Coassembly is a homotopy limit map

We prove a claim by Williams that the coassembly map is a homotopy limit map. As an application, we show that the homotopy limit map for the coarse version of equivariant $A$-theory agrees with the coassembly map for bivariant $A$-theory that appears in the statement of the topological Riemann-Roch theorem.Comment: Accepted version. Several improvements from the referee, including a more elegant proof of Lemma 3.

Equivariant infinite loop space theory, I. The space level story

We rework and generalize equivariant infinite loop space theory, which shows how to construct G-spectra from G-spaces with suitable structure. There is a naive version which gives naive G-spectra for any topological group G, but our focus is on the construction of genuine G-spectra when G is finite. We give new information about the Segal and operadic equivariant infinite loop space machines, supplying many details that are missing from the literature, and we prove by direct comparison that the two machines give equivalent output when fed equivalent input. The proof of the corresponding nonequivariant uniqueness theorem, due to May and Thomason, works for naive G-spectra for general G but fails hopelessly for genuine G-spectra when G is finite. Even in the nonequivariant case, our comparison theorem is considerably more precise, giving a direct point-set level comparison. We have taken the opportunity to update this general area, equivariant and nonequivariant, giving many new proofs, filling in some gaps, and giving some corrections to results in the literature.Comment: 94 page

Categorical Models for Equivariant Classifying Spaces

Starting categorically, we give simple and precise models for classifying spaces of equivariant principal bundles. We need these models for work in progress in equi- variant infinite loop space theory and equivariant algebraic K–theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory

On the functoriality of the space of equivariant smooth hh-cobordisms

We construct an $(\infty,1)$-functor that takes each smooth $G$-manifold with corners $M$ to the space of equivariant smooth $h$-cobordisms $H_{\mathrm{Diff}}(M)$. We also give a stable analogue $H^U_{\mathrm{Diff}}(M)$ where the manifolds are stabilized with respect to representation discs. The functor structure is subtle to construct, and relies on several new ideas. In particular, for $G=e$, we get an $(\infty,1)$-functor structure on the smooth $h$-cobordism space $H_{\mathrm{Diff}}(M)$. This agrees with previous constructions as a functor to the homotopy category.Comment: 65 pages. Comments welcome

Unbased calculus for functors to chain complexes

Recently, the Johnson-McCarthy discrete calculus for homotopy functors was extended to include functors from an unbased simplicial model category to spectra. This paper completes the constructions needed to ensure that there exists a discrete calculus tower for functors from an unbased simplicial model category to chain complexes over a fixed commutative ring. Much of the construction of the Taylor tower for functors to spectra carries over to this context. However, one of the essential steps in the construction requires proving that a particular functor is part of a cotriple. For this, one needs to prove that certain identities involving homotopy limits hold up to isomorphism, rather than just up to weak equivalence. As the target category of chain complexes is not a simplicial model category, the arguments for functors to spectra need to be adjusted for chain complexes. In this paper, we take advantage of the fact that we can construct an explicit model for iterated fibers, and prove that the functor is a cotriple directly. We use related ideas to provide concrete infinite deloopings of the first terms in the resulting Taylor towers when evaluated at the initial object in the source category.Comment: 20 page

Categorical models for equivariant classifying spaces

Starting categorically, we give simple and precise models of equivariant classifying spaces. We need these models for work in progress in equivariant infinite loop space theory and equivariant algebraic K-theory, but the models are of independent interest in equivariant bundle theory and especially equivariant covering space theory.Comment: 29 pages. Revised version, to appear in AGT. Considerable changes of notation and organization and other changes aimed at making the paper more user friendl