Algebraic Topology : VIASM 2012–2015 / / edited by H.V. Hưng Nguyễn, Lionel Schwartz

• Record Nr.: UNINA9910257380603321
• Pubbl/distr/stampa: Cham : , : Springer International Publishing : , : Imprint : Springer, , 2017
• ISBN: 3-319-69434-0
• Edizione: [1st ed. 2017.]
• Descrizione fisica: 1 online resource (VII, 180 p. 5 illus., 2 illus. in color.)
• Collana: Lecture Notes in Mathematics, , 0075-8434 ; ; 2194
• Disciplina: 514.2
• Soggetti: Algebraic topology; Category theory (Mathematics); Homological algebra; Algebraic Topology; Category Theory, Homological Algebra
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Nota di contenuto: Intro -- Introduction -- Contents -- 1 Hodge Filtration and Operations in Higher Hochschild (Co)homology and Applications to Higher String Topology -- 1.1 Introduction and Overview -- 1.2 Notations, Conventions and a Few Standard Facts -- 1.3 Higher Hochschild (Co)homology -- 1.3.1 -Modules and Hochschild (Co)chain Complexes over Spaces -- 1.3.2 Combinatorial Higher Hochschild (Co)chains -- 1.3.3 Derived Hochschild (Co)chains -- 1.4 Hodge Filtration and λ-Operations on Hochschild (Co)homology over Spheres and Suspensions -- 1.4.1 γ-Rings and Lambda Operations -- 1.4.2 Edgewise Subdivision and Simplicial Approach to λ-Operations -- 1.4.3 Hodge Filtration for Hochschild Cochains over Spheres and Suspensions -- 1.4.4 Hodge Filtration on Hochschild Cochains on the Standard Model -- 1.4.5 Hodge Filtration and λ-Operations for Hochschild Chains over Spheres and Suspensions -- 1.4.6 Hodge Filtration and the Eilenberg-Zilber Model for Hochschild Cochains of Suspensions and Products -- 1.5 Additional Ring Structures for Higher Hochschild Cohomology -- 1.5.1 The Wedge and Cup Product -- 1.5.2 The Universal En-Algebra Structure Lifting the Cup-Product -- 1.5.2.1 The En-Structure of Hochschild (Co)homology over Sn -- 1.5.2.2 The Combinatorial ; Description of the Centralizer of CDGA Maps -- 1.5.3 The O(d)-Equivariance of the Universal Ed Algebra Structure on Hochschild Cochomology over Spheres -- 1.6 Applications of Higher Hochschild-Kostant-Rosenberg Theorem -- 1.6.1 Statement of HKR Theorem -- 1.6.2 HKR Isomorphism and Hodge Decomposition -- 1.6.3 Compatibility of Hodge Decomposition with the Algebra Structure in Cohomology and Induced Poisn+1-Algebra Structure -- 1.6.4 Applications to Poisn-Algebras (Co)homology -- 1.7 Applications to Brane Topology -- 1.7.1 Higher Hochschild (Co)homology as a Model for Mapping Spaces.; 1.7.2 Models for Brane Topology in Characteristic Zero -- References -- 2 On the Derived Functors of Destabilization and of Iterated Loop Functors -- 2.1 Introduction -- 2.2 Background -- 2.2.1 The Steenrod Algebra as a Quadratic Algebra -- 2.2.2 The Category of A-Modules -- 2.2.3 Unstable Modules and Destabilization -- 2.2.4 Derived Functors -- 2.2.5 Motivation for Studying Derived Functors of Destabilization and of Iterated Loop Functors -- 2.3 First Results on Derived Functors of Destabilization and of Iterated Loops -- 2.3.1 Derived Functors of Ω -- 2.3.2 Applications of Ω and Ω1 -- 2.3.3 Interactions Between Loops and Destabilization -- 2.3.4 Connectivity for Ds -- 2.3.5 Comparing Ds and Ωts -- 2.4 Singer Functors -- 2.4.1 The Unstable Singer Functors Rs -- 2.4.2 Singer Functors for M -- 2.4.3 The Singer Differential -- 2.5 Constructing Chain Complexes -- 2.5.1 Destabilization -- 2.5.2 Iterated Loops -- 2.5.3 The Lannes-Zarati Homomorphism -- 2.6 Perspectives -- 2.6.1 The Spherical Class Conjecture and Related Problems -- 2.6.2 Generalizations of the Lannes-Zarati Homomorphism -- References -- 3 A Mini-Course on Morava Stabilizer Groups and Their Cohomology -- 3.1 Introduction -- 3.2 Bousfield Localization and the Chromatic Set Up -- 3.2.1 Bousfield Localization -- 3.2.2 Morava K-Theories -- 3.2.3 LK(n)S0 as Homotopy Fixed Point Spectrum -- 3.3 Resolutions of K(n)-Local Spheres -- 3.3.1 The Example n=1 and p>2 -- 3.3.2 The Case That p-1 Does Not Divide n -- 3.3.3 The Example n=2 and p>3 -- 3.3.4 The Example n=1 and p=2 -- 3.3.5 The General Case p-1 Divides n -- 3.3.6 The Example n=2 and p=3 -- 3.3.7 Permutation Resolutions and Realizations -- 3.3.8 Applications and Work in Progress -- 3.3.8.1 The Case n=2 and p=3 -- 3.3.8.2 The Case n=2 and p>3 -- 3.3.8.3 The Case n=p=2 -- 3.4 The Morava Stabilizer Groups: First Properties.; 3.4.1 The Morava Stabilizer Group as a Profinite Group -- 3.4.2 The Associated Mixed Lie Algebra of Sn -- 3.4.3 Torsion in the Morava Stabilizer Groups -- 3.5 On the Cohomology of the Stabilizer Groups with Trivial Coefficients -- 3.5.1 H1: The Stabilizer Group Made Abelian -- 3.5.2 The Cohomology of S1 -- 3.5.3 Structural Properties of H*(Sn,Z/p) -- 3.5.4 The Reduced Norm and a Decomposition of Sn -- 3.5.5 Cohomology in Case n=2 and p>2 -- 3.5.5.1 The Case p>3 -- 3.5.5.2 The Case p=3 -- 3.5.5.3 The Case p=2 -- 3.6 Cohomology with Non-trivial Coefficients and Resolutions -- 3.6.1 The Case n=1 -- 3.6.1.1 The Case p>2 -- 3.6.1.2 The Case p=2 -- 3.6.2 Some Comments on the Case n=2 -- References.
• Sommario/riassunto: Held during algebraic topology special sessions at the Vietnam Institute for Advanced Studies in Mathematics (VIASM, Hanoi), this set of notes consists of expanded versions of three courses given by G. Ginot, H.-W. Henn and G. Powell. They are all introductory texts and can be used by PhD students and experts in the field. Among the three contributions, two concern stable homotopy of spheres: Henn focusses on the chromatic point of view, the Morava K(n)-localization and the cohomology of the Morava stabilizer groups. Powell’s chapter is concerned with the derived functors of the destabilization and iterated ; loop functors and provides a small complex to compute them. Indications are given for the odd prime case. Providing an introduction to some aspects of string and brane topology, Ginot’s contribution focusses on Hochschild homology and its generalizations. It contains a number of new results and fills a gap in the literature. .