Author: Jonathan Hillman
Publisher:
ISBN: 9781935107118
Category : Mathematics
Languages : en
Pages : 0
Book Description
Poincaré duality is central to the understanding of manifold topology. Dimension 3 is critical in various respects, being between the known territory of surfaces and the wilderness manifest in dimensions >= 4. The main thrust of 3-manifold topology for the past half century has been to show that aspherical closed 3-manifolds are determined by their fundamental groups. Relatively little attention has been given to the question of which groups arise. This book is the first comprehensive account of what is known about PD3-complexes, which model the homotopy types of closed 3-manifolds, and PD3-groups, which correspond to aspherical 3-manifolds. In the first half we show that every P2-irreducible PD3-complex is a connected sum of indecomposables, which are either aspherical or have virtually free fundamental group, and largely determine the latter class. The picture is much less complete in the aspherical case. We sketch several possible approaches for tackling the central question, whether every PD3-group is a 3-manifold group, and then explore properties of subgroups of PD3-groups, unifying many results of 3-manifold topology. We conclude with an appendix listing over 60 questions. Our general approach is to prove most assertions which are specifically about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results. The new edition adds new sections in chapters 2, 3, 5 and 7, and a new chapter on pairs with compressible boundary. The author has also improved the exposition in numerous minor ways.
Poincaré Duality in Dimension 3, 2nd Edition
Author: Jonathan Hillman
Publisher:
ISBN: 9781935107118
Category : Mathematics
Languages : en
Pages : 0
Book Description
Poincaré duality is central to the understanding of manifold topology. Dimension 3 is critical in various respects, being between the known territory of surfaces and the wilderness manifest in dimensions >= 4. The main thrust of 3-manifold topology for the past half century has been to show that aspherical closed 3-manifolds are determined by their fundamental groups. Relatively little attention has been given to the question of which groups arise. This book is the first comprehensive account of what is known about PD3-complexes, which model the homotopy types of closed 3-manifolds, and PD3-groups, which correspond to aspherical 3-manifolds. In the first half we show that every P2-irreducible PD3-complex is a connected sum of indecomposables, which are either aspherical or have virtually free fundamental group, and largely determine the latter class. The picture is much less complete in the aspherical case. We sketch several possible approaches for tackling the central question, whether every PD3-group is a 3-manifold group, and then explore properties of subgroups of PD3-groups, unifying many results of 3-manifold topology. We conclude with an appendix listing over 60 questions. Our general approach is to prove most assertions which are specifically about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results. The new edition adds new sections in chapters 2, 3, 5 and 7, and a new chapter on pairs with compressible boundary. The author has also improved the exposition in numerous minor ways.
Publisher:
ISBN: 9781935107118
Category : Mathematics
Languages : en
Pages : 0
Book Description
Poincaré duality is central to the understanding of manifold topology. Dimension 3 is critical in various respects, being between the known territory of surfaces and the wilderness manifest in dimensions >= 4. The main thrust of 3-manifold topology for the past half century has been to show that aspherical closed 3-manifolds are determined by their fundamental groups. Relatively little attention has been given to the question of which groups arise. This book is the first comprehensive account of what is known about PD3-complexes, which model the homotopy types of closed 3-manifolds, and PD3-groups, which correspond to aspherical 3-manifolds. In the first half we show that every P2-irreducible PD3-complex is a connected sum of indecomposables, which are either aspherical or have virtually free fundamental group, and largely determine the latter class. The picture is much less complete in the aspherical case. We sketch several possible approaches for tackling the central question, whether every PD3-group is a 3-manifold group, and then explore properties of subgroups of PD3-groups, unifying many results of 3-manifold topology. We conclude with an appendix listing over 60 questions. Our general approach is to prove most assertions which are specifically about Poincaré duality in dimension 3, but otherwise to cite standard references for the major supporting results. The new edition adds new sections in chapters 2, 3, 5 and 7, and a new chapter on pairs with compressible boundary. The author has also improved the exposition in numerous minor ways.
Poincaré Duality Pairs of Dimension Three
Author: Beatrice Bleile
Publisher:
ISBN:
Category : Duality theory (Mathematics)
Languages : en
Pages : 136
Book Description
Publisher:
ISBN:
Category : Duality theory (Mathematics)
Languages : en
Pages : 136
Book Description
On the Decomposition of 3-dimensional Poincaré Duality Complexes
Author: J. Crisp
Publisher:
ISBN:
Category :
Languages : en
Pages : 12
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 12
Book Description
Fields Medallists' Lectures, 2nd Edition
Author: Michael Atiyah
Publisher: World Scientific
ISBN: 9814486809
Category : Mathematics
Languages : en
Pages : 824
Book Description
Although the Fields Medal does not have the same public recognition as the Nobel Prizes, they share a similar intellectual standing. It is restricted to one field — that of mathematics — and an age limit of 40 has become an accepted tradition. Mathematics has in the main been interpreted as pure mathematics, and this is not so unreasonable since major contributions in some applied areas can be (and have been) recognized with Nobel Prizes.A list of Fields Medallists and their contributions provides a bird's-eye view of mathematics over the past 60 years. It highlights the areas in which, at various times, greatest progress has been made. This volume does not pretend to be comprehensive, nor is it a historical document. On the other hand, it presents contributions from Fields Medallists and so provides a highly interesting and varied picture.The second edition of Fields Medallists' Lectures features additional contributions from the following Medallists: Kunihiko Kodaira (1954), Richard E Borcherds (1998), William T Gowers (1998), Maxim Kontsevich (1998), Curtis T McMullen (1998) and Vladimir Voevodsky (2002).
Publisher: World Scientific
ISBN: 9814486809
Category : Mathematics
Languages : en
Pages : 824
Book Description
Although the Fields Medal does not have the same public recognition as the Nobel Prizes, they share a similar intellectual standing. It is restricted to one field — that of mathematics — and an age limit of 40 has become an accepted tradition. Mathematics has in the main been interpreted as pure mathematics, and this is not so unreasonable since major contributions in some applied areas can be (and have been) recognized with Nobel Prizes.A list of Fields Medallists and their contributions provides a bird's-eye view of mathematics over the past 60 years. It highlights the areas in which, at various times, greatest progress has been made. This volume does not pretend to be comprehensive, nor is it a historical document. On the other hand, it presents contributions from Fields Medallists and so provides a highly interesting and varied picture.The second edition of Fields Medallists' Lectures features additional contributions from the following Medallists: Kunihiko Kodaira (1954), Richard E Borcherds (1998), William T Gowers (1998), Maxim Kontsevich (1998), Curtis T McMullen (1998) and Vladimir Voevodsky (2002).
Some Questions on Subgroups of 3-dimensional Poincaré Duality Groups
Author: Jonathan A. Hillman
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 15
Book Description
Finite Dimensional Algebras and Quantum Groups
Author: Bangming Deng
Publisher: American Mathematical Soc.
ISBN: 0821841866
Category : Mathematics
Languages : en
Pages : 790
Book Description
"The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak{gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature."--Publisher's website.
Publisher: American Mathematical Soc.
ISBN: 0821841866
Category : Mathematics
Languages : en
Pages : 790
Book Description
"The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak{gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature."--Publisher's website.
Groups '93 Galway/St Andrews: Volume 1
Author: C. M. Campbell
Publisher: Cambridge University Press
ISBN: 0521477492
Category : Mathematics
Languages : en
Pages : 320
Book Description
Representing the wealth and diversity of group theory for experienced researchers as well as new postgraduates, this two-volume book contains selected papers from the international conference which was held at University College Galway in August 1993.
Publisher: Cambridge University Press
ISBN: 0521477492
Category : Mathematics
Languages : en
Pages : 320
Book Description
Representing the wealth and diversity of group theory for experienced researchers as well as new postgraduates, this two-volume book contains selected papers from the international conference which was held at University College Galway in August 1993.
Cohomology of Groups
Author: Kenneth S. Brown
Publisher: Springer Science & Business Media
ISBN: 1468493272
Category : Mathematics
Languages : en
Pages : 318
Book Description
Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.
Publisher: Springer Science & Business Media
ISBN: 1468493272
Category : Mathematics
Languages : en
Pages : 318
Book Description
Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.
Lectures on the Topology of 3-manifolds
Author: Nikolai Saveliev
Publisher: Walter de Gruyter
ISBN: 9783110162721
Category : Mathematics
Languages : en
Pages : 220
Book Description
Publisher: Walter de Gruyter
ISBN: 9783110162721
Category : Mathematics
Languages : en
Pages : 220
Book Description
Equivariant Poincaré Duality on G-Manifolds
Author: Alberto Arabia
Publisher: Springer Nature
ISBN: 3030704408
Category : Mathematics
Languages : en
Pages : 383
Book Description
This book carefully presents a unified treatment of equivariant Poincaré duality in a wide variety of contexts, illuminating an area of mathematics that is often glossed over elsewhere. The approach used here allows the parallel treatment of both equivariant and nonequivariant cases. It also makes it possible to replace the usual field of coefficients for cohomology, the field of real numbers, with any field of arbitrary characteristic, and hence change (equivariant) de Rham cohomology to the usual singular (equivariant) cohomology . The book will be of interest to graduate students and researchers wanting to learn about the equivariant extension of tools familiar from non-equivariant differential geometry.
Publisher: Springer Nature
ISBN: 3030704408
Category : Mathematics
Languages : en
Pages : 383
Book Description
This book carefully presents a unified treatment of equivariant Poincaré duality in a wide variety of contexts, illuminating an area of mathematics that is often glossed over elsewhere. The approach used here allows the parallel treatment of both equivariant and nonequivariant cases. It also makes it possible to replace the usual field of coefficients for cohomology, the field of real numbers, with any field of arbitrary characteristic, and hence change (equivariant) de Rham cohomology to the usual singular (equivariant) cohomology . The book will be of interest to graduate students and researchers wanting to learn about the equivariant extension of tools familiar from non-equivariant differential geometry.