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 Poincaré and Heinz Hopf in the early 20th 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.

chapter 1|38 pages

Goodwillie calculus

ByGregory Arone, Michael Ching

chapter 2|63 pages

A factorization homology primer

ByDavid Ayala, John Francis

chapter 3|42 pages

Polyhedral products and features of their homotopy theory

ByAnthony Bahri, Martin Bendersky, Frederick R. Cohen

chapter 4|18 pages

A guide to tensor-triangular classification

ByPaul Balmer

chapter 5|58 pages

Chromatic structures in stable homotopy theory

ByTobias Barthel, Agnès Beaudry

chapter 6|41 pages

Topological modular and automorphic forms

ByMark Behrens

chapter 7|34 pages

A survey of models for (∞, n)-categories

ByJulia E. Bergner

chapter 8|33 pages

Persistent homology and applied homotopy theory

ByGunnar Carlsson

chapter 9|38 pages

Algebraic models in the homotopy theory of classifying spaces

ByNatàlia Castellana

chapter 10|36 pages

Floer homotopy theory, revisited

ByRalph L. Cohen

chapter 12|43 pages

Moduli spaces of manifolds: a user's guide

BySøren Galatius, Oscar Randal-Williams

chapter 13|62 pages

An introduction to higher categorical algebra

ByDavid Gepner

chapter 14|69 pages

A short course on ∞-categories

ByMoritz Groth

chapter 15|38 pages

Topological cyclic homology

ByLars Hesselholt, Thomas Nikolaus

chapter 16|42 pages

Lie algebra models for unstable homotopy theory

ByGijs Heuts

chapter 17|58 pages

Equivariant stable homotopy theory

ByMichael A. Hill

chapter 18|35 pages

Motivic stable homotopy groups

ByDaniel C. Isaksen, Paul Arne Østvær

chapter 19|57 pages

E n -spectra and Dyer-Lashof operations

ByTyler Lawson

chapter 20|40 pages

Assembly maps

ByWolfgang Lück

chapter 21|39 pages

Lubin-Tate theory, character theory, and power operations

ByNathaniel Stapleton

chapter 22|42 pages

Unstable motivic homotopy theory

ByKirsten Wickelgren, Ben Williams