Braidings of Self-Equivalences and Bordism

Let $M$ be a closed, smooth or topological $n$-manifold, with $n\ge 4$. We construct a homotopy highly cartesian square relating the space $\mathcal{E}(M(\mathrm{\ell }))$ of homotopy self-equivalences of $M$ (in a suitable range) over the Postnikov $\mathrm{\ell }$-sections of its stable normal microbundle, and an $(\mathrm{\infty }+n)$-fold loop space representing an associated (normal) bordism theory. This implies the existence of braids of interlocking exact sequences involving the homotopy groups of $\mathcal{E}(M(\mathrm{\ell }))$ and certain Lashof bordism groups, leading to a conceptual explanation and broad generalization of earlier work of Hambleton–Kreck for closed, oriented $4$-manifolds.