  We study a connection between mapping spaces of bimodules and of
infinitesimal bimodules over an operad. As main application and motivation of
our work, we produce an explicit delooping of the manifold calculus tower
associated to the space of smooth maps $D^{m}\rightarrow D^{n}$ of discs,
$n\geq m$, avoiding any given multisingularity and coinciding with the standard
inclusion near $\partial D^{m}$. In particular, we give a new proof of the
delooping of the space of disc embeddings in terms of little discs operads maps
with the advantage that it can be applied to more general mapping spaces.
