Fiedorowicz’s finite monoid realization conjecture

Fiedorowicz conjectured that every finite simply connected CW complex is homotopy equivalent to the classifying space of a finite monoid, after exhibiting a $5$-element monoid realizing $S^2$.

Known results

• McDuff (1979): every connected CW complex is the classifying space of a discrete monoid, generally an infinite one.
• Fiedorowicz (1984): realized $S^2$ using a $5$-element monoid.

November 16, 2025 progress

Martínian and Steinberg report that finite wedges of simply connected Moore spaces are realizable by finite semigroups. Their Theorem B gives finite-monoid realizations for finite simply connected CW complexes whose Hurewicz homomorphisms split in every degree, but does not settle the full conjecture.

Current status (as of September 2026): substantial classes, including spaces with split Hurewicz homomorphisms in every degree, are claimed to be realizable, but realization for every finite simply connected CW complex remains open.