# Parametrized homotopy theory

@inproceedings{May2006ParametrizedHT, title={Parametrized homotopy theory}, author={Jon P. May and J. Sigurdsson}, year={2006} }

Prologue Point-set topology, change functors, and proper actions: Introduction to Part I The point-set topology of parametrized spaces Change functors and compatibility relations Proper actions, equivariant bundles and fibrations Model categories and parametrized spaces: Introduction to Part II Topologically bicomplete model categories Well-grounded topological model categories The $qf$-model structure on $\mathcal{K}_B$ Equivariant $qf$-type model structures Ex-fibrations and ex…

## 241 Citations

### Twisted differential generalized cohomology theories and their Atiyah–Hirzebruch spectral sequence

- MathematicsAlgebraic & Geometric Topology
- 2019

We construct the Atiyah-Hirzebruch spectral sequence (AHSS) for twisted differential generalized cohomology theories. This generalizes to the twisted setting the authors' corresponding earlier…

### Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map

- MathematicsGeometry & Topology
- 2018

We introduce a general theory of parametrized objects in the setting of infinity categories. Although spaces and spectra parametrized over spaces are the most familiar examples, we establish our…

### MULTIPLICATIVE PARAMETRIZED HOMOTOPY THEORY VIA SYMMETRIC SPECTRA IN RETRACTIVE SPACES

- MathematicsForum of Mathematics, Sigma
- 2020

In order to treat multiplicative phenomena in twisted (co)homology, we introduce a new point-set-level framework for parametrized homotopy theory. We provide a convolution smash product that descends…

### Twisted spin cobordism and positive scalar curvature

- MathematicsJournal of Topology
- 2020

We show how a suitably twisted spin cobordism spectrum connects to the question of existence of metrics of positive scalar curvature on closed, smooth manifolds by building on fundamental work of…

### The transfer map of free loop spaces

- MathematicsTransactions of the American Mathematical Society
- 2018

For any perfect fibration $E \rightarrow B$, there is a "free loop transfer map" $LB_+ \rightarrow LE_+$, defined using topological Hochschild homology. We prove that this transfer is compatible with…

### Equivariant Ordinary Homology and Cohomology

- Mathematics
- 2017

Poincare duality lies at the heart of the homological theory of manifolds. In the presence of the action of a group it is well-known that Poincare duality fails in Bredon's ordinary, integer-graded…

### Bundles of spectra and algebraic K-theory

- Mathematics
- 2016

A parametrized spectrum E is a family of spectra E_x continuously parametrized by the points x of a topological space X. We take the point of view that a parametrized spectrum is a bundle-theoretic…

### Twists of K-theory and TMF

- Mathematics
- 2010

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…

### Fixed Point Theory and Trace for Bicategories

- Mathematics
- 2010

The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in…

### Units of ring spectra and Thom spectra

- Mathematics
- 2008

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…

## References

SHOWING 1-10 OF 156 REFERENCES

### Fibrewise Homotopy Theory

- Mathematics
- 1998

Part 1: A Survey of Fibrewise Homotopy Theory. Introduction to Fibrewise Homotopy Theory. The Pointed Theory. Applications Part 2: An Introduction to Fibrewise Stable Homotopy Theory. Foundations.…

### Axiomatic stable homotopy theory

- Mathematics
- 1997

Introduction and definitions Smallness, limits and constructibility Bousfield localization Brown representability Nilpotence and thick subcategories Noetherian stable homotopy categories Connective…

### Strong cofibrations and fibrations in enriched categories

- Mathematics
- 2002

We define strong cofibrations and fibrations in suitably enriched categories using the relative homotopy extension resp. lifting property. We prove a general pairing result, which for topological…

### Equivariant Orthogonal Spectra and S-Modules

- Mathematics
- 2002

Introduction Orthogonal spectra and $S$-modules Equivariant orthogonal spectra Model categories of orthogonal $G$-spectra Orthogonal $G$-spectra and $S_G$-modules ""Change"" functors for orthogonal…

### Isomorphisms between left and right adjoints

- Mathematics
- 2002

There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being…

### The Additivity of Traces in Triangulated Categories

- Mathematics
- 2001

Abstract We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric…