Francisco Javier González-Acuña
Encyclopedia
Francisco Javier González-Acuña (nickname "Fico") is a mathematician in the UNAM
's institute of mathematics and CIMAT, specializing in low-dimensional topology
.
He did his graduate studies at Princeton University
, obtaining his Ph.D. in 1970. His thesis, written under the supervision of Ralph Fox
, was titled On homology spheres.
An early result of González-Acuña is that a group
G is the homomorphic image of some knot group
if and only if G is finitely generated
and has weight at most one. This result (a "remarkable theorem", as Lee Neuwirth called it in his review),
was published in 1975 in the highly respected journal, Annals of Mathematics
. In 1978, together with José María Montesinos, he answered a question posed by Fox, proving the existence of 2-knots whose groups have infinitely many ends.
With Hamish Short, González-Acuña proposed and worked on the cabling conjecture: the only knot
s in the 3-sphere
which admit a reducible Dehn surgery
, i.e. a surgery which results in a reducible 3-manifold, are the cable knots. This conjecture is one of the most relevant, unresolved questions in the theory of Dehn surgery on knots in the 3-sphere.
González-Acuña has made other significant contributions, which have been published in journals such as Transactions of the American Mathematical Society
, Topology
and Mathematical Proceedings of the Cambridge Philosophical Society
.
Unam
UNAM or UNaM may refer to:* National University of Misiones, a National University in Posadas, Argentina*National Autonomous University of Mexico , the large public autonomous university based in Mexico City...
's institute of mathematics and CIMAT, specializing in low-dimensional topology
Low-dimensional topology
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. It can be regarded as a part of geometric topology.A number of...
.
He did his graduate studies at Princeton University
Princeton University
Princeton University is a private research university located in Princeton, New Jersey, United States. The school is one of the eight universities of the Ivy League, and is one of the nine Colonial Colleges founded before the American Revolution....
, obtaining his Ph.D. in 1970. His thesis, written under the supervision of Ralph Fox
Ralph Fox
Ralph Hartzler Fox was an American mathematician. As a professor at Princeton University, he taught and advised many of the contributors to the Golden Age of differential topology, and he played an important role in the modernization and main-streaming of knot theory.Ralph Fox attended Swarthmore...
, was titled On homology spheres.
An early result of González-Acuña is that a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
G is the homomorphic image of some knot group
Knot group
In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,\pi_1....
if and only if G is finitely generated
Finitely generated
In mathematics, finitely generated may refer to:* Finitely generated group* Finitely generated monoid* Finitely generated abelian group* Finitely generated module* Finitely generated ideal* Finitely generated algebra* Finitely generated space...
and has weight at most one. This result (a "remarkable theorem", as Lee Neuwirth called it in his review),
was published in 1975 in the highly respected journal, Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
. In 1978, together with José María Montesinos, he answered a question posed by Fox, proving the existence of 2-knots whose groups have infinitely many ends.
With Hamish Short, González-Acuña proposed and worked on the cabling conjecture: the only knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
s in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
which admit a reducible Dehn surgery
Dehn surgery
In topology, a branch of mathematics, a Dehn surgery, named after Max Dehn, is a specific construction used to modify 3-manifolds. The process takes as input a 3-manifold together with a link...
, i.e. a surgery which results in a reducible 3-manifold, are the cable knots. This conjecture is one of the most relevant, unresolved questions in the theory of Dehn surgery on knots in the 3-sphere.
González-Acuña has made other significant contributions, which have been published in journals such as Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society is a monthly mathematics journal published by the American Mathematical Society. It started in 1900...
, Topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and Mathematical Proceedings of the Cambridge Philosophical Society
Mathematical Proceedings of the Cambridge Philosophical Society
Mathematical Proceedings of the Cambridge Philosophical Society is a mathematical journal published by Cambridge University Press. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, formerly entitled Proceedings of the Cambridge Philosophical...
.
External links
- http://arxiv.org/abs/0908.4009 Unsolvability of word problems with knot groups, at arXiv-2010 and L'Enseignement Mathematique.