Decoupling configurations from moduli spaces of manifolds

Thesis

We consider moduli spaces of d-dimensional manifolds with embedded particles and discs. In such a moduli space, the location of the particles and discs is constrained by the d-dimensional manifold. We will compare this moduli space with the moduli space of d-dimensional manifolds in which the location of such decorations is no longer constrained, i.e. the decorations are decoupled. We generalise work by Bodigheimer–Tillmann for oriented surfaces and obtain new results for surfaces with different tangential structures as well as to higher dimensional manifolds. We also provide a generalisation of this result to moduli spaces with more general submanifold decorations and specialise in the case of decorations being unparameterised unlinked circles.

The moduli space of oriented surfaces with one boundary component can be made into a topological monoid with the operation of gluing surfaces along a section of their boundary. This operation can be extended to the spaces of configurations on surfaces with labels. We show that the group completion of this monoid splits as a product of the group completion of the moduli space of oriented surfaces and labelled configuration spaces of points in infinite Euclidean space. We further generalise this result for the configuration spaces on surfaces with partially summable labels, where particles are allowed to collide whenever their labels are summable.

The final project consists of endowing normalized cacti with the structure of an infinity-operad by showing that its existing composition laws are associative up to all higher homotopies. The higher homotopies are encoded by a new topological operad of bracketed trees which we relate both to an enrichment of the dendroidal category and to the Boardman-Vogt W-construction on the operad of operads.

Authors: Basualdo Bonatto, L

• DOI: 10.5287/ora-pdgkmw8qy
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Homological stability; Configuration spaces; Infinity operads; Mapping class groups; Operads; Moduli spaces of manifolds
• Subjects: Topology