Bordism and Cobordism
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Cohomological Descent on the Overconvergent Site
COHOMOLOGICAL DESCENT ON THE OVERCONVERGENT SITE DAVID ZUREICK-BROWN Abstract. We prove that cohomological descent holds for finitely presented crystals on the overconvergent site with respect to proper or fppf hypercovers. 1. Introduction Cohomological descent is a robust computational and theoretical tool, central to p-adic cohomology and its applications. On one hand, it facilitates explicit calculations (analo- gous to the computation of coherent cohomology in scheme theory via Cechˇ cohomology); on another, it allows one to deduce results about singular schemes (e.g., finiteness of the cohomology of overconvergent isocrystals on singular schemes [Ked06]) from results about smooth schemes, and, in a pinch, sometimes allows one to bootstrap global definitions from local ones (for example, for a scheme X which fails to embed into a formal scheme smooth near X, one actually defines rigid cohomology via cohomological descent; see [lS07, comment after Proposition 8.2.17]). The main result of the series of papers [CT03,Tsu03,Tsu04] is that cohomological descent for the rigid cohomology of overconvergent isocrystals holds with respect to both flat and proper hypercovers. The barrage of choices in the definition of rigid cohomology is burden- some and makes their proofs of cohomological descent very difficult, totaling to over 200 pages. Even after the main cohomological descent theorems [CT03, Theorems 7.3.1 and 7.4.1] are proved one still has to work a bit to get a spectral sequence [CT03, Theorem 11.7.1]. Actually, even to state what one means by cohomological descent (without a site) is subtle. The situation is now more favorable. -
Homological Algebra
Homological Algebra Donu Arapura April 1, 2020 Contents 1 Some module theory3 1.1 Modules................................3 1.6 Projective modules..........................5 1.12 Projective modules versus free modules..............7 1.15 Injective modules...........................8 1.21 Tensor products............................9 2 Homology 13 2.1 Simplicial complexes......................... 13 2.8 Complexes............................... 15 2.15 Homotopy............................... 18 2.23 Mapping cones............................ 19 3 Ext groups 21 3.1 Extensions............................... 21 3.11 Projective resolutions........................ 24 3.16 Higher Ext groups.......................... 26 3.22 Characterization of projectives and injectives........... 28 4 Cohomology of groups 32 4.1 Group cohomology.......................... 32 4.6 Bar resolution............................. 33 4.11 Low degree cohomology....................... 34 4.16 Applications to finite groups..................... 36 4.20 Topological interpretation...................... 38 5 Derived Functors and Tor 39 5.1 Abelian categories.......................... 39 5.13 Derived functors........................... 41 5.23 Tor functors.............................. 44 5.28 Homology of a group......................... 45 1 6 Further techniques 47 6.1 Double complexes........................... 47 6.7 Koszul complexes........................... 49 7 Applications to commutative algebra 52 7.1 Global dimensions.......................... 52 7.9 Global dimension of -
Lecture 15. De Rham Cohomology
Lecture 15. de Rham cohomology In this lecture we will show how differential forms can be used to define topo- logical invariants of manifolds. This is closely related to other constructions in algebraic topology such as simplicial homology and cohomology, singular homology and cohomology, and Cechˇ cohomology. 15.1 Cocycles and coboundaries Let us first note some applications of Stokes’ theorem: Let ω be a k-form on a differentiable manifold M.For any oriented k-dimensional compact sub- manifold Σ of M, this gives us a real number by integration: " ω : Σ → ω. Σ (Here we really mean the integral over Σ of the form obtained by pulling back ω under the inclusion map). Now suppose we have two such submanifolds, Σ0 and Σ1, which are (smoothly) homotopic. That is, we have a smooth map F : Σ × [0, 1] → M with F |Σ×{i} an immersion describing Σi for i =0, 1. Then d(F∗ω)isa (k + 1)-form on the (k + 1)-dimensional oriented manifold with boundary Σ × [0, 1], and Stokes’ theorem gives " " " d(F∗ω)= ω − ω. Σ×[0,1] Σ1 Σ1 In particular, if dω =0,then d(F∗ω)=F∗(dω)=0, and we deduce that ω = ω. Σ1 Σ0 This says that k-forms with exterior derivative zero give a well-defined functional on homotopy classes of compact oriented k-dimensional submani- folds of M. We know some examples of k-forms with exterior derivative zero, namely those of the form ω = dη for some (k − 1)-form η. But Stokes’ theorem then gives that Σ ω = Σ dη =0,sointhese cases the functional we defined on homotopy classes of submanifolds is trivial. -
The Relation of Cobordism to K-Theories
The Relation of Cobordism to K-Theories PhD student seminar winter term 2006/07 November 7, 2006 In the current winter term 2006/07 we want to learn something about the different flavours of cobordism theory - about its geometric constructions and highly algebraic calculations using spectral sequences. Further we want to learn the modern language of Thom spectra and survey some levels in the tower of the following Thom spectra MO MSO MSpin : : : Mfeg: After that there should be time for various special topics: we could calculate the homology or homotopy of Thom spectra, have a look at the theorem of Hartori-Stong, care about d-,e- and f-invariants, look at K-, KO- and MU-orientations and last but not least prove some theorems of Conner-Floyd type: ∼ MU∗X ⊗MU∗ K∗ = K∗X: Unfortunately, there is not a single book that covers all of this stuff or at least a main part of it. But on the other hand it is not a big problem to use several books at a time. As a motivating introduction I highly recommend the slides of Neil Strickland. The 9 slides contain a lot of pictures, it is fun reading them! • November 2nd, 2006: Talk 1 by Marc Siegmund: This should be an introduction to bordism, tell us the G geometric constructions and mention the ring structure of Ω∗ . You might take the books of Switzer (chapter 12) and Br¨oker/tom Dieck (chapter 2). Talk 2 by Julia Singer: The Pontrjagin-Thom construction should be given in detail. Have a look at the books of Hirsch, Switzer (chapter 12) or Br¨oker/tom Dieck (chapter 3). -
Algebraic Cobordism
Algebraic Cobordism Marc Levine January 22, 2009 Marc Levine Algebraic Cobordism Outline I Describe \oriented cohomology of smooth algebraic varieties" I Recall the fundamental properties of complex cobordism I Describe the fundamental properties of algebraic cobordism I Sketch the construction of algebraic cobordism I Give an application to Donaldson-Thomas invariants Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Naive algebraic analogs: Algebraic topology Algebraic geometry ∗ ∗ Singular homology H (X ; Z) $ Chow ring CH (X ) ∗ alg Topological K-theory Ktop(X ) $ Grothendieck group K0 (X ) Complex cobordism MU∗(X ) $ Algebraic cobordism Ω∗(X ) Marc Levine Algebraic Cobordism Algebraic topology and algebraic geometry Refined algebraic analogs: Algebraic topology Algebraic geometry The stable homotopy $ The motivic stable homotopy category SH category over k, SH(k) ∗ ∗;∗ Singular homology H (X ; Z) $ Motivic cohomology H (X ; Z) ∗ alg Topological K-theory Ktop(X ) $ Algebraic K-theory K∗ (X ) Complex cobordism MU∗(X ) $ Algebraic cobordism MGL∗;∗(X ) Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Complex cobordism is special Complex cobordism MU∗ is distinguished as the universal C-oriented cohomology theory on differentiable manifolds. We approach algebraic cobordism by defining oriented cohomology of smooth algebraic varieties, and constructing algebraic cobordism as the universal oriented cohomology theory. Marc Levine Algebraic Cobordism Cobordism and oriented cohomology Oriented cohomology What should \oriented cohomology of smooth varieties" be? Follow complex cobordism MU∗ as a model: k: a field. Sm=k: smooth quasi-projective varieties over k. An oriented cohomology theory A on Sm=k consists of: D1. -
New Ideas in Algebraic Topology (K-Theory and Its Applications)
NEW IDEAS IN ALGEBRAIC TOPOLOGY (K-THEORY AND ITS APPLICATIONS) S.P. NOVIKOV Contents Introduction 1 Chapter I. CLASSICAL CONCEPTS AND RESULTS 2 § 1. The concept of a fibre bundle 2 § 2. A general description of fibre bundles 4 § 3. Operations on fibre bundles 5 Chapter II. CHARACTERISTIC CLASSES AND COBORDISMS 5 § 4. The cohomological invariants of a fibre bundle. The characteristic classes of Stiefel–Whitney, Pontryagin and Chern 5 § 5. The characteristic numbers of Pontryagin, Chern and Stiefel. Cobordisms 7 § 6. The Hirzebruch genera. Theorems of Riemann–Roch type 8 § 7. Bott periodicity 9 § 8. Thom complexes 10 § 9. Notes on the invariance of the classes 10 Chapter III. GENERALIZED COHOMOLOGIES. THE K-FUNCTOR AND THE THEORY OF BORDISMS. MICROBUNDLES. 11 § 10. Generalized cohomologies. Examples. 11 Chapter IV. SOME APPLICATIONS OF THE K- AND J-FUNCTORS AND BORDISM THEORIES 16 § 11. Strict application of K-theory 16 § 12. Simultaneous applications of the K- and J-functors. Cohomology operation in K-theory 17 § 13. Bordism theory 19 APPENDIX 21 The Hirzebruch formula and coverings 21 Some pointers to the literature 22 References 22 Introduction In recent years there has been a widespread development in topology of the so-called generalized homology theories. Of these perhaps the most striking are K-theory and the bordism and cobordism theories. The term homology theory is used here, because these objects, often very different in their geometric meaning, Russian Math. Surveys. Volume 20, Number 3, May–June 1965. Translated by I.R. Porteous. 1 2 S.P. NOVIKOV share many of the properties of ordinary homology and cohomology, the analogy being extremely useful in solving concrete problems. -
Floer Homology, Gauge Theory, and Low-Dimensional Topology
Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. -
On the Bordism Ring of Complex Projective Space Claude Schochet
proceedings of the american mathematical society Volume 37, Number 1, January 1973 ON THE BORDISM RING OF COMPLEX PROJECTIVE SPACE CLAUDE SCHOCHET Abstract. The bordism ring MUt{CPa>) is central to the theory of formal groups as applied by D. Quillen, J. F. Adams, and others recently to complex cobordism. In the present paper, rings Et(CPm) are considered, where E is an oriented ring spectrum, R=7rt(£), andpR=0 for a prime/». It is known that Et(CPcc) is freely generated as an .R-module by elements {ßT\r^0}. The ring structure, however, is not known. It is shown that the elements {/VI^O} form a simple system of generators for £t(CP°°) and that ßlr=s"rß„r mod(/?j, • • • , ßvr-i) for an element s e R (which corre- sponds to [CP"-1] when E=MUZV). This may lead to information concerning Et(K(Z, n)). 1. Introduction. Let E he an associative, commutative ring spectrum with unit, with R=n*E. Then E determines a generalized homology theory £* and a generalized cohomology theory E* (as in G. W. White- head [5]). Following J. F. Adams [1] (and using his notation throughout), assume that E is oriented in the following sense : There is given an element x e £*(CPX) such that £*(S2) is a free R- module on i*(x), where i:S2=CP1-+CP00 is the inclusion. (The Thom-Milnor spectrum MU which yields complex bordism theory and cobordism theory satisfies these hypotheses and is of seminal interest.) By a spectral sequence argument and general nonsense, Adams shows: (1.1) E*(CP ) is the graded ring of formal power series /?[[*]]. -
Algebraic Topology
Algebraic Topology John W. Morgan P. J. Lamberson August 21, 2003 Contents 1 Homology 5 1.1 The Simplest Homological Invariants . 5 1.1.1 Zeroth Singular Homology . 5 1.1.2 Zeroth deRham Cohomology . 6 1.1.3 Zeroth Cecˇ h Cohomology . 7 1.1.4 Zeroth Group Cohomology . 9 1.2 First Elements of Homological Algebra . 9 1.2.1 The Homology of a Chain Complex . 10 1.2.2 Variants . 11 1.2.3 The Cohomology of a Chain Complex . 11 1.2.4 The Universal Coefficient Theorem . 11 1.3 Basics of Singular Homology . 13 1.3.1 The Standard n-simplex . 13 1.3.2 First Computations . 16 1.3.3 The Homology of a Point . 17 1.3.4 The Homology of a Contractible Space . 17 1.3.5 Nice Representative One-cycles . 18 1.3.6 The First Homology of S1 . 20 1.4 An Application: The Brouwer Fixed Point Theorem . 23 2 The Axioms for Singular Homology and Some Consequences 24 2.1 The Homotopy Axiom for Singular Homology . 24 2.2 The Mayer-Vietoris Theorem for Singular Homology . 29 2.3 Relative Homology and the Long Exact Sequence of a Pair . 36 2.4 The Excision Axiom for Singular Homology . 37 2.5 The Dimension Axiom . 38 2.6 Reduced Homology . 39 1 3 Applications of Singular Homology 39 3.1 Invariance of Domain . 39 3.2 The Jordan Curve Theorem and its Generalizations . 40 3.3 Cellular (CW) Homology . 43 4 Other Homologies and Cohomologies 44 4.1 Singular Cohomology . -
On the Motivic Spectra Representing Algebraic Cobordism and Algebraic K-Theory
Documenta Math. 359 On the Motivic Spectra Representing Algebraic Cobordism and Algebraic K-Theory David Gepner and Victor Snaith Received: September 9, 2008 Communicated by Lars Hesselholt Abstract. We show that the motivic spectrum representing alge- braic K-theory is a localization of the suspension spectrum of P∞, and similarly that the motivic spectrum representing periodic alge- braic cobordism is a localization of the suspension spectrum of BGL. In particular, working over C and passing to spaces of C-valued points, we obtain new proofs of the topological versions of these theorems, originally due to the second author. We conclude with a couple of applications: first, we give a short proof of the motivic Conner-Floyd theorem, and second, we show that algebraic K-theory and periodic algebraic cobordism are E∞ motivic spectra. 2000 Mathematics Subject Classification: 55N15; 55N22 1. Introduction 1.1. Background and motivation. Let (X, µ) be an E∞ monoid in the ∞ category of pointed spaces and let β ∈ πn(Σ X) be an element in the stable ∞ homotopy of X. Then Σ X is an E∞ ring spectrum, and we may invert the “multiplication by β” map − − ∞ Σ nβ∧1 Σ nΣ µ µ(β):Σ∞X ≃ Σ∞S0 ∧ Σ∞X −→ Σ−nΣ∞X ∧ Σ∞X −→ Σ−nΣ∞X. to obtain an E∞ ring spectrum −n β∗ Σ β∗ Σ∞X[1/β] := colim{Σ∞X −→ Σ−nΣ∞X −→ Σ−2nΣ∞X −→···} with the property that µ(β):Σ∞X[1/β] → Σ−nΣ∞X[1/β] is an equivalence. ∞ ∞ In fact, as is well-known, Σ X[1/β] is universal among E∞ Σ X-algebras A in which β becomes a unit. -
Topology - Wikipedia, the Free Encyclopedia Page 1 of 7
Topology - Wikipedia, the free encyclopedia Page 1 of 7 Topology From Wikipedia, the free encyclopedia Topology (from the Greek τόπος , “place”, and λόγος , “study”) is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation. Ideas that are now classified as topological were expressed as early as 1736. Toward the end of the 19th century, a distinct A Möbius strip, an object with only one discipline developed, which was referred to in Latin as the surface and one edge. Such shapes are an geometria situs (“geometry of place”) or analysis situs object of study in topology. (Greek-Latin for “picking apart of place”). This later acquired the modern name of topology. By the middle of the 20 th century, topology had become an important area of study within mathematics. The word topology is used both for the mathematical discipline and for a family of sets with certain properties that are used to define a topological space, a basic object of topology. Of particular importance are homeomorphisms , which can be defined as continuous functions with a continuous inverse. For instance, the function y = x3 is a homeomorphism of the real line. Topology includes many subfields. The most basic and traditional division within topology is point-set topology , which establishes the foundational aspects of topology and investigates concepts inherent to topological spaces (basic examples include compactness and connectedness); algebraic topology , which generally tries to measure degrees of connectivity using algebraic constructs such as homotopy groups and homology; and geometric topology , which primarily studies manifolds and their embeddings (placements) in other manifolds. -
Math 535B Final Project: Lecture and Paper
Math 535b Final Project: Lecture and Paper 1. Overview and timeline The goal of the Math 535b final project is to explore some advanced aspect of Riemannian, symplectic, and/or K¨ahlergeometry (most likely chosen from a list below, but you are welcome to select and plan out a different topic, with instructor approval). You will then give two forms of exposition of this topic: • A 50 minute lecture, delivered to the class. • A 5+ page paper exposition of the topic. Your paper must be typeset (and I strongly recommend LATEX) The timeline for these assignments is: • Your lectures will be scheduled during the last two weeks of class, April 15-19 and 22- 26 - both during regular class time and during our make-up lecture slot, Wednesday between 4:30 and 6:30pm. We will schedule precise times later this week. If you have any scheduling constraints, please let me know ASAP. • Your final assignment will be due the last day of class, April 26. If it is necessary, this can be extended somewhat, but you will need to e-mail me in advance to schedule a final (hard) deadline. Some general requirements and/or suggestions: (1) In your paper, you are required to make reference to, in some non-trivial fashion, at least one original research paper. (you could also mention any insights you might have learned in your lecture though this is not necessary. Although many of the topics will be \textbook topics," i.e., they are now covered as advanced chapters in textbooks, there will be some component of the topic for which one or more original research papers are still the \best" or \most definitive" references.