arXiv subjects

Andrew J. Blumberg

Publications and source records attributed to Andrew J. Blumberg.

62 records · Page 4Linked to original sources

arXiv2010

Derived Koszul Duality and Involutions in the Algebraic K-Theory of Spaces

We interpret different constructions of the algebraic $K$-theory of spaces as an instance of derived Koszul (or bar) duality and also as an instance of Morita equivalence. We relate the interplay between these two descriptions to the homotopy involution. We define a geometric analog of the Swan theory $G^{\bZ}(\bZ[π])$ in terms of $Σ^{\infty}_{+} ΩX$ and show that it is the algebraic $K$-theory of the $E_{\infty}$ ring spectrum $DX=S^{X_{+}}$.

math.KT↗

arXiv2010

Algebraic K-theory and abstract homotopy theory

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra.

math.KT↗

arXiv2010

Twists of K-theory and TMF

We explore an approach to twisted generalized cohomology from the point of view of stable homotopy theory and quasicategory theory provided by arXiv:0810.4535. We explain the relationship to the twisted K-theory provided by Fredholm bundles. We show how our approach allows us to twist elliptic cohomology by degree four classes, and more generally by maps to the four-stage Postnikov system BO<0...4>. We also discuss Poincare duality and umkehr maps in this setting.

math.AT↗

arXiv2009

Units of ring spectra and Thom spectra

We review and extend the theory of Thom spectra and the associated obstruction theory for orientations. We recall (from May, Quinn, and Ray) that a commutative ring spectrum A has a spectrum of units gl(A). To a map of spectra f: b -> bgl(A), we associate a commutative A-algebra Thom spectrum Mf, which admits a commutative A-algebra map to R if and only if b -> bgl(A) -> bgl(R) is null. If A is an associative ring spectrum, then to a map of spaces f: B -> BGL(A) we associate an A-module Thom spectrum Mf, which admits an R-orientation if and only if B -> BGL(A) -> BGL(R) is null. We also note that BGL(A) classifies the twists of A-theory. We develop and compare two approaches to the theory of Thom spectra. The first involves a rigidified model of A-infinity and E-infinity spaces. Our second approach is via infinity categories. In order to compare these approaches to one another and to the classical theory, we characterize the Thom spectrum functor from the perspective of Morita theory.

math.AT↗

arXiv2009

Open-closed field theories, string topology, and Hochschild homology

In this expository paper we discuss a project regarding the string topology of a manifold, that was inspired by recent work of Moore-Segal, Costello, and Hopkins and Lurie, on "open-closed topological conformal field theories". Given a closed, oriented manifold M, we describe the "string topology category" S_M, which is enriched over chain complexes over a fixed field k. The objects of S_M are connected, closed, oriented submanifolds N of M, and the complex of morphisms between N_1 and N_2 is a chain complex homotopy equivalent to the singular chains C_*(P_{N_1, N_2}), where C_*(P_{N_1, N_2}) is the space of paths in M that start in N_1 and end in N_2. The composition pairing in this category is a chain model for the open string topology operations of Sullivan and expanded upon by Harrelson, and Ramirez. We will describe a calculation yielding that the Hochschild homology of the category S_M is the homology of the free loop space, LM. Another part of the project is to calculate the Hochschild cohomology of the open string topology chain algebras C_*(P_{N,N}) when M is simply connected, and relate the resulting calculation to H_*(LM). We also discuss a spectrum level analogue of the above results and calculations, as well as their relations to various Fukaya categories of the cotangent bundle T^*M.

math.AT↗

arXiv2008

THH of Thom spectra that are E_\infty ring spectra

We identify the topological Hochschild homology (THH) of the Thom spectrum associated to an E_\infty classifying map X -> BG, for G an appropriate group or monoid (e.g. U, O, and F). We deduce the comparison from the observation of McClure, Schwanzl, and Vogt that THH of a cofibrant commutative S-algebra (E_\infty ring spectrum) R can be described as an indexed colimit together with a verification that the Lewis-May operadic Thom spectrum functor preserves indexed colimits. We prove a splitting result THH(Mf) \htp Mf \sma BX_+ which yields a convenient description of THH(MU). This splitting holds even when the classifying map f: X -> BG is only a homotopy commutative A_\infty map, provided that the induced multiplication on Mf extends to an E_\infty ring structure; this permits us to recover Bokstedt's calculation of THH(HZ).

math.AT↗

arXiv2007

The localization sequence for the algebraic K-theory of topological K-theory

We prove a conjecture of Rognes by establishing a localization cofiber sequence of spectra, K(Z) to K(ku) to K(KU) to Sigma K(Z), for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a devissage theorem identifying the K-theory of the Waldhausen category of Postnikov towers of modules over a connective A-infinity ring spectrum R with the Quillen K-theory of the abelian category of finitely generated pi_0(R)-modules.

math.KT↗

arXiv2004

A discrete model of $S^1$-homotopy theory

We construct a discrete model of the homotopy theory of $S^1$-spaces. We define a category $\sP$ with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. $\sP$ inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between $\sP$ and the model category of $S^1$-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets.

math.AT↗