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Author : Andrew J. Blumberg
Publisher : Cambridge University Press
Page : 441 pages
File Size : 25,86 MB
Release : 2022-07-21
Category : Mathematics
ISBN : 1009123297
A graduate-level introduction to the homotopical technology in use at the forefront of modern algebraic topology.
Author : Andrew Baker
Publisher : Cambridge University Press
Page : 246 pages
File Size : 43,7 MB
Release : 2004-11-18
Category : Mathematics
ISBN : 9780521603058
This book contains some important new contributions to the theory of structured ring spectra.
Author : John Rognes
Publisher : American Mathematical Soc.
Page : 154 pages
File Size : 15,9 MB
Release : 2008
Category : Mathematics
ISBN : 0821840762
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 : Anthony D. Elmendorf
Publisher : American Mathematical Soc.
Page : 265 pages
File Size : 24,39 MB
Release : 1997
Category : Mathematics
ISBN : 0821843036
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 : David Barnes
Publisher : Cambridge University Press
Page : 432 pages
File Size : 45,36 MB
Release : 2020-03-26
Category : Mathematics
ISBN : 1108672671
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 : Stefan Schwede
Publisher : Cambridge University Press
Page : 847 pages
File Size : 49,18 MB
Release : 2018-09-06
Category : Mathematics
ISBN : 110842581X
A comprehensive, self-contained approach to global equivariant homotopy theory, with many detailed examples and sample calculations.
Author : Akira Kōno
Publisher : American Mathematical Soc.
Page : 276 pages
File Size : 21,16 MB
Release : 2006
Category : Mathematics
ISBN : 9780821835142
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 : Robert R. Bruner
Publisher : Springer
Page : 396 pages
File Size : 45,16 MB
Release : 2006-11-14
Category : Mathematics
ISBN : 3540397787
Author : M. A. Mandell
Publisher : American Mathematical Soc.
Page : 132 pages
File Size : 31,34 MB
Release : 2002-08-19
Category : Mathematics
ISBN : 9780821864777
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 : Michael A. Hill
Publisher : Cambridge University Press
Page : 881 pages
File Size : 17,70 MB
Release : 2021-07-29
Category : Mathematics
ISBN : 1108831443
A complete and definitive account of the authors' resolution of the Kervaire invariant problem in stable homotopy theory.