Andrews–Curtis conjecture

In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews 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.

Although it is believed that the Andrews–Curtis conjecture is false, there are no counter-examples known, nor are there many good ideas for possible counter-examples.^[1] It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.^[2]

References

Andrews, J. J.; Curtis, M. L. (1965), "Free groups and handlebodies", Proceedings of the American Mathematical Society (American Mathematical Society) 16 (2): 192–195, doi:10.2307/2033843, JSTOR 2033843, MR 0173241
Hazewinkel, Michiel, ed. (2001), "Low-dimensional topology, problems in", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=l/l120170

^ Open problems in combinatorial group theory
^ Hazewinkel, Michiel, ed. (2001), "Collapsibility", Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://www.encyclopediaofmath.org/index.php?title=c/c110310