DOCSLIB.ORG

Fibrewise Stable Rational Homotopy

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fibrewise Stable Rational Homotopy Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 Fibrewise Stable Rational Homotopy Yves F´elix, Aniceto Murillo and Daniel Tanr´e Abstract In this paper, for a given space B we establish a correspondence between differential graded ∗ modules over C (B; Q) and fibrewise rational stable spaces over B. This correspondence opens the door for topological translations of algebraic constructions made with modules over a commutative differential graded algebra. More precisely, given fibrations E → B and ′ E → B, the set of stable rational homotopy classes of maps over B is isomorphic to ∗ ∗ ′ ∗ ExtC∗(B;Q)`C (E ; Q),C (E; Q)´. In particular, a nilpotent, finite type CW-complex X is a q rational Poincar´ecomplex if there exist non trivial stable maps over XQ from (X × S )Q to q+N (X ∨ S )Q for exactly one N. 1. Introduction Sullivan’s approach to rational homotopy theory is based on an adjoint pair of functors between the category of simplicial sets and the category CDGA of commutative differential graded algebras over Q (henceforth called cdga’s ). This correspondence induces an equivalence between the homotopy category of rational (nilpotent and with finite Betti numbers) simplicial sets and a homotopy category of cdga’s. The fact that any algebraic construction or property in CDGA has a topological translation is the most important feature of rational homotopy theory. Until now, however, there was no known procedure for realizing differential graded modules (henceforth called dgm’s) over cdga’s topologically, even though a number of important topological invariants have been interpreted in terms of dgm’s over the years. For instance, by [4], the rational cohomology of a nilpotent space X satisfies Poincar´eduality if and only ∗ if the module invariant ExtC∗(X;Q)(Q, C (X; Q)) has dimension one. This fact is one of the motivations of the present paper, in which we establish a general method for a topological realization of the category DGM of dgm’s. We consider maps over a fixed space B. The map is called pointed if it has a fixed section. For each map p: E → B, we denote by p: E+ → B the associated pointed map with E+ = E ∐ B. q The q-th fibrewise suspension of the map p: E → B is the map ΣBE → B, with q q q−1 ΣBE = (E × D ) ∪(E×Sq−1 ) (B × S ). The augmented q-th fibrewise suspension of p is the q-th fibrewise suspension of the map E+ → B, therefore q q q ΣBE+ = (E × D ) ∪(E×Sq−1 ) (B × D ). If p is a fibration of fibre F , then ΣBE → B is a fibration of fibre ΣF and ΣBE+ → B a fibration of fibre ΣF+, cf. [18, Lemma 6]. Given maps p: E → B and p′ : E′ → B, we use the following notation: ′ ′ • [E, E ]B is the set of homotopy classes of maps over B, i.e., maps f : E → E satisfying p′ ◦ f = p, with homotopy over B, 2000 Mathematics Subject Classification 55P62 (primary), 18G15, 55P42, 55R70, 55M30 (secondary). The three authors are partially supported by the MICINN grant MTM2010-18089 Page 2 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´ ′ q q ′ B ′ • [Σ E+, Σ E ] , for q, q > 0, is the set of pointed homotopy classes of maps over B B + B ′ q q ′ ′ ′ ′ B, i.e., maps f : ΣBE+ → ΣB E+ satisfying p ◦ f = p and f ◦ σ = σ , where σ, σ are the canonical sections of the augmented fibrewise suspensions, with homotopy over and under B. By taking iterated augmented fibrewise suspension, we get a sequence of sets of homotopy classes ′ Σ ′ B Σ 2 2 ′ B [E, E ]B → [ΣBE+, ΣBE+]B → [ΣBE+, ΣBE+]B →··· The set of stable fibrewise homotopy classes of maps, of degree r, from E to E′ is the limit ′ r q q+r B E, E = lim [Σ E+, Σ E+]B . B q→+∞ B B In particular, two fibrations, p: E → B and p′ : E′ → B, are called stably rationally equivalent, 0 if they are connected by an isomorphism in E, E′ . B Stable fibrewise homotopy is the subject of the book of Crabb and James [3]. In [3, Proposition 15.8], with some restrictions on the involved spaces, the authors prove that the rational fibrewise stable theory over B can be identified as a fibrewise rational cohomology. In this text, we describe it in terms of an Ext-group in the homotopy category of modules over a cdga model of B. Our main result can be stated as follows. Theorem 1.1. Let p: E → B and p′ : E′ → B be two fibrations of fibres F and F ′ respectively. We suppose that E, E′ and B are nilpotent spaces with finite Betti numbers ′ and that the fundamental group of B acts locally nilpotently in each Hi(F ; Q) and Hi(F ; Q). Then, we have a bijection ′ r r ∗ ′ ∗ EQ, E =∼ Ext ∗ C (E ; Q), C (E; Q) , Q B C (B;Q) ∗ where C (−; Q) denotes the usual cochain algebra with rational coefficients and −Q the rationalization functor. The definition of a locally nilpotent action is given in [9], see also [5, Page 197]. In particular, the previous hypothesis are fulfilled if all spaces are simply connected with finite Betti numbers. As a consequence of Theorem 1.1, the two fibrations, p and p′, are stably rationally equivalent if, and only if, there is an isomorphism of C∗(B; Q)-modules between C∗(E; Q) and C∗(E′; Q). This can be stated as follows. Corollary 1.2. Two fibrations p: E → B and p′ : E′ → B are stably rationally equiva- lent, if, and only if, C∗(E; Q) and C∗(E′; Q) have isomorphic semifree models as modules over C∗(B; Q). Lusternik-Schnirelmann category (LS category henceforth) is another invariant which has been characterized by B. Jessup [11] and K. Hess [10] in the homotopy category of DGM. Later on, it was topologically reinterpreted in terms of fibrewise symmetric products [16]. Here, in section 3, we give a new description of stable LS category in terms of fibrewise suspensions. There are two particular fibrations to which we apply Theorem 1.1: the trivial fibration ∗ → X → X, whose augmented q-th fibrewise suspension is the trivial fibration Sq → X × Sq → X, and the path fibration ΩX → PX → X, whose augmented q-th fibrewise suspension is the fibration ΩX ⋉ Sq−1 → X ∨ Sq → X. FIBREWISE STABLE RATIONAL HOMOTOPY Page 3 of 16 Theorem 1.3. Let X be a nilpotent space with finite Betti numbers. Then there is an isomorphism, r ∗ ∼ r ∼ q q+r XQ ExtC∗(X;Q) Q, C (X; Q) = XQ,PXQ = lim [XQ × S ,XQ ∨ S ]X XQ q→+∞ Q which fits in the commutative diagram ∼ r = / r ∗ XQ,PXQ / Ext ∗ Q Q, C (X; Q) XQ C (X; ) OO ll OO lll OOO lll θ OOO lll ev O' ulll Hr(X; Q). Here, ev is the usual evaluation map [4, 6, 14] and θ associates to a map f : X × Sq → q+r ∗ q+r q X ∨ S the slant product of f [S ] with the generator of Hq(S ). In particular, Theorem 1.1 provides a new perspective on Poincar´eduality. Theorem 1.4. Let X be a nilpotent CW-complex of finite LS category and with finite Betti numbers. Then, X is a rational Poincar´eduality complex if, and only if, there is an integer N such that: (i) for r 6= N, all pointed stable maps over X, X × Sq → X ∨ Sq+r, are rationally homotopically trivial, (ii) there is, up to a multiple, only one non trivial pointed stable rational map over X, X × Sq → X ∨ Sq+N . In that case, N is the dimension of X. Theorem 1.5. Let X be a nilpotent CW-complex, with finite Betti numbers and for which πq(X) ⊗ Q =0 for q large enough. Then, there is an integer N such that: (i) for r 6= N, all pointed stable maps over X, X × Sq → X ∨ Sq+r, are rationally homotopically trivial, (ii) there is, up to a multiple, only one non trivial pointed stable rational map over X, X × Sq → X ∨ Sq+N . Finally, we point out that fibrewise suspensions appear also in the study of embeddings. 1 2 Let M and X be compact n-dimensional manifolds and let j : M × [ 3 , 3 ] ֒→ X × [0, 1] be an embedding with complement W . Then the projection W ֒→ X × [0, 1] → X makes W a pointed 1 space over X. The composite ν+ : M ∐ X ֒→ (M ×{ 3 }) ∐ (X ×{0}) ֒→ W is a map of pointed X 12 spaces over X. So [ν+] ∈ [M+, W ]X . In [ ], J. Klein shows that [ν+] = 0 if, and only if, there is an embedding of M in X which induces the given embedding j. Our program is carried out in Sections 1-7 below, whose headings are self explanatory. Contents 1. Introduction ................ 1 2. Fibrewise suspensions .............. 4 3. Lusternik-Schnirelmann category ........... 6 4. Differential modules over differential algebras ........ 6 5. Sullivan models of fibrewise suspensions ......... 8 6. Topology of the functor Ext ............ 11 7. Applications to Poincar´eduality ........... 13 Page 4 of 16 YVES FELIX,´ ANICETO MURILLO AND DANIEL TANRE´ References ................. 15 In this text, all spaces are compactly generated and of the homotopy type of a CW-complex. p By a fibration F → E → B, we mean a map p whose homotopy fibre is F . 2. Fibrewise suspensions The q-th suspension of a space X, ΣqX, is the pushout q q q−1 Σ X = (X × D ) ∪X×Sq−1 ({∗} × S ).
Load more

Recommended publications
  • 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.
    [Show full text]
  • Remarks on Combinatorial and Accessible Model Categories
    Theory and Applications of Categories, Vol. 37, No. 9, 2021, pp. 266{275. REMARKS ON COMBINATORIAL AND ACCESSIBLE MODEL CATEGORIES JIRˇ´I ROSICKY´ Abstract. Using full images of accessible functors, we prove some results about com- binatorial and accessible model categories. In particular, we give an example of a weak factorization system on a locally presentable category which is not accessible. 1. Introduction Twenty years ago, M. Hovey asked for examples of model categories which are not cofi- brantly generated. This is the same as asking for examples of weak factorization systems which are not cofibrantly generated. One of the first examples was given in [1]: it is a weak factorization system (L; R) on the locally presentable category of posets where L consists of embeddings. The reason is that posets injective to embeddings are precisely complete lattices which do not form an accessible category. Hence L is not cofibrantly generated. Since then, the importance of accessible model categories and accessible weak factorization systems has emerged, and, the same question appears again, i.e., to give an example of a weak factorization system (L; R) on a locally presentable category which is not accessible. Now, L-injective objects do not necessarily form an accessible category but only a full image of an accessible functor. Such full images are accessible only under quite restrictive assumptions (see [5]). But, for an accessible weak factorization system, L-injective objects form the full image of a forgetful functor from algebraically L-injective objects (see Corollary 3.3). Moreover, such full images are closed under reduced prod- ucts modulo κ-complete filters for some regular cardinal κ (see Theorem 2.5).
    [Show full text]
  • CURRICULUM VITAE Jonathan Andrew Scott
    CURRICULUM VITAE Jonathan Andrew Scott [email protected] Address: 6-53 Maclaren Street, Ottawa ON K2P 0K3 Canada Current Affiliation: Department of Mathematics and Statistics, University of Ottawa Telephone (office): (613) 562-5800 ext. 2082 Telephone (home): (613) 236-4812 Education 2000 Ph.D. in Mathematics, University of Toronto. Supervisor: Steve Halperin. 1994 M.Sc. in Mathematics, University of Toronto. 1993 B.Sc. (Honours) in Mathematical Physics, Queen’s University at Kingston. Research Interests Operads, category theory, E∞ algebras, cohomology operations, loop space homology, Bockstein spectral sequence. Publications 1. K. Hess, P.-E. Parent, and J. Scott, A chain coalgebra model for the James map, Homology, Homotopy Appl. 9 (2007) 209–231, arXiv:math.AT/0609444. 2. K. Hess, P.-E. Parent, J. Scott, and A. Tonks, A canonical enriched Adams-Hilton model for simplicial sets, Adv. Math. 207 (2006) 847–875, arXiv:math.AT/0408216. 3. J. Scott, Hopf algebras up to homotopy and the Bockstein spectral sequence, Algebr. Geom. Topol. 5 (2005) 119–128, arXiv:math.AT/0412207. 4. J. Scott, A torsion-free Milnor-Moore theorem, J. London Math. Soc. (2) 67 (2003) 805–816, arXiv:math.AT/0103223. 5. J. Scott, Algebraic structure in the loop space homology Bockstein spectral sequence, Trans. Amer. Math. Soc. 354 (2002) 3075–3084, arXiv: math.AT/9912106. 6. J. Scott, A factorization of the homology of a differential graded Lie algebra, J. Pure Appl. Alg. 167 (2002) 329–340. Submitted Manuscripts 1. K. Hess and J. Scott, Homotopy morphisms and Koszul resolutions of operads 2. K. Hess, P.-E. Parent, and J.
    [Show full text]
  • 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.
    [Show full text]
  • A Monoidal Dold-Kan Correspondence for Comodules
    A MONOIDAL DOLD-KAN CORRESPONDENCE FOR COMODULES MAXIMILIEN PEROUX´ Abstract. We present the notion of fibrantly generated model categories. Cofibrantly generated model categories are generalizations of CW-approximations which provide an inductive cofibrant replacement. We provide examples of inductive fibrant replacements constructed as Postnikov towers. These provide new types of arguments to compute homotopy limits in model categories. We provide examples for simplicial and differential graded comodules. Our main application is to show that simplicial comodules and connective differential graded comodules are Quillen equivalent and their derived cotensor products correspond. 1. Introduction Let R be a commutative ring. The Dold-Kan correspondence established in [Dol58, Kan58] is an equiv- ≥0 alence between the category sModR of simplicial R-modules and the category ChR of non-negative chain ≥0 complexes over R. The normalization functor N : sModR → ChR is symmetric monoidal with respect to the graded tensor product of chains, the levelwise tensor product of simplicial modules and the Eilenberg-Zilber ≥0 map. The inverse equivalence Γ : ChR → sModR also has a (non-symmetric) monoidal structure coming from the Alexander-Whitney map. The categories are not isomorphic as symmetric monoidal categories. ≥0 Nevertheless, the homotopy categories of sModR and ChR have isomorphic symmetric monoidal structures when endowed with their respective derived tensor product. Formally, it was shown in [SS03] that there is a weak symmetric monoidal Quillen equivalence between the symmetric monoidal model categories of sModR ≥0 and ChR . The equivalence lifts to the associated categories of algebras and modules. In [P´er20a, 4.1], ≥0 we proved that the underlying ∞-categories of sModR and ChR (in the sense of [Lur17, 1.3.4.15, 4.1.7.4]) are equivalent as symmetric monoidal ∞-categories.
    [Show full text]
  • Normal and Conormal Maps in Homotopy Theory 3
    NORMAL AND CONORMAL MAPS IN HOMOTOPY THEORY EMMANUEL D. FARJOUN AND KATHRYN HESS Abstract. Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of monoids and of conormality for maps of comonoids in M. These notions generalize both principal bundles and crossed modules and are preserved by nice enough monoidal functors, such as the normaliized chain complex functor. We provide several explicit classes of examples of homotopy-normal and of homotopy-conormal maps, when M is the category of simplicial sets or the category of chain complexes over a commutative ring. Introduction In a recent article [3], it was observed that a crossed module of groups could be viewed as a up-to-homotopy version of the inclusion of a normal subgroup. Mo- tivated by this observation, the authors of [3] formulated a definition of h-normal (homotopy normal) maps of simplicial groups, noting in particular that the con- jugation map from a simplicial group into its simplicial automorphism group is h-normal. The purpose of this paper is to provide a general, homotopical framework for the results in [3]. In particular, given a monoidal category M that is also a ho- motopical category [2], under reasonable compatability conditions between the two structures, we define and study notions of h-normality for maps of monoids and of h-conormality for maps of comonoids (Definition 2.4), observing that the two notions are dual in a strong sense.
    [Show full text]
  • Homotopy Completion and Topological Quillen Homology of Structured Ring Spectra
    HOMOTOPY COMPLETION AND TOPOLOGICAL QUILLEN HOMOLOGY OF STRUCTURED RING SPECTRA JOHN E. HARPER AND KATHRYN HESS Abstract. Working in the context of symmetric spectra, we describe and study a homotopy completion tower for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra). We prove a strong convergence theorem that for 0-connected algebras and modules over a (−1)-connected operad, the homotopy completion tower interpolates (in a strong sense) between topological Quillen homology and the identity functor. By systematically exploiting strong convergence, we prove several theo- rems concerning the topological Quillen homology of algebras and modules over operads. These include a theorem relating finiteness properties of topo- logical Quillen homology groups and homotopy groups that can be thought of as a spectral algebra analog of Serre's finiteness theorem for spaces and H.R. Miller's boundedness result for simplicial commutative rings (but in re- verse form). We also prove absolute and relative Hurewicz theorems and a corresponding Whitehead theorem for topological Quillen homology. Further- more, we prove a rigidification theorem, which we use to describe completion with respect to topological Quillen homology (or TQ-completion). The TQ- completion construction can be thought of as a spectral algebra analog of Sullivan's localization and completion of spaces, Bousfield-Kan’s completion of spaces with respect to homology, and Carlsson's and Arone-Kankaanrinta's completion and localization of spaces with respect to stable homotopy. We prove analogous results for algebras and left modules over operads in un- bounded chain complexes.
    [Show full text]
  • Wahlen Und Ernennungen Des ETH-Rats
    MEDIA RELEASE Meeting of the ETH Board on 10/11 July 2019 21 new professors appointed at the two Federal Institutes of Technology Bern, 11 July 2019 – At its meeting of 10/11 July 2019 and upon application of the President of ETH Zurich, Professor Joël Mesot, and the President of EPFL, Professor Martin Vetterli, the ETH Board appointed a total of 21 professors. It also took note of the resignations of 2 professors and thanked them for their services. Appointments at ETH Zurich Professor Florian Dörfler (*1982), currently Tenure Track Assistant Professor at ETH Zurich, as Associate Professor of Complex Systems Control. Florian Dörfler’s research interests lie in the analysis, control design and security of networked systems for controlling cyber-physical processes. His applications focus on robust smart grids and the associated optimisation of power flows. Florian Dörfler’s special field thus fits perfectly into the existing areas of teaching and research in the Department of Information Technology and Electrical Engineering and at ETH Zurich. Professor Roger Gassert (*1976), currently Associate Professor at ETH Zurich, as Full Professor of Rehabilitation Engineering. Roger Gassert conducts research at the interface between engineering, neuroscience and movement science. He and his team develop mechatronic systems to explore the neuromechanical generation of movement and haptic perception and to perform quantitative evaluation of the movement quality and rehabilitation progress of people with sensorimotor deficits. Roger Gassert uses this interdisciplinary approach to forge links between research-based neuroscience and movement science, and application-based engineering science, as well as with hospitals. Dr Robert Katzschmann (*1986), currently Chief Technology Officer in the private sector, as Tenure Track Assistant Professor of Robotics.
    [Show full text]
  • Long Knots and Maps Between Operads
    LONG KNOTS AND MAPS BETWEEN OPERADS WILLIAM DWYER AND KATHRYN HESS Abstract. We identify the space of tangentially straightened long knots in Rm, m ≥ 4, as the double loops on the space of derived operad maps from the associative operad into a version of the lit- tle m-disk operad. This verifies a conjecture of Kontsevich, Lam- brechts, and Turchin. 1. Introduction A long knot in Euclidean m-space Rm is a smooth embedding R → Rm which agrees with inclusion of the first coordinate axis on the com- plement of some compact set in R; a tangential straightening for the knot is a null homotopy (constant near ∞) of the map R → Sm−1 ob- tained by taking the unit tangent vector of the knot at each point. See [23, 5.1] for more details. Starting from work of Goodwillie, Klein, and Weiss [24] [10] [8] [9] on embedding spaces, Sinha [23] has proved that the space of tangentially straightened long knots in Rm is equivalent to the homotopy limit of an explicit cosimplicial space constructed from the m’th Kontsevich operad Km. Let A denote the associative operad. The cosimplicial space Sinha considers is derived by formulas of McClure and Smith [19] from an operad map A→Km; more generally, McClure and Smith build a cosimplicial space O• from any operad O of spaces and operad map A → O. Our main theorem gives an independent formula for the arXiv:1006.0874v1 [math.AT] 4 Jun 2010 homotopy limit of this construction. Say that the operad O is reduced if O0 and O1 are weakly contractible.
    [Show full text]
  • RATIONAL HOMOTOPY THEORY: a BRIEF INTRODUCTION Kathryn
    RATIONAL HOMOTOPY THEORY: A BRIEF INTRODUCTION Kathryn Hess Ecole Polytechnique F´ed´erale de Lausanne December 2005 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 [17] 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 foundations of the 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 extremely 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 [17], 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 [17] or [23].
    [Show full text]
  • WALDHAUSEN K-THEORY of SPACES VIA COMODULES 11 Where Retx : Sset∗ → RX Sends a Pointed Simplicial Set Y with Basepoint Y0 To
    WALDHAUSEN K-THEORY OF SPACES VIA COMODULES KATHRYN HESS AND BROOKE SHIPLEY Abstract. Let X be a simplicial set. We construct a novel adjunction be- X X tween the categories RX of retractive spaces over and ComodX+ of +- comodules, then apply recent work on left-induced model category structures [5], [16] to establish the existence of a left proper, simplicial model category structure on ComodX+ with respect to which the adjunction is a Quillen equiv- alence after localization with respect to some generalized homology theory E∗. We show moreover that this model category structure on ComodX+ stabilizes, giving rise to a model category structure on ComodΣ∞X , the category of ∞ + Σ X+-comodule spectra. It follows that the Waldhausen K-theory of X, A(X), is naturally weakly K hf equivalent to the Waldhausen -theory of ComodΣ∞X , the category of ho- ∞ + motopically finite Σ X+-comodule spectra, where the weak equivalences are given by twisted homology. For X simply connected, we exhibit explicit, nat- hf ural weak equivalences between the K-theory of Comod ∞ and that of the Σ X+ ∞ category of homotopically finite Σ (ΩX)+-modules, a more familiar model for A(X). For X not necessarily simply connected, we have E∗-local versions of these results for any generalized homology theory E∗. For H a simplicial monoid, ComodΣ∞H admits a monoidal structure and + ∞ induces a model structure on the category Alg ∞ of Σ H -comodule al- Σ H+ + gebras. This provides a setting for defining homotopy coinvariants of the ∞ ∞ coaction of Σ H+ on a Σ H+-comodule algebra, which is essential for ho- motopic Hopf-Galois extensions of ring spectra as originally defined by Rognes [27] and generalized in [15].
    [Show full text]
  • Arxiv:1108.4627V1 [Math.AT]
    THE COSMIC GALOIS GROUP AS KOSZUL DUAL TO WALDHAUSEN’S A(∗) JACK MORAVA Abstract. The world is so full of a number of things I’m sure we shall all be as happy as kings. Robert Louis Stevenson, A Child’s Garden of Verses 1. Basic questions 1.1 Existence: Why is there something, rather than nothing? This does not seem very accessible by current methods. A more realistic goal may be Classification: Given that there’s something, what could it be? This suggests a Program: If things fall into categories (A, B,. ), hopefully small and stable enough to be manageable, techniques from K-theory may be useful. 1.2 In fact Blumberg, Gepner, and Tabuada ([4], see also [10]) have con- perf structed a Cartesian closed category Cat∞ of small stable ∞-categories, eg A, B, Funex(A, B),... and there is then a (similarly Cartesian closed) big spectral category of pre-motives: with objects as above, and morphism arXiv:1108.4627v1 [math.AT] 23 Aug 2011 objects ex HomMot(A, B) := K(Fun (A, B)) ∈ K($) − Mod enriched over Waldhausen’s A-theory spectrum. [The superscript ‘ex’ signi- fies functors which preserve finite limits and colimits, and the objects of the category are taken to be idempotent complete (ie, the category is suitably localized with respect to Morita equivalence).] Date: 23 August 2011. 2010 Mathematics Subject Classification. 11G, 19F, 57R, 81T. This work was supported by the NSF. 1 2 JACK MORAVA Such a category has a functorial completion to a pre - triangulated category Mot ([6 §4.5]: ie, whose homotopy category is triangulated); this involves enlarging the set of objects by adjoining suitable cofibers, generalizing the classical Karoubification in Grothendieck’s original construction of a cate- gory of pure motives.
    [Show full text]