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Realisability problems in Algebraic Topology Ph. D. Dissertation David Méndez Martínez Advisors: Cristina Costoya & Antonio Viruel Tesis Doctoral Departamento de Álgebra, Geometría y Topología Programa de Doctorado en Matemáticas Facultad de Ciencias Universidad de Málaga AUTOR: David Méndez Martínez http://orcid.org/0000-0003-4023-172X EDITA: Publicaciones y Divulgación Científica. Universidad de Málaga Esta obra está bajo una licencia de Creative Commons Reconocimiento-NoComercial- SinObraDerivada 4.0 Internacional: http://creativecommons.org/licenses/by-nc-nd/4.0/legalcode Cualquier parte de esta obra se puede reproducir sin autorización pero con el reconocimiento y atribución de los autores. No se puede hacer uso comercial de la obra y no se puede alterar, transformar o hacer obras derivadas. Esta Tesis Doctoral está depositada en el Repositorio Institucional de la Universidad de Málaga (RIUMA): riuma.uma.es Esta tesis ha sido parcialmente financiada por la Ayuda de Formación de Profesorado Universitario FPU14/05137 del antiguo Ministerio de Educación, Cultura y Deporte, los Proyectos de Investigación MTM2013-41768-P, MTM2016-78647-P y MTM2016-79661-P del Ministerio de Economía y Competitividad (fondos FEDER incluidos), el Proyecto Consoli- dado FQM-213 de la Junta de Andalucía y el Proyecto Emergente EM2013/16 de la Xunta de Galicia. Agradecementos – Agradecimientos – Acknowledgements Moitas grazas aos meus pais e á miña irmá por transmitirme as inquedanzas e a ambición que en boa parte me levaron ata aquí, e polo voso constante apoio e cariño. Nada disto sería posíbel sen o enorme esforzo que empregastes para apoiarme, nin sen a confianza que depositastes en min. Gracias también a todos los amigos, viejos y nuevos, que me habéis apoyado a lo largo de estos años. A Álex, por ser una inagotable y constante fuente de apoyo, ánimo y cariño a lo largo de esta etapa y de todas las que están por venir. A Pepe, Kiko, Luis, Marco, Oihana, Alicia, Alejandro y Mario por tantos inolvidables momentos en ese pequeño cubículo que llamamos despacho y que en cierto modo todos terminamos considerando también nuestro hogar. Muchas gracias a mis directores de tesis Antonio y Cristina, por la confianza, el apoyo y las conversaciones productivas y, en resumen, por enseñarme qué son realmente las matemáticas. Esta tesis es tan vuestra como mía. I would like to thank Kathryn Hess, Jérôme Scherer and the rest of the topology group at the EPFL: Lyne, Haoqing, Stefania, Aras, Daniela, Gard, Nikki... for your amazing hospitality during my stay at Lausanne, the seminars and the chats about maths. I really learnt a lot during those few months. I am also thankful to Pascal Lambrechts, Yves Félix, Miradain Atontsa and Daniel Zimmer for your warm welcome at Louvain-la-Neuve, the seminars and the amazing working group. Contents Introduction 11 1 Preliminaries 21 1.1 Notation and conventions . 21 1.2 The classical group realisability problem in Graph Theory . 22 1.3 Coalgebras . 26 1.4 CDGAs and Rational Homotopy Theory . 29 2 1.5 A classification of the homotopy types of An-polyhedra . 35 2 Realisability problems in Graph Theory 39 2.1 Realisability in the arrow category of binary relational systems . 39 2.2 Realisability of permutation representations in binary relational systems . 47 2.3 Arrow replacement: from binary relational systems to simple graphs . 49 3 Realisability problems in coalgebras 57 3.1 A faithful functor from digraphs to coalgebras . 57 3.2 Generalised realisability problems in coalgebras . 63 3.3 The isomorphism problem for groups through coalgebra representations . 64 4 Realisability problems in CDGAs and spaces 67 4.1 Highly connected rigid CDGAs . 68 4.2 A family of almost fully faithful functors from digraphs to CDGAs . 70 4.3 Generalised realisability problems in CDGAR and HoT op . 77 5 Further applications to the family of functors from digraphs to CDGAs 81 5.1 Representation of categories in CDGAR and HoT op . 81 5.2 The isomorphism problem for groups through CDGA representations . 83 5.3 Highly connected inflexible and strongly chiral manifolds . 84 5.4 On a lower bound for the LS-category . 88 6 A non rational approach 91 2 6.1 Some general results on self-homotopy equivalences of An-polyhedra . 92 6.2 Obstructions to the realisability of groups . 94 Resumen en español 101 References 115 9 10 Contents Introduction In this thesis we are interested in realisability problems, which are very natural and quite easy to state and generally hard to solve. One such problem is the so-called inverse Galois problem, that asks whether any finite group may appear as the Galois group of a finite Galois extension of Q or not. The first to study this problem in depth was Hilbert in the late 19th century, [52], and to this day it remains open. In Algebraic Topology, where algebraic structures play a key role, realisability questions have been raised and deeply studied in many settings. A classical example is the problem of realisability of cohomological algebras, raised by Steenrod in 1961, [73]. This problem asks for a characterisation of the algebras that arise as the cohomology algebra of a space, and it has been treated in different surveys, [1, 3]. One more example is the G-Moore space problem, also raised by Steenrod, [59, Problem 51]. For a given group G, it asks if any ZG-module appears as the homology of a simply connected G-Moore space. It was solved in the negative in 1981, [21], and a characterization of the groups G for which every ZG-module is realisable appeared in 1987, [78]. In this thesis we focus on the so-called group realisability problem, which on its most general setting may be stated as follows: given a category C, does every group arise as the group of automorphisms of an object in C? This problem has been studied in different areas of combinatorics, appearing in different surveys, [7, 8, 54], and we make use of their terminology. Namely, if a group G happens to be the automorphism group of an object X ∈ Ob(C), i.e., ∼ if G = AutC(X), we say that X realises G, and we say that G is realisable in C. A category C where every finite group is realisable is said to be finitely universal, and if every group is realisable in C we say that it is universal. Our interest in combinatorics goes beyond terminology, since the solution to the histori- cally important group realisability problem in the category of graphs is a key element in many of our constructions. König raised this problem as early as 1936, [58], and barely three years later Frucht proved that the category of (finite) graphs is finitely universal, [43]. However, a solution to the general case had to wait for more than twenty additional years, until de Groot, in 1959, [32], and Sabidussi, in 1960, [72], independently proved that the category of graphs is indeed universal. Nonetheless, the main problem motivating this thesis is the group realisability problem in the category HoT op, the homotopy category of pointed topological spaces. It was proposed by Kahn in the sixties and asks if every group appears as the group of automorphisms of an object in HoT op. It has received significant attention, having appeared in different surveys and lists of open problems, [4, 5, 37, 55, 56, 71]. Recall that, given a space X, the group AutHoT op(X) is usually denoted E(X) and receives the name of group of self-homotopy equivalences of X. Its elements are the homotopy classes of continuous self-maps of X that have a homotopy inverse. Immediate examples that come to mind when trying to realise groups as self-homotopy 11 12 Realisability problems in Algebraic Topology equivalences of spaces can be found in Eilenberg-MacLane spaces. Indeed, if H is an abelian group and n ≥ 2, EK(H, n) =∼ Aut(H). However, this procedure does not give a full positive answer to the realisability in the category HoT op, given that not every group G is isomorphic to Aut(H) for some other group H (for example, Zp is not realisable as the automorphism group of any other group, when p is odd). The difficulty of this problem arises precisely from the fact that, other than making use of Eilenberg-MacLane spaces as described above, there is no obvious procedure to obtain spaces realising a certain given group. Consequently, the tools for seriously digging into Kahn’s question were at the beginning insufficient, and for over decades this problem had only been studied using ad-hoc procedures for certain families of groups, [15, 16, 36, 62, 65]. This impasse ended with a general method obtained by Costoya and Viruel, [27], that gives a positive answer to Kahn’s realisability problem in the case of finite groups. Namely, for every finite group G, there exists a topological space X (in fact, an infinite number of them) such that G =∼ E(X) ([27, Theorem 1.1]). They provide such a method by combining Frucht’s solution for the group realisability problem in the category of graphs with the remarkable computational power of Rational Homotopy Theory. Our main goal in this thesis is to expand on that sort of techniques to study further realisability problems. On the one hand, we refine the results in [27] (see Chapter 4) and, on the other hand, we extend these techniques to realisability problems in categories other than HoT op (see Chapter 3). Notice that Rational Homotopy Theory deals with spaces which are not of finite type over Z (see Definition 1.36), and thus they are not geometrically simple.
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