Decoupling decorations on moduli spaces of manifolds

TL;DR

This paper compares constrained and unconstrained moduli spaces of d-dimensional manifolds with embedded decorations, generalizing previous work to higher dimensions and more complex submanifold decorations.

Contribution

It extends Bödigheimer--Tillmann's results to higher dimensions and various tangential structures, analyzing the impact of decoration constraints on moduli spaces.

Findings

Established equivalence between constrained and decoupled moduli spaces for certain decorations.

Generalized results to higher-dimensional manifolds with complex submanifold decorations.

Provided new insights into the topology of moduli spaces with unlinked circle decorations.

Abstract

We consider moduli spaces of $d$-dimensional manifolds with embedded particles and discs. In this 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 B\"odigheimer--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 unparametrised unlinked circles.