Uniform K-homology theory

We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C^∗-algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C^∗-algebras. We show that our theory has a Mayer-Vietoris sequence. We prove that for a torsion-free countable discrete group Γ, the direct limit of the uniform K-homology of the Rips complexes of Γ, , is isomorphic to , the left-hand side of the Baum-Connes conjecture with coefficients in ?^∞Γ. In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras.     

The Baum-Connes assembly map, delocalization and the Chern character

For a discrete group Γ, we explicitly describe the rational Baum-Connes assembly map in “homological degree ?2” and show that in this domain it factors through the algebraic K-theory of the complex group ring of Γ. We also state and prove a delocalization property for , namely expressing it rationally in terms of the Novikov assembly map. Finally, we give a handicrafted construction of the delocalized equivariant Chern character (in the analytic language) and prove that it coincides with the equivariant Chern character of Lück (Invent. Math. 149 (2002) 123-152) (defined in the topological framework).     

Functional calculus and ∗-regularity of a class of Banach algebras II

In this article, we define a natural Banach ∗-algebra for a C^∗-dynamical system (A,G,α) which is slightly bigger than L¹(G;A) (they are the same if A is finite-dimensional). We will show that this algebra is ∗-regular if G has polynomial growth. The main result in this article extends the two main results in [C.W. Leung, C.K. Ng, Functional calculus and ∗-regularity of a class of Banach algebras, Proc. Amer. Math. Soc., in press].     

Equivariant–Bivariant Chern Character for Profinite Groups

For the action of a locally compact and totally disconnected group G on a pair of locally compact spaces X and Y we construct, by sheaf theoretic means, a new equivariant and bivariant cohomology theory. If we take for the first space Y an universal proper G-action then we obtain for the second space its delocalized equivariant homology. This is in exact formal analogy to the definition of equivariant K-homology by Baum, Connes, Higson starting from the bivariant equivariant Kasparov KK-theory. Under certain basic finiteness conditions on the first space Y we conjecture the existence of a Chern character from the equivariant Kasparov KK-theory of Y and X into our cohomology theory made two-periodic which becomes an isomorphism upon tensoring the KK-theory with the complex numbers. This conjecture is proved for profinite groups G. An essential role in our construction is played by a bivariant version of Segal localization which we establish for KK-theory.     

Spreading speed and traveling waves for a multi-type SIS epidemic model

The theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows is applied to a multi-type SIS epidemic model to obtain the spreading speed c^∗, and the nonexistence of traveling waves with wave speed c<c^∗. Then the method of upper and lower solutions is used to establish the existence of monotone traveling waves connecting the disease-free and endemic equilibria for c?c^∗. This shows that the spreading speed coincides with the minimum wave speed for monotone traveling waves. We also give an affirmative answer to an open problem presented by Rass and Radcliffe [L. Rass, J. Radcliffe, Spatial Deterministic Epidemics, Math. Surveys Monogr. 102, Amer. Math. Soc., Providence, RI, 2003].     

Twisted K-theory of differentiable stacks

In this paper, we develop twisted K-theory for stacks, where the twisted class is given by an S¹-gerbe over the stack. General properties, including the Mayer-Vietoris property, Bott periodicity, and the product structure are derived. Our approach provides a uniform framework for studying various twisted K-theories including the usual twisted K-theory of topological spaces, twisted equivariant K-theory, and the twisted K-theory of orbifolds. We also present a Fredholm picture, and discuss the conditions under which twisted K-groups can be expressed by so-called “twisted vector bundles”.Our approach is to work on presentations of stacks, namely groupoids, and relies heavily on the machinery of K-theory (KK-theory) of C^∗-algebras.     

The higher fixed point theorem for foliations I. Holonomy invariant currents

In this paper, we prove a higher Lefschetz formula for foliations in the presence of a closed Haefliger current. To this end, we associate with such a current an equivariant cyclic cohomology class of Connes' C^∗-algebra of the foliation, and compute its pairing with the localized equivariant K-theory in terms of local contributions near the fixed points.     

Higher index theory for certain expanders and Gromov monster groups,I

In this paper, the first of a series of two, we continue the study of higher index theory for expanders. We prove that if a sequence of graphs is an expander and the girth of the graphs tends to infinity, then the coarse Baum–Connes assembly map is injective, but not surjective, for the associated metric space X.Expanders with this girth property are a necessary ingredient in the construction of the so-called ‘Gromov monster’ groups that (coarsely) contain expanders in their Cayley graphs. We use this connection to show that the Baum–Connes assembly map with certain coefficients is injective but not surjective for these groups. Using the results of the second paper in this series, we also show that the maximal Baum–Connes assembly map with these coefficients is an isomorphism.     

Euler characteristics and Gysin sequences for group actions on boundaries

Let G be a locally compact group, let X be a universal proper G-space, and let be a G-equivariant compactification of X that is H-equivariantly contractible for each compact subgroup . Let . Assuming the Baum-Connes conjecture for G with coefficients and C(?X), we construct an exact sequence that computes the map on K-theory induced by the embedding . This exact sequence involves the equivariant Euler characteristic of X, which we study using an abstract notion of Poincaré duality in bivariant K-theory. As a consequence, if G is torsion-free and the Euler characteristic is non-zero, then the unit element of is a torsion element of order . Furthermore, we get a new proof of a theorem of Lück and Rosenberg concerning the class of the de Rham operator in equivariant K-homology.     

Topology and combinatorics of partitions of masses by hyperplanes

An old problem in combinatorial geometry is to determine when one or more measurable sets in Rd admit an equipartition by a collection of k hyperplanes [B. Grünbaum, Partitions of mass-distributions and convex bodies by hyperplanes, Pacific J. Math. 10 (1960) 1257-1261]. A related topological problem is the question of (non)existence of a map , equivariant with respect to the Weyl group Wk=Bk:=(Z/2)^⊕k?Sk, where U is a representation of Wk and S(U)⊂U the corresponding unit sphere. We develop general methods for computing topological obstructions for the existence of such equivariant maps. Among the new results is the well-known open case of 5 measures and 2 hyperplanes in R⁸ [E.A. Ramos, Equipartitions of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996) 147-167]. The obstruction in this case is identified as the element 2Xab∈H₁(D₈;Z)≅Z/4, where Xab is a generator, which explains why this result cannot be obtained by the parity count formulas of Ramos [loc. cit.] or the methods based on either Stiefel-Whitney classes or ideal valued cohomological index theory [E. Fadell, S. Husseini, An ideal-valued cohomological index theory with applications to Borsuk-Ulam and Bourgin-Yang theorems, Ergodic Theory Dynam. Systems ^8* (1988) 73-85].     

Combinatorial aspects of the Lascoux-Schützenberger tree

In 1982, Richard Stanley introduced the formal series F_σ(X) in order to enumerate reduced decompositions of a given permutation σ. Stanley (European J. Combin. 5(4) (1984) 359) not only showed F_σ(X) to be symmetric, but in certain cases, F_σ(X) was a Schur function. Stanley conjectured that for arbitrary was always Schur positive. Edelman and Greene subsequently proved this fact (Combinatories and Algebra (Boulder, CO, 1983), Amer. Math. Soc., Providence RI, 1984, pp. 155-162; Adv. in Math. 63(1) (1987) 42). Using the techniques of Lascoux and Schützenberger (Lett. Math. Phys. 10(2-3) (1985) 111) for computing Littlewood-Richardson coefficients, we will exhibit a new bijective proof of the Schur positivity of F_σ(X).     

Uniqueness of complex structure and real hereditarily indecomposable Banach spaces

There exists a real hereditarily indecomposable Banach space X=X(C) (respectively X=X(H)) such that the algebra L(X)/S(X) is isomorphic to C (respectively to the quaternionic division algebra H).Up to isomorphism, X(C) has exactly two complex structures, which are conjugate, totally incomparable, and both hereditarily indecomposable. So there exist two Banach spaces which are isometric as real spaces but totally incomparable as complex spaces. This extends results of J. Bourgain and S. Szarek [J. Bourgain, Real isomorphic complex Banach spaces need not be complex isomorphic, Proc. Amer. Math. Soc. 96 (2) (1986) 221-226; S. Szarek, On the existence and uniqueness of complex structure and spaces with “few” operators, Trans. Amer. Math. Soc. 293 (1) (1986) 339-353; S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444], and proves that a theorem of G. Godefroy and N.J. Kalton [G. Godefroy, N.J. Kalton, Lipschitz-free Banach spaces, Studia Math. 159 (1) (2003) 121-141] about isometric embeddings of separable real Banach spaces does not extend to the complex case.The quaternionic example X(H), on the other hand, has unique complex structure up to isomorphism; other examples with a unique complex structure are produced, including a space with an unconditional basis and non-isomorphic to l₂. This answers a question of S. Szarek in [S. Szarek, A superreflexive Banach space which does not admit complex structure, Proc. Amer. Math. Soc. 97 (3) (1986) 437-444].     

An alternative definition of coarse structures

Roe [J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol. 31, Amer. Math. Soc., Providence, RI, 2003] introduced coarse structures for arbitrary sets X by considering subsets of X×X. In this paper we introduce large scale structures on X via the notion of uniformly bounded families and we show their equivalence to coarse structures on X. That way all basic concepts of large scale geometry (asymptotic dimension, slowly oscillating functions, Higson compactification) have natural definitions and basic results from metric geometry carry over to coarse geometry.     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .     

Chern character for twisted K-theory of orbifolds

For an orbifold X and α∈H³(X,Z), we introduce the twisted cohomology and prove that the non-commutative Chern character of Connes-Karoubi establishes an isomorphism between the twisted K-groups and the twisted cohomology . This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Adem-Ruan regarding the Chern character isomorphism of twisted orbifold K-theory when the orbifold is a global quotient by a finite group and the twist is a special torsion class, as well as Mathai-Stevenson's theorem regarding the Chern character isomorphism of twisted K-theory of a compact manifold.     

On rigidity of Roe algebras

Roe algebras are C^?$C^{?}$-algebras built using large scale (or ‘coarse’) aspects of a metric space (X,d)$(X,d)$. In the special case that X=Γ$X=\Gamma$ is a finitely generated group and d   is a word metric, the simplest Roe algebra associated to (Γ,d)$(\Gamma ,d)$ is isomorphic to the crossed product C^?$C^{?}$-algebra l^∞(Γ)_?rΓ$l^{\infty }(\Gamma ){?}_r\Gamma$.     

A generalization of the Auslander transpose and the generalized Gorenstein dimension

Let R be a left and right Noetherian ring and C a semidualizing R-bimodule. We introduce a transpose Tr _c M of an R-module M with respect to C which unifies the Auslander transpose and Huang’s transpose, see Z.Y.Huang, On a generalization of the Auslander-Bridger transpose, Comm. Algebra 27 (1999), 5791–5812, in the two-sided Noetherian setting, and use Tr _c M to develop further the generalized Gorenstein dimension with respect to C. Especially, we generalize the Auslander-Bridger formula to the generalized Gorenstein dimension case. These results extend the corresponding ones on the Gorenstein dimension obtained by Auslander in M. Auslander, M. Bridger, Stable Module Theory, Mem. Amer. Math. Soc. vol. 94, Amer. Math. Soc., Providence, RI, 1969.     

A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation

It is known that every skew-polynomial ring with generating set X and binomial relations in the sense of Gateva-Ivanova (Trans. Amer. Math. Soc. 343 (1994) 203) is an Artin-Schelter regular domain of global dimension |X|. Moreover, every such ring gives rise to a non-degenerate unitary set-theoretical solution of the quantum Yang-Baxter equation which fixes the diagonal of X². Gateva-Ivanova's conjecture (Talk at the International Algebra Conference, Miskolc, Hungary, 1996) states that conversely, every such solution R comes from a skew-polynomial ring with binomial relations. An equivalent conjecture (Duke Math. J. 100 (1999) 169) says that the underlying set X is R-decomposable. We prove these conjectures and construct an indecomposable solution R with |X|=∞ which shows that an extension to infinite X is false.     

K-theory of equivariant quantization

Using an equivariant version of Connes? Thom isomorphism, we prove that equivariant K-theory is invariant under strict deformation quantization for a compact Lie group action.     

The E-theoretic descent functor for groupoids

The paper establishes, for a wide class of locally compact groupoids Γ, the E-theoretic descent functor at the C^∗-algebra level, in a way parallel to that established for locally compact groups by Guentner, Higson and Trout. Section 2 shows that Γ-actions on a C₀(X)-algebra B, where X is the unit space of Γ, can be usefully formulated in terms of an action on the associated bundle B^?. Section 3 shows that the functor B→C^∗(Γ,B) is continuous and exact, and uses the disintegration theory of J. Renault. Section 4 establishes the existence of the descent functor under a very mild condition on Γ, the main technical difficulty involved being that of finding a Γ-algebra that plays the role of Cbcont(T,B) in the group case.