  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.

Algebraic Topology;Geometric Topology