Allen Hatcher

Allen Edward Hatcher (born 1944) is an American topologist and author.

He received his Ph.D. under the supervision 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.

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.

Perhaps among his most recognized results in 3-manifolds concern the classification of incompressible surfaces 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.

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