reader, and since readability is one of the ﬁrst priorities of the book, this homotopic interpretation of homology and cohomology is described only after the latter theories have been developed independently of homotopy theory. Preceding the four main chapters there is a preliminary Chapter 0 introducing. Aug 27, · There are a ton of relationships. Let me touch on two. Motivic homotopy theory is one such relationship: this is supposed to be the “homotopy theory of schemes”. (Mike Hopkins has a nice lecture here: Michael Hopkins - Algebraic and motivic vector. Homotopy, homotopy equivalence, the categories of based and unbased space. Week 2. Higher homotopy groups, weak homotopy equivalence, CW complex. Week 3. Cofibrations and the Homotopy Extension Property. Week 4. Relative homotopy groups, homotopy fiber, long exact sequence in homotopy, Whitehead theorem. Week 5. Cellular and CW approximation. Book Description: This book presents a coherent account of the current status of etale homotopy theory, a topological theory introduced into abstract algebraic geometry by M. Artin and B. Mazur. Eric M. Friedlander presents many of his own applications of this theory to algebraic topology, finite Chevalley groups, and algebraic geometry.

A distinguishing feature is a thematic focus on Eckmann-Hilton duality. this book offers an attractive option for a course or self-study, fitting a niche between the introductory texts of Munkres, Massey and Thatcher and the comprehensive treatments of homotopy theory by Spanier and walkingshops.com: Martin Arkowitz. (P. Goerss) Gross-Hopkins duality. Brown-Comenetz duality and its variants are something of a curiosity in stable homotopy theory, but it is an insight of Hopkins that in the K (n)-local category it is much more like Serre-Grothendieck duality. Flesh out that statement. and analysis: homotopy theories are everywhere, along with functorial methods of relating them. This book is, however, not quite so cosmological in scope. The theory has broad applications in many areas, but it has always been quite a sharp tool within ordinary . Key words and phrases. ex-space, parametrized spectrum, parametrized homotopy theory, equivariant homotopy theory, parametrized stable homotopy theory, equivariant stable homotopy theory, transfer, twisted K-theory, Poincar´e duality, Atiyah duality, Thom spectrum, bicategory, model category May was partially supported by the NSF.

7 Homotopy Theory of Fibrations 9 Poincar e Duality, Intersection theory, and Linking numbers Poincar e Duality, the \shriek map", and the Thom isomorphism Duality The subject of much of this book is the topology of manifolds. n-dimensional. Ambidexterity in K(n)-Local Stable Homotopy Theory. Joint with Mike Hopkins. Investigates some surprising duality phenomena in the world of K(n)-local homotopy theory. Mostly finished, though it is a bit rough in places. Last update: December pdf: Representability Theorems. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces. Algebraic topology in general and homotopy theory in particular is in an exciting period of growth and transformation, driven in part by strong interactions with algebraic geometry, mathematical.