Whitehead's Asphericity Conjecture

Whitehead asphericity conjecture: every connected subcomplex of an aspherical 2-dimensional CW complex is itself aspherical — i.e. has vanishing second homotopy group \(\pi_2 = 0\).

Whitehead asked it in 1941. It is the combinatorial shadow of asphericity questions across low-dimensional topology: labeled oriented tree (LOT) complexes — spines of ribbon disc complements — are the critical test cases, and Bestvina–Brady (1997) showed Whitehead and Eilenberg–Ganea cannot both hold. Recent claimed proofs (Kawauchi 2024) have not carried the field; the conjecture stands open after eight decades.