Symmetric K-Theory Spectra of C*-Categories

We define K-theory groups and symmetric K-theory spectra associated to ₂-graded C ^*-categories and show that the exterior product of K-theory groups can be expressed in terms of the smash product of symmetric spectra.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

Rational isomorphisms between K-theories and cohomology theories

The well known isomorphism relating the rational algebraic K-theory groups and the rational motivic cohomology groups of a smooth variety over a field of characteristic 0 is shown to be realized by a map (the Segre map) of infinite loop spaces. Moreover, the associated Chern character map on rational homotopy groups is shown to be a ring isomorphism. A technique is introduced that establishes a useful general criterion for a natural transformation of functors on quasi-projective complex varieties to induce a homotopy equivalence of semi-topological singular complexes. Since semi-topological K-theory and morphic cohomology can be formulated as the semi-topological singular complexes associated to algebraic K-theory and motivic cohomology, this criterion provides a rational isomorphism between the semi-topological K-theory groups and the morphic cohomology groups of a smooth complex variety. Consequences include a Riemann-Roch theorem for the Chern character on semi-topological K-theory and an interpretation of the topological filtration on singular cohomology groups in K-theoretic terms.     

Techniques,computations, and conjectures for semi-topological <Emphasis Type="Italic">K</Emphasis>-theory

We establish the existence of an Atiyah-Hirzebruch-like spectral sequence relating the morphic cohomology groups of a smooth, quasi-projective complex variety to its semi-topological K-groups. This spectral sequence is compatible with (and, indeed, is built from) the motivic spectral sequence that relates the motivic cohomology and algebraic K-theory of varieties, and it is also compatible with the classical Atiyah-Hirzebruch spectral sequence in algebraic topology. In the second part of this paper, we use this spectral sequence in conjunction with another computational tool that we introduce — namely, a variation on the integral weight filtration of the Borel-Moore (singular) homology of complex varieties introduced by H. Gillet and C. Soulé – to compute the semi-topological K-theory of a large class of varieties. In particular, we prove that for curves, surfaces, toric varieties, projective rational three-folds, and related varieties, the semi-topological K-groups and topological K-groups are isomorphic in all degrees permitted by cohomological considerations. We also formulate integral conjectures relating semi-topological K-theory to topological K-theory analogous to more familiar conjectures (namely, the Quillen-Lichtenbaum and Beilinson-Lichtenbaum Conjectures) concerning mod-n algebraic K-theory and motivic cohomology. In particular, we prove a local vanishing result for morphic cohomology which enables us to formulate precisely a conjectural identification of morphic cohomology by A. Suslin. Our computations verify that these conjectures hold for the list of varieties above.Mathematics Subject Classification (2000): 19E20, 19E15, 14F43The first author was partially supported by the NSF and the NSAThe second author was supported by the Helen M. Galvin Fellowship of Northwestern UniversityThe third author was partially supported by the NSF and the NSA     

Elliptic Operators in Subspaces and the Eta Invariant

In this paper we obtain a formula for the fractional part of the -invariant for elliptic self-adjoint operators in topological terms. The computation of the -invariant is based on the index theorem for elliptic operators in subspaces obtained by Savin and Sternin. We also apply the K-theory with coefficients _n . In particular, it is shown that the group K(T ^* M, _n ) is realized by elliptic operators (symbols) acting in appropriate subspaces.     

ApproximatingK*(?) through degree five

We approximate theK-theory spectrum of the integers using a spectrum level rank filtration. By means of a certain poset spectral sequence, we explicitly compute the first three subquotients of this filtration. Assuming a conjecture about the filtration's rate of convergence, we conclude thatK ₄()=0 andK ₅() is a copy of (the Borel summand) plus two-torsion of order at most eight.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

On a Theorem of Grothendieck

A smooth projective morphism p : T S to a smooth variety S is considered. In particular, the following result is proved. The total direct image Rp _*(/n) of the constant étale sheaf /n is locally (in Zariski topology) quasiisomorphic to a bounded complex on S that consists of locally constant, constructible étale sheaves of /n-modules. Bibliography: 2 titles.     

Computations of K- and L-Theory of Cocompact Planar Groups

The verification of the isomorphism conjectures of Baum and Connes and Farrell and Jones for certain classes of groups is used to compute the algebraic K- and L-theory and the topological K-theory of cocompact planar groups (=cocompact N.E.C-groups) and of groups G appearing in an extension where is a finite group and the conjugation -action on ⁿ is free outside . These computations apply, for instance, to two-dimensional crystallographic groups and cocompact Fuchsian groups.     

Controlled algebraic K-theory of integral group ring of SL(3, ?)

We calculate the lower Controlled Algebraic K-theory of any finitely generated infinite subgroup of SL(3,Z), the group of 3×3 integral matrices of determinant 1.     

On finite groups acting on ℤ<Subscript>2</Subscript>-homology 3-spheres

We consider finite groups G admitting orientation-preserving actions on homology 3-spheres (arbitrary, i.e. not necessarily free actions), concentrating on the case of nonsolvable groups. It is known that every finite group G admits actions on rational homology 3-spheres (and even free actions). On the other hand, the class of groups admitting actions on integer homology 3-spheres is very restricted (and close to the class of finite subgroups of the orthogonal group SO(4), acting on the 3-sphere). In the present paper, we consider the intermediate case of ₂-homology 3-spheres (i.e., with the ₂-homology of the 3-sphere where ₂ denote the integers mod two; we note that these occur much more frequently in 3-dimensional topology than the integer ones). Our main result is a list of finite nonsolvable groups G which are the candidates for orientation-preserving actions on ₂-homology 3-spheres. From this we deduce a corresponding list for the case of integer homology 3-spheres. In the integer case, the groups of the list are closely related to the dodecahedral group or the binary dodecahedral group most of these groups are subgroups of the orthogonal group SO(4) and hence admit actions on S³. Roughly, in the case of ₂-homology 3-spheres the groups PSL(2,5) and SL(2,5) get replaced by the groups PSL(2,q) and SL(2,q), for an arbitrary odd prime power q. We have many examples of actions of the groups PSL(2,q) and SL(2,q) on ₂-homology 3-spheres, for various small values of q (constructed as regular coverings of suitable hyperbolic 3-orbifolds and 3-manifolds, using computer-supported methods to calculate the homology of the coverings). We think that all of them occur but have no method to prove this at present (in particular, the exact classification of the finite nonsolvable groups admitting actions on ₂-homology 3-spheres remains still open).     

Rieffel projections in certain transformation group C*-algebras

LetA be the transformation groupC ^*-algebra associated with an arbitrary orientation-preserving homeomorphism of . ThisC ^*-algebra contains an infinite family of projections, called Rieffel projections, each of which generates theK ₀-groupK ₀(A). Although these projections must beK-theoretically equivalent, it is easy to see that most are not Murray-von Neumann equivalent. The mystery of how large the matrix algebra must be to implement theK-theory equivalence, is solved by explicitly constructing the equivalence in the smallest possible algebra:A with unit adjoined.Partially supported by NSF Grant DMS 8901923.     

On Cyclotomic Numbers and the Reduction Map for the K-Theory of the Integers

We apply the recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map K _2n+1() K_{2n+1}(F_p)is nontrivial. Here n is an even, positive integer and F_p denotes the field of p elements. In the proof we use arithmetic of cyclotomic numbers which come from Soulé elements. Divisibility properties of the numbers are related to the Vandiver conjecture on the class group of cyclotomic fields. Using the K-theory of the integers, we compute an upper bound on the divisibility of these cyclotomic numbers.     

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

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.     

Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

Summary We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ^d or ^d × with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ^d or on a slab ^d × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=²×{0,1} or larger.A.G. was partially supported from AFOSR through grant no. 90-0090     

Coniveau of classes of flat bundles trivialized on a finite smooth covering of a complex manifold

On a smooth algebraic complex varietyX, we show that the classes of a flat bundle, which is trivialized on a finite cover ofX, with values in the odd-dimensional cohomology of the underlying complex manifold with / (i), are living in the bottom part of Grothendieck's coniveau filtration. This answers positively when the basis is smooth complex a question of Bruno Kahn [K-Theory (1992), conjecture 2].     

Some Criteria for Multivalently Starlikeness

Let S_n(p)(p, n N) be the class of functions f() = ^p + a_p+n^p+n + which are p-valently starlike in the unit disk. Some sufficient conditions for a function f() to be in the class S_n(p) are given.AMS Subject Classification (2000): primary 30C45