The Serre Spectral Sequence: Construction and First Examples
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The Leray-Serre Spectral Sequence
THE LERAY-SERRE SPECTRAL SEQUENCE REUBEN STERN Abstract. Spectral sequences are a powerful bookkeeping tool, used to handle large amounts of information. As such, they have become nearly ubiquitous in algebraic topology and algebraic geometry. In this paper, we take a few results on faith (i.e., without proof, pointing to books in which proof may be found) in order to streamline and simplify the exposition. From the exact couple formulation of spectral sequences, we ∗ n introduce a special case of the Leray-Serre spectral sequence and use it to compute H (CP ; Z). Contents 1. Spectral Sequences 1 2. Fibrations and the Leray-Serre Spectral Sequence 4 3. The Gysin Sequence and the Cohomology Ring H∗(CPn; R) 6 References 8 1. Spectral Sequences The modus operandi of algebraic topology is that \algebra is easy; topology is hard." By associating to n a space X an algebraic invariant (the (co)homology groups Hn(X) or H (X), and the homotopy groups πn(X)), with which it is more straightforward to prove theorems and explore structure. For certain com- putations, often involving (co)homology, it is perhaps difficult to determine an invariant directly; one may side-step this computation by approximating it to increasing degrees of accuracy. This approximation is bundled into an object (or a series of objects) known as a spectral sequence. Although spectral sequences often appear formidable to the uninitiated, they provide an invaluable tool to the working topologist, and show their faces throughout algebraic geometry and beyond. ∗;∗ ∗;∗ Loosely speaking, a spectral sequence fEr ; drg is a collection of bigraded modules or vector spaces Er , equipped with a differential map dr (i.e., dr ◦ dr = 0), such that ∗;∗ ∗;∗ Er+1 = H(Er ; dr): That is, a bigraded module in the sequence comes from the previous one by taking homology. -
Some Extremely Brief Notes on the Leray Spectral Sequence Greg Friedman
Some extremely brief notes on the Leray spectral sequence Greg Friedman Intro. As a motivating example, consider the long exact homology sequence. We know that if we have a short exact sequence of chain complexes 0 ! C∗ ! D∗ ! E∗ ! 0; then this gives rise to a long exact sequence of homology groups @∗ ! Hi(C∗) ! Hi(D∗) ! Hi(E∗) ! Hi−1(C∗) ! : While we may learn the proof of this, including the definition of the boundary map @∗ in beginning algebraic topology and while this may be useful to know from time to time, we are often happy just to know that there is an exact homology sequence. Also, in many cases the long exact sequence might not be very useful because the interactions from one group to the next might be fairly complicated. However, in especially nice circumstances, we will have reason to know that certain homology groups vanish, or we might be able to compute some of the maps here or there, and then we can derive some consequences. For example, if ∼ we have reason to know that H∗(D∗) = 0, then we learn that H∗(E∗) = H∗−1(C∗). Similarly, to preserve sanity, it is usually worth treating spectral sequences as \black boxes" without looking much under the hood. While the exact details do sometimes come in handy to specialists, spectral sequences can be remarkably useful even without knowing much about how they work. What spectral sequences look like. In long exact sequence above, we have homology groups Hi(C∗) (of course there are other ways to get exact sequences). -
The Grothendieck Spectral Sequence (Minicourse on Spectral Sequences, UT Austin, May 2017)
The Grothendieck Spectral Sequence (Minicourse on Spectral Sequences, UT Austin, May 2017) Richard Hughes May 12, 2017 1 Preliminaries on derived functors. 1.1 A computational definition of right derived functors. We begin by recalling that a functor between abelian categories F : A!B is called left exact if it takes short exact sequences (SES) in A 0 ! A ! B ! C ! 0 to exact sequences 0 ! FA ! FB ! FC in B. If in fact F takes SES in A to SES in B, we say that F is exact. Question. Can we measure the \failure of exactness" of a left exact functor? The answer to such an obviously leading question is, of course, yes: the right derived functors RpF , which we will define below, are in a precise sense the unique extension of F to an exact functor. Recall that an object I 2 A is called injective if the functor op HomA(−;I): A ! Ab is exact. An injective resolution of A 2 A is a quasi-isomorphism in Ch(A) A ! I• = (I0 ! I1 ! I2 !··· ) where all of the Ii are injective, and where we think of A as a complex concentrated in degree zero. If every A 2 A embeds into some injective object, we say that A has enough injectives { in this situation it is a theorem that every object admits an injective resolution. So, for A 2 A choose an injective resolution A ! I• and define the pth right derived functor of F applied to A by RpF (A) := Hp(F (I•)): Remark • You might worry about whether or not this depends upon our choice of injective resolution for A { it does not, up to canonical isomorphism. -
HE WANG Abstract. a Mini-Course on Rational Homotopy Theory
RATIONAL HOMOTOPY THEORY HE WANG Abstract. A mini-course on rational homotopy theory. Contents 1. Introduction 2 2. Elementary homotopy theory 3 3. Spectral sequences 8 4. Postnikov towers and rational homotopy theory 16 5. Commutative differential graded algebras 21 6. Minimal models 25 7. Fundamental groups 34 References 36 2010 Mathematics Subject Classification. Primary 55P62 . 1 2 HE WANG 1. Introduction One of the goals of topology is to classify the topological spaces up to some equiva- lence relations, e.g., homeomorphic equivalence and homotopy equivalence (for algebraic topology). In algebraic topology, most of the time we will restrict to spaces which are homotopy equivalent to CW complexes. We have learned several algebraic invariants such as fundamental groups, homology groups, cohomology groups and cohomology rings. Using these algebraic invariants, we can seperate two non-homotopy equivalent spaces. Another powerful algebraic invariants are the higher homotopy groups. Whitehead the- orem shows that the functor of homotopy theory are power enough to determine when two CW complex are homotopy equivalent. A rational coefficient version of the homotopy theory has its own techniques and advan- tages: 1. fruitful algebraic structures. 2. easy to calculate. RATIONAL HOMOTOPY THEORY 3 2. Elementary homotopy theory 2.1. Higher homotopy groups. Let X be a connected CW-complex with a base point x0. Recall that the fundamental group π1(X; x0) = [(I;@I); (X; x0)] is the set of homotopy classes of maps from pair (I;@I) to (X; x0) with the product defined by composition of paths. Similarly, for each n ≥ 2, the higher homotopy group n n πn(X; x0) = [(I ;@I ); (X; x0)] n n is the set of homotopy classes of maps from pair (I ;@I ) to (X; x0) with the product defined by composition. -
Arxiv:Math/0305187V1 [Math.AT] 13 May 2003 Oehn Aiir Ieapiigo Iglrchmlg G Th Subtleties Cohomology Are Singular There of Time
MULTIPLICATIVE STRUCTURES ON HOMOTOPY SPECTRAL SEQUENCES II DANIEL DUGGER Contents 1. Introduction 1 2. Sign conventions in singular cohomology 1 3. Spectral sequences for filtered spaces 3 4. The Postnikov/Whitehead spectral sequence 8 5. Bockstein spectral sequences 13 6. Thehomotopy-fixed-pointspectralsequence 15 7. Spectralsequencesfromopencoverings 16 References 19 1. Introduction This short paper is a companion to [D1]. Here the main results of that paper are used to establish multiplicative structures on a few standard spectral sequences. The applications consist of (a) applying [D1, Theorem 6.1] to obtain a pairing of spectral sequences, and (b) identifying the pairing on the E1- or E2-term with something familiar, like a pairing of singular cohomology groups. Most of the arguments are straightforward, but there are subtleties that appear from time to time. Originally the aim was just to record a careful treatment of pairings on Post- nikov/Whitehead towers, but in the end other examples have been included because it made sense to do so. These examples are just the ones that I personally have needed to use at some point over the years, and so of course it is a very limited selection. arXiv:math/0305187v1 [math.AT] 13 May 2003 In this paper all the notation and conventions of [D1] remain in force. In particu- lar, the reader is referred to [D1, Appendix C] for our standing assumptions about the category of spaces and spectra, and for basic results about signs for bound- ary maps. The symbol ∧ denotes the derived functor of ∧, and W⊥ denotes an ‘augmented tower’ as in [D1, Section 6]. -
Picard Groupoids and $\Gamma $-Categories
PICARD GROUPOIDS AND Γ-CATEGORIES AMIT SHARMA Abstract. In this paper we construct a symmetric monoidal closed model category of coherently commutative Picard groupoids. We construct another model category structure on the category of (small) permutative categories whose fibrant objects are (permutative) Picard groupoids. The main result is that the Segal’s nerve functor induces a Quillen equivalence between the two aforementioned model categories. Our main result implies the classical result that Picard groupoids model stable homotopy one-types. Contents 1. Introduction 2 2. The Setup 4 2.1. Review of Permutative categories 4 2.2. Review of Γ- categories 6 2.3. The model category structure of groupoids on Cat 7 3. Two model category structures on Perm 12 3.1. ThemodelcategorystructureofPermutativegroupoids 12 3.2. ThemodelcategoryofPicardgroupoids 15 4. The model category of coherently commutatve monoidal groupoids 20 4.1. The model category of coherently commutative monoidal groupoids 24 5. Coherently commutative Picard groupoids 27 6. The Quillen equivalences 30 7. Stable homotopy one-types 36 Appendix A. Some constructions on categories 40 AppendixB. Monoidalmodelcategories 42 Appendix C. Localization in model categories 43 Appendix D. Tranfer model structure on locally presentable categories 46 References 47 arXiv:2002.05811v2 [math.CT] 12 Mar 2020 Date: Dec. 14, 2019. 1 2 A. SHARMA 1. Introduction Picard groupoids are interesting objects both in topology and algebra. A major reason for interest in topology is because they classify stable homotopy 1-types which is a classical result appearing in various parts of the literature [JO12][Pat12][GK11]. The category of Picard groupoids is the archetype exam- ple of a 2-Abelian category, see [Dup08]. -
Rational Homotopy Theory: a Brief Introduction
Contemporary Mathematics Rational Homotopy Theory: A Brief Introduction Kathryn Hess Abstract. These notes contain a brief introduction to rational homotopy theory: its model category foundations, the Sullivan model and interactions with the theory of local commutative rings. Introduction This overview of rational homotopy theory consists of an extended version of lecture notes from a minicourse based primarily on the encyclopedic text [18] of F´elix, Halperin and Thomas. With only three hours to devote to such a broad and rich subject, it was difficult to choose among the numerous possible topics to present. Based on the subjects covered in the first week of this summer school, I decided that the goal of this course should be to establish carefully the founda- tions of rational homotopy theory, then to treat more superficially one of its most important tools, the Sullivan model. Finally, I provided a brief summary of the ex- tremely fruitful interactions between rational homotopy theory and local algebra, in the spirit of the summer school theme “Interactions between Homotopy Theory and Algebra.” I hoped to motivate the students to delve more deeply into the subject themselves, while providing them with a solid enough background to do so with relative ease. As these lecture notes do not constitute a history of rational homotopy theory, I have chosen to refer the reader to [18], instead of to the original papers, for the proofs of almost all of the results cited, at least in Sections 1 and 2. The reader interested in proper attributions will find them in [18] or [24]. The author would like to thank Luchezar Avramov and Srikanth Iyengar, as well as the anonymous referee, for their helpful comments on an earlier version of this article. -
Fibrations I
Fibrations I Tyrone Cutler May 23, 2020 Contents 1 Fibrations 1 2 The Mapping Path Space 5 3 Mapping Spaces and Fibrations 9 4 Exercises 11 1 Fibrations In the exercises we used the extension problem to motivate the study of cofibrations. The idea was to allow for homotopy-theoretic methods to be introduced to an otherwise very rigid problem. The dual notion is the lifting problem. Here p : E ! B is a fixed map and we would like to known when a given map f : X ! B lifts through p to a map into E E (1.1) |> | p | | f X / B: Asking for the lift to make the diagram to commute strictly is neither useful nor necessary from our point of view. Rather it is more natural for us to ask that the lift exist up to homotopy. In this lecture we work in the unpointed category and obtain the correct conditions on the map p by formally dualising the conditions for a map to be a cofibration. Definition 1 A map p : E ! B is said to have the homotopy lifting property (HLP) with respect to a space X if for each pair of a map f : X ! E, and a homotopy H : X×I ! B starting at H0 = pf, there exists a homotopy He : X × I ! E such that , 1) He0 = f 2) pHe = H. The map p is said to be a (Hurewicz) fibration if it has the homotopy lifting property with respect to all spaces. 1 Since a diagram is often easier to digest, here is the definition exactly as stated above f X / E x; He x in0 x p (1.2) x x H X × I / B and also in its equivalent adjoint formulation X B H B He B B p∗ EI / BI (1.3) f e0 e0 # p E / B: The assertion that p is a fibration is the statement that the square in the second diagram is a weak pullback. -
Notes on Spectral Sequence
Notes on spectral sequence He Wang Long exact sequence coming from short exact sequence of (co)chain com- plex in (co)homology is a fundamental tool for computing (co)homology. Instead of considering short exact sequence coming from pair (X; A), one can consider filtered chain complexes coming from a increasing of subspaces X0 ⊂ X1 ⊂ · · · ⊂ X. We can see it as many pairs (Xp; Xp+1): There is a natural generalization of a long exact sequence, called spectral sequence, which is more complicated and powerful algebraic tool in computation in the (co)homology of the chain complex. Nothing is original in this notes. Contents 1 Homological Algebra 1 1.1 Definition of spectral sequence . 1 1.2 Construction of spectral sequence . 6 2 Spectral sequence in Topology 12 2.1 General method . 12 2.2 Leray-Serre spectral sequence . 15 2.3 Application of Leray-Serre spectral sequence . 18 1 Homological Algebra 1.1 Definition of spectral sequence Definition 1.1. A differential bigraded module over a ring R, is a collec- tion of R-modules, fEp;qg, where p, q 2 Z, together with a R-linear mapping, 1 H.Wang Notes on spectral Sequence 2 d : E∗;∗ ! E∗+s;∗+t, satisfying d◦d = 0: d is called the differential of bidegree (s; t). Definition 1.2. A spectral sequence is a collection of differential bigraded p;q R-modules fEr ; drg, where r = 1; 2; ··· and p;q ∼ p;q ∗;∗ ∼ p;q ∗;∗ ∗;∗ p;q Er+1 = H (Er ) = ker(dr : Er ! Er )=im(dr : Er ! Er ): In practice, we have the differential dr of bidegree (r; 1 − r) (for a spec- tral sequence of cohomology type) or (−r; r − 1) (for a spectral sequence of homology type). -
We Begin with a Review of the Classical Fibration Replacement Construction
Contents 11 Injective model structure 1 11.1 The existence theorem . 1 11.2 Injective fibrant models and descent . 13 12 Other model structures 18 12.1 Injective model structure for simplicial sheaves . 18 12.2 Intermediate model structures . 21 11 Injective model structure 11.1 The existence theorem We begin with a review of the classical fibration replacement construction. 1) Suppose that f : X ! Y is a map of Kan com- plexes, and form the diagram I f∗ I d1 X ×Y Y / Y / Y s f : d0∗ d0 / X f Y Then d0 is a trivial fibration since Y is a Kan com- plex, so d0∗ is a trivial fibration. The section s of d0 (and d1) induces a section s∗ of d0∗. Then (d1 f∗)s∗ = d1(s f ) = f 1 Finally, there is a pullback diagram I f∗ I X ×Y Y / Y (d0∗;d1 f∗) (d0;d1) X Y / Y Y × f ×1 × and the map prR : X ×Y ! Y is a fibration since X is fibrant, so that prR(d0∗;d1 f∗) = d1 f∗ is a fibra- tion. I Write Z f = X ×Y Y and p f = d1 f∗. Then we have functorial replacement s∗ d0∗ X / Z f / X p f f Y of f by a fibration p f , where d0∗ is a trivial fibra- tion such that d0∗s∗ = 1. 2) Suppose that f : X ! Y is a simplicial set map, and form the diagram j ¥ X / Ex X q f s∗ f∗ # f ˜ / Z f Z f∗ p f∗ ¥ { / Y j Ex Y 2 where the diagram ˜ / Z f Z f∗ p˜ f p f∗ ¥ / Y j Ex Y is a pullback. -
Etale Cohomology of Curves
ETALE COHOMOLOGY OF CURVES SIDDHARTH VENKATESH Abstract. These are notes for a talk on the etale cohomology of curves given in a grad student seminar on etale cohomology held in Spring 2016 at MIT. The talk follows Chapters 13 and 14 of Milne ([Mil3]). Contents 1. Outline of Talk 1 2. The Kummer Exact Sequence 2 3. Cohomology of Gm 4 4. Cohomology of µn 7 4.1. For Projective Curves 7 4.2. Lefschetz Trace Formula and Rationality of Zeta Function 7 4.3. For Affine Curves 10 5. Compactly Supported Cohomology for Curves 10 6. Poincare Duality 11 6.1. Local Systems in the Etale Topology 11 6.2. Etale Fundamental Group 11 6.3. The Duality Statement 13 7. Curves over Finite Fields 17 7.1. Hochschild-Serre Spectral Sequence 17 7.2. Poincare Duality 17 References 18 1. Outline of Talk The main goal of this talk is to prove a Poincare duality result for smooth curves over algebraically closed fields. Eventually, we will want to get a similar result for higher dimensional varieties as well but for now, we only have the machinery to deal with curves. Let me begin by recalling how Poincare duality works for smooth varieties of dimension n over C. In the analytic topology, we have a perfect pairing r 2n−r 2n H (X; Z) × Hc (X; Z) ! H (X; Z) = Z: More generally, you can replace the sheaf Z with any locally constant sheaf F to get a perfect pairing r 2n−r H (X; F) × H (X; Fˇ(1)) ! Z where Fˇ(1) = Hom(F; Z): This cannot be done in the Zariski topology because the topolgy is contractible (if the variety is irreducible). -
Lecture Notes on Homotopy Theory and Applications
LAURENTIUMAXIM UNIVERSITYOFWISCONSIN-MADISON LECTURENOTES ONHOMOTOPYTHEORY ANDAPPLICATIONS i Contents 1 Basics of Homotopy Theory 1 1.1 Homotopy Groups 1 1.2 Relative Homotopy Groups 7 1.3 Homotopy Extension Property 10 1.4 Cellular Approximation 11 1.5 Excision for homotopy groups. The Suspension Theorem 13 1.6 Homotopy Groups of Spheres 13 1.7 Whitehead’s Theorem 16 1.8 CW approximation 20 1.9 Eilenberg-MacLane spaces 25 1.10 Hurewicz Theorem 28 1.11 Fibrations. Fiber bundles 29 1.12 More examples of fiber bundles 34 1.13 Turning maps into fibration 38 1.14 Exercises 39 2 Spectral Sequences. Applications 41 2.1 Homological spectral sequences. Definitions 41 2.2 Immediate Applications: Hurewicz Theorem Redux 44 2.3 Leray-Serre Spectral Sequence 46 2.4 Hurewicz Theorem, continued 50 2.5 Gysin and Wang sequences 52 ii 2.6 Suspension Theorem for Homotopy Groups of Spheres 54 2.7 Cohomology Spectral Sequences 57 2.8 Elementary computations 59 n 2.9 Computation of pn+1(S ) 63 3 2.10 Whitehead tower approximation and p5(S ) 66 Whitehead tower 66 3 3 Calculation of p4(S ) and p5(S ) 67 2.11 Serre’s theorem on finiteness of homotopy groups of spheres 70 2.12 Computing cohomology rings via spectral sequences 74 2.13 Exercises 76 3 Fiber bundles. Classifying spaces. Applications 79 3.1 Fiber bundles 79 3.2 Principal Bundles 86 3.3 Classification of principal G-bundles 92 3.4 Exercises 97 4 Vector Bundles. Characteristic classes. Cobordism. Applications. 99 4.1 Chern classes of complex vector bundles 99 4.2 Stiefel-Whitney classes of real vector bundles 102 4.3 Stiefel-Whitney classes of manifolds and applications 103 The embedding problem 103 Boundary Problem.