Allen Hatcher
Encyclopedia
Allen Edward Hatcher is an American
topologist and also a noted author. His book Algebraic Topology, which is the first in a series, is considered by many to be one of the best introductions to the subject.
He received his Ph.D.
under the advisorship of Hans Samelson
at Stanford University
in 1971. He went on to become a professor in UCLA. Since 1983 he has been a professor at Cornell University
. His students include Ulrich Oertel, Kiyoshi Igusa, Mark Brittenham, Charles Delman, and Rachel Roberts.
and important results in the theory of surface
s and 3-manifold
s.
s in certain 3-manifolds and their boundary slopes. Bill Floyd
and Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. Bill Thurston
and Hatcher classified the incompressible surfaces in 2-bridge
knot
complements
. As corollaries, this gave more examples of non-Haken
, non-Seifert fibered
, irreducible
3-manifolds and extended the techniques and line of investigation started in Thurston's Princeton lecture notes. Hatcher also showed that irreducible, boundary-irreducible 3-manifolds with toral
boundary have at most "half" of all possible boundary slopes resulting from essential surfaces. In the case of one torus boundary, one can conclude that the number of slopes given by essential surfaces is finite.
Hatcher is also a pioneer in the theory of essential laminations in 3-manifolds. He invented the notion of "end-incompressibility" and several of his students, such as Mark Brittenham, Charles Delman, and Rachel Roberts, have made important contributions to the theory.
of a closed
, orientable surface
. Their work relied on the notion of a cut system and moves that relate any two systems.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
topologist and also a noted author. His book Algebraic Topology, which is the first in a series, is considered by many to be one of the best introductions to the subject.
He received his Ph.D.
Doctor of Philosophy
Doctor of Philosophy, abbreviated as Ph.D., PhD, D.Phil., or DPhil , in English-speaking countries, is a postgraduate academic degree awarded by universities...
under the advisorship of Hans Samelson
Hans Samelson
Hans Samelson was a German American mathematician who worked in differential geometry, topology and the theory of Lie groups and Lie algebras—important in describing the symmetry of analytical structures....
at Stanford University
Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is a private research university on an campus located near Palo Alto, California. It is situated in the northwestern Santa Clara Valley on the San Francisco Peninsula, approximately northwest of San...
in 1971. He went on to become a professor in UCLA. Since 1983 he has been a professor at Cornell University
Cornell University
Cornell University is an Ivy League university located in Ithaca, New York, United States. It is a private land-grant university, receiving annual funding from the State of New York for certain educational missions...
. His students include Ulrich Oertel, Kiyoshi Igusa, Mark Brittenham, Charles Delman, and Rachel Roberts.
Mathematical contributions
His contributions include a proof of the Smale conjecture for the 3-sphere3-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...
and important results in the theory of surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s and 3-manifold
3-manifold
In mathematics, a 3-manifold is a 3-dimensional manifold. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds.Phenomena in three dimensions...
s.
3-manifolds
Perhaps among his most recognized results in 3-manifolds concern the classification of incompressible surfaceIncompressible surface
In mathematics, an incompressible surface, in intuitive terms, is a surface, embedded in a 3-manifold, which has been simplified as much as possible while remaining "nontrivial" inside the 3-manifold....
s in certain 3-manifolds and their boundary slopes. Bill Floyd
William Floyd (mathematician)
William J. Floyd is an American mathematician specializing in topology. He is currently a professor at Virginia Polytechnic Institute and State University....
and Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. Bill Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
and Hatcher classified the incompressible surfaces in 2-bridge
2-bridge knot
In the mathematical field of knot theory, a 2-bridge knot is a knot which can be isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points....
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...
complements
Knot complement
In mathematics, the knot complement of a tame knot K is the complement of the interior of the embedding of a solid torus into the 3-sphere. To make this precise, suppose that K is a knot in a three-manifold M. Let N be a thickened neighborhood of K; so N is a solid torus...
. As corollaries, this gave more examples of non-Haken
Haken manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface...
, non-Seifert fibered
Seifert fiber space
A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a S^1-bundle over a 2-dimensional orbifold...
, irreducible
Irreducible (mathematics)
In mathematics, the concept of irreducibility is used in several ways.* In abstract algebra, irreducible can be an abbreviation for irreducible element; for example an irreducible polynomial....
3-manifolds and extended the techniques and line of investigation started in Thurston's Princeton lecture notes. Hatcher also showed that irreducible, boundary-irreducible 3-manifolds with toral
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
boundary have at most "half" of all possible boundary slopes resulting from essential surfaces. In the case of one torus boundary, one can conclude that the number of slopes given by essential surfaces is finite.
Hatcher is also a pioneer in the theory of essential laminations in 3-manifolds. He invented the notion of "end-incompressibility" and several of his students, such as Mark Brittenham, Charles Delman, and Rachel Roberts, have made important contributions to the theory.
Surfaces
Hatcher and Thurston exhibited an algorithm to produce a presentation of the mapping class groupMapping class group
In mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
of a closed
Closed
Closed may refer to:Math* Closure * Closed manifold* Closed orbits* Closed set* Closed differential form* Closed map, a function that is closed.Other* Cloister, a closed walkway* Closed-circuit television...
, orientable surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
. Their work relied on the notion of a cut system and moves that relate any two systems.
Papers
- Hatcher and Thurston, A presentation for the mapping class group of a closed orientable surface. Topology 19 (1980), no. 3, 221—237.
- Hatcher, On the boundary curves of incompressible surfaces. Pacific J. Math. 99 (1982), no. 2, 373—377.
- Floyd and Hatcher, Incompressible surfaces in punctured-torus bundles. Topology Appl. 13 (1982), no. 3, 263—282.
- Hatcher and Thurston, Incompressible surfaces in $2$-bridge knot complements. Invent. Math. 79 (1985), no. 2, 225—246.
- Hatcher, A proof of the Smale conjecture, ${\rm Diff}(S\sp{3})\simeq {\rm O}(4)$. Ann. of Math. (2) 117 (1983), no. 3, 553—607.
Books
- Hatcher, Allen, Algebraic topology. Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0