TUM Library

The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems, and is proving itself of increasing importance across many other mathematical disciplines. The origins of the subject date back to work of Henri Poincar and Heinz Hopf in the early20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

Cover -- Half Title -- Series Page -- Title Page -- Copyright Page -- Contents -- Preface -- 1. Goodwillie calculus -- 1.1 Polynomial Approximation and the Taylor Tower -- 1.2 The Classification of Homogeneous Functors -- 1.3 The Taylor tower of the identity functor for based spaces -- 1.4 Operads and Tate data: the Classification of Taylor towers -- 1.5 Applications and calculations in algebraic K-theory -- 1.6 Taylor towers of infinity-categories -- 1.7 The manifold and orthogonal calculi -- 1.8 Further directions -- Bibliography -- 2. A factorization homology primer -- 2.1 Introduction

2.2 Manifolds with tangential structure -- 2.2.1 Manifolds and embeddings -- 2.2.2 Sheaves on n-manifolds -- 2.2.3 Tangent classifier -- 2.2.4 B-framed manifolds -- 2.2.5 Examples and discussion of B-framings -- 2.2.6 Weiss sheaves on n-manifolds -- 2.2.7 Disks -- 2.2.8 Manifolds with boundary -- 2.2.9 Localizing with respect to isotopy equivalences -- 2.3 Homology theories for manifolds -- 2.3.1 Disk algebras -- 2.3.2 Factorization algebras -- 2.3.3 Factorization homology over oriented 1-manifolds with boundary -- 2.3.4 Homology theories: definition -- 2.3.5 Pushforward

2.3.6 Homology theories: characterization -- 2.4 Nonabelian Poincaré duality -- 2.5 Calculations -- 2.5.1 Factorization homology for direct sum -- 2.5.2 Factorization homology with coefficients in commutative algebras -- 2.5.3 Factorization homology from Lie algebras -- 2.5.4 Factorization homology of free DiskBn-algebras -- 2.6 Filtrations -- 2.6.1 Cardinality filtrations -- 2.6.2 Goodwillie filtrations -- 2.7 Poincaré/Koszul duality -- 2.8 Factorization homology for singular manifolds -- 2.8.1 Singular manifolds -- 2.8.2 Homology theories for structured singular manifolds

2.8.3 Characterizing some Disk(DU)-algebras -- Bibliography -- 3. Polyhedral products and features of their homotopy theory -- 3.1 Introduction -- 3.2 The origin of polyhedral products in toric topology -- 3.3 The introduction of moment-angle complexes -- 3.4 Moment-angle complexes as intersections of quadrics -- 3.5 The cohomology of moment-angle complexes -- 3.6 The exponentiation property of polyhedral products -- 3.7 Fibrations -- 3.8 Unstable and stable decompositions of the polyhedral product -- 3.9 Equivariance of the stable splitting and an application to number theory

3.10 The case that Ai = ? for all i -- 3.11 The cohomology of polyhedral products and a spectral sequence -- 3.12 A geometric approach to the cohomology of polyhedral products -- 3.13 Polyhedral products and the Golodness of monomial ideal rings -- 3.14 Higher Whitehead products and loop spaces -- Bibliography -- 4. A guide to tensor-triangular classification -- 4.1 Introduction -- 4.2 The tt-spectrum and the classification of tt-ideals -- 4.3 Topology -- 4.4 Commutative algebra and algebraic geometry -- 4.5 Modular representation theory and related topics

OCLC-licensed vendor bibliographic record.