LEADER 06165nam 2200577 450 001 996466408503316 005 20231110220357.0 010 $a3-030-78977-2 035 $a(CKB)4100000012037898 035 $a(MiAaPQ)EBC6737913 035 $a(Au-PeEL)EBL6737913 035 $a(OCoLC)1272991765 035 $a(PPN)257920285 035 $a(EXLCZ)994100000012037898 100 $a20220628d2021 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 00$aHomotopy theory and arithmetic geometry $emotivic and diophantine aspects, LMS-CMI Research School, London, July 2018 /$fedited by Frank Neumann and Ambrus Pa?l 210 1$aCham, Switzerland :$cSpringer,$d[2021] 210 4$d©2021 215 $a1 online resource (223 pages) 225 1 $aLecture Notes in Mathematics ;$vv.2292 300 $aIncludes index. 311 $a3-030-78976-4 327 $aIntro -- Preface -- Contents -- 1 Homotopy Theory and Arithmetic Geometry-Motivic and Diophantine Aspects: An Introduction -- 1.1 Overview of Themes -- 1.2 Summaries of Individual Contributions -- References -- 2 An Introduction to A1-Enumerative Geometry -- 2.1 Introduction -- 2.2 Preliminaries -- 2.2.1 Enriching the Topological Degree -- 2.2.2 The Grothendieck-Witt Ring -- 2.2.3 Lannes' Formula -- 2.2.4 The Unstable Motivic Homotopy Category -- 2.2.5 Colimits -- 2.2.6 Purity -- 2.3 A1-enumerative Geometry -- 2.3.1 The Eisenbud-Khimshiashvili-Levine Signature Formula -- 2.3.2 Sketch of Proof for Theorem 4 -- 2.3.3 A1-Milnor Numbers -- 2.3.4 An Arithmetic Count of the Lines on a Smooth Cubic Surface -- 2.3.5 An Arithmetic Count of the Lines Meeting 4Lines in Space -- Notation Guide -- References -- 3 Cohomological Methods in Intersection Theory -- 3.1 Introduction -- 3.2 Étale Motives -- 3.2.1 The h-topology -- 3.2.2 Construction of Motives, After Voevodsky -- 3.2.3 Functoriality -- 3.2.4 Representability Theorems -- 3.3 Finiteness and Euler Characteristic -- 3.3.1 Locally Constructible Motives -- 3.3.2 Integrality of Traces and Rationality of ?-Functions -- 3.3.3 Grothendieck-Verdier Duality -- 3.3.4 Generic Base Change: A Motivic Variation on Deligne's Proof -- 3.4 Characteristic Classes -- 3.4.1 Künneth Formula -- 3.4.2 Grothendieck-Lefschetz Formula -- References -- 4 Étale Homotopy and Obstructions to Rational Points -- 4.1 Introduction -- 4.2 ?-Categories -- 4.2.1 Motivation -- 4.2.2 Quasi-Categories -- 4.2.3 ?-Groupoids and the Homotopy Hypothesis -- 4.2.4 Quasi-Categories from Topological Categories -- 4.2.5 ?-Category Theory -- 4.2.6 The Homotopy Category -- 4.2.7 ?-Categories and Homological Algebra -- 4.2.8 Stable ?-Categories -- 4.2.9 Localization -- 4.3 ?-Topoi -- 4.3.1 Definitions -- 4.3.2 The Shape of an ?-Topos. 327 $a4.4 Obstruction Theory -- 4.4.1 Obstruction Theory for Homotopy Types -- 4.4.2 For ?-Topoi and Linear(ized) Versions -- 4.5 Étale Homotopy and Rational Points -- 4.5.1 The étale ?-Topos -- 4.5.2 Rational Points -- 4.5.3 The Local-to-Global Principle -- 4.6 Galois Theory and Embedding Problems -- 4.6.1 Topoi and Embedding Problems -- References -- 5 A1-homotopy Theory and Contractible Varieties: A Survey -- 5.1 Introduction: Topological and Algebro-Geometric Motivations -- 5.1.1 Open Contractible Manifolds -- 5.1.2 Contractible Algebraic Varieties -- 5.2 A User's Guide to A1-homotopy Theory -- 5.2.1 Brief Topological Motivation -- 5.2.2 Homotopy Functors in Algebraic Geometry -- 5.2.3 The Unstable A1-homotopy Category: Construction -- Spaces -- Nisnevich and cdh Distinguished Squares -- Localization -- 5.2.4 The Unstable A1-homotopy Category: Basic Properties -- Motivic Spheres -- Representability Statements -- Representability of Chow Groups -- The Purity Isomorphism -- Comparison of Nisnevich and cdh-local A1-weak Equivalences -- 5.2.5 A Snapshot of the Stable Motivic Homotopy Category -- Stable Representablity of Algebraic K-theory -- Milnor-Witt K-theory -- 5.3 Concrete A1-weak Equivalences -- 5.3.1 Constructing A1-weak Equivalences of Smooth Schemes -- 5.3.2 A1-weak Equivalences vs. Weak Equivalences -- 5.3.3 Cancellation Questions and A1-weak Equivalences -- 5.3.4 Danielewski Surfaces and Generalizations -- 5.3.5 Building Quasi-Affine A1-contractible Varieties -- Unipotent Quotients -- Other Quasi-Affine A1-contractible Varieties -- 5.4 Further Computations in A1-homotopy Theory -- 5.4.1 A1-homotopy Sheaves -- Basic Definitions -- A1-rigid Varieties Embed into H(k) -- 5.4.2 A1-connectedness and Geometry -- A1-connectedness and Rationality Properties -- 5.4.3 A1-homotopy Sheaves Spheres and Brouwer Degree -- 5.4.4 A1-homotopy at Infinity. 327 $aOne-point Compactifications -- Stable End Spaces -- 5.5 Cancellation Questions and A1-contractibility -- 5.5.1 The Biregular Cancellation Problem -- 5.5.2 A1-contractibility vs Topological Contractibility -- Affine Lines on Topologically Contractible Surfaces -- Chow Groups and Vector Bundles on Topologically Contractible Surfaces -- 5.5.3 Cancellation Problems and the Russell Cubic -- The Russell Cubic and Equivariant K-theory -- Higher Chow Groups and Stable A1-contractibility -- 5.5.4 A1-contractibility of the Koras-Russell Threefold -- 5.5.5 Koras-Russell Fiber Bundles -- References -- Index. 410 0$aLecture Notes in Mathematics 606 $aArithmetical algebraic geometry$vCongresses 606 $aHomotopy theory$vCongresses 606 $aTeoria de l'homotopia$2thub 606 $aGeometria algebraica aritmètica$2thub 608 $aCongressos$2thub 608 $aLlibres electrònics$2thub 615 0$aArithmetical algebraic geometry 615 0$aHomotopy theory 615 7$aTeoria de l'homotopia 615 7$aGeometria algebraica aritmètica 676 $a514.24 702 $aNeumann$b Frank$c(Mathematician), 702 $aPa?l$b Ambrus 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a996466408503316 996 $aHomotopy theory and arithmetic geometry$92891588 997 $aUNISA