Two-vector bundles and forms of elliptic cohomology (original) (raw)

Two-vector bundles define a form of elliptic cohomology, available at arxiv:math.KT/0706.0531

2012

Abstract. We prove that for well-behaved small commutative rig categories (aka. symmetric bimoidal categories) R the algebraic K-theory space of the K-theory spectrum, HR, of R is equivalent to K0(π0R) × |BGL(R) | + where GL(R) is the monoidal category of weakly invertible matrices over R. In particular, this proves the conjecture from [BDR] that K(ku) is the K-theory of the 2-category of complex 2-vector spaces. Hence, the work of the fourth author and Christian Ausoni on K(ku) [A, AR] shows that the theory of virtual 2-vector bundles as in [BDR, Theorem 4.10] qualifies as a form of elliptic cohomology theory.

Two-Vector Bundles Define a Form of Elliptic Cohomology

We prove that for well-behaved small rig categories R (also known as bimonoidal categories) the algebraic K-theory space, K(HR), of the K-theory ring spectrum of R is equivalent to K(R) ≃ Z × |BGL(R)|+, where GL(R) is the monoidal category of weakly invertible matrices over R. To achieve this, we solve the long-standing problem of group completing within the context of rig categories. More precisely, we construct an additive group completion ¯ R of R that retains the multiplicative structure, i.e., that remains a rig category. In particular, this proves the conjecture from (BDR) that K(ku) is the K-theory of the 2-category of complex 2-vector spaces. Hence, the work of Christian Ausoni and the fourth author on K(ku) (AR, A) shows that the theory of virtual 2-vector bundles as in (BDR, Theorem 4.10) qualifies as a form of elliptic cohomology. In telescopic complexity 0, 1 and ∞ there are cohomology theories that possess a geometric definition: de Rham cohomology of manifolds is given...

An algebraic description of the elliptic cohomology of classifying spaces

Journal of Pure and Applied Algebra, 1998

Let G be a finite group of order (GI odd and let 6Y~*(-)~?@[l/lGl] denote elliptic cohomology tensored by Z[l/lGl]. Then we give a description of &Y*(E(N,G) x ,vX) @ Z[l/lGl], where N is a normal subgroup of G, E(N, G) is the universal N-free G space and X is any finite G-CW complex where N acts freely. We explain how some of the results of Hopkins-Kuhn-Ravenel can be recovered for our results.

Differential K-Theory: A Survey

Springer Proceedings in Mathematics, 2011

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics.

Structured vector bundles define differential K-theory

2008

A equivalence relation, preserving the Chern-Weil form, is defined between connections on a complex vector bundle. Bundles equipped with such an equivalence class are called Structured Bundles, and their isomorphism classes form an abelian semi-ring. By applying the Grothedieck construction one obtains the ringK, elements of which, modulo a complex torus of dimension the sum of the odd Betti numbers of the base, are uniquely determined by the corresponding element of ordinary K and the Chern-Weil form. This construction provides a simple model of differential K-theory, c.f. Hopkins-Singer (2005), as well as a useful codification of vector bundles with connection.