Michael J. Hopkins

Generalized group characters and complex oriented cohomology theories

Let BG be the classifying space of a finite group G. Given a multiplicative cohomology theory E ⁄ , the assignment G 7i! E ⁄ (BG) is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories E ⁄ , using the theory of complex representations of finite groups as a model for what one would like to know. An analogue of Artins Theorem is proved for all complex oriented E ⁄ : the abelian subgroups of G serve as a detecting family for E ⁄ (BG), modulo torsion dividing the order of G. When E ⁄ is a complete local ring, with residue field of characteristic p and associated formal group of height n, we construct a character ring of class functions that computes 1 E ⁄ (BG). The domain of the characters is Gn,p, the set of n-tuples of elements in G each of which has order a power of p. A formula for induction is also found. The ideas we use are related to the Lubin Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, E ⁄ n- theory, etc. The nth Morava K-theory Euler characteristic for BG is computed to be the number of G-orbits in Gn.p. For various groups G, including all symmetric groups, we prove that K(n)⁄(BG) concentrated in even degrees. Our results about E⁄(BG) extend to theorems about E⁄(EG◊GX), where X is a finite G-CW complex.

Loop groups and twisted K-theory I

This is the first in a series of papers investigating the relationship between the twisted equivariant K-theory of a compact Lie group G and the “Verlinde ring” of its loop group. In this paper we set up the foundations of twisted equivariant K-groups, and more generally twisted K-theory of groupoids. We establish enough basic properties to make effective computations. We determine the twisted equivariant K-groups of a compact connected Lie group G with torsion free fundamental group. We relate this computation to the representation theory of the loop group at a level related to the twisting.

On Ramond-Ramond fields and K-theory

A recent paper by Moore and Witten [1] explained that Ramond-Ramond fields in type-II superstring theory have a global meaning in K-theory. In this note we amplify and generalize some points raised in that paper. In particular, we express the coupling of the Ramond-Ramond fields to D-branes in a K-theoretic framework and show that the anomaly in this coupling exactly cancels the anomaly from the fermions on the brane, both in type IIA and type IIB.

Homotopy fixed point spectra for closed subgroups of the Morava stabilizer groups

Abstract Let G be a closed subgroup of the semi-direct product of the nth Morava stabilizer group Sn with the Galois group of the field extension F p n / F p . We construct a “homotopy fixed point spectrum” EnhG whose homotopy fixed point spectral sequence involves the continuous cohomology of G. These spectra have the expected functorial properties and agree with the Hopkins-Miller fixed point spectra when G is finite.

Twisted equivariant K-theory with complex coefficients

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact group acting on itself by conjugation and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlindes formula is also discussed in this context.

Topological Modular Forms, the Witten Genus, and the Theorem of the Cube

There is a rick mathematical structure attached to the cobordism invariants of manifolds. In the cases described by the index theorem, a generalized cohomology theory is used to express the global properties of locally defined analytic objects. Hirzebruch’s theory of multiplicative sequences gives an expression for these invariants in terms of characteristic classes, and brings to focus their remarkable arithmetic properties. Quillen’s theory of formal groups and complex oriented cohomology theories illuminates the generalized cohomology theories themselves.

An ∞-categorical approach to R-line bundles, R-module Thom spectra, and twisted R-homology

We develop a generalization of the theory of Thom spectra using the language of infinity categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parametrized spectra, and our definition is motivated by the geometric definition of Thom spectra of May-Sigurdsson. For an associative ring spectrum

Loop groups and twisted

R

K

, we associate a Thom spectrum to a map of infinity categories from the infinity groupoid of a space

-theory II

X