah-Patodi-Singer rho-invariant associated to a pair of finite dimensional unitary representations. - the L2-rho invariant of Cheeger-Gromov - the delocalized eta invariant of Lott for a finite conjugacy class of G. We prove that all these rho-invariants vanish if the group G is torsion-free and the Baum-Connes map for the maximal group C^*-algebra is bijective. For the delocalized invariant we only assume the validity of the Baum-Connes conjecture for the reduced C^*-algebra. In particular, the three rho-invariants associated to the signature operator are, for such groups, homotopy invariant. For the APS and the Cheeger-Gromov rho-invariants the latter result had been established by Navin Keswani. Our proof re-establishes this result and also extends it to the delocalized eta-invariant of Lott. Our method also gives some information about the eta-invariant itself (a much more saddle object than the rho-invariant).
v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian ... more v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible. This condition is used in the proof of Lemma 4.62.
High-Dimensional Manifold Topology, 2003