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[1] where he had been a faculty member since 1967.

Stallings joined the University of California at Berkeley as a faculty member in 1967 where he remained until his retirement in 1994.

Stallings delivered an invited address as the International Congress of Mathematicians in Nice in 1970[4] and a James K. Whittemore Lecture at Yale University in 1969.

[5] Stallings received the Frank Nelson Cole Prize in Algebra from the American Mathematical Society in 1970.

[6] 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.

[7] In 2002 a special issue of the journal Geometriae Dedicata was dedicated to Stallings on the occasion of his 65th birthday.

An early significant result of Stallings is his 1960 proof[10] of the Poincaré conjecture in dimensions greater than six.

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.

The Stallings group is a key object in the version of discrete Morse theory for cubical complexes developed by Mladen Bestvina and Noel Brady[14] and in the study of subgroups of direct products of limit groups.

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

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,[24][25][26] including a connection to CAT(0) cubical complexes.

[27] A comprehensive survey discussing, in particular, numerous applications and generalizations of Stallings' theorem, is given in a 2003 paper of C. T. C.

[30] Stallings' paper put forward a topological approach based on the methods of covering space theory that also used a simple graph-theoretic framework.

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 [31]).

[32][33][34][35] Stallings subgroup graphs can also be viewed as finite-state automata[31] and they have also found applications in semigroup theory and in computer science.

This is an important structural result in the theory of Haken manifolds that engendered many alternative proofs, generalizations and applications (e.g.[50][51][52][53] ), including a higher-dimensional analog.

The paper began with a humorous admission: "I have committed the sin of falsely proving Poincaré's Conjecture.

"[1][55] Despite its ironic title, Stallings' paper informed much of the subsequent research on exploring the algebraic aspects of the Poincaré conjecture (see, for example,[56][57][58][59]).