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 G = ( S ∣ R ) {\displaystyle G=(S\mid R)} is called aspherical if the two-dimensional CW complex K ( S ∣ R ) {\displaystyle K(S\mid R)} associated with this presentation is aspherical or, equivalently, if π 2 ( K ( S ∣ R ) ) = 0 {\displaystyle \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. (1941). "On adding relations to homotopy groups". Annals of Mathematics. 2nd Ser. 42 (2): 409–428. doi:10.2307/1968907. JSTOR 1968907. MR 0004123.
  • Bestvina, Mladen; Brady, Noel (1997). "Morse theory and finiteness properties of groups". Inventiones Mathematicae. 129 (3): 445–470. Bibcode:1997InMat.129..445B. doi:10.1007/s002220050168. MR 1465330. S2CID 120422255.
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