Author: Andrew BakerPublisher: Cambridge University Press
ISBN: 9780521603058
Category : Mathematics
Languages : en
Pages : 246
Book Description
This book contains some important new contributions to the theory of structured ring spectra.
Structured Ring Spectra
Author: Andrew BakerPublisher: Cambridge University Press
ISBN: 9780521603058
Category : Mathematics
Languages : en
Pages : 246
Book Description
This book contains some important new contributions to the theory of structured ring spectra.
Rings, Modules, and Algebras in Stable Homotopy Theory
Author: Anthony D. ElmendorfPublisher: American Mathematical Soc.
ISBN: 0821843036
Category : Mathematics
Languages : en
Pages : 265
Book Description
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
Stable Categories and Structured Ring Spectra
Author: Andrew J. BlumbergPublisher: Cambridge University Press
ISBN: 1009123297
Category : Mathematics
Languages : en
Pages : 441
Book Description
A graduate-level introduction to the homotopical technology in use at the forefront of modern algebraic topology.
Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups
Author: John RognesPublisher: American Mathematical Soc.
ISBN: 0821840762
Category : Mathematics
Languages : en
Pages : 154
Book Description
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 $E$.
H Ring Spectra and Their Applications
Author: Robert R. BrunerPublisher: Springer
ISBN: 3540397787
Category : Mathematics
Languages : en
Pages : 396
Book Description
Proceedings of the International Congress of Mathematicians 2010 (icm 2010) (in 4 Volumes) - Vol. I: Plenary Lectures and Ceremonies, Vols. Ii-iv: Invited Lectures
Author: Publisher: World Scientific
ISBN: 9814324353
Category :
Languages : en
Pages : 814
Book Description
Global Homotopy Theory
Author: Stefan SchwedePublisher: Cambridge University Press
ISBN: 110842581X
Category : Mathematics
Languages : en
Pages : 847
Book Description
A comprehensive, self-contained approach to global equivariant homotopy theory, with many detailed examples and sample calculations.
Equivariant Orthogonal Spectra and S-modules
Author: M. A. MandellPublisher: American Mathematical Soc.
ISBN: 9780821864777
Category : Mathematics
Languages : en
Pages : 132
Book Description
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.
Generalized Cohomology
Author: Akira KōnoPublisher: American Mathematical Soc.
ISBN: 9780821835142
Category : Mathematics
Languages : en
Pages : 276
Book Description
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.
Stable Homotopy and Generalised Homology
Author: John Frank AdamsPublisher: University of Chicago Press
ISBN: 0226005240
Category : Mathematics
Languages : en
Pages : 384
Book Description
J. Frank Adams, the founder of stable homotopy theory, gave a lecture series at the University of Chicago in 1967, 1970, and 1971, the well-written notes of which are published in this classic in algebraic topology. The three series focused on Novikov's work on operations in complex cobordism, Quillen's work on formal groups and complex cobordism, and stable homotopy and generalized homology. Adams's exposition of the first two topics played a vital role in setting the stage for modern work on periodicity phenomena in stable homotopy theory. His exposition on the third topic occupies the bulk of the book and gives his definitive treatment of the Adams spectral sequence along with many detailed examples and calculations in KU-theory that help give a feel for the subject.