Whitehead Asphericity Conjecture

Whitehead's asphericity conjecture says that every connected subcomplex of a two-dimensional aspherical CW complex is itself aspherical [Whitehead1941AddingRelations]. Since a two-complex has no cells above dimension two, this is equivalently the assertion that the subcomplex has trivial second homotopy group. The hypothesis that the ambient complex is aspherical is essential.

Two recent preprints claim complete proofs by different routes: Pasku uses identities among relations and crossed-module methods [Pasku2021Whitehead], while Kawauchi proposes a geometric argument through links and homology [Kawauchi2024Whitehead]. Neither claim has yet acquired an independently checked, broadly accepted proof, so both remain on the proof-claim watchlist rather than changing the catalog status.

Contemporary work still treats the classical integral conjecture as open. Mikhovich proves related rational and pro-$p$ asphericity results while explicitly distinguishing them from the unresolved Whitehead problem [Mikhovich2025Whitehead]. The remaining gap is therefore the full subcomplex statement, not one of these completed analogues.