In mathematics, the Andrews–Curtis conjecture states that every balanced

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

of the

trivial group In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...

can be transformed into a trivial presentation by a sequence of

Nielsen transformation In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and ...

s on the relators together with conjugations of relators, named after

James J. Andrews James J. Andrews (c. 1829 – June 7, 1862) was a Kentucky civilian who worked for the Union Army during the early years of the American Civil War. He led a daring raid behind enemy lines on the Western and Atlantic Railroad, known as the ...

and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not. It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the

Zeeman conjecture In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex K, the space K\times ,1/math> is collapsible. The conjecture, due to Christopher Zeeman, implies the Po ...

collapsibility In topology, a branch of mathematics, a collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were invented by J. H. C. Whitehead. Collapses find appli ...

implies the Andrews–Curtis conjecture.

References

* * Combinatorial group theory Conjectures Unsolved problems in mathematics {{algebra-stub