John Robert Stallings Jr. (July 22, 1935 – November 24, 2008) was a mathematician known for his seminal contributions to geometric group theory and 3-manifold topology. Stallings was a Professor Emeritus in the Department of Mathematics at the University of California at Berkeley Mathematician John Stallings died last year at 73. UC Berkeley press release, January 12, 2009. Accessed January 26, 2009 where he had been a faculty member since 1967. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds. Stallings' most important contributions include a proof, in a 1960 paper, of the Poincaré Conjecture in dimensions greater than six and a proof, in a 1971 paper, of the Stallings theorem about ends of groups.
Biography
John Stallings was born on July 22, 1935, in Morrilton, Arkansas.
Stallings received his B.Sc. from University of Arkansas in 1956 (where he was one of the first two graduates in the university's Honors program)All things academic. Volume 3, Issue 4; November 2002. and he received a Ph.D. in Mathematics from Princeton University in 1959 under the direction of Ralph Fox.
After completing his PhD, Stallings held a number of postdoctoral and faculty positions, including being an NSF postdoctoral fellow at the University of Oxford as well as an instructorship and a faculty appointment at Princeton. Stallings joined the University of California at Berkeley as a faculty member in 1967 where he remained until his retirement in 1994. Even after his retirement, Stallings continued supervising UC Berkeley graduate students until 2005. Stallings was an Alfred P. Sloan Research fellow from 1962 to 1965 and a Miller Institute fellow from 1972 to 1973. Over the course of his career, Stallings had 22 doctoral students including Marc Culler, Stephen M. Gersten, and J. Hyam Rubinstein and has over 140 doctoral descendants. He published over 50 papers, predominantly in the areas of geometric group theory and the topology of 3-manifolds.
Stallings delivered an invited address as the International Congress of Mathematicians in Nice in 1970John R. Stallings. Group theory and 3-manifolds. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 165–167. Gauthier-Villars, Paris, 1971. and a James K. Whittemore Lecture at Yale University in 1969.John Stallings. Group theory and three-dimensional manifolds. A James K. Whittemore Lecture in Mathematics given at Yale University, 1969. Yale Mathematical Monographs, 4. Yale University Press, New Haven, Conn.–London, 1971.
Stallings received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in 1970.Frank Nelson Cole Prize in Algebra. American Mathematical Society.
The conference "Geometric and Topological Aspects of Group Theory", held at the Mathematical Sciences Research Institute in Berkeley in May 2000, was dedicated to the 65th birthday of Stallings.Geometric and Topological Aspects of Group Theory, conference announcement , atlas-conferences.com In 2002 a special issue of the journal Geometriae Dedicata was dedicated to Stallings on the occasion of his 65th birthday. Stallings died from prostate cancer on November 24, 2008.. Accessed January 26, 2009.Professor Emeritus John Stallings of the UC Berkeley Mathematics Department has died. Announcement at the website of the Department of Mathematics of the University of California at Berkeley. Accessed December 4, 2008
Mathematical contributions
Most of Stallings' mathematical contributions are in the areas of geometric group theory and low-dimensional topology (particularly the topology of 3-manifolds) and on the interplay between these two areas.
An early significant result of Stallings is his 1960 proofJohn Stallings. Polyhedral homotopy spheres. Bulletin of the American Mathematical Society, vol. 66 (1960), pp. 485–488. of the Poincaré conjecture in dimensions greater than six. (Stallings' proof was obtained independently from and shortly after the different proof of Stephen Smale who established the same result in dimensions bigger than fourStephen Smale. Generalized Poincaré's conjecture in dimensions greater than four. Annals of Mathematics (2nd Ser.), vol. 74 (1961), no. 2, pp. 391–406).
Using "engulfing" methods similar to those in his proof of the Poincaré conjecture for n > 6, Stallings proved that ordinary Euclidean n-dimensional space has a unique piecewise linear, hence also smooth, structure, if n is not equal to 4. This took on added significance when, as a consequence of work of Michael Freedman and Simon Donaldson in 1982, it was shown that 4-space has exotic smooth structures, in fact uncountably many inequivalent ones.
In a 1963 paper Stallings constructed an example of a finitely presented group with infinitely generated 3-dimensional integral homology group and, moreover, not of the type , that is, not admitting a classifying space with a finite 3-skeleton. This example came to be called the Stallings group and is a key example in the study of homological finiteness properties of groups. Robert Bieri later showedRobert Bieri. "Homological dimension of discrete groups." Queen Mary College Mathematical Notes. Queen Mary College, Department of Pure Mathematics, London, 1976. that the Stallings group is exactly the kernel of the homomorphism from the direct product of three copies of the free group to the additive group of integers that sends to the six elements coming from the choice of free bases for the three copies of . Bieri also showed that the Stallings group fits into a sequence of examples of groups of type but not of type . The Stallings group is a key object in the version of discrete Morse theory for cubical complexes developed by Mladen Bestvina and Noel Brady and in the study of subgroups of direct products of limit groups.Martin R. Bridson, James Howie, Charles F. Miller, and Hamish Short. "The subgroups of direct products of surface groups". Geometriae Dedicata, vol. 92 (2002), pp. 95–103.Martin R. Bridson, and James Howie. "Subgroups of direct products of elementarily free groups." Geometric and Functional Analysis, vol. 17 (2007), no. 2, pp. 385–403Martin R. Bridson, and James Howie. Subgroups of direct products of two limit groups. Mathematical Research Letters, vol. 14 (2007), no. 4, 547–558.
Stallings' most famous theorem in group theory is an algebraic characterization of groups with more than one end (that is, with more than one "connected component at infinity"), which is now known as Stallings' theorem about ends of groups. Stallings proved that a finitely generated group G has more than one end if and only if this group admits a nontrivial splitting as an amalgamated free product or as an HNN extension over a finite group (that is, in terms of Bass–Serre theory, if and only if the group admits a nontrivial action on a tree with finite edge stabilizers). More precisely, the theorem states that a finitely generated group G has more than one end if and only if either G admits a splitting as an amalgamated free product , where the group C is finite and , , or G admits a splitting as an HNN extension where are finite subgroups of H.
Stallings proved this result in a series of works, first dealing with the torsion-free case (that is, a group with no nontrivial elements of finite order)John R. Stallings. On torsion-free groups with infinitely many ends. Annals of Mathematics (2), vol. 88 (1968), pp. 312–334. and then with the general case.John Stallings. "Groups of cohomological dimension one." Applications of Categorical Algebra (Proc. Sympos. Pure Math., Vol. XVIII, New York, 1968) pp. 124–128. American Mathematical Society, Providence, R.I, 1970. Stalling's theorem yielded a positive solution to the long-standing open problem about characterizing finitely generated groups of cohomological dimension one as exactly the free groups.John R. Stallings. Groups of dimension 1 are locally free. Bulletin of the American Mathematical Society, vol. 74 (1968), pp. 361–364 Stallings' theorem about ends of groups is considered one of the first results in geometric group theory proper since it connects a geometric property of a group (having more than one end) with its algebraic structure (admitting a splitting over a finite subgroup). Stallings' theorem spawned many subsequent alternative proofs by other mathematicians (e.g.Martin J. Dunwoody. "Cutting up graphs." Combinatorica 2 (1982), no. 1, pp. 15–23.Warren Dicks and Martin J. Dunwoody. Groups acting on graphs. Cambridge Studies in Advanced Mathematics, 17. Cambridge University Press, Cambridge, 1989. ) as well as many applications (e.g.Peter Scott. "A new proof of the annulus and torus theorems." American Journal of Mathematics, vol. 102 (1980), no. 2, pp. 241–277). The theorem also motivated several generalizations and relative versions of Stallings' result to other contexts, such as the study of the notion of relative ends of a group with respect to a subgroup,Gadde A. Swarup. "Relative version of a theorem of Stallings." Journal of Pure and Applied Algebra, vol. 11 (1977/78), no. 1–3, pp. 75–82Martin J. Dunwoody and E. L. Swenson. "The algebraic torus theorem." Inventiones Mathematicae, vol. 140 (2000), no. 3, pp. 605–637G. Peter Scott, and Gadde A. Swarup. An algebraic annulus theorem. Pacific Journal of Mathematics, vol. 196 (2000), no. 2, pp. 461–506 including a connection to CAT(0) cubical complexes.Michah Sageev. Proceedings of the London Mathematical Society (3), vol. 71 (1995), no. 3, pp. 585–617 A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of C. T. C. Wall.
Another influential paper of Stallings is his 1983 article "Topology of finite graphs".John R. Stallings. "Topology of finite graphs." Inventiones Mathematicae, vol. 71 (1983), no. 3, pp. 551–565 Traditionally, the algebraic structure of subgroups of free groups has been studied in combinatorial group theory using combinatorial methods, such as the Schreier rewriting method and Nielsen transformations.Roger C. Lyndon and Paul E. Schupp. Combinatorial Group Theory. Springer–Verlag, New York, 2001. "Classics in Mathematics" series, reprint of the 1977 edition. Stallings' paper put forward a topological approach based on the methods of covering space theory that also used a simple graph-theoretic framework. The paper introduced the notion of what is now commonly referred to as Stallings subgroup graph for describing subgroups of free groups, and also introduced a foldings technique (used for approximating and algorithmically obtaining the subgroup graphs) and the notion of what is now known as a Stallings folding. Most classical results regarding subgroups of free groups acquired simple and straightforward proofs in this set-up and Stallings' method has become the standard tool in the theory for studying the subgroup structure of free groups, including both the algebraic and algorithmic questions (see Ilya Kapovich and Alexei Myasnikov. "Stallings foldings and subgroups of free groups." Journal of Algebra, vol. 248 (2002), no. 2, 608–668). In particular, Stallings subgroup graphs and Stallings foldings have been the used as a key tools in many attempts to approach the Hanna Neumann conjecture.J. Meakin, and P. Weil. Subgroups of free groups: a contribution to the Hanna Neumann conjecture. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000). Geometriae Dedicata, vol. 94 (2002), pp. 33–43.Bilal Khan. Positively generated subgroups of free groups and the Hanna Neumann conjecture. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 155–170, Contemp. Math., 296, American Mathematical Society, Providence, RI, 2002;