Whitehead conjecture
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.
A group presentation [math]\displaystyle{ G=(S\mid R) }[/math] is called aspherical if the two-dimensional CW complex [math]\displaystyle{ K(S\mid R) }[/math] associated with this presentation is aspherical or, equivalently, if [math]\displaystyle{ \pi_2(K(S\mid R))=0 }[/math]. 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. (1941). "On adding relations to homotopy groups". Annals of Mathematics. 2nd Ser. 42 (2): 409–428. doi:10.2307/1968907.
- Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae 129 (3): 445–470. doi:10.1007/s002220050168. Bibcode: 1997InMat.129..445B.
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