Bjørn Ian Dundas

The local structure of algebraic K-theory

Algebraic K-theory.- Gamma-spaces and S-algebras.- Reductions.- Topological Hochschild Homology.- The Trace K --> THH.- Topological Cyclic Homology.- The Comparison of K-theory and TC.

Topology, Geometry and Quantum Field Theory: Two-vector bundles and forms of elliptic cohomology

In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology.

Topological Hochschild homology of ring functors and exact categories

Abstract In analogy with Hochschild-Mitchell homology for linear categories topological Hochschild and cyclic homology (THH and TC) are defined for ring functors on a category β. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring functor on an exact category is treated separately, and is compared with algebraic K-theory via a Dennis-Bokstedt trace map. Calling THH and TC applied to these ring functors simply THH( ) and TC( ), we get that the iteration of Waldhausens S construction yields spectra {THH(S(n) )} and {TC(S(n) )}, and the maps from K-theory become maps of spectra. If is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH0(S(n) ) ⊆ THH(S(n) ) is a stable equivalence, and it is shown how this leads to a weak resolution theorem for THH. If ℘A is the category of finitely generated projective modules over a unital and associative ring A, we get that THH(A) THH(℘A) and TC(A) TC(℘A).

Stable bundles over rig categories

The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku), the algebraic K-theory of topological K-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology. The main technical step is showing that for well-behaved small rig categories R (also known as bimonoidal categories), the algebraic K-theory space, K(HR), of the ring spectrum HR associated to R is equivalent to K(R) � Z ×| BGL(R)| + ,w here GL(R) is the monoidal category of weakly invertible matrices over R. The title refers to the sharper result that BGL(R) is equivalent to BGL(HR). If π0R is a ring, this is almost formal, and our approach is to replace R by ar ing completed version,¯ R, provided

Ring completion of rig categories

Abstract We offer a solution to the long-standing problem of group completing within the context of rig categories (also known as bimonoidal categories). Given a rig category ℛ we construct a natural additive group completion ℛ̅ that retains the multiplicative structure, hence has become a ring category. If we start with a commutative rig category ℛ (also known as a symmetric bimonoidal category), the additive group completion ℛ̅ will be a commutative ring category. In an accompanying paper we show how to use this construction to prove the conjecture that the algebraic K-theory of the connective topological K-theory ring spectrum ku is equivalent to the algebraic K-theory of the rig category 𝒱 of complex vector spaces.

THE CYCLOTOMIC TRACE FOR S -ALGEBRAS

A functorial and categorical defined cyclotomic trace is given, extending the usual one for rings to ring spectra. There are two ingredients to this: first a cyclotomic trace is needed that accepts a categorical input with few restrictive assumptions. This is important in its own right, since this allows one to transport rich structures through the cyclotomic trace. Secondly, a sufficiently nice model is needed for the category of finitely generated free modules which is functorial in the ring spectrum.

K-theory theorems in topological cyclic homology

Abstract Topological cyclic homology serves as an approximation to algebraic K-theory. It is more accessible to calculations, but how well does it reflect the structural properties of K-theory? In addition to confinality, resolution and finite products which have been previously discussed for topological Hochschild homology, this paper addresses localization and Devissage. Its main result is a Devissage theorem and a “vanishing of nil-terms” result for topological Hochschild homology and topological cyclic homology.

CONTINUITY OF K-THEORY : AN EXAMPLE IN EQUAL CHARACTERISTICS

If k is a perfect field of characteristic p > 0, we show that the Quillen K-groups Ki(k[[t]]) are uniquely p-divisible for i = 2, 3. In fact, the Milnor K-groups KM n (k((t))) are uniquely p-divisible for all n > 1. This implies that K(A)→ holim←− nK(A/m) is 4-connected after profinite completion for A a complete discrete valuation ring with perfect residue field. Let A be a complete discrete valuation ring with maximal ideal m. Let K(A) = holim ←− n K(A/m) We say that K-theory is continuous (at A) if it commutes with the (inverse) limit, in the sense that K(A)̂→ Ktop(A)̂ is an equivalence, where X → X̂denotes profinite completion. This question of continuity has acquired new relevance since the fibers of K(A/mn)̂→ K(A/m)̂ are now better understood, and have been shown by McCarthy [Mc] to agree with the corresponding fibers in topological cyclic homology. Hence we are in a position where we sometimes can calculate K(A). One situation where we have an affirmative answer is the theorem of Suslin and Panin [Su], [P], which says that if A is a Henselian discrete valuation ring with maximal ideal m, then K(A)̂̀ → holim ←− n K(A/mn)̂̀ is an equivalence for all primes ` different from the characteristic of (the field of fractions of) A. So, if A is of characteristic zero, then K-theory is continuous at A. This theorem was used critically in Bökstedt and Madsen’s calculation [BM] of the K-theory of the p-adic integers in order to get the correspondence with topological cyclic homology (here the situation was a bit special, as a similar statement holds for TC). Received by the editors October 17, 1996. 1991 Mathematics Subject Classification. Primary 11S70; Secondary 13J05, 19D45, 19D50.

Cubical and cosimplicial descent

We prove that algebraic K-theory, topological Hochschild homology and topological cyclic homology satisfy cubical and cosimplicial descent at connective structured ring spectra along 1-connected maps of such ring spectra.

A Short Course in Differential Topology

Manifolds are abound in mathematics and physics, and increasingly in cybernetics and visualization where they often reflect properties of complex systems and their configurations. Differential topology gives us the tools to study these spaces and extract information about the underlying systems. This book offers a concise and modern introduction to the core topics of differential topology for advanced undergraduates and beginning graduate students. It covers the basics on smooth manifolds and their tangent spaces before moving on to regular values and transversality, smooth flows and differential equations on manifolds, and the theory of vector bundles and locally trivial fibrations. The final chapter gives examples of local-to-global properties, a short introduction to Morse theory and a proof of Ehresmanns fibration theorem. The treatment is hands-on, including many concrete examples and exercises woven into the text, with hints provided to guide the student.