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Algorithmic Construction of the Postnikov Tower for Diagrams of Simplicial Sets

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Algorithmic Construction of the Postnikov Tower for Diagrams of Simplicial Sets Masaryk University Faculty of Science Algorithmic construction of the Postnikov tower for diagrams of simplicial sets Doctoral Thesis Marek Filakovský Brno, 2015 Declaration Hereby I declare, that this paper is my original authorial work, which I have worked out by my own. All sources, references and literature used or excerpted during elaboration of this work are properly cited and listed in complete reference to the due source. Marek Filakovský Advisor: doc. RNDr. Martin Čadek, CSc. iii Acknowledgement I would like to thank my supervisor M. Čadek for his support and many useful discus- sions, suggestions and comments. The course in algebraic topology he taught inspired me to continue in this field. I can hardly imagine a teacher that could be more generous with his time and knowledge. I am also indebted to L. Vokřínek, who took the unpaid job of being my unofficial secondary supervisor. Many of the results of this thesis follow from ideas originally developed by him. Trying to keep up with his thoughts encouraged me to learn more about model categories, homotopy theory and simplicial sets. This work also benefited from results achieved by M. Krčál, J. Matoušek, F.Ser- geraert and U. Wagner. Finally, I would like to thank my family for their endless material and emotional support. I dedicate this thesis to my wife, Martina. v Abstract The aim of the thesis is to provide an algorithm that given a nonnegative integer n and a finite diagram of simplicial sets Y : I! sSet, where Y (i) is simply connected for all i 2 I, constructs the n-stage Postnikov tower for Y . Given a finite simplicial set Y with an action of a finite group G, the Elmendorf’s op theorem provides a finite diagram of simplicial sets Y : OG ! sSet, where the spaces are fixed points Y H for subgroups H ≤ G. The diagram Y further reflects the homotopy properties of space Y . Therefore, in the case the set of fixed points Y H is simply connected for every subgroup H ≤ G, the algorithm constructs the n-stage Postnikov tower for Y , which, informally speaking, represents the n-stage Postnikov tower for Y as a G-simplicial set. Further, we present an algorithm that decides if a simplicial map f : X ! Y between finite simplicial sets X; Y is homotopic to a trivial map under the assumption that Y is simply connected. Keywords simplicial set, Postnikov tower, chain complex, effective homology, equivariant algebraic topology, model category Abstrakt Hlavním cílem této práce je popis algoritmu, který pro konečný diagram simpliciálních množin Y : I! sSet, kde Y (i) je jednoduše souvislý prostor pro každé i 2 I, a pro libovolné nezáporné n 2 Z, zkonstruuje n-patrovou Postnikovovu věž pro diagram Y . Podle Elmendorfovy věty, lze každé konečné simpliciální množině Y s akcí grupy op G přiřadit diagram simpliciálních množin Y : OG ! sSet. Prvky v tomto diagramu jsou prostory pevných bodů Y H pro podgrupy H ≤ G. Diagram Y dále zachovává homotopické vlastnosti prostoru Y . Proto v případě kdy je každý prostor Y H , H ≤ G jednoduše souvislý, algoritmus konstruuje n-patrovou Postnikovovu věž pro diagram Y , která, neformálně řečeno, zodpovídá n-patrové Postnikovově věži pro G-simpliciální množinu Y . Dále uvádíme algoritmus, který pro dané simpliciální zobrazení f : X ! Y mezi konečnými simpliciálními prostory, kde Y je jednoduše souvislý prostor, rozhoduje, zda je f homotopické s triviálním zobrazením. Klíčová slova simpliciální množina, Postnikovova věž, řetězcový komplex, efektivní homologie, ekvi- variantní algebraická topologie, modelová kategorie vii Contents Foreword ..................................... xi 1 Motivation ....................................1 1.1 Representing simplicial sets and simplicial maps in a computer . .3 Finite simplicial sets . .3 Locally effective simplicial sets . .4 1.2 Postnikov tower for simplicial sets . .4 1.3 Effective homology . .6 1.4 Our motivation . .7 1.5 Category of orbits . .7 Simplicial sets with a group action . .8 Effective homology for diagrams . .9 1.6 Postnikov tower for diagrams . 10 2 Tools ....................................... 13 2.1 Simplicial sets . 13 Fibrations, cofibrations and weak equivalences . 14 Twisted products . 14 2.2 Diagrams of simplicial sets . 15 Homotopy and homology . 15 Model structure . 16 Homotopy left Kan extension . 17 2.3 Effective homology of chain complexes . 19 Effective chain complexes, reductions and strong equivalences . 20 Perturbation Lemmas . 21 2.4 Effective homology of twisted products . 22 Effective chain complex for twisted product . 24 Vector fields . 26 Twisted division . 28 2.5 Effective homology for diagrams . 28 Constructions with effective homology . 31 Perturbation lemmas for diagrams . 32 Filtrations . 34 2.6 Homotopy colimit and cofibrant replacement have effective homology . 35 Functorial cofibrant replacement . 36 2.7 Effective abelian groups . 37 2.8 Polycyclic groups . 39 Computations with fully effective polycyclic groups . 40 2.9 Eilenberg–MacLane spaces and diagrams . 42 Evaluation maps . 43 Simplicial maps to E(휋; k) and K(휋; k) ................. 45 Representing a map of diagrams by an effective cocycle . 48 A pullback from a fibration of Eilenberg–MacLane diagrams . 48 Effective homology for E(휋; n) and K(휋; n) .............. 49 3 Postnikov tower for diagrams ........................ 51 3.1 Reformulation of Theorem A . 51 3.2 Description of the algorithm . 52 3.3 Correctness of the algorithm . 53 ix ef The cochain 휅k−1 is a cocycle. 54 0 0 The map 'k takes values in Pk ..................... 54 Pk and 'k satisfy the properties of the Postnikov system . 55 3.4 Computing equivariant cohomology operations . 56 4 How to decide if a map is homotopically trivial ............ 59 Relative statement . 59 4.1 Computations with Postnikov towers . 60 4.2 Maps out of suspensions . 61 Homotopy concatenation . 61 4.3 Deciding the existence of a homotopy . 62 An exact sequence associated with a fibration . 62 Proof of Theorem D . 62 Proof of (poly)n−1 + (null)n−1 ! (poly)n .............. 63 x Foreword This thesis contains the results of my research during my PhD studies. My initial assignment was to deal with the effective homology of twisted cartesian products, namely to generalize F Seregraert’s previous results [39]. The generalization was needed for the paper [8]. The work on this issue turned to be relatively straightforward and after a year, I published my results in [16]. In this thesis these results are contained in Section 2.4 and the main result is stated as Corollary 2.21. Together with L. Vokřínek, we used the methods introduced in [8] to give an al- gorithm that decides whether two simplicial maps are homotopic. Our result can be found in [15] and is contained here in a simplified version as Chapter 4. Afterwards, my advisor and L. Vokřínek suggested a particular road map that would lead us to generalize one of the main results of [8] - an algorithm that for given finite simplicial sets X; Y with an action of a finite group G computes the set [X; Y ]G of equivariant homotopy classes of maps, whereas [8] deals with the situation where the group G acts only freely. Our general aim was to use Elmendorf’s theorem on an equivalence of the category of G-simplicial sets with a certain category of diagrams of simplicial sets. Hence our attention turned to working with diagrams of simplicial sets. Following an idea of L. Vokřínek, I summed up some introductory technical results in article [17]. These are also utilized here in Section 2.5 and Section 3.4. The main result of this thesis describes an algorithm that given a finite diagram of 1-connected simplicial sets Y and a positive integer n, constructs the n-stage Postnikov system for Y . This serves as a generalization of [7] and is proved in Chapter 3. xi 1 Motivation In this introductory chapter, we will focus on algorithms that compute solutions of classical problems in algebraic topology. We will mainly concentrate on the following problems: decide whether topological spaces X; Y are homotopy equivalent, describe the structure of the set of homotopy classes [X; Y ] of maps from X to Y and given the following diagram of spaces A; B; X; Y and maps i; p; f; g, g A 8/ Y (1.1) fe i p f X / B determine whether there is a lift, i.e. the dotted arrow fe making the diagram commut- ative and classify all such lifts up to homotopy. The last problem is known as the lifting–extension problem. If we set B = * (a point) then this is an extension problem and if A = ;, this problem is called a lifting problem. We will also deal with corresponding equivariant versions of tasks listed above, where the spaces are topological spaces with an action of a group G and maps are equivariant. Classical approach of algebraic topology is to solve these problems using algebraic invariants such as homology and cohomology groups, K–theory, homotopy groups etc. However, we can look at these problems from a computational and algorithmic perspective: Given a description of spaces X; Y , we ask whether there is an algorithm that computes [X; Y ] and similarly for the lifting–extension problem, we ask whether there exists an algorithm that for given f; g; i; p decides the existence of an extension fe and that computes all such extensions up to homotopy. The first paper with this point of view was the paper [2] by E. H. Brown jr.Inhis work, he assumed that the spaces X; Y are represented as finite simplicial complexes and he then provided the following algorithms: ∙ Given X; Y simply connected simplicial complexes with finite homology groups, an algorithm decides whether they are homotopy equivalent. ∙ Given a finite subcomplex A ⊆ X and a map f : A ! Y , where Y has finite homology groups an algorithm decides whether f can be extended to a map fe: X ! Y .
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