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Handbook of Homotopy Theory
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.
The Goodwillie Tower and the EHP Sequence
The author studies the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime $2$. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. He relates the Goodwillie filtration to the $P$ map, and the Goodwillie differentials to the $H$ map. Furthermore, he studies an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. He shows that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. He uses his theory to recompute the $2$-primary unstable stems through the Toda range (up to the $19$-stem). He also studies the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod $2$ stable homology of the Goodwillie layers of any functor from spaces to spaces.
New Developments in Topology
Eleven of the fourteen invited speakers at a symposium held by the Oxford Mathematical Institute in June 1972 have revised their contributions and submitted them for publication in this volume. The present papers do not necessarily closely correspond with the original talks, as it was the intention of the volume editor to make this book of mathematical rather than historical interest. The contributions will be of value to workers in topology in universities and polytechnics.
The Local Structure of Algebraic K-Theory
Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the the...
The Unstable Adams Spectral Sequence for Free Iterated Loop Spaces
The semi-stable homotopy groups of a topological space [italic]X are the unstable homotopy groups [lowercase Greek]Pi [subscript]*[capital Greek]Sigma[superscript]n[italic]X, [italic]n [greater than symbol] 0, of the suspensions of [italic]X. This monograph is concerned with computing these semi-stable homotopy groups using the unstable Adams spectral sequence for the free iterated loop spaces [capital Greek]Omega[superscript italic]n [capital Greek]Sigma[superscript italic]n [italic]X generated by [italic]X.
Cohomological Methods in Homotopy Theory
This book contains a collection of articles summarizing the state of knowledge in a large portion of modern homotopy theory. A call for articles was made on the occasion of an emphasis semester organized by the Centre de Recerca Matemtica in Bellaterra (Barcelona) in 1998. The main topics treated in the book include abstract features of stable and unstable homotopy, homotopical localizations, p-compact groups, H-spaces, classifying spaces for proper actions, cohomology of discrete groups, K-theory and other generalized cohomology theories, configuration spaces, and Lusternik-Schnirelmann category. The book is addressed to all mathematicians interested in homotopy theory and in geometric aspects of group theory. New research directions in topology are highlighted. Moreover, this informative and educational book serves as a welcome reference for many new results and recent methods.
Stable Homotopy Around the Arf-Kervaire Invariant
Were I to take an iron gun, And ?re it o? towards the sun; I grant ‘twould reach its mark at last, But not till many years had passed. But should that bullet change its force, And to the planets take its course, ‘Twould never reach the nearest star, Because it is so very far. from FACTS by Lewis Carroll [55] Let me begin by describing the two purposes which prompted me to write this monograph. This is a book about algebraic topology and more especially about homotopy theory. Since the inception of algebraic topology [217] the study of homotopy classes of continuous maps between spheres has enjoyed a very exc- n n tional, central role. As is well known, for homotopy classes of maps f : S ?? S with n? 1 the sole homotopy invariant is the degree, which characterises the homotopy class completely. The search for a continuous map between spheres of di?erent dimensions and not homotopic to the constant map had to wait for its resolution until the remarkable paper of Heinz Hopf [111]. In retrospect, ?nding 3 an example was rather easy because there is a canonical quotient map from S to 3 1 1 2 theorbitspaceofthe freecircleactionS /S =CP = S .
