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 proof 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 four). 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 F_3 , 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 showed that the Stallings group is exactly the kernel of the homomorphism from the direct product of three copies of the
free group F_2 to the additive group \Z of integers that sends to 1\in \Z the six elements coming from the choice of free bases for the three copies of F_2. Bieri also showed that the Stallings group fits into a sequence of examples of groups of type F_n but not of type F_{n+1} . 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. 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 G=A\ast_C B, where the group
C is finite and C\ne A, C\ne B, or
G admits a splitting as an HNN extension G=\langle H, t | t^{-1}Kt=L\rangle where K, L \le H 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) and then with the general case. 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. 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.) as well as many applications (e.g.). 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, including a connection to
CAT(0) cubical complexes. 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". 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. 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 ). 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. Stallings subgroup graphs can also be viewed as
finite-state automata Stallings' foldings method has been generalized and applied to other contexts, particularly in
Bass–Serre theory for approximating group actions on
trees and studying the subgroup structure of the
fundamental groups of graphs of groups. The first paper in this direction was written by Stallings himself, with several subsequent generalizations of Stallings' folding methods in the
Bass–Serre theory context by other mathematicians. Stallings' 1991 paper "Non-positively curved triangles of groups" introduced and studied the notion of a
triangle of groups. This notion was the starting point for the theory of
complexes of groups (a higher-dimensional analog of
Bass–Serre theory), developed by
André Haefliger and others. Stallings' work pointed out the importance of imposing some sort of "non-positive curvature" conditions on the complexes of groups in order for the theory to work well; such restrictions are not necessary in the one-dimensional case of Bass–Serre theory. Among Stallings' contributions to
3-manifold topology, the most well-known is the
Stallings fibration theorem. The theorem states that if
M is a compact irreducible
3-manifold whose
fundamental group contains a
normal subgroup, such that this subgroup is
finitely generated and such that the
quotient group by this subgroup is
infinite cyclic, then
M fibers over a circle. This is an important structural result in the theory of
Haken manifolds that engendered many alternative proofs, generalizations and applications (e.g. ), including a higher-dimensional analog. A 1965 paper of Stallings
"How not to prove the Poincaré conjecture" gave a
group-theoretic reformulation of the famous
Poincaré conjecture. The paper began with a humorous admission: "I have committed the sin of falsely proving Poincaré's Conjecture. But that was in another country; and besides, until now, no one has known about it."). Stallings was also interested in languages, and wrote one of the very few mathematical research papers in the constructed language
Interlingua. == Selected works ==