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Author : Andrew J. Blumberg,Teena Gerhardt,Michael A. Hill
Publisher : Cambridge University Press
Page : 441 pages
File Size : 50,6 Mb
Release : 2022-07-21
Category : Mathematics
ISBN : 9781009123297
A graduate-level introduction to the homotopical technology in use at the forefront of modern algebraic topology.
Author : John Rognes
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 48,9 Mb
Release : 2008
Category : Commutative algebra
ISBN : 9780821840764
The author introduces the notion of a Galois extension of commutative $S$-algebras ($E_\infty$ ring spectra), often localized with respect to a fixed homology theory. There are numerous examples, including some involving Eilenberg-Mac Lane spectra of commutative rings, real and complex topological $K$-theory, Lubin-Tate spectra and cochain $S$-algebras. He establishes the main theorem of Galois theory in this generality. Its proof involves the notions of separable and etale extensions of commutative $S$-algebras, and the Goerss-Hopkins-Miller theory for $E_\infty$ mapping spaces. He shows that the global sphere spectrum $S$ is separably closed, using Minkowski's discriminant theorem, and he estimates the separable closure of its localization with respect to each of the Morava $K$-theories. He also defines Hopf-Galois extensions of commutative $S$-algebras and studies the complex cobordism spectrum $MU$ as a common integral model for all of the local Lubin-Tate Galois extensions. The author extends the duality theory for topological groups from the classical theory for compact Lie groups, via the topological study by J. R. Klein and the $p$-complete study for $p$-compact groups by T. Bauer, to a general duality theory for stably dualizable groups in the $E$-local stable homotopy category, for any spectrum $E$.
Author : Andrew Baker,Birgit Richter
Publisher : Cambridge University Press
Page : 246 pages
File Size : 40,8 Mb
Release : 2004-11-18
Category : Mathematics
ISBN : 0521603056
This book contains some important new contributions to the theory of structured ring spectra.
Author : Anthony D. Elmendorf
Publisher : American Mathematical Soc.
Page : 265 pages
File Size : 41,5 Mb
Release : 1997
Category : Groups of points
ISBN : 9780821843031
This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $S$, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of ``$S$-modules'' whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\wedge SR \longrightarrow R$. These notions are essentially equivalent to the earlier notions of $A {\infty $ and $E {\infty $ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\wedge SM\longrightarrow M$. When $R$ is commutative, the category of $R$-modules also has a
Author : M. A. Mandell,J. Peter May
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 43,5 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821829363
The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of $S$-modules and the category of symmetric spectra. We focus on the category of orthogonal spectra, which enjoys some of the best features of $S$-modules and symmetric spectra and which is particularly well-suited to equivariant generalization. We first complete the nonequivariant theory by comparing orthogonal spectra to $S$-modules. We then develop the equivariant theory.For a compact Lie group $G$, we construct a symmetric monoidal model category of orthogonal $G$-spectra whose homotopy category is equivalent to the classical stable homotopy category of $G$-spectra. We also complete the theory of $S_G$-modules and compare the categories of orthogonal $G$-spectra and $S_G$-modules. A key feature is the analysis of change of universe, change of group, fixed point, and orbit functors in these two highly structured categories for the study of equivariant stable homotopy theory.
Author : Akira Kōno,Dai Tamaki
Publisher : American Mathematical Soc.
Page : 276 pages
File Size : 44,8 Mb
Release : 2006
Category : Mathematics
ISBN : 0821835149
Aims to give an exposition of generalized (co)homology theories that can be read by a group of mathematicians who are not experts in algebraic topology. This title starts with basic notions of homotopy theory, and introduces the axioms of generalized (co)homology theory. It also discusses various types of generalized cohomology theories.
Author : Stefan Schwede
Publisher : Cambridge University Press
Page : 847 pages
File Size : 52,8 Mb
Release : 2018-09-06
Category : Mathematics
ISBN : 9781108425810
A comprehensive, self-contained approach to global equivariant homotopy theory, with many detailed examples and sample calculations.
Author : David Barnes,Constanze Roitzheim
Publisher : Cambridge University Press
Page : 432 pages
File Size : 53,7 Mb
Release : 2020-03-26
Category : Mathematics
ISBN : 9781108672672
The beginning graduate student in homotopy theory is confronted with a vast literature on spectra that is scattered across books, articles and decades. There is much folklore but very few easy entry points. This comprehensive introduction to stable homotopy theory changes that. It presents the foundations of the subject together in one place for the first time, from the motivating phenomena to the modern theory, at a level suitable for those with only a first course in algebraic topology. Starting from stable homotopy groups and (co)homology theories, the authors study the most important categories of spectra and the stable homotopy category, before moving on to computational aspects and more advanced topics such as monoidal structures, localisations and chromatic homotopy theory. The appendix containing essential facts on model categories, the numerous examples and the suggestions for further reading make this a friendly introduction to an often daunting subject.
Author : J. Peter May,Johann Sigurdsson
Publisher : American Mathematical Soc.
Page : 456 pages
File Size : 52,8 Mb
Release : 2006
Category : Homotopy equivalences
ISBN : 9780821839225
This book develops rigorous foundations for parametrized homotopy theory, which is the algebraic topology of spaces and spectra that are continuously parametrized by the points of a base space. It also begins the systematic study of parametrized homology and cohomology theories. The parametrized world provides the natural home for many classical notions and results, such as orientation theory, the Thom isomorphism, Atiyah and Poincare duality, transfer maps, the Adams and Wirthmuller isomorphisms, and the Serre and Eilenberg-Moore spectral sequences. But in addition to providing a clearer conceptual outlook on these classical notions, it also provides powerful methods to study new phenomena, such as twisted $K$-theory, and to make new constructions, such as iterated Thom spectra. Duality theory in the parametrized setting is particularly illuminating and comes in two flavors. One allows the construction and analysis of transfer maps, and a quite different one relates parametrized homology to parametrized cohomology. The latter is based formally on a new theory of duality in symmetric bicategories that is of considerable independent interest. The text brings together many recent developments in homotopy theory. It provides a highly structured theory of parametrized spectra, and it extends parametrized homotopy theory to the equivariant setting. The theory of topological model categories is given a more thorough treatment than is available in the literature. This is used, together with an interesting blend of classical methods, to resolve basic foundational problems that have no nonparametrized counterparts.
Author : Robert R. Bruner,J. Peter May,James E. McClure,Mark Steinberger
Publisher : Springer
Page : 396 pages
File Size : 46,7 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540397786
Author : Michael A. Hill,Michael J. Hopkins,Douglas C. Ravenel
Publisher : Cambridge University Press
Page : 881 pages
File Size : 48,9 Mb
Release : 2021-07-29
Category : Mathematics
ISBN : 9781108831444
A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.
Author : John Rognes
Publisher : American Mathematical Society(RI)
Page : 137 pages
File Size : 41,9 Mb
Release : 2014-09-11
Category : Commutative algebra
ISBN : 1470405040
Galois Extensions of Structured Ring Spectra: Abstract Introduction Galois extensions in algebra Closed categories of structured module spectra Galois extensions in topology Examples of Galois extensions Dualizability and alternate characterizations Galois theory I Pro-Galois extensions and the Amitsur complex Separable and etale extensions Mapping spaces of commutative $S$-algebras Galois theory II Hopf-Galois extensions in topology References Stably Dualizable Groups: Abstract Introduction The dualizing spectrum Duality theory Computations Norm and transfer maps References Index.
Author : J. Peter May,M. Cole
Publisher : American Mathematical Soc.
Page : 384 pages
File Size : 53,7 Mb
Release : 1996
Category : Mathematics
ISBN : 9780821803196
This volume introduces equivariant homotopy, homology, and cohomology theory, along with various related topics in modern algebraic topology. It explains the main ideas behind some of the most striking recent advances in the subject. The works begins with a development of the equivariant algebraic topology of spaces culminating in a discussion of the Sullivan conjecture that emphasizes its relationship with classical Smith theory. The book then introduces equivariant stable homotopy theory, the equivariant stable homotopy category, and the most important examples of equivariant cohomology theories. The basic machinery that is needed to make serious use of equivariant stable homotopy theory is presented next, along with discussions of the Segal conjecture and generalized Tate cohomology. Finally, the book gives an introduction to "brave new algebra", the study of point-set level algebraic structures on spectra and its equivariant applications. Emphasis is placed on equivariant complex cobordism, and related results on that topic are presented in detail.
Author : Eric Peterson
Publisher : Cambridge University Press
Page : 421 pages
File Size : 50,7 Mb
Release : 2019
Category : Mathematics
ISBN : 9781108428033
Delivers a broad, conceptual introduction to chromatic homotopy theory, focusing on contact with arithmetic and algebraic geometry.
Author : Bertrand Toen,Bertrand Toën,Gabriele Vezzosi
Publisher : American Mathematical Soc.
Page : 242 pages
File Size : 53,7 Mb
Release : 2008
Category : Algebra, Homological
ISBN : 9780821840993
This is the second part of a series of papers called "HAG", devoted to developing the foundations of homotopical algebraic geometry. The authors start by defining and studying generalizations of standard notions of linear algebra in an abstract monoidal model category, such as derivations, étale and smooth morphisms, flat and projective modules, etc. They then use their theory of stacks over model categories to define a general notion of geometric stack over a base symmetric monoidal model category $C$, and prove that this notion satisfies the expected properties.
