## Derived Categories: A Quick Tour

*D. Huybrechts*

### in Fourier-Mukai Transforms in Algebraic Geometry

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter briefly outlines the main steps in the construction of the derived category of an arbitrary abelian category. The homotopy category of complexes is considered as an intermediate step, ... More

## Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.001.0001
- Subject:
- Mathematics, Geometry / Topology

Since its introduction by the author in the 1970s, the algebraic K-theory of spaces has been recognized as the main tool for studying parametrized phenomena in the theory of manifolds. However, a ... More

## On simple maps

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

### in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with simple maps of finite simplicial sets, along with some of their formal properties. It begins with a discussion of simple maps of simplicial sets, presenting a proposition for ... More

## Links as Sasakian Manifolds

*Charles P. Boyer and Krzysztof Galicki*

### in Sasakian Geometry

- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198564959
- eISBN:
- 9780191713712
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198564959.003.0010
- Subject:
- Mathematics, Geometry / Topology

This chapter is devoted to the geometry of links of isolated hypersurface singularities, as well as a review of the differential topology of homotopy spheres a la Kervaire and Milnor. The ... More

## Liquid Crystals and Harmonic Maps in Polyhedral Domains

*Apala Majumdar, Jonathan Robbins, and Maxim Zyskin*

### in Analysis and Stochastics of Growth Processes and Interface Models

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0014
- Subject:
- Mathematics, Probability / Statistics, Analysis

This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet ... More

## SURGERY ON MAPS

*Andrew Ranicki*

### in Algebraic and Geometric Surgery

- Published in print:
- 2002
- Published Online:
- September 2007
- ISBN:
- 9780198509240
- eISBN:
- 9780191708725
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509240.003.0010
- Subject:
- Mathematics, Geometry / Topology

This chapter gives the definition of a normal map. Killing elements in the relative homotopy group by surgery on a normal map is discussed. The kernel modulesand relative regular homotopy groups of ... More

## Introduction

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

### in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0001
- Subject:
- Mathematics, Geometry / Topology

This book presents a proof of the stable parametrized h-cobordism theorem, which deals with the existence of a natural homotopy equivalence for each compact CAT manifold. In this theorem, a stable ... More

##
The stable parametrized *h*-cobordism theorem

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

### in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0002
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with the stable parametrized h-cobordism theorem. It begins with a discussion of the manifold part; here DIFF is written for the category of Csuperscript infinity smooth manifolds, ... More

## The non-manifold part

*Friedhelm Waldhausen, Bjørn Jahren, and John Rognes*

### in Spaces of PL Manifolds and Categories of Simple Maps (AM-186)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157757
- eISBN:
- 9781400846528
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157757.003.0004
- Subject:
- Mathematics, Geometry / Topology

Abstract and Keywords to be supplied.

## An equivalence of categories

*Ehud Hrushovski and François Loeser*

### in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691161686
- eISBN:
- 9781400881222
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161686.003.0013
- Subject:
- Mathematics, Geometry / Topology

This chapter deduces from Theorem 11.1.1 an equivalence of categories between a certain homotopy category of definable subsets of quasi-projective varieties over a given valued field and a suitable ... More

## Modal Homotopy Type Theory: The Prospect of a New Logic for Philosophy

*David Corfield*

- Published in print:
- 2020
- Published Online:
- March 2020
- ISBN:
- 9780198853404
- eISBN:
- 9780191888069
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853404.001.0001
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

In[KF1] 1914, in an essay entitled ‘Logic as the Essence of Philosophy’, Bertrand Russell promised to revolutionize philosophy by introducing there the ‘new logic’ of Frege and Peano: “The old logic ... More

## Topological Classification Of Defects

*VOLOVIK GRIGORY E.*

### in The Universe in a Helium Droplet

- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199564842
- eISBN:
- 9780191709906
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564842.003.0013
- Subject:
- Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

The effective metric and effective gauge fields are simulated in superfluids by the inhomogeneity of the superfluid vacuum. In superfluids, many inhomogeneous configurations of the vacuum are stable ... More

## Vortices In 3He-B

*VOLOVIK GRIGORY E.*

### in The Universe in a Helium Droplet

- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199564842
- eISBN:
- 9780191709906
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564842.003.0014
- Subject:
- Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

This chapter discusses the topology of singular topological defects — defects with singular core of coherence length size — in 3He-B and in quantum chromodynamics, such as conventional mass vortices, ... More

## Continuous Structures

*VOLOVIK GRIGORY E.*

### in The Universe in a Helium Droplet

- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199564842
- eISBN:
- 9780191709906
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564842.003.0016
- Subject:
- Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

When several distinct energy scales are involved, the vacuum symmetry is different for different length scales: the larger the length scale, the more the symmetry is reduced. The interplay of ... More

## Monopoles and Boojums

*VOLOVIK GRIGORY E.*

### in The Universe in a Helium Droplet

- Published in print:
- 2009
- Published Online:
- January 2010
- ISBN:
- 9780199564842
- eISBN:
- 9780191709906
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199564842.003.0017
- Subject:
- Physics, Condensed Matter Physics / Materials, Particle Physics / Astrophysics / Cosmology

This chapter is devoted to point defects such as hedgehogs, Dirac magnetic monopoles, 't Hooft–Polyakov monopole, and nexus. Some of these objects represent composite defects resulting from a ... More

## Differential geometry, topology and fibre bundles

*Reinhold A. Bertlmann*

### in Anomalies in Quantum Field Theory

- Published in print:
- 2000
- Published Online:
- February 2010
- ISBN:
- 9780198507628
- eISBN:
- 9780191706400
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507628.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces all necessary mathematical concepts. Section 2.1 briefly summarizes some topological definitions. Section 2.2 explains the homotopy of maps and the homotopy of groups. Section ... More

## Chern–Simons form, homotopy operator and anomaly

*Reinhold A. Bertlmann*

### in Anomalies in Quantum Field Theory

- Published in print:
- 2000
- Published Online:
- February 2010
- ISBN:
- 9780198507628
- eISBN:
- 9780191706400
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507628.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The Chern–Simons form and the homotopy operator plays an important role in connection with anomalies. In fact, the anomaly can be calculated on pure algebraic grounds from a variation of the ... More

## Topological Methods for Description of Quantum Many-Body Systems

*Lucjan Jacak, Piotr Sitko, Konrad Wieczorek, and Arkadiusz Wojs*

### in Quantum Hall systems: Braid groups, composite fermions and fractional charge

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198528708
- eISBN:
- 9780191713477
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528708.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

This chapter presents a topological description of quantum many-body systems within the Feynman path integration formalism. Configuration spaces for the system of two particles on the one-dimensional ... More

## Cohomology of Homotopy 2-types

*Graham Ellis*

### in An Invitation to Computational Homotopy

- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198832973
- eISBN:
- 9780191871375
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198832973.003.0005
- Subject:
- Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter introduces some of the basic ingredients in the classification of homotopy 2-types and describes datatypes and algorithms for implementing them on a computer. These are illustrated using ... More

## The main theorem

*Ehud Hrushovski and François Loeser*

### in Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691161686
- eISBN:
- 9781400881222
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691161686.003.0011
- Subject:
- Mathematics, Geometry / Topology

This chapter introduces the main theorem, which states: Let V be a quasi-projective variety over a valued field F and let X be a definable subset of V x Γsuperscript Script Small l subscript ... More