In a nutshell, motivic homotopy theory is a particular homotopy theory of algebraic varieties. Basic concepts, constructing topologies, connectedness, separation axioms and the hausdorff property, compactness and its relatives, quotient spaces, homotopy, the fundamental group and some application, covering spaces and classification of covering space. Homotopy theory and algebraic geometry fall 2015 wism462. Voevodsky uses extremely simple leading ideas to extend topological methods of homotopy into number theory and abstract algebra. My main research interests are in stable homotopy theory. Handbook of homotopy theory 1st edition haynes miller. Motivic homotopy theory is the homotopy theory of algebraic varieties schemes over a eld. This does not work in our general setting, and it fails already in the concrete setting of monoid schemes. The talk takes the simple ideas right up to the edge of some extremely re ned applications. The idea of the subject is to apply techniques from algebraic topology to solve problems in algebraic geometry. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science university of pennsylvania philadelphia, pa 19104, usa email.
A major part of this work relies on highly structured models for motivic stable homotopy theory. Fabien morel, an introduction to 1 \mathbba1 homotopy theory, ictp trieste july 2002 directory, pdf, ps fabien morel, on the motivic of the sphere spectrum ps peter arndt, abstract motivic homotopy theory, thesis 2017 web, pdf, pdf. Beginning with an introduction to the homotopy theory of simplicial sets and. A topologists introduction to the motivic homotopy theory 67 where we have regarded the ordered set n. After all, the new developments in motivic cohomology were affecting our knowledge of the k theory of fields and varieties. Ams proceedings of the american mathematical society. As background material, we recommend the lectures of dundas dun and levine lev in this volume. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect a 1 homotopy theory is at least as complicated as classical homotopy theory.
A topologists introduction to the motivic homotopy theory for transformation group theorists 1 by norihiko minami. Xy is a universal homeomorphism if it induces a homeomorphism on underlying topological spaces after any base change, or equivalently, if it. The aim of this book is to be an accessible introduction to stable homotopy theory that novices, particularly graduate students, can use to learn the fundamentals of the subject. For any noetherian and separated base scheme of finite krull dimension these frameworks give rise to a homotopy theoretic meaningful study of modules over motivic cohomology. The theory of algebraic cycles, higher algebraic k theory, and motivic homotopy theory are modern versions of grothendiecks legacy. Category theory provides structure for the mathematical world and is seen everywhere in modern mathematics. In recent years it has seen some spectacular developments, on which we want to build further. We present some recent results in a1algebraic topology, which means both in a1 homotopy theory of schemes and its relationship with algebraic geometry. The handbook of homotopy theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to longstanding open problems, and is proving itself of increasing importance across many other mathematical disciplines.
Its primary object is to study algebraic varieties from a homotopy theoretic viewpoint. The electronic computational homotopy theory seminar is an international research seminar on the topic of computational homotopy theory. Rational homotopy theory 3 it is clear that for all r, sn r is a strong deformation retract of xr, which implies that hkxr 0 if k 6 0,n. Download motivic homotopy theory ebook in pdf, epub, mobi.
Oliver rondigs, markus spitzweck and paul arne ostv. By this time 1995, the k theory landscape had changed, and with it my vision of what my k theory book should be. Norm functors in unstable motivic homotopy theory were rst introduced by voevod. Then 0 is the unique proper ideal in f, and is prime since f is an integral domain ab 0, a6 0. The aim is to provide readers with some background to. To do this, one rst has to nd a notion of homotopy which works for algebraic varieties. All schemes will implicitly be assumed to be quasicompact and quasiseparated. Advances in the computation of the basic structure constants of motivic homotopy theory, that is, the stable and unstable motivic homotopy groups of spheres, with applications to. Abstract an introductory survey of the motivic homotopy theory for topologisits is given, by focusing upon the algebraic k theory representability and the homotopy purity. I completed my doctoral work at mit in 2018 under the supervision of haynes miller. An introductory reference to motivic homotopy theory is voevodsky.
The last part, containing the most innovative results, assumes some knowledge of motivic homotopy theory, although precise statements and references are given. In chapter 2, we construct the formalism of six operations on the stable motivic homotopy category. This entry is a detailed introduction to stable homotopy theory, hence to the stable homotopy category and to its key computational tool, the adams spectral sequence. Motives and modules over motivic cohomology sciencedirect. Jardine september 23, 1999 introduction let smj k nisbe the smooth nisnevich site for a eld k, and let sshvsmj k nis respectively shvsmj k nis denote the category of simplicial sheaves respec tively sheaves on that site.
Among other applications, we prove grothendieckverdier duality in this. This leads to a theory of motivic spheres s p,q with two indices. Mathematics department, princeton university the first stable homotopy groups of motivic spheres authors. We now describe these two main sections in reverse order. Inspired by classical results in algebraic topology, we present new. With this book, the author bridges the gap between pure category theory and its numerous applications in homotopy theory, providing the necessary background information to make the subject accessible to graduate students or researchers with a background in algebraic topology and algebra. From categories to homotopy theory cambridge studies in. The intent of the course was to bring graduate students who had completed a first course in algebraic topology. The result, an introduction to homological algebra, took over five years to write. The basic framework of equivariant motivic homotopy theory.
We will begin with an introduction to stable motivic homotopy theory. Specifically, we prove the convergence of motivic analogues of the adams and adams. In the accompanying seminar we consider applications to cobordism theory and complex oriented cohomology such as to converge in. Summer course in motivic homotopy theory 3 examples 1. In forthcoming papers, we apply the theory developed here to produce a theory of spanierwhitehead duality and beckergottlieb transfer in this framework, and explore various applications of the transfer. For instance, in homotopy theory, a solid ball of any dimension and a point are considered as equivalent, also a solid torus and a circle are equivalent from the point of view of homotopy theory. Topics include any part of homotopy theory that has a computational flavor, including but not limited to stable homotopy theory, unstable homotopy theory, chromatic homotopy theory, equivariant homotopy theory, motivic homotopy theory, and k theory.
A topologists introduction to the motivic homotopy theory. Presheaves with transfers homotopy invariant presheaves 17 lecture 3. This text deals with a 1 homotopy theory over a base field, i. In this expository article, we give the foundations, basic facts, and first examples of unstable motivic homotopy theory with a view towards the approach of. Cech, introduction of abstract homotopy groups, 1932 hurewicz, higher homotopy groups and homotopy equivalence, 1935 eilenberg and obstruction theory, 1940 isabel vogt a brief history of homotopy theory. Motivic homotopy theory over classical base schemes was introduced in mv99, using the language of model categories. Rims kokyuroku bessatsu b39 20, 063107 a topologist s introduction to the motivic homotopy theory for transformation group theorists. Ktheory, algebraic cycles and motivic homotopy theory. We prove a topological invariance statement for the morelvoevodsky motivic homotopy category up to inverting exponential characteristics of residue fields. A gentle introduction to homology, cohomology, and sheaf. Remarks on motivic homotopy theory over algebraically closed fields po hu, igor kriz and kyle ormsby abstract. Then we will use the motivic version of the adams spectral sequence to compute stable motivic homotopy groups.
In this spirit, this paper adapts homotopical galois theory to the context of motivic spaces and spectra. Introduction the goal of this paper is to develop the formalism of norm functors in motivic homotopy theory, to study the associated notion of normed motivic spectrum, and to construct many examples. It is a natural sequel to the foundational paper on a 1 homotopy theory written together with v. Perfection in motivic homotopy theory elmanto 2020. The localization theorem 3 the main goal of this paper is to prove the analogue of this theorem in spectral algebraic geometry. For more background and motivation, the reader may wish to consult the introduction of by18.
Motivic homotopical galois extensions sciencedirect. When a cohomology theory is representable as a motivic spectrum, we may view cohomology classes as. Lecture 1 an introduction to motivic homotopy theory. For the experts we hope to have provided a useful compendium of results across the main areas of stable homotopy theory. This text is a report from voevodskys summer school lectures on motivic homotopy in nordfjordeid.
Indeed, giving this concrete homotopy requires both an addition and a. Motivic homotopy theory in derived algebraic geometry. Research my current research focuses on computational aspects of classical and motivic stable homotopy theory. Furthermore, the homomorphism induced in reduced homology by the inclusion xr. Informally, the theory enriches the category of smooth schemes over a base eld so it also admits simplicial constructions, and then imposes a homotopy theoretic structure in which the a ne line a1 plays the role of the unit. To that end we introduce the modern tools, such as model categories and highly structured ring spectra. Mixed weil cohomologies are also representable by motivic spectra see cd12. This implies in particular that sh 1 p of characteristic p 0 schemes is invariant under passing to perfections. Over a base eld k, motivic homotopy theory is built from the category of kspaces, denoted spck, which is the category of simplicial presheaves on smk, the category of. Along the way, we will encounter some new results about classical stable homotopy groups. Many classical results of homotopy theory have since been translated to the motivic world. Motivic homotopy theory also available for read online in mobile and kindle. We will discuss a number of open questions concerning these computations. It is designed to be a reference work and could also be useful outside motivic homotopy theory.
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