Whitehead conjecture explained

The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in algebraic topology. It was formulated by J. H. C. Whitehead in 1941. It states that every connected subcomplex of a two-dimensional aspherical CW complex is aspherical.

G = (S\mid R) is called aspherical if the two-dimensional CW complex K(S\mid R) associated with this presentation is aspherical or, equivalently, if \pi _{2(K(S\mid} R)) =0 . The Whitehead conjecture is equivalent to the conjecture that every sub-presentation of an aspherical presentation is aspherical.

In 1997, Mladen Bestvina and Noel Brady constructed a group G so that either G is a counterexample to the Eilenberg–Ganea conjecture, or there must be a counterexample to the Whitehead conjecture; in other words, it is not possible for both conjectures to be true.

References

• Whitehead. J. H. C.. J. H. C. Whitehead. On adding relations to homotopy groups. Annals of Mathematics. 2nd Ser. . 42. 1941. 2. 409–428. 10.2307/1968907. 1968907. 0004123.
• Bestvina. Mladen. Mladen Bestvina. Brady. Noel. Morse theory and finiteness properties of groups. Inventiones Mathematicae. 129 . 1997. 3. 445–470. 10.1007/s002220050168. 1997InMat.129..445B. 1465330. 120422255.