6,137 research outputs found
Splitting homomorphisms and the Geometrization Conjecture
This paper gives an algebraic conjecture which is shown to be equivalent to
Thurston's Geometrization Conjecture for closed, orientable 3-manifolds. It
generalizes the Stallings-Jaco theorem which established a similar result for
the Poincare Conjecture. The paper also gives two other algebraic conjectures;
one is equivalent to the finite fundamental group case of the Geometrization
Conjecture, and the other is equivalent to the union of the Geometrization
Conjecture and Thurston's Virtual Bundle Conjecture.Comment: 11 pages, Some typos are correcte
End sums of irreducible open 3-manifolds
An end sum is a non-compact analogue of a connected sum. Suppose we are given
two connected, oriented -manifolds and . Recall that to form
their connected sum one chooses an -ball in each , removes its
interior, and then glues together the two boundary components thus
created by an orientation reversing homeomorphism. Now suppose that and
are also open, i.e. non-compact with empty boundary. To form an end sum
of and one chooses a halfspace (a manifold \homeo\ to ) embedded in , removes its interior, and then
glues together the two resulting boundary components by an
orientation reversing homeomorphism. In order for this space to be an
-manifold one requires that each be {\bf end-proper} in in the
sense that its intersection with each compact subset of is compact. Note
that one can regard as a regular neighborhood of an end-proper ray (a
1-manifold \homeo\ to ) \ga_i in
Contractible open 3-manifolds which non-trivially cover only non-compact 3-manifolds
Suppose is a closed, connected, orientable, \irr\ \3m\ such that
is infinite. One consequence of Thurston's geometrization
conjecture is that the universal covering space of must be
\homeo\ to \RRR. This has been verified directly under several different
additional assumptions on . (See, for example, \cite{2}, \cite{3}, \cite{6},
\cite{19}.
On covering translations and homeotopy groups of contractible open n-manifolds
This paper gives a new proof of a result of Geoghegan and Mihalik which
states that whenever a contractible open -manifold which is not
homeomorphic to is a covering space of an -manifold and
either or and is irreducible, then the group of covering
translations injects into the homeotopy group of .Comment: 4 pages, LaTeX, amsart styl
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