Miscellaneous problems 3 — This one is due to Mike Hopkins.

Definitions and mathematical statement

Let $E$ be an $A_\infty$ ring spectrum. Choose the intended space $\operatorname{Aut}(E)$ of self-equivalences with its composition structure, and form a homotopy-coherent classifying space $B\operatorname{Aut}(E)$. The problem, attributed to Mike Hopkins, is to construct a generalized Thom spectrum from each map \[ f:X\longrightarrow B\operatorname{Aut}(E), \] and give the precise twisted half-smash interpretation suggested by the source.

The meaning of self-equivalence must be fixed. Equivalences of $A_\infty$ ring spectra, equivalences of underlying spectra and equivalences of rank-one $E$-modules give different automorphism spaces. The unit-space version uses $\operatorname{GL}_1(E)$, the invertible components of $\Omega^\infty E$, interpreted as module self-equivalences with the chosen side convention; it is not the same as the ring-automorphism space.

A precise local-system formulation would associate to $f$ an $E$-fiber local system $\mathcal E_f$ and define its Thom object by a homotopy colimit over $X$, in spectra or $E$-modules according to the automorphism choice. For a trivial local system the expected normalization is $E\wedge\Sigma^\infty_+X$; a nontrivial one is a twisted construction, not automatically that ordinary smash. Specify functoriality, coherence, coefficient/module target and comparison to classical Thom spectra. The source explicitly says Hopkins's more detailed description was forgotten, so these choices are not supplied by the excerpt.

The summary reports a rigorous units-based construction in parametrized/infinity-categorical work. That report does not resolve which of the above literal self-equivalence meanings the original wording intended.

Short English statement

Construct Thom spectra for coherent bundles of copies of an $A_\infty$ ring spectrum, clarifying the automorphism space and the twisted half-smash construction, especially the units/module version.

Sources

• Mark Hovey, Algebraic Topology Problem List, "Miscellaneous problems," item 3, explicitly attributed to Mike Hopkins, surviving mirror https://www-users.cse.umn.edu/~tlawson/hovey/. Recorded access 2026-07-29; extraction html-ordered-list.
• The supplied summary credits Ando--Blumberg--Gepner--Hopkins--Rezk with parametrized/infinity-categorical realization, but gives no exact paper title or bibliography entry.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11100065$, exact source code catalogue entry; supplied solved/SOLVED-IN-LITERATURE status.