John Stallings

John Robert Stallings junior (born 1935 in Morrilton , Arkansas , † November 24, 2008 ) was an American mathematician who studied geometric topology and algebra .

Life

Stallings studied at Princeton University (one of his fellow students was John Milnor ) and received his doctorate there in 1959 under Ralph Fox ( Some Topological Proofs and Extensions of Grushko's Theorem ). He was a professor at Berkeley University . In 1961/62 and 1971 he was at the Institute for Advanced Study .

In 1960, independently of Stephen Smale , he proved the Poincaré conjecture for dimensions greater than 6. His proof was extended to dimensions 5 and 6 by Erik Christopher Zeeman in 1962 . Stallings also formulated purely algebraic (group-theoretical) conjectures that are equivalent to the Poincaré conjecture (as he proved with Jaco).

According to Stallings, the Poincaré conjecture is equivalent to the following theorem (conjecture from Stallings):

Be an orientable two-dimensional manifold ( surface ) by sex , and free groups of rank and a surjective homomorphism from the fundamental group on . Then there is a non-trivial element of the kernel of represented by a simple closed curve on . ${\ displaystyle T}$ ${\ displaystyle n \ geq 1}$ ${\ displaystyle F_ {1}}$ ${\ displaystyle F_ {2}}$ ${\ displaystyle n}$ ${\ displaystyle \ nu}$ ${\ displaystyle \ pi _ {1} T}$ ${\ displaystyle \, F_ {1} \ times F_ {2}}$ ${\ displaystyle \ nu}$ ${\ displaystyle T}$

In 1970 he received the Cole Prize in Algebra with Richard Swan for the proof that finitely generated free groups are characterized by the fact that they have cohomological dimension 1 (the Stallings or Stallings-Swan theorem ).

In 1970 he was invited speaker at the International Congress of Mathematicians in Nice ( Group theory and 3-manifolds ) and in 1962 in Stockholm ( Topological unknottedness of certain spheres ).

Fonts

with Stephen M. Gersten: Combinatorial Group Theory and Topology. Princeton University Press 1987, ISBN 0-691-08409-2 .
Group Theory and Three-dimensional Manifolds. Yale University Press 1971, ISBN 0-300-01397-3 .
Topology of finite graphs , Inventiones Mathematicae, Volume 71, 1983, pp. 551-565

Web links

Remarks

↑ John Stallings: Polyhedral homotopy spheres . Bulletin American Mathematical Society, Vol. 66, 1960, pp. 485-488.
↑ Stallings reports on this in his article How not to prove the Poincaré conjecture on his homepage. In purely algebraic terms, this is the "Conjecture D."
↑ since the Poincaré conjecture has since been proven
↑ that is, without colons
^ John Stallings: On torsion-free groups with infinitely many ends . Annals of Mathematics, Vol. 88, 1968, pp. 312-334.

personal data
SURNAME	Stallings, John
ALTERNATIVE NAMES	Stallings, John Robert Junior (full name)
BRIEF DESCRIPTION	American mathematician
DATE OF BIRTH	1935
PLACE OF BIRTH	Morrilton , Arkansas
DATE OF DEATH	November 24, 2008

[1] John Stallings: Polyhedral homotopy spheres . Bulletin American Mathematical Society, Vol. 66, 1960, pp. 485-488.

[2] Stallings reports on this in his article How not to prove the Poincaré conjecture on his homepage. In purely algebraic terms, this is the "Conjecture D."

[3] since the Poincaré conjecture has since been proven

[4] that is, without colons

[5] John Stallings: On torsion-free groups with infinitely many ends . Annals of Mathematics, Vol. 88, 1968, pp. 312-334.