# Allen Hatcher

Allen Edward Hatcher (born October 23, 1944) is an American topologist.

Allen E. Hatcher
Allen Hatcher
BornOctober 23, 1944
NationalityAmerican
Alma materStanford University
Scientific career
FieldsMathematics
InstitutionsCornell University
Doctoral students

## Biography

Hatcher received his Ph.D. under the supervision of Hans Samelson at Stanford University in 1971. He went on to become a professor at the University of California, Los Angeles. Since 1983 he has been a professor at Cornell University.

## Mathematical contributions

He has worked in geometric topology, both in high dimensions, relating pseudoisotopy to algebraic K-theory, and in low dimensions: surfaces and 3-manifolds, such as proving the Smale conjecture for the 3-sphere.

### 3-manifolds

Perhaps among his most recognized results in 3-manifolds concern the classification of incompressible surfaces in certain 3-manifolds and their boundary slopes. William Floyd and Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle. William 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 has made contributions to the so-called 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 group of a closed, orientable surface. Their work relied on the notion of a cut system and moves that relate any two systems.

## Selected publications

### Papers

• Allen Hatcher and William Thurston, A presentation for the mapping class group of a closed orientable surface, Topology 19 (1980), no. 3, 221—237.
• Allen Hatcher, On the boundary curves of incompressible surfaces, Pacific Journal of Mathematics 99 (1982), no. 2, 373—377.
• William Floyd and Allen Hatcher, Incompressible surfaces in punctured-torus bundles, Topology and its Applications 13 (1982), no. 3, 263—282.
• Allen Hatcher and William Thurston, Incompressible surfaces in ${\displaystyle \scriptstyle 2}$-bridge knot complements, Inventiones Mathematicae 79 (1985), no. 2, 225—246.
• Allen Hatcher, A proof of the Smale conjecture, ${\displaystyle \scriptstyle {\mathrm {Diff} }(S^{3})\simeq {\mathrm {O} }(4)}$, Annals of Mathematics (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