603 research outputs found
Small knots and large handle additions
We construct a hyperbolic 3-manifold (with totally geodesic)
which contains no essential closed surfaces, but for any even integer
there are infinitely many separating slopes on so that ,
the 3-manifold obtained by attaching 2-handle to along , contains an
essential separating closed surface of genus and is still hyperbolic. The
result contrasts sharply with those known finiteness results for the cases
. Our 3-manifold is the complement of a simple small knot in a
handlebody.Comment: 25 pages, 14 figure
Handle additions producing essential surfaces
We construct a small, hyperbolic 3-manifold such that, for any integer
, there are infinitely many separating slopes in so
that , the 3-manifold obtained by attaching a 2-handle to along ,
is hyperbolic and contains an essential separating closed surface of genus .
The result contrasts sharply with those known finiteness results on Dehn
filling, and it also contrasts sharply with the known finiteness result on
handle addition for the cases . Our 3-manifold is the complement of
a hyperbolic, small knot in a handlebody of genus 3.Comment: 28 pages, 22 figure
Non-zero degree maps between -manifolds
Thom-Pontrjagin constructions are used to give a computable necessary and
sufficient condition when a homomorphism can be
realized by a map of degree for closed -connected
-manifolds and , . A corollary is that each -connected
-manifold admits selfmaps of degree larger than 1, .
In the most interesting case of dimension 4, with the additional surgery
arguments we give a necessary and sufficient condition for the existence of a
degree map from a closed orientable 4-manifold to a closed simply
connected 4-manifold in terms of their intersection forms, in particular
there is a map of degree 1 if and only if the intersection form of
is isomorphic to a direct summand of that of .Comment: 18 page
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