The Hopf Invariant One Problem
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The Kervaire Invariant in Homotopy Theory
The Kervaire invariant in homotopy theory Mark Mahowald and Paul Goerss∗ June 20, 2011 Abstract In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the image of the J-homomorphism in the EHP sequence. One of the central projects of algebraic topology is to calculate the homotopy classes of maps between two finite CW complexes. Even in the case of spheres – the smallest non-trivial CW complexes – this project has a long and rich history. Let Sn denote the n-sphere. If k < n, then all continuous maps Sk → Sn are null-homotopic, and if k = n, the homotopy class of a map Sn → Sn is detected by its degree. Even these basic facts require relatively deep results: if k = n = 1, we need covering space theory, and if n > 1, we need the Hurewicz theorem, which says that the first non-trivial homotopy group of a simply-connected space is isomorphic to the first non-vanishing homology group of positive degree. The classical proof of the Hurewicz theorem as found in, for example, [28] is quite delicate; more conceptual proofs use the Serre spectral sequence. n Let us write πiS for the ith homotopy group of the n-sphere; we may also n write πk+nS to emphasize that the complexity of the problem grows with k. n n ∼ Thus we have πn+kS = 0 if k < 0 and πnS = Z. Given the Hurewicz theorem and knowledge of the homology of Eilenberg-MacLane spaces it is relatively simple to compute that 0, n = 1; n ∼ πn+1S = Z, n = 2; Z/2Z, n ≥ 3. -
Algebraic Hopf Invariants and Rational Models for Mapping Spaces
Algebraic Hopf Invariants and Rational Models for Mapping Spaces Felix Wierstra Abstract The main goal of this paper is to define an invariant mc∞(f) of homotopy classes of maps f : X → YQ, from a finite CW-complex X to a rational space YQ. We prove that this invariant is complete, i.e. mc∞(f)= mc∞(g) if an only if f and g are homotopic. To construct this invariant we also construct a homotopy Lie algebra structure on certain con- volution algebras. More precisely, given an operadic twisting morphism from a cooperad C to an operad P, a C-coalgebra C and a P-algebra A, then there exists a natural homotopy Lie algebra structure on HomK(C,A), the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to models mapping spaces. More precisely, suppose that C is a C∞-coalgebra model for a simply-connected finite CW- complex X and A an L∞-algebra model for a simply-connected rational space YQ of finite Q-type, then HomK(C,A), the space of linear maps from C to A, can be equipped with an L∞-structure such that it becomes a rational model for the based mapping space Map∗(X,YQ). Contents 1 Introduction 1 2 Conventions 3 I Preliminaries 3 3 Twisting morphisms, Koszul duality and bar constructions 3 4 The L∞-operad and L∞-algebras 5 5 Model structures on algebras and coalgebras 8 6 The Homotopy Transfer Theorem, P∞-algebras and C∞-coalgebras 9 II Algebraic Hopf invariants -
Adams Operations and Symmetries of Representation Categories Arxiv
Adams operations and symmetries of representation categories Ehud Meir and Markus Szymik May 2019 Abstract: Adams operations are the natural transformations of the representation ring func- tor on the category of finite groups, and they are one way to describe the usual λ–ring structure on these rings. From the representation-theoretical point of view, they codify some of the symmetric monoidal structure of the representation category. We show that the monoidal structure on the category alone, regardless of the particular symmetry, deter- mines all the odd Adams operations. On the other hand, we give examples to show that monoidal equivalences do not have to preserve the second Adams operations and to show that monoidal equivalences that preserve the second Adams operations do not have to be symmetric. Along the way, we classify all possible symmetries and all monoidal auto- equivalences of representation categories of finite groups. MSC: 18D10, 19A22, 20C15 Keywords: Representation rings, Adams operations, λ–rings, symmetric monoidal cate- gories 1 Introduction Every finite group G can be reconstructed from the category Rep(G) of its finite-dimensional representations if one considers this category as a symmetric monoidal category. This follows from more general results of Deligne [DM82, Prop. 2.8], [Del90]. If one considers the repre- sentation category Rep(G) as a monoidal category alone, without its canonical symmetry, then it does not determine the group G. See Davydov [Dav01] and Etingof–Gelaki [EG01] for such arXiv:1704.03389v3 [math.RT] 3 Jun 2019 isocategorical groups. Examples go back to Fischer [Fis88]. The representation ring R(G) of a finite group G is a λ–ring. -
Suspension of Ganea Fibrations and a Hopf Invariant
Topology and its Applications 105 (2000) 187–200 Suspension of Ganea fibrations and a Hopf invariant Lucile Vandembroucq 1 URA-CNRS 0751, U.F.R. de Mathématiques, Université des Sciences et Techniques de Lille, 59655 Villeneuve d’Ascq Cedex, France Received 24 August 1998; received in revised form 4 March 1999 Abstract We introduce a sequence of numerical homotopy invariants σ i cat;i2 N, which are lower bounds for the Lusternik–Schnirelmann category of a topological space X. We characterize, with dimension restrictions, the behaviour of σ i cat with respect to a cell attachment by means of a Hopf invariant. Furthermore we establish for σ i cat a product formula and deduce a sufficient C condition, in terms of the Hopf invariant, for a space X [ ep 1 to satisfy the Ganea conjecture, i.e., C C cat..X [ ep 1/ × Sm/ D cat.X [ ep 1/ C 1. This extends a recent result of Strom and a concrete example of this extension is given. 2000 Elsevier Science B.V. All rights reserved. Keywords: LS-category; Product formula; Hopf invariant AMS classification: 55P50; 55Q25 The Lusternik–Schnirelmann category of a topological space X, denoted cat X,isthe least integer n such that X can be covered by n C 1 open sets each of which is contractible in X. (If no such n exists, one sets cat X D1.) Category was originally introduced by Lusternik and Schnirelmann [18] to estimate the minimal number of critical points of differentiable functions on manifolds. They showed that a smooth function on a smooth manifold M admits at least cat M C 1 critical points. -
Realising Unstable Modules As the Cohomology of Spaces and Mapping Spaces
Realising unstable modules as the cohomology of spaces and mapping spaces G´eraldGaudens and Lionel Schwartz UMR 7539 CNRS, Universit´eParis 13 LIA CNRS Formath Vietnam Abstract This report discuss the question whether or not a given unstable module is the mod-p cohomology of a space. One first discuss shortly the Hopf invariant 1 problem and the Kervaire invariant 1 problem and give their relations to homotopy theory and geometric topology. Then one describes some more qualitative results, emphasizing the use of the space map(BZ=p; X) and of the structure of the category of unstable modules. 1 Introduction Let p be a prime number, in all the sequel H∗X will denote the the mod p singular cohomology of the topological space X. All spaces X will be supposed p-complete and connected. The singular mod-p cohomology is endowed with various structures: • it is a graded Fp-algebra, commutative in the graded sense, • it is naturally a module over the algebra of stable cohomology operations which is known as the mod p Steenrod algebra and denoted by Ap. The Steenrod algebra is generated by element Sqi of degree i > 0 if p = 2, β and P i of degree 1 and 2i(p − 1) > 0 if p > 2. These elements satisfy the Adem relations. In the mod-2 case the relations write [a=2] X b − t − 1 SqaSqb = Sqa+b−tSqt a − 2t 0 There are two types of relations for p > 2 [a=p] X (p − 1)(b − t) − 1 P aP b = (−1)a+t P a+b−tP t a − pt 0 1 for a; b > 0, and [a=p] [(a−1)=p] X (p − 1)(b − t) X (p − 1)(b − t) − 1 P aβP b = (−1)a+t βP a+b−tP t+ (−1)a+t−1 P a+b−tβP t a − pt a − pt − 1 0 0 for a; b > 0. -
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. -
Cochain Operations and Higher Cohomology Operations Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 42, No 4 (2001), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES STEPHAN KLAUS Cochain operations and higher cohomology operations Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 4 (2001), p. 261-284 <http://www.numdam.org/item?id=CTGDC_2001__42_4_261_0> © Andrée C. Ehresmann et les auteurs, 2001, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET Volume XLII-4 (2001) GEOMETRIE DIFFERENTIELLE CATEGORIQ UES COCHAIN OPERATIONS AND HIGHER COHOMOLOGY OPERATIONS By Stephan KLAUS RESUME. Etendant un programme initi6 par Kristensen, cet article donne une construction alg6brique des operations de cohomologie d’ordre sup6rieur instables par des operations de cochaine simpliciale. Des pyramides d’op6rations cocycle sont consid6r6es, qui peuvent 6tre utilisées pour une seconde construction des operations de cohomologie d’ordre superieur. 1. Introduction In this paper we consider the relation between cohomology opera- tions and simplicial cochain operations. This program was initialized by L. Kristensen in the case of (stable) primary, secondary and tertiary cohomology operations. The method is strong enough that Kristensen obtained sum, prod- uct and evaluation formulas for secondary cohomology operations by skilful combinatorial computations with special cochain operations ([8], [9], [10]). As significant examples of applications we mention the inde- pendent proof for the Hopf invariant one theorem by the computation of Kristensen of Massey products in the Steenrod algebra [11], the ex- amination of the /3-family in stable homotopy by L. -
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 -
ADAMS OPERATIONS on HIGHER K-THEORY Daniel R. Grayson
February 6, 1995 ADAMS OPERATIONS ON HIGHER K-THEORY Daniel R. Grayson University of Illinois at Urbana-Champaign Abstract. We construct Adams operations on higher algebraic K-groups induced by oper- ations such as symmetric powers on any suitable exact category, by constructing an explicit map of spaces, combinatorially defined. The map uses the S-construction of Waldhausen, and deloops (once) earlier constructions of the map. 1. Introduction. Let P be an exact category with a suitable notion of tensor product M ⊗ N, symmetric power SkM,andexteriorpowerΛkM. For example, we may take P to be the category P(X) of vector bundles on some scheme X.OrwemaytakeP to be the category P(R)of finitely generated projective R-modules, where R is a commutative ring. Or we may fix a group Γ and a commutative ring R and take P to be the category P(R, Γ) of representations of Γ on projective finitely generated R-modules. We impose certain exactness requirements on these functors, so that in particular the tensor product is required to be bi-exact, and this prevents us from taking for P a category such as the category M(R) of finitely generated R-modules. In a previous paper [5] I showed how to use the exterior power operations on modules to construct the lambda operations λk on the higher K-groups as a map of spaces λk : |GP| → |GkP| in a combinatorial fashion. Here the simplicial set GP, due to Gillet and me [3], provides an alternate definition for the K-groups of any exact category, KiP = πiGP, which has the advantage that the Grothendieck group appears as π0 and is not divorced from the higher K-groups. -
Local SU-Bordism
On K(1)-local SU-bordism Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) der Fakult¨at f¨urMathematik der Ruhr-Universit¨at Bochum vorgelegt von Dipl.-Math. Holger Reeker Mai 2009 Contents Chapter 1. Introduction and statement of results 5 Chapter 2. Some homotopical algebra 9 1. Generalized cohomology theories and spectra 9 2. Symmetric spectra over topological spaces 11 3. Complex oriented theories and computational methods 14 4. Bousfield localization of spectra 15 5. Algebraic manipulations of spectra 18 6. A resolution of the K(1)-local sphere 20 7. Thom isomorphism 21 Chapter 3. The algebraic structure of K(1)-local E∞ ring spectra 23 1. Operads 23 2. Dyer-Lashof operations for K(1)-local E∞ spectra 25 3. The θ-algebra structure of π0K ∧ MU 26 4. The θ-algebra structure of π0K ∧ MSU 29 Chapter 4. Splitting off an E∞ summand Tζ 31 1. The image of MSU∗ → MU∗ 32 2. Formal group laws and Miscenkos formula 34 3. Construction of an SU-manifold with Aˆ = 1 38 4. Construction of an Artin-Schreier class 41 5. Construction of an E∞ map Tζ → MSU 41 6. Split map - direct summand argument 42 0 Chapter 5. Detecting free E∞ summands TS 45 1. Introduction to Adams operations in K-theory 45 ∞ 2. Calculating operations on K∗CP 47 3. Bott’s formula and cannibalistic classes 52 4. Spherical classes in K∗MSU 55 5. Umbral calculus 57 Chapter 6. Open questions and concluding remarks 61 Bibliography 63 3 CHAPTER 1 Introduction and statement of results One of the highlights in algebraic topology was the invention of generalized homology and cohomology theories by Whitehead and Brown in the 1960s. -
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. -
Adams Operations in Differential Algebraic K-Theory
Adams Operations in Differential Algebraic K-theory Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der Fakult¨atf¨urMathematik der Universit¨atRegensburg vorgelegt von Christoph Schrade aus Leipzig im Jahr 2018 Promotionsgesuch eingereicht am: 29.05.2018 Die Arbeit wurde angeleitet von: • Prof. Dr. Ulrich Bunke • Dr. Georg Tamme 1 Contents 1. Introduction 3 2. 1-categorical background 7 2.1. General theory 7 2.2. Higher commutative algebra 25 2.3. Additive and stable 1-categories 41 3. Multiplicative Adams operations on algebraic K-theory 50 3.1. The Snaith model for the motivic K-theory spectrum 51 3.2. Construction of the Adams operations 61 4. Adams operations on differential algebraic K-theory 66 4.1. The Beilinson regulator as a map of ring spectra 67 4.2. Construction of the differential refinement of the Adams operations 79 Appendix A. Segre embeddings and commutative algebra structure on P1 95 References 101 3 1. Introduction Adams operations first appeared in the context of K-theory of topo- logical spaces. For a finite CW-complex X there is the zeroth topolog- ical K-group K0(X) constructed by group completion of the monoid Vect(X) of isomorphism classes of finite rank vector bundles over X with the direct sum operation. In fact there is even a ring structure on K0(X) which is induced by the tensor product operation of vector bundles. For a natural number k, the k-th Adams operation is then a ring homomorphism k : K0(X) ! K0(X) which is characterized by the property that it sends the class of a line bundle to the class of its k-th tensor power.