Delooping the functor calculus tower

Abstract

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 → 𝐷 𝑛 of discs, 𝑛 ⩾ 𝑚, avoiding any given multisingularity and coinciding with the standard inclusion near the boundary 𝜕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. As a spin-off result, we discover a homotopy recurrence relation on the components of the little discs operads.