The Whitehead conjecture (also known as the Whitehead asphericity conjecture) is a claim in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

. It was formulated by

J. H. C. Whitehead John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princet ...

in 1941. It states that every

connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...

subcomplex of a two-dimensional aspherical

CW complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...

is aspherical. A

group presentation In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...

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 The Eilenberg–Ganea conjecture is a claim in algebraic topology. It was formulated by Samuel Eilenberg and Tudor Ganea in 1957, in a short, but influential paper. It states that if a group ''G'' has cohomological dimension 2, then it h ...

, or there must be a counterexample to the Whitehead conjecture; in other words, it is not possible for both conjectures to be true.

References

* * Algebraic topology Conjectures Unsolved problems in mathematics {{topology-stub