Finite type of the classifying space of homotopy self-equivalences

Determine whether the classifying space \(B{M}\) of the group-like \(H\)-space of based, locally orientation-preserving homotopy self-equivalences of a closed manifold \(M\) with finite fundamental group is of finite type.

Background

Dror, Dwyer, and Kan established finite-type results for classifying spaces of topological monoids associated with connected virtually nilpotent finite CW complexes. The paper explains that applying a similar strategy here would require control of the Postnikov terms of the auxiliary spaces B{B}, in addition to the bordism groups appearing in the braid.

The space M{M} in this paper is a group-like HH-space rather than a strictly associative and unital topological monoid, so the classifying space is understood after strictifying M{M} to a topological monoid. The authors explicitly ask whether this classifying space has finite type when MM has finite fundamental group.

References

If M has finite fundamental group, does it hold that B{M} is of finite type?

— Braidings of Self-Equivalences and Bordism  (2610.11110 - Hambleton et al., 8 Oct 2026) in Section 9, subsection “Finite type of the classifying space of \({M}\),” Question

A long-standing and open problem in rational homotopy theory was raised by Schlessinger in 1976 (see p. 519), which asks to characterize the space realized as the classifying space $aut (X)$ up to rational homotopy equivalence.

— A complete classification of the rational realization of gap-two products of Eilenberg--MacLane spaces as classifying spaces  (2610.11671 - Bai et al., 8 Oct 2026) in Introduction