An Introduction to Spectra
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L22 Adams Spectral Sequence
Lecture 22: Adams Spectral Sequence for HZ=p 4/10/15-4/17/15 1 A brief recall on Ext We review the definition of Ext from homological algebra. Ext is the derived functor of Hom. Let R be an (ordinary) ring. There is a model structure on the category of chain complexes of modules over R where the weak equivalences are maps of chain complexes which are isomorphisms on homology. Given a short exact sequence 0 ! M1 ! M2 ! M3 ! 0 of R-modules, applying Hom(P; −) produces a left exact sequence 0 ! Hom(P; M1) ! Hom(P; M2) ! Hom(P; M3): A module P is projective if this sequence is always exact. Equivalently, P is projective if and only if if it is a retract of a free module. Let M∗ be a chain complex of R-modules. A projective resolution is a chain complex P∗ of projec- tive modules and a map P∗ ! M∗ which is an isomorphism on homology. This is a cofibrant replacement, i.e. a factorization of 0 ! M∗ as a cofibration fol- lowed by a trivial fibration is 0 ! P∗ ! M∗. There is a functor Hom that takes two chain complexes M and N and produces a chain complex Hom(M∗;N∗) de- Q fined as follows. The degree n module, are the ∗ Hom(M∗;N∗+n). To give the differential, we should fix conventions about degrees. Say that our differentials have degree −1. Then there are two maps Y Y Hom(M∗;N∗+n) ! Hom(M∗;N∗+n−1): ∗ ∗ Q Q The first takes fn to dN fn. -
Generalized Cohomology Theories
Lecture 4: Generalized cohomology theories 1/12/14 We've now defined spectra and the stable homotopy category. They arise naturally when considering cohomology. Proposition 1.1. For X a finite CW-complex, there is a natural isomorphism 1 ∼ r [Σ X; HZ]−r = H (X; Z). The assumption that X is a finite CW-complex is not necessary, but here is a proof in this case. We use the following Lemma. Lemma 1.2. ([A, III Prop 2.8]) Let F be any spectrum. For X a finite CW- 1 n+r complex there is a natural identification [Σ X; F ]r = colimn!1[Σ X; Fn] n+r On the right hand side the colimit is taken over maps [Σ X; Fn] ! n+r+1 n+r [Σ X; Fn+1] which are the composition of the suspension [Σ X; Fn] ! n+r+1 n+r+1 n+r+1 [Σ X; ΣFn] with the map [Σ X; ΣFn] ! [Σ X; Fn+1] induced by the structure map of F ΣFn ! Fn+1. n+r Proof. For a map fn+r :Σ X ! Fn, there is a pmap of degree r of spectra Σ1X ! F defined on the cofinal subspectrum whose mth space is ΣmX for m−n−r m ≥ n+r and ∗ for m < n+r. This pmap is given by Σ fn+r for m ≥ n+r 0 n+r and is the unique map from ∗ for m < n+r. Moreover, if fn+r; fn+r :Σ X ! 1 Fn are homotopic, we may likewise construct a pmap Cyl(Σ X) ! F of degree n+r 1 r. -
On Relations Between Adams Spectral Sequences, with an Application to the Stable Homotopy of a Moore Space
Journal of Pure and Applied Algebra 20 (1981) 287-312 0 North-Holland Publishing Company ON RELATIONS BETWEEN ADAMS SPECTRAL SEQUENCES, WITH AN APPLICATION TO THE STABLE HOMOTOPY OF A MOORE SPACE Haynes R. MILLER* Harvard University, Cambridge, MA 02130, UsA Communicated by J.F. Adams Received 24 May 1978 0. Introduction A ring-spectrum B determines an Adams spectral sequence Ez(X; B) = n,(X) abutting to the stable homotopy of X. It has long been recognized that a map A +B of ring-spectra gives rise to information about the differentials in this spectral sequence. The main purpose of this paper is to prove a systematic theorem in this direction, and give some applications. To fix ideas, let p be a prime number, and take B to be the modp Eilenberg- MacLane spectrum H and A to be the Brown-Peterson spectrum BP at p. For p odd, and X torsion-free (or for example X a Moore-space V= So Up e’), the classical Adams E2-term E2(X;H) may be trigraded; and as such it is E2 of a spectral sequence (which we call the May spectral sequence) converging to the Adams- Novikov Ez-term E2(X; BP). One may say that the classical Adams spectral sequence has been broken in half, with all the “BP-primary” differentials evaluated first. There is in fact a precise relationship between the May spectral sequence and the H-Adams spectral sequence. In a certain sense, the May differentials are the Adams differentials modulo higher BP-filtration. One may say the same for p=2, but in a more attenuated sense. -
Dylan Wilson March 23, 2013
Spectral Sequences from Sequences of Spectra: Towards the Spectrum of the Category of Spectra Dylan Wilson March 23, 2013 1 The Adams Spectral Sequences As is well known, it is our manifest destiny as 21st century algebraic topologists to compute homotopy groups of spheres. This noble venture began even before the notion of homotopy was around. In 1931, Hopf1 was thinking about a map he had encountered in geometry from S3 to S2 and wondered whether or not it was essential. He proved that it was by considering the linking of the fibers. After Hurewicz developed the notion of higher homotopy groups this gave the first example, aside from the self-maps of spheres, of a non-trivial higher homotopy group. Hopf classified maps from S3 to S2 and found they were given in a manner similar to degree, generated by the Hopf map, so that 2 π3(S ) = Z In modern-day language we would prove the nontriviality of the Hopf map by the following argument. Consider the cofiber of the map S3 ! S2. By construction this is CP 2. If the map were nullhomotopic then the cofiber would be homotopy equivalent to a wedge S2 _ S3. But the cup-square of the generator in H2(CP 2) is the generator of H4(CP 2), so this can't happen. This gives us a general procedure for constructing essential maps φ : S2n−1 ! Sn. Cook up fancy CW- complexes built of two cells, one in dimension n and another in dimension 2n, and show that the square of the bottom generator is the top generator. -
Introduction to the Stable Category
AN INTRODUCTION TO THE CATEGORY OF SPECTRA N. P. STRICKLAND 1. Introduction Early in the history of homotopy theory, people noticed a number of phenomena suggesting that it would be convenient to work in a context where one could make sense of negative-dimensional spheres. Let X be a finite pointed simplicial complex; some of the relevant phenomena are as follows. n−2 • For most n, the homotopy sets πnX are abelian groups. The proof involves consideration of S and so breaks down for n < 2; this would be corrected if we had negative spheres. • Calculation of homology groups is made much easier by the existence of the suspension isomorphism k Hen+kΣ X = HenX. This does not generally work for homotopy groups. However, a theorem of k Freudenthal says that if X is a finite complex, we at least have a suspension isomorphism πn+kΣ X = k+1 −k πn+k+1Σ X for large k. If could work in a context where S makes sense, we could smash everything with S−k to get a suspension isomorphism in homotopy parallel to the one in homology. • We can embed X in Sk+1 for large k, and let Y be the complement. Alexander duality says that k−n HenY = He X, showing that X can be \turned upside-down", in a suitable sense. The shift by k is unpleasant, because the choice of k is not canonical, and the minimum possible k depends on X. Moreover, it is unsatisfactory that the homotopy type of Y is not determined by that of X (even after taking account of k). -
Spectra and Stable Homotopy Theory
Spectra and stable homotopy theory Lectures delivered by Michael Hopkins Notes by Akhil Mathew Fall 2012, Harvard Contents Lecture 1 9/5 x1 Administrative announcements 5 x2 Introduction 5 x3 The EHP sequence 7 Lecture 2 9/7 x1 Suspension and loops 9 x2 Homotopy fibers 10 x3 Shifting the sequence 11 x4 The James construction 11 x5 Relation with the loopspace on a suspension 13 x6 Moore loops 13 Lecture 3 9/12 x1 Recap of the James construction 15 x2 The homology on ΩΣX 16 x3 To be fixed later 20 Lecture 4 9/14 x1 Recap 21 x2 James-Hopf maps 21 x3 The induced map in homology 22 x4 Coalgebras 23 Lecture 5 9/17 x1 Recap 26 x2 Goals 27 Lecture 6 9/19 x1 The EHPss 31 x2 The spectral sequence for a double complex 32 x3 Back to the EHPss 33 Lecture 7 9/21 x1 A fix 35 x2 The EHP sequence 36 Lecture 8 9/24 1 Lecture 9 9/26 x1 Hilton-Milnor again 44 x2 Hopf invariant one problem 46 x3 The K-theoretic proof (after Atiyah-Adams) 47 Lecture 10 9/28 x1 Splitting principle 50 x2 The Chern character 52 x3 The Adams operations 53 x4 Chern character and the Hopf invariant 53 Lecture 11 8/1 x1 The e-invariant 54 x2 Ext's in the category of groups with Adams operations 56 Lecture 12 10/3 x1 Hopf invariant one 58 Lecture 13 10/5 x1 Suspension 63 x2 The J-homomorphism 65 Lecture 14 10/10 x1 Vector fields problem 66 x2 Constructing vector fields 70 Lecture 15 10/12 x1 Clifford algebras 71 x2 Z=2-graded algebras 73 x3 Working out Clifford algebras 74 Lecture 16 10/15 x1 Radon-Hurwitz numbers 77 x2 Algebraic topology of the vector field problem 79 x3 The homology of -
LECTURE 2: WALDHAUSEN's S-CONSTRUCTION 1. Introduction
LECTURE 2: WALDHAUSEN'S S-CONSTRUCTION AMIT HOGADI 1. Introduction In the previous lecture, we saw Quillen's Q-construction which associated a topological space to an exact category. In this lecture we will see Waldhausen's construction which associates a spectrum to a Waldhausen category. All exact categories are Waldhausen categories, but there are other important examples (see2). Spectra are more easy to deal with than topological spaces. In the case, when a topological space is an infinite loop space (e.g. Quillen's K-theory space), one does not loose information while passing from topological spaces to spectra. 2. Waldhausen's S construction 1. Definition (Waldhausen category). A Waldhausen category is a small cate- gory C with a distinguished zero object 0, equipped with two subcategories of morphisms co(C) (cofibrations, to be denoted by ) and !(C) (weak equiva- lences) such that the following axioms are satisfied: (1) Isomorphisms are cofibrations as well as weak equivalences. (2) The map from 0 to any object is a cofibration. (3) Pushouts by cofibration exist and are cofibrations. (4) (Gluing) Given a commutative diagram C / A / / B ∼ ∼ ∼ C0 / A0 / / B0 where vertical arrows are weak equivalences and the two right horizontal maps are cofibrations, the induced map on pushouts [ [ C B ! C0 B0 A A0 is a weak equivalence. By a cofibration sequence in C we will mean a sequence A B C such that A B is a cofibration and C is the cokernel (which always exists since pushout by cofibrations exist). 1 2 AMIT HOGADI 2. Example. Every exact category is a Waldhausen category where cofibrations are inflations and weak equivalences are isomorphisms. -
THE ORIENTED COBORDISM RING Contents Introduction 1 1
THE ORIENTED COBORDISM RING ARAMINTA GWYNNE Abstract. We give an exposition of the computation of the oriented cobor- SO dism ring Ω∗ using the Adam's spectral sequence. Our proof follows Pengel- ley [15]. The unoriented and complex cobordism rings are also computed in a similar fashion. Contents Introduction 1 1. Homology of classifying spaces 4 2. The stable Hurewicz map and rational data 5 3. Steenrod algebra structures 5 4. Main theorem: statement and remarks 6 5. Proof of the main theorem 8 6. A spectrum level interpretation 11 7. Adams spectral sequence 12 8. The easier torsion 13 8.1. The MO case 14 8.2. The MU and MSO cases 14 9. The harder torsion 15 10. Relation to other proofs 17 11. Consequences 21 Acknowledgments 22 References 22 Introduction In his now famous paper [19], Thom showed that cobordism rings are isomor- phic to stable homotopy groups of Thom spectra, and used this isomorphism to compute certain cobordism rings. We'll use the notation N∗ for the unoriented U SO cobordism ring, Ω∗ for the complex cobordism ring, and Ω∗ for the oriented ∼ U ∼ cobordism ring. Then Thom's theorem says that N∗ = π∗(MO), Ω∗ = π∗(MU), SO ∼ and Ω∗ = π∗(MSO). Thom's theorem is one of the main examples of a general approach to problems in geometric topology: use classifying spaces to translate the problem into the world of algebraic topology, and then use algebraic tools to compute. For the majority of this paper, we will focus on the algebraic side of the computation. -
Equivariant Stable Homotopy Theory
EQUIVARIANT STABLE HOMOTOPY THEORY J.P.C. GREENLEES AND J.P. MAY Contents Introduction 1 1. Equivariant homotopy 2 2. The equivariant stable homotopy category 10 3. Homologyandcohomologytheoriesandfixedpointspectra 15 4. Change of groups and duality theory 20 5. Mackey functors, K(M,n)’s, and RO(G)-gradedcohomology 25 6. Philosophy of localization and completion theorems 30 7. How to prove localization and completion theorems 34 8. Examples of localization and completion theorems 38 8.1. K-theory 38 8.2. Bordism 40 8.3. Cohomotopy 42 8.4. The cohomology of groups 45 References 46 Introduction The study of symmetries on spaces has always been a major part of algebraic and geometric topology, but the systematic homotopical study of group actions is relatively recent. The last decade has seen a great deal of activity in this area. After giving a brief sketch of the basic concepts of space level equivariant homo- topy theory, we shall give an introduction to the basic ideas and constructions of spectrum level equivariant homotopy theory. We then illustrate ideas by explain- ing the fundamental localization and completion theorems that relate equivariant to nonequivariant homology and cohomology. The first such result was the Atiyah-Segal completion theorem which, in its simplest terms, states that the completion of the complex representation ring R(G) at its augmentation ideal I is isomorphic to the K-theory of the classifying space ∧ ∼ BG: R(G)I = K(BG). A more recent homological analogue of this result describes 1 2 J.P.C. GREENLEES AND J.P. MAY the K-homology of BG. -
A New Proof of the Bott Periodicity Theorem
Topology and its Applications 119 (2002) 167–183 A new proof of the Bott periodicity theorem Mark J. Behrens Department of Mathematics, University of Chicago, Chicago, IL 60615, USA Received 28 February 2000; received in revised form 27 October 2000 Abstract We give a simplification of the proof of the Bott periodicity theorem presented by Aguilar and Prieto. These methods are extended to provide a new proof of the real Bott periodicity theorem. The loop spaces of the groups O and U are identified by considering the fibers of explicit quasifibrations with contractible total spaces. 2002 Elsevier Science B.V. All rights reserved. AMS classification: Primary 55R45, Secondary 55R65 Keywords: Bott periodicity; Homotopy groups; Spaces and groups of matrices 1. Introduction In [1], Aguilar and Prieto gave a new proof of the complex Bott periodicity theorem based on ideas of McDuff [4]. The idea of the proof is to use an appropriate restriction of the exponential map to construct an explicit quasifibration with base space U and contractible total space. The fiber of this map is seen to be BU ×Z. This proof is compelling because it is more elementary and simpler than previous proofs. In this paper we present a streamlined version of the proof by Aguilar and Prieto, which is simplified by the introduction of coordinate free vector space notation and a more convenient filtration for application of the Dold–Thom theorem. These methods are then extended to prove the real Bott periodicity theorem. 2. Preliminaries We shall review the necessary facts about quasifibrations that will be used in the proof of the Bott periodicity theorem, as well as prove a technical result on the behavior of the E-mail address: [email protected] (M.J. -
The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres
The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres Paul Goerss∗ Abstract These are notes for a five lecture series intended to uncover large-scale phenomena in the homotopy groups of spheres using the Adams-Novikov Spectral Sequence. The lectures were given in Strasbourg, May 7–11, 2007. May 21, 2007 Contents 1 The Adams spectral sequence 2 2 Classical calculations 5 3 The Adams-Novikov Spectral Sequence 10 4 Complex oriented homology theories 13 5 The height filtration 21 6 The chromatic decomposition 25 7 Change of rings 29 8 The Morava stabilizer group 33 ∗The author were partially supported by the National Science Foundation (USA). 1 9 Deeper periodic phenomena 37 A note on sources: I have put some references at the end of these notes, but they are nowhere near exhaustive. They do not, for example, capture the role of Jack Morava in developing this vision for stable homotopy theory. Nor somehow, have I been able to find a good way to record the overarching influence of Mike Hopkins on this area since the 1980s. And, although, I’ve mentioned his name a number of times in this text, I also seem to have short-changed Mark Mahowald – who, more than anyone else, has a real and organic feel for the homotopy groups of spheres. I also haven’t been very systematic about where to find certain topics. If I seem a bit short on references, you can be sure I learned it from the absolutely essential reference book by Doug Ravenel [27] – “The Green Book”, which is not green in its current edition. -
Chromatic Structures in Stable Homotopy Theory
CHROMATIC STRUCTURES IN STABLE HOMOTOPY THEORY TOBIAS BARTHEL AND AGNES` BEAUDRY Abstract. In this survey, we review how the global structure of the stable homotopy category gives rise to the chromatic filtration. We then discuss computational tools used in the study of local chromatic homo- topy theory, leading up to recent developments in the field. Along the way, we illustrate the key methods and results with explicit examples. Contents 1. Introduction1 2. A panoramic view of the chromatic landscape4 3. Local chromatic homotopy theory 14 4. K(1)-local homotopy theory 29 5. Finite resolutions and their spectral sequences 35 6. Chromatic splitting, duality, and algebraicity 46 References 57 1. Introduction At its core, chromatic homotopy theory provides a natural approach to 0 the computation of the stable homotopy groups of spheres π∗S . Histori- cally, the first few of these groups were computed geometrically through the classification of stably framed manifolds, using the Pontryagin{Thom iso- 0 fr morphism π∗S ∼= Ω∗ . However, beginning with the work of Serre, it soon arXiv:1901.09004v2 [math.AT] 29 Apr 2019 turned out that algebraic tools were more effective, both for the computation of specific low-degree values as well as for establishing structural results. In 0 particular, Serre proved that π∗S is a degreewise finitely generated abelian 0 group with π0S ∼= Z and that all higher groups are torsion. Serre's method was essentially inductive: starting with the knowledge of 0 0 0 the first n groups π0S ; : : : ; πn−1S , one can in principle compute πnS . Said 0 differently, Serre worked with the Postnikov filtration of π∗S , in which the 0 (n + 1)st filtration quotient is given by πnS .