D. Huybrechts
- 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, ...
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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
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.
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 ...
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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 full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.Less
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 full proof of the equivalence relating the two areas has not appeared until now. This book presents such a proof, essentially completing the author's program from more than thirty years ago. The main result is a stable parametrized h-cobordism theorem, derived from a homotopy equivalence between a space of PL h-cobordisms on a space X and the classifying space of a category of simple maps of spaces having X as deformation retract. The smooth and topological results then follow by smoothing and triangulation theory. The proof has two main parts. The essence of the first part is a “desingularization,” improving arbitrary finite simplicial sets to polyhedra. The second part compares polyhedra with PL manifolds by a thickening procedure. Many of the techniques and results developed should be useful in other connections.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- 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 ...
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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
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.
Charles P. Boyer and Krzysztof Galicki
- 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 ...
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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
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.
Apala Majumdar, Jonathan Robbins, and Maxim Zyskin
- 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 ...
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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
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.
Stanley Chang
- Published in print:
- 2019
- Published Online:
- May 2021
- ISBN:
- 9780691160498
- eISBN:
- 9780691200354
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160498.001.0001
- Subject:
- Mathematics, Geometry / Topology
Surgery theory, a subfield of geometric topology, is the study of the classifications of manifolds. This book offers a modern look at this important mathematical discipline and some of its ...
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Surgery theory, a subfield of geometric topology, is the study of the classifications of manifolds. This book offers a modern look at this important mathematical discipline and some of its applications. The book explains some of the triumphs of surgery theory during the past three decades, from both an algebraic and geometric point of view. It also provides an extensive treatment of basic ideas, main theorems, active applications, and recent literature. The authors methodically cover all aspects of surgery theory, connecting it to other relevant areas of mathematics, including geometry, homotopy theory, analysis, and algebra. Later chapters are self-contained, so readers can study them directly based on topic interest. Of significant use to high-dimensional topologists and researchers in noncommutative geometry and algebraic K-theory, the book serves as an important resource for the mathematics community.Less
Surgery theory, a subfield of geometric topology, is the study of the classifications of manifolds. This book offers a modern look at this important mathematical discipline and some of its applications. The book explains some of the triumphs of surgery theory during the past three decades, from both an algebraic and geometric point of view. It also provides an extensive treatment of basic ideas, main theorems, active applications, and recent literature. The authors methodically cover all aspects of surgery theory, connecting it to other relevant areas of mathematics, including geometry, homotopy theory, analysis, and algebra. Later chapters are self-contained, so readers can study them directly based on topic interest. Of significant use to high-dimensional topologists and researchers in noncommutative geometry and algebraic K-theory, the book serves as an important resource for the mathematics community.
Andrew Ranicki
- 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 ...
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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
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.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- 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 ...
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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
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.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- 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, ...
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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
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.
Friedhelm Waldhausen, Bjørn Jahren, and John Rognes
- 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.
Abstract and Keywords to be supplied.
Loring W. Tu
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0007
- Subject:
- Mathematics, Educational Mathematics
This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by ...
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This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by rotation. The method of the chapter only gives the module structure of equivariant cohomology. Suppose a topological group G acts on the left on a topological space M. Let EG → BG be a universal G-bundle. The homotopy quotient MG fits into Cartan's mixing diagram. One can then apply Leray's spectral sequence of the fiber bundle MG → BG to compute the equivariant cohomology from the cohomology of M and the cohomology of the classifying space BG.Less
This chapter shows how to use the spectral sequence of a fiber bundle to compute equivariant cohomology. As an example, it computes the equivariant cohomology of S2 under the action of S1 by rotation. The method of the chapter only gives the module structure of equivariant cohomology. Suppose a topological group G acts on the left on a topological space M. Let EG → BG be a universal G-bundle. The homotopy quotient MG fits into Cartan's mixing diagram. One can then apply Leray's spectral sequence of the fiber bundle MG → BG to compute the equivariant cohomology from the cohomology of M and the cohomology of the classifying space BG.
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 ...
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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 put thought in fetters, while the new logic gives it wings.” A century later, this book proposes a comparable revolution with a newly emerging logic, modal homotopy type theory. Russell’s prediction turned out to be accurate. Frege’s first-order logic, along with its extension to modal logic, is to be found throughout anglophone analytic philosophy. This book provides a considerable array of evidence for the claim that philosophers working in metaphysics, as well as those treating language, logic or mathematics, would be much better served with the new ‘new logic’. It offers an introduction to this new logic, thoroughly motivated by intuitive explanations of the need for all of its component parts—the discipline of a type theory, the flexibility of type dependency, the more refined homotopic notion of identity and a powerful range of modalities. Innovative applications of the calculus are given, including analysis of the distinction between objects and events, an intrinsic treatment of structure and a conception of modality both as a form of general variation and as allowing constructions in modern geometry. In this way, we see how varied are the applications of this powerful new language—modal homotopy type theory.Less
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 put thought in fetters, while the new logic gives it wings.” A century later, this book proposes a comparable revolution with a newly emerging logic, modal homotopy type theory. Russell’s prediction turned out to be accurate. Frege’s first-order logic, along with its extension to modal logic, is to be found throughout anglophone analytic philosophy. This book provides a considerable array of evidence for the claim that philosophers working in metaphysics, as well as those treating language, logic or mathematics, would be much better served with the new ‘new logic’. It offers an introduction to this new logic, thoroughly motivated by intuitive explanations of the need for all of its component parts—the discipline of a type theory, the flexibility of type dependency, the more refined homotopic notion of identity and a powerful range of modalities. Innovative applications of the calculus are given, including analysis of the distinction between objects and events, an intrinsic treatment of structure and a conception of modality both as a form of general variation and as allowing constructions in modern geometry. In this way, we see how varied are the applications of this powerful new language—modal homotopy type theory.
Ehud Hrushovski and François Loeser
- 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 ...
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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
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.
Loring W. Tu
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0009
- Subject:
- Mathematics, Educational Mathematics
This chapter assesses the general properties of equivariant cohomology. Both the homotopy quotient and equivariant cohomology are functorial constructions. Equivariant cohomology is particularly ...
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This chapter assesses the general properties of equivariant cohomology. Both the homotopy quotient and equivariant cohomology are functorial constructions. Equivariant cohomology is particularly simple when the action is free. Throughout the chapter, by a G-space, it means a left G-space. Let G be a topological group and consider the category of G-spaces and G-maps. A morphism of left G-spaces is a G-equivariant map (or G-map). Such a morphism induces a map of homotopy quotients. The map in turn induces a ring homomorphism in cohomology. The chapter then looks at the coefficient ring of equivariant cohomology, as well as the equivariant cohomology of a disjoint union.Less
This chapter assesses the general properties of equivariant cohomology. Both the homotopy quotient and equivariant cohomology are functorial constructions. Equivariant cohomology is particularly simple when the action is free. Throughout the chapter, by a G-space, it means a left G-space. Let G be a topological group and consider the category of G-spaces and G-maps. A morphism of left G-spaces is a G-equivariant map (or G-map). Such a morphism induces a map of homotopy quotients. The map in turn induces a ring homomorphism in cohomology. The chapter then looks at the coefficient ring of equivariant cohomology, as well as the equivariant cohomology of a disjoint union.
Loring W. Tu
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0018
- Subject:
- Mathematics, Educational Mathematics
This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a ...
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This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.Less
This chapter investigates differential graded algebras. Throughout the chapter, G will be a Lie group with Lie algebra g. On a manifold M, the de Rham complex is a differential graded algebra, a graded algebra that is also a differential complex. If the Lie group G acts smoothly on M, then the de Rham complex Ω(M) is more than a differential graded algebra. It has in addition two actions of the Lie algebra: interior multiplication and the Lie derivative. A differential graded algebra Ω with an interior multiplication and a Lie derivative satisfying Cartan's homotopy formula is called a g-differential graded algebra. To construct an algebraic model for equivariant cohomology, the chapter first constructs an algebraic model for the total space EG of the universal G-bundle. It is a g-differential graded algebra called the Weil algebra.
VOLOVIK GRIGORY E.
- 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 ...
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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
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).
VOLOVIK GRIGORY E.
- 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, ...
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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
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.
VOLOVIK GRIGORY E.
- 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 ...
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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
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.
VOLOVIK GRIGORY E.
- 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 ...
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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
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.
Reinhold A. Bertlmann
- 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 ...
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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
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.