Smooth space-level model for the homotopy-cartesian square

Construct an analogue, for smooth manifolds, of the homotopy-cartesian square relating spaces of homotopy self-equivalences over Postnikov sections of the stable normal bundle to the corresponding Lashof bordism spaces.

Background

The main theorem constructs the homotopy-cartesian square in the topological category using semi-simplicial models for Lashof bordism theories. The paper obtains the smooth braid of exact sequences indirectly by comparison with the topological setting.

The authors explicitly state that a smooth analogue of the space-level theorem is expected but is not currently available because suitable semi-simplicial models for smooth bordism theories remain under development.

References

We expect that there is an analogue of Theorem A for smooth manifolds, but the techniques for modelling smooth bordism theories by using, for example, semi-simplicial sets, are still under development (see for recent work on the Quinn spectra \S 6).

— Braidings of Self-Equivalences and Bordism  (2610.11110 - Hambleton et al., 8 Oct 2026) in Remark 1.1, immediately following Theorem A

We currently do not know how to produce such maps of spaces in the smooth case, but nevertheless may produce the corresponding ``long exact sequence of homotopy groups'' needed for the proof of Theorem B in the smooth case by hand.

— Braidings of Self-Equivalences and Bordism  (2610.11110 - Hambleton et al., 8 Oct 2026) in Section 8, subsection “Homotopy groups and the maps \(\alpha_*\) and \(\beta_*\),” opening paragraph