05410 - Group Theory and Three-dimensional Manifolds [Stallings].pdf

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I
James
K.
Whittemore· Lectures in Mathematics
given at Yale University
GROUP
THEORY AND THREE-DIMENSIONAL
MANIFOLDS
by
John Stallings
New Haven and London, Yale University Press, 1971
I BACKGROUND AND SIGNIFICANCE OF
THREE-DIMENSIONAL MANIFOLDS
I.A Introduction
The
study
of
three-dimensional manifolds
has
often interacted with a
certain stream of group theory, which is concerned with free groups, free
products, finite presentations of groups, and simUar combinatorial mat-
ters.
Thus, Kneser's fundamental paper [4] had latent implications toward
Grushko's Theorem [8]; one
of
the sections of
that
paper dealt with the
theorem that,
if
a manifold's fundamental group is a fre,e product, then the
manifold exhibits this geometrically, being divided into two regions by a
sphere with appropriate properties. Kneser's proof is fraught
with
geo-
metric hazards, but one of the steps, consisting of modifying the I-skele-
ton of the manifold and dividing it up, contains obscurely something like
Grusbko's ,Theorem:
that
a set of
tors
of
the factors.
Similarly, in the sequence
of
theorems by PapakyriakopoUlos, the Loop
Theorem [14], Debn's Lemma,
and
the Sphere Theorem [15], there are
,
.
..
implicit facts about group theory, which form the
\~tn.
subject
of
these
!it"
chapt~rs.
"
'
)
~enerators
of a free product can be
modified
in
a certain st'mple
way
so as to
be
the union of sets
of
genera-
Philosophically speaking, the depth and beauty of
3~manifold
theory
Is,
it seems to me, mainly due
to
the fact that its theorems have offshoots that
eventually blossom in a different subject, namely group theory. Thus I
A James
K.
Whittemore Lecture,
october
1969.

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