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The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper. The proof is long and involved, using many sophisticated tools of modern (equivariant) stable homotopy theory that are unfamiliar to non-experts. This book presents the proof together with a full development of all the background material to make it accessible to a graduate student with an elementary algebraic topology knowledge. There are explicit examples of constructions used in solving the problem. Also featuring a motivating history of the problem and numerous conceptual and expository improvements on the proof, this is the definitive account of the resolution of the Kervaire invariant problem.

Homotopy theory.

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"We construct a Goodwillie tower of categories which interpolates between the category of pointed spaces and the category of spectra. This tower of categories refines the Goodwillie tower of the identity functor in a precise sense. More generally, we construct such a tower for a large class of -categories C and classify such Goodwillie towers in terms of the derivatives of the identity functor of C. As a particular application we show how this provides a model for the homotopy theory of simply-connected spaces in terms of coalgebras in spectra with Tate diagonals. Our classification of Goodwillie towers simplifies considerably in settings where the Tate cohomology of the symmetric groups vanishes. As an example we apply our methods to rational homotopy theory. Another application identifies the homotopy theory of p-local spaces with homotopy groups in a certain finite range with the homotopy theory of certain algebras over Ching's spectral version of the Lie operad. This is a close analogue of Quillen's results on rational homotopy"--

Homotopy groups. --- Algebraic topology. --- Spectral sequences (Mathematics) --- Class field towers. --- Algebraic topology -- Homotopy theory -- None of the above, but in this section. --- Algebraic topology -- Homotopy theory -- Classification of homotopy type. --- Algebraic topology -- Homotopy theory -- Homotopy functors. --- Algebraic topology -- Applied homological algebra and category theory -- Abstract and axiomatic homotopy theory. --- Algebraic topology -- Applied homological algebra and category theory -- Topological categories, foundations of homotopy theory.

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Teoria de l'homotopia --- Geometria algebraica aritmètica --- Geometria algèbrica aritmètica --- Geometria diofàntica --- Geometria algebraica --- Teoria de nombres --- Punts racionals (Geometria) --- Varietats de Shimura --- Deformacions contínues --- Homotopia --- Teoria homotòpica --- Topologia --- Transformacions (Matemàtica) --- Cirurgia (Topologia) --- Equivalències d'homotopia --- Grups d'homotopia --- Teoria de la forma (Topologia) --- Teoria de la localització --- Arithmetical algebraic geometry --- Homotopy theory --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory