Thomas G. Goodwillie

Calculus III: Taylor Series

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors, meaning n-excisive functors with trivial (n 1)-excisive part, can be classied: they correspond to symmetric functors of n variables that are reduced and 1-excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic K -theory.

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.

On the general linear group and Hochschild homology

Our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring. Before stating the result we establish some notation and conventions. Rings are associative and with unit. If A is a ring then GL(A)= Uk?0GLk(A) is its infinite general linear group. An A-bimodule is an abelian group B which is both a left A-module and a right A-module and satisfies (alb)a2 = al(ba2) for ai eA, be B (for example B = A). If B is an A-bimodule, then M(B) = Uk?oMk(B) is the infinite additive group of matrices with entries in B. Conjugation defines an action (the adjoint action) of GL(A) on M(B). Note that an A ? Q-bimodule is just an A-bimodule which is also a rational vector space. If B is an A ? Q-bimodule, then HJ(A C) Q; B) denotes the Hochschild homology of A ? Q with coefficients in B. Our main result (it appears in slightly more detailed form as Theorem V.3) is:

A Haefliger style description of the embedding calculus tower

Abstract Let M and N be smooth manifolds. The calculus of embeddings produces, for every k⩾1, a best degree ⩽k polynomial approximation to the cofunctor taking an open V⊂M to the space of embeddings from V to N. In this paper, a description of these polynomial approximations in terms of equivariant mapping spaces is given, for k⩾2. The description is new only for k⩾3. In the case k=2 we recover Haefligers approximation and the known result that it is the best degree ⩽2 approximation.

Multiple disjunction for spaces of smooth embeddings

We obtain multirelative connectivity statements about spaces of smooth embeddings, deducing these from analogous results about spaces of Poincare embeddings that were established in our previous paper.

A remark on the homology of cosimplicial spaces

Abstract The (second-quadrant) mod p homology spectral sequence determined by a cosimplicial space always converges to zero in negative dimensions. More precisely, for every s and t with t − s r such that E − s , t r = E − s , t ∞ = 0.

An equivariant version of Hatcher's G/O construction

We explicitly construct generators of the rational homotopy groups of the space of stable h-cobordisms of the classifying space of a cyclic group of order n by generalizing a construction of Hatcher. This result will be used in a separate paper by the third author to classify axiomatic higher twisted torsion invariants.

Memories of Raoul Bott

In 1973–74 I had the good fortune to be introduced to topology by Raoul Bott. Let me try to convey something of what that was like and how the experience has stayed with me over the last forty years. Twice a week we would take our seats in the classroom, and there he would be: standing tall in front of us with a piece of chalk in his hand and a bit of a twinkle in his eye. He had a commanding presence, but never a formidable one. Above all it was a presence: he was so much with us! His love of the subject was unmistakable. Not that he talked about the power and the glory and the beauty of mathematics: he simply talked about mathematics, and it was beautiful and glorious and powerful.

Topological Hochschild Homology

Topological Hochschild homology is introduced along with its natural extension to enriched model categories. Several basic properties such as Morita invariance and invariance under weak equivalences of enriched categories are demonstrated.

Topological Cyclic Homology

This chapter introduces the (integral) topological cyclic homology via a pullback diagram involving invariants of topological Hochschild homology. A lift of the trace to topological cyclic homology is given. Considerations of what occurs when completing at a prime is given special attention, as are connections to more algebraic counterparts like negative cyclic homology.