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Derived Categories: A Quick Tour

D. Huybrechts

in Fourier-Mukai Transforms in Algebraic Geometry

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

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, which is then localized with respect to quasi-isomorphisms. Left and right derived functors are explained in general, and particular examples are studied in more detail. Spectral sequences are treated in a separate section.Less

On simple maps

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

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

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

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 the conditions that qualify a map of finite simplicial sets as a simple map. In particular, it considers a simple map as a weak homotopy equivalence. Weak homotopy equivalences have the 2-out-of-3 property, which combines the composition, right cancellation and left cancellation properties. The chapter proceeds by defining some relevant terms, such as Euclidean neighborhood retract, absolute neighborhood retract, Čech homotopy type, and degeneracy operator. It also describes normal subdivision of simplicial sets, geometric realization and subdivision, the reduced mapping cylinder, how to make simplicial sets non-singular, and the approximate lifting property.Less

Links as Sasakian Manifolds

Charles P. Boyer and Krzysztof Galicki

in Sasakian Geometry

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

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 differential topology of links is a beautiful piece of mathematics, and the chapter offers a hands-on ‘user's guide’ approach with much emphasis on the famous work of Brieskorn in determining the difieomorphism types of certain homotopy spheres. This includes a presentation of the well known Brieskorn graph theorem as well as the geometry of Brieskorn-Pham links. When the singularities arise from weighted homogeneous polynomials, the links have a natural Sasakian structure with either definite (positive or negative) or null basic first Chern class. Emphasis is given to the positive case which corresponds to having positive Ricci curvature.Less

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

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

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 energy of homo-topy classes of such harmonic maps, subject to tangent boundary conditions, and investigate lower and upper bounds for this Dirichlet energy on each homotopy class; local minimisers of this energy correspond to equilibrium and metastable configurations. A lower bound for the infimum Dirichlet energy for a given homotopy class is obtained as a sum of minimal connections between fractional defects at the vertices. In certain cases, this lower bound can be improved. For a rectangular prism, upper bounds are obtained from locally conformal solutions of the Euler-Lagrange equations, with the ratio of the upper and lower bounds bounded independently of homotopy type.Less

SURGERY ON MAPS

Andrew Ranicki

in Algebraic and Geometric Surgery

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

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 immersions is provided.Less

Introduction

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

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

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

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 CAT h-cobordism space is defined in terms of manifolds, whereas a CAT Whitehead space is defined in terms of algebraic K-theory. This is a stable range extension to parametrized families of the classical hand s-cobordism theorems first stated by A. E. Hatcher, but his proofs were incomplete. This book provides a full proof of this key result, which provides the link between the geometric topology of high-dimensional manifolds and their automorphisms, as well as the algebraic K-theory of spaces and structured ring spectra.Less

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)

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

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, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. CAT is generically written for any one of these geometric categories. Relevant terms such as stabilization map, simple map, pullback map, PL Serre fibrations, weak homotopy equivalence, PL Whitehead space, and cofibration are also defined. The chapter proceeds by describing the non-manifold part, the algebraic K-theory of spaces, and the relevance of simple maps to the study of PL homeomorphisms of manifolds.Less

The non-manifold part

Friedhelm Waldhausen, Bjørn Jahren, and John Rognes

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

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)

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

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 homotopy category of definable spaces over the o-minimal Γ‎. The chapter introduces three categories that can be viewed as ind-pro definable and admit natural functors to the category TOP of topological spaces with continuous maps. The discussion is often limited to the subcategory consisting of A-definable objects and morphisms. The morphisms are factored out by (strong) homotopy equivalence. The chapter presents the proof of the equivalence of categories before concluding with remarks on homotopies over imaginary base sets.Less

Topological Classification Of Defects

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

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

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 and thus can be experimentally investigated in detail, since they are protected by r-space topology. In particular, the effect of the chiral anomaly has been verified using such topologically stable objects as vortex-skyrmions in 3He-A and quantized vortices in 3He-B. Other topological objects can produce non-trivial effective metrics. In addition, many topological defects have almost direct analogs in some relativistic quantum field theory. Topological defects are results of spontaneously broken symmetry. This chapter discusses the spontaneous symmetry breaking both in 3He-A and 3He-B, which is responsible for topologically stable objects in these phases, and analogous ‘superfluid’ phases in high-energy physics, such as chiral and color superfluidity in quantum chromodynamics (QCD).Less

Vortices In 3He-B

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

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

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, spin vortices, axion and pion strings, hedgehogs, monopoles, and vortons. The Casimir force between spin and mass vortices in 3He-B leads to formation of a composite defect — spin-mass vortex, which is stabilized in rotating cryostat. Spin-mass vortex serves as string terminating topological soliton. Topological confinement of two spin-mass vortices by soliton leads to another composite object observed in 3He-B — doubly quantized vortex. The chapter also discusses the symmetry of defects, the interplay of symmetry and topology, the symmetry of hedgehogs and monopoles, spherically symmetric objects in superfluids, enhanced superfluidity in the core of hedgehog, spontaneously broken parity, and axial symmetry in the core of 3He-B vortices. Vortex with spontaneously broken parity in the core represents analog of Witten superconducting cosmic string. The observed twist of the core corresponds to supercurrent along the Witten string.Less

Continuous Structures

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

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

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 topologies on different length scales gives rise to many different types of topological objects, which are described by relative homotopy groups. This chapter discusses the continuous structures generated by relative homotopy groups, such as soliton terminating on a half-quantum vortex, skyrmion — the doubly quantized vortex in 3He-A, meron — the fraction of skyrmion, continuous structures in spinor Bose condensate and superconductors, semilocal strings in the Standard Model of particle physics, and the vortex sheet. The vortex sheet is the chain of alternating circular and hyperbolic merons concentrated inside the topological soliton in 3He-A and the chain of kinks in the domain wall in chiral superconductors. The chapter also discusses topological transitions between continuous textures, which are mediated by singular topological defects. For example, destruction of topological soliton in 3He-A occurs via creation of the loop of half-quantum vortex.Less

Monopoles and Boojums

VOLOVIK GRIGORY E.

in The Universe in a Helium Droplet

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

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 hierarchy of energy scales with different symmetries. Examples are the hedgehog-monopole, which serves as a termination point of vortex-string, and nexus which represents the magnetic pole with emanating flux. In chiral superconductors the magnetic flux to the nexus can be supplied by four half-quantum vortices. Due to topological confinement chiral superconductor represents the natural trap for 't Hooft–Polyakov monopole. The chapter also discusses topology of boojums — point and line defects living at surfaces of the ordered system or at the interface between different vacua. Examples are provided by defects at the interface between 3He-A and 3He-B, including Alice string. In many cases these defects represent composite objects. For example, boojum on the A-phase side of the interface is the termination point of the vortex living on the B-phase side. Vortex sheet is discussed which is formed at the interface between 3He-A and 3He-B in rotating cryostat. It separates the vortex lattice in 3He-A, which experiences the solid body rotation, and the vortex free 3He-B.Less

Differential geometry, topology and fibre bundles

Reinhold A. Bertlmann

in Anomalies in Quantum Field Theory

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

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 2.3 introduces the concept of differentiable manifolds while Section 2.4 presents the differential forms together with their Hodge duals, along with the differentiation and integration. Section 2.5 discusses homology and de Rham cohomology. Section 2.6 explains important concepts such as pullback of a differential form the Lie derivative, the Lie group, and the Lie algebra. Finally, Section 2.7 constructs fibre bundles including connection and curvature, which turn out to be a suitable mathematical concept to describe the physics of gauge theories.Less

Chern–Simons form, homotopy operator and anomaly

Reinhold A. Bertlmann

in Anomalies in Quantum Field Theory

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

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 Chern–Simons form using a homotopy operator. This chapter begins with a discussion of a symmetric invariant polynomial of fields, which is the starting point for deriving the Chern–Simons form and ‘transgression formula’. It then proves the important Poincaré lemma and introduces in this connection a homotopy operator. A generalization of the ‘transgression’ — the Cartan homotopy formula — follows. The homotopy formula is applied to a Chern–Simons form with gauge transformed fields, and the non-Abelian anomaly is derived in this way. Finally, the chapter presents a general formula for the variation of the Chern–Simons form, which expresses the anomaly.Less

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

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

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 manifolds (line and circle) and two- and three-dimensional Euclidean spaces are described. Braid group — the first homotopy group of the configuration space of the quantum system — is defined. Generators and defining relations of the full and pure braid groups for some manifolds (sphere, torus, Euclidean plane, and three-dimensional Euclidean space) and their geometrical illustrations are presented. The generic deep difference between exchanges of particles positions in two-dimensional and three-dimensional spaces is indicated.Less

Cohomology of Homotopy 2-types

Graham Ellis

in An Invitation to Computational Homotopy

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

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 computer examples involving: the fundamental crossed modules of a CW-complex, cat-1-groups, simplicial groups, Moore complexes, the Dold-Kan correspondence, integral homology of simplicial groups, homological perturbation theory. A manual classification of homotopy classes of maps from a surface to the projective plane is also included.Less

The main theorem

Ehud Hrushovski and François Loeser

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

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

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 infinity over some base set V ⊂ VF ∪ Γ‎, with F = VF(A). Then there exists an A-definable deformation retraction h : I × unit vector X → unit vector X with image an iso-definable subset definably homeomorphic to a definable subset of Γ‎superscript w subscript Infinity, for some finite A-definable set w. The chapter presents several preliminary reductions to essentially reduce to a curve fibration. It then constructs a relative curve homotopy and a liftable base homotopy, along with a purely combinatorial homotopy in the Γ‎-world. It also constructs the homotopy retraction by concatenating the previous three homotopies together with an inflation homotopy. Finally, it describes a uniform version of the main theorem with respect to parameters.Less