21 research outputs found
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 -theory agrees with the coassembly map for bivariant
-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 -cobordisms
We construct an -functor that takes each smooth -manifold with
corners to the space of equivariant smooth -cobordisms
. We also give a stable analogue
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 , we get an -functor structure on the smooth
-cobordism space . 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