A mod 2 index theorem for pin<Superscript>−</Superscript> manifolds

We establish a mod 2 index theorem for real vector bundles over 8k + 2 dimensional compact pin^? manifolds. The analytic index is the reduced η invariant of (twisted) Dirac operators and the topological index is defined through KO-theory. Our main result extends the mod 2 index theorem of Atiyah and Singer (1971) to non-orientable manifolds.     

The -theory of toric manifolds

Toric manifolds, a topological generalization of smooth projective toric varieties, are determined by an -dimensional simple convex polytope and a function from the set of codimension-one faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of Danilov-Jurkiewicz in the toric variety case. Recently it has been shown by Buchstaber and Ray that they generate the complex cobordism ring. We use the Adams spectral sequence to compute the -theory of all toric manifolds and certain singular toric varieties.

     

Real C*-Algebras, United KK-Theory, and the Universal Coefficient Theorem

We define united KK-theory for real C*-algebras A and B such that A is separable and B is -unital, extending united K-theory in the sense that KK^CRT( , B) = K^CRT(B). United KK-theory combines real, complex, and self-conjugate KK-theory; but unlike unaugmented KK-theory for real C*-algebras, it admits a Universal Coefficient Theorem. For all separable A and B in which the complexification of A is in the bootstrap category, KK^CRT(A,B) appears as the middle term of a short exact sequence whose outer terms involve the united K-theory of A and B. As a corollary, we prove that united K-theory classifies KK-equivalence for real C*-algebras whose complexification is in the bootstrap category.Mathematics Subject Classification (2000): 19K35, 46L80.     

Representable K-Theory for σ-C*-algebras

We define a version of K-theory on the category of -C ^*-algebras (countable inverse limits of C ^*-algebras). Our theory is homotopy invariant, has long exact sequences and a Milnor sequence, and satisfies Bott periodicity. On C ^*-algebras it gives the ordinary K-theory, and on the space of continuous functions on a countable direct limit X of compact Hausdorff spaces, it gives the representable K-theory of X. (We do not claim that our theory is in general a representable functor.) We also define an equivariant version, and discuss several related groups.Partially supported by a National Science Foundation Postdoctoral Fellowship.     

Abstract Bott periodicity inKK-theory

We show that the universalC*-algebras KqA and K²A are homotopy equivalent and define abstract analogues of the Bott elements inKK-theory.     

Classification of Bott towers by matrix

A criterion for the classification of Bott towers is presented, i.e., two Bott towers B _*(A) and B _*(A′) are isomorphic if and only if the matrices A and A′ are equivalent. The equivalence relation is defined by two operations on matrices. And it is based on the observation that any Bott tower B _*(A) is uniquely determined by its structure matrix A, which is a strictly upper triangular integer matrix. The classification of Bott towers is closely related to the cohomological rigidity problem for both Bott towers and Bott manifolds.     

Foliations, Groupoids, and the Baum―Connes Conjecture

The BaumConnes conjecture establishes, for foliated manifolds, an analog of the well-known isomorphism between the topological K-theory of a locally compact space M and the analytic K-theory of the C ^*-algebra of continuous functions on M vanishing at infinity. In this work, we describe the principal notions involved in the statement of the conjecture and indicate its contemporary status. Bibliography: 11 titles.     

Spin structures and spectra of ℤ<Subscript>2</Subscript><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-manifolds

We give necessary and sufficient conditions for the existence of pin^± and spin structures on Riemannian manifolds with holonomy group ₂^k. For any n4 (resp. n6) we give examples of pairs of compact manifolds (resp. compact orientable manifolds) M₁, M₂, non homeomorphic to each other, that are Laplace isospectral on functions and on p-forms for any p and such that M₁ admits a pin^± (resp. spin) structure whereas M₂ does not.Mathematics Subject Classification (2000):58J53, 57R15, 20H15Partially supported by Conicet and grants from SecytUNC, Foncyt and AgCba.     

Holomorphic Lefschetz formula for manifolds with boundary

The classical Lefschetz fixed point formula expresses the number of fixed points of a continuous map f:MM in terms of the transformation induced by f on the cohomology of M. In 1966 Atiyah and Bott extended this formula to elliptic complexes over a compact closed manifold. In particular, they presented a holomorphic Lefschetz formula for compact complex manifolds without boundary, a result, in the framework of algebraic geometry due to Eichler (1957) for holomorphic curves. On compact complex manifolds with boundary the Dolbeault complex is not elliptic, hence the Atiyah-Bott theory is no longer applicable. To get rid of the difficulties related to the boundary behaviour of the Dolbeault cohomology, Donelli and Fefferman (1986) derived a fixed point formula for the Bergman metric. The purpose of this paper is to present a holomorphic Lefschetz formula on a strictly convex domain in ⁿ, n>1.Mathematics Subject Classification (2000):32S50; 58J20*Supported by the Deutsche Forschungsgemeinschaft and the RFFI grant 02–01–00167.**Supported by the Deutsche Forschungsgemeinschaft and the RFFI grant 02–01–00167.     

Groupoid C *-Algebras and Index Theory on Manifolds with Singularities

The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification M M × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S ¹, we show how to attach to such a space a noncommutative C ^*-algebra that captures the extra structure. We then use this C ^*-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S ¹ singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.     

Real C*-Algebras, United K-Theory, and the Künneth Formula

We define united K-theory for real C^*-algebras, generalizing Bousfield's topological united K-theory. United K-theory incorporates three functors – real K-theory, complex K-theory, and self-conjugate K-theory – and the natural transformations among them. The advantage of united K-theory over ordinary K-theory lies in its homological algebraic properties, which allow us to construct a Künneth-type, nonsplitting, short exact sequence whose middle term is the united K-theory of the tensor product of two real C^*-algebras A and B which holds as long as the complexification of A is in the bootstrap category . Since united K-theory contains ordinary K-theory, our sequence provides a way to compute the K-theory of the tensor product of two real C^*-algebras. As an application, we compute the united K-theory of the tensor product of two real Cuntz algebras. Unlike in the complex case, it turns out that the isomorphism class of the tensor product is not determined solely by the greatest common divisor of K and l. Hence, we have examples of nonisomorphic, simple, purely infinite, real C^*-algebras whose complexifications are isomorphic.     

Semi-Topological K-Homology and Thomason''s Theorem

In this paper, we introduce the 'semi-topological K-homology' of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasi-projective complex variety Y coincides with the connective topological K-homology of the associated analytic space Y ^an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the 'Bott inverted' semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of X ^an. In combination with a result of Friedlander and the author, this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason's celebrated theorem that 'Bott inverted' algebraic K-theory with /n coefficients coincides with topological K-theory with /n coefficients.     

Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian T -isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology H* _T (M; ?) with a point â in H ²(M;?) \ {0} up to weak H*(BT; ?)-algebra isomorphism preserving â.     

Classification of torus manifolds with codimension one extended actions

The purpose of this paper is to classify torus manifolds (M ²ⁿ , T ⁿ ) with codimension one extended G-actions (M ²ⁿ , G) up to essential isomorphism, where G is a compact, connected Lie group whose maximal torus is T ⁿ . For technical reasons, we do not assume torus manifolds are orientable. We prove that there are seven types of such manifolds. As a corollary, if a nonsingular toric variety or a quasitoric manifold has a codimension one extended action then such manifold is a complex projective bundle over a product of complex projective spaces.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

Para-tt*-bundles on the tangent bundle of an almost para-complex manifold

In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with ${\nabla=D + S}In this paper we study para-tt ^*-bundles (TM, D, S) on the tangent bundle of an almost para-complex manifold (M, τ). We characterise those para-tt ^*-bundles with induced by the one-parameter family of connections given by and prove a uniqueness result for solutions with a para-complex connection D. Flat nearly para-K?hler manifolds and special para-complex manifolds are shown to be such solutions. We analyse which of these solutions admit metric or symplectic para-tt ^*-bundles. Moreover, we give a generalisation of the notion of a para-pluriharmonic map to maps from almost para-complex manifolds (M, τ) into pseudo-Riemannian manifolds and associate to the above metric and symplectic para-tt ^*-bundles generalised para-pluriharmonic maps into , respectively, into SO ₀(n,n)/U ^π(C ⁿ ), where U ^π(C ⁿ ) is the para-complex analogue of the unitary group.      

Isoperimetric inequalities,isometric actions and the higher Newman numbers

M will be a compact connected n-dimensional Riemannian manifold. If M contains a closed connected k-dimensional, 2 k < n, minimal immersed submanifold of M, we define the kth isoperimetric number of M, Ñ _k (M), as the infimum of the volumes of all such submanifolds. We obtain a number of interesting estimates for Ñ _k (M), for both general and special manifolds, which appear to be new.Next we turn to isometric actions and a 1931 theorem of M. H. A. Newman involving the size of orbits of group actions on manifolds. We introduce the higher Newman numbers N _k (M), 1 k n. Roughly speaking, if M admits isometric actions of compact connected Lie groups with k-dimensional principal orbits, N _k (M) is defined as the infimum over all such actions of the maximum volume of all maximal dimensional orbits. We observe that N _k (M) Ñ _k (M), 2 k < n, provided N _k (M) is defined; hence our prior estimates for the isoperimetric numbers of M apply directly to the higher Newman numbers.As a best possible candidate we conjecture that N _k (M) vol S ^k (i(M)/), 1 k n, where i(M) denotes the radius of injectivity of M and S ^k (i(M)/) denotes the standard k-sphere of radius i(M)/. We verify the conjecture for various special cases. We conclude the paper by studying Newman's theorem for compact connected Lie groups with invariant metrics and obtaining a lower bound for the size of small subgroups.     

On multilinear operators commuting with Lie derivatives

LetE ₁, ...,E _k andE be natural vector bundles defined over the categoryMf _m ⁺ of smooth orientedm-dimensional manifolds and orientation preserving local diffeomorphisms, withm2. LetM be an object ofMf _m ⁺ which is connected. We give a complete classification of all separately continuousk-linear operatorsD : _c(E ₁ M) × ... × _c(E _k M) (EM) defined on sections with compact supports, which commute whith Lie derivatives, i.e., which satisfy