<p>Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is compact because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces.</p>

In this thesis we study the homological behaviour of configuration spaces as the number of objects in the configuration goes to infinity. For unordered configurations of distinct points (possibly equipped with some internal parameters) in a connected, open manifold it is a well-known result, going back to G. Segal and D. McDuff in the 1970s, that these spaces enjoy the property of homological stability. In Chapter 2 we prove that this property also holds for so-called oriented configuration spaces, in which the points of a configuration are equipped with an ordering up to even permutations. There are two important differences from the unordered setting: the rate (or slope) of stabilisation is strictly slower, and the stabilisation maps are not in general split-injective on homology. This can be seen by some explicit calculations of Guest-Kozlowski-Yamaguchi in the case of surfaces. In Chapter 3 we refine their calculations to show that, for an odd prime p, the difference between the mod-p homology of the oriented and the unordered configuration spaces on a surface is zero in a stable range whose slope converges to 1 as p goes to infinity. In Chapter 4 we prove that unordered configuration spaces satisfy homological stability with respect to finite-degree twisted coefficient systems, generalising the corresponding result of S. Betley for the symmetric groups. We deduce this from a general “twisted stability from untwisted stability” principle, which also applies to the configuration spaces studied in the next chapter. In Chapter 5 we study configuration spaces of submanifolds of a background manifold M. Roughly, these are spaces of pairwise unlinked, mutually isotopic copies of a fixed closed, connected manifold P in M. We prove that if the dimension of P is at most (dim(M)−3)/2 then these configuration spaces satisfy homological stability w.r.t. the number of copies of P in the configuration. If P is a sphere this upper bound on its dimension can be increased to dim(M)−3.

We give a definition of the scanning map for configuration spaces that is equivariant under the action of the diffeomorphism group of the underlying manifold. We use this to extend the Bödigheimer-Madsen result for the stable splittings of the Borel constructions of certain mapping spaces from compact Lie group actions to all smooth actions. Moreover, we construct a stable splitting of configuration spaces which is equivariant under smooth group actions, completing a zig-zag of equivariant stable homotopy equivalences between mapping spaces and certain wedge sums of spaces. Finally we generalise these results to configuration spaces with twisted labels (labels in a fibre bundle subject to certain conditions) and extend the Bödigheimer-Madsen result to more mapping spaces.

Dada uma superfície M, definiremos os grupos de tranças de M, denotado por \'B IND. n\' (M), geometricamente e usando a noção de espaços de confiuração. Mostraremos a equivalência das definições. Na mesma linha de raciocínio, definiremos os grupos de tranças puras de superfícies \'P IND. n\' (M). Apresentaremos as propriedades mais importantes dos grupos de tranças do plano e mostraremos que \'B IND. n\' (\'R POT. 2\') injeta em \'B IND. n\' (M), para muitas superfícies M. Mais detalhadamente, obteremos a apresentação de \'B IND. n\' (\'RP POT. 2\' ) e \'P IND. n\'(\'RP POT. 2\') / For a surface M, we define the braid groups of M, \'B IND. n\'(M), geometricaly and using the notion of configuration spaces. We show the equivalence of these definitions. In the sequence, we define the pure braid group of M, \'P IND. n\' (M). We present the most important properties of braid groups of the plane and we show that \'B IND. n\'\'(\'R POT. 2\') embedds in \'B IND. n\' (M), for almost all M. In a more detailed fashion, we present \'B IND. n\' (\'RP POT. 2\') and \'P IND. n\' (\'RP POT. 2)

Dada uma superfície M, definiremos os grupos de tranças de M, denotado por \'B IND. n\' (M), geometricamente e usando a noção de espaços de confiuração. Mostraremos a equivalência das definições. Na mesma linha de raciocínio, definiremos os grupos de tranças puras de superfícies \'P IND. n\' (M). Apresentaremos as propriedades mais importantes dos grupos de tranças do plano e mostraremos que \'B IND. n\' (\'R POT. 2\') injeta em \'B IND. n\' (M), para muitas superfícies M. Mais detalhadamente, obteremos a apresentação de \'B IND. n\' (\'RP POT. 2\' ) e \'P IND. n\'(\'RP POT. 2\') / For a surface M, we define the braid groups of M, \'B IND. n\'(M), geometricaly and using the notion of configuration spaces. We show the equivalence of these definitions. In the sequence, we define the pure braid group of M, \'P IND. n\' (M). We present the most important properties of braid groups of the plane and we show that \'B IND. n\'\'(\'R POT. 2\') embedds in \'B IND. n\' (M), for almost all M. In a more detailed fashion, we present \'B IND. n\' (\'RP POT. 2\') and \'P IND. n\' (\'RP POT. 2)

O objetivo principal deste trabalho será apresentar um estudo detalhado dos espaços de configurações. Dissertaremos sobre: espaços de configurações clássicos, invariância do bordo, espaço de configurações para superfícies, fibração de Fadell e Neuwirth e espaços de configurações do espaço Euclideano, da esfera e do espaço projetivo complexo. / The main objective of this work will be to present a detailed study of the configuration spaces. We will study: classical configuration spaces, invariance of the boundary, configuration spaces of surfaces, Fadell and Neuwirth fibration and configuration spaces of the Euclidean space and spheres.

O objetivo principal deste trabalho será apresentar um estudo detalhado dos espaços de configurações. Dissertaremos sobre: espaços de configurações clássicos, invariância do bordo, espaço de configurações para superfícies, fibração de Fadell e Neuwirth e espaços de configurações do espaço Euclideano, da esfera e do espaço projetivo complexo. / The main objective of this work will be to present a detailed study of the configuration spaces. We will study: classical configuration spaces, invariance of the boundary, configuration spaces of surfaces, Fadell and Neuwirth fibration and configuration spaces of the Euclidean space and spheres.

Made available in DSpace on 2016-06-02T20:28:22Z (GMT). No. of bitstreams: 1 DissFPM.pdf: 979275 bytes, checksum: 1b13e7e3772ecbeac26224804b180369 (MD5) Previous issue date: 2007-06-27 / Universidade Federal de Sao Carlos / In this work, we study the Artin braid group, B(n), and the confguration spaces (ordered and unordered) of a path connected manifold of dimension ¸ 2. The fundamental group of confguration space (unordered) of IR2 is identifed with the Artin braid group. This identifcation is used to conclude that the confguration space of IR2 is an Eilenberg-MacLane space of type K(B(n), 1). Therefore, it can be proved that the braid group B(n) contains no nontrivial element of the finite order. We use this fact to prove a generalization of a 2&#8722;dimensional version of the Borsuk-Ulam theorem presented by Connett [3]. / Neste trabalho, apresentamos o grupo de tranças de Artin, B(n), e os espaços de configurações (ordenado e não ordenado) de uma variedade conexa por caminhos de dimensão ¸ 2, a fim de identificar o grupo fundamental do espaço de configurações (não ordenado) de IR2 com o grupo de tranças de Artin. Usamos este fato para concluir que o espaço de configurações de IR2 é um espaço de Eilenberg-MacLane do tipo K(B(n), 1). Deste modo pode ser provado que o grupo de tranças B(n) não possui elementos não triviais de ordem finita, e usamos este fato na demonstração de uma generalização da versão bi-dimensional do teorema de Borsuk-Ulam apresentado por Connett [3].

Pour X un espace topologique, l'algèbre de Lie de Malcev de son groupe fondamental (ou algèbre de Lie de Malcev de X) fait partie des invariants étudiés en homotopie rationnelle. Un espace est dit 1-formel si cette algèbre de Lie est quadratique. Les connexions de type Knizhnik-Zamolodochikov peuvent permettre d'établir des résultats de "formalité " des espaces de configurations de points sur les surfaces. On s'intéresse à une famille d'espaces X qui sont des espaces de configurations de points sur la sphère, tordus par l'action d'un groupe fini d'homographies. On étudie le groupe fondamental de X et on construit une connexion de type Knizhnik-Zamolodochikov qui permet de calculer l'algèbre de Lie de Malcev de X et de démontrer sa 1-formalité. / The Malcev Lie algebra of the fundamental group of X (or Macev Lie algebra of X) is an algebraic invariant of the space X studied in rational homotopy theory. The space X is 1-formal if its Malcev algebra is quadratic. One can use Knizhnik–Zamolodchikov-type connections to obtain "formality" (1-formality or filtered formality) results for configuration spaces of surfaces. In the thesis we consider a family of orbit configuration spaces X of the complex projective line associated to finite finite groups of homographies. We study the fundamental group of X and constuct Knizhnik– Zamolodchikov-type connections. This allows us to give a presentation of the Malcev Lie algebra of X and to prove the 1-formality of X.