His contributions include a proof of the Smale conjecture for the 3-sphere and important results in the theory of surfaces and 3-manifolds.

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. 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.

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.

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 a 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

Books in progress

• Vector Bundles and K-Theory
• Spectral Sequences in Algebraic Topology
• Basic Topology of 3-Manifolds