5,783 research outputs found
Topology of multiple log transforms of 4-manifolds
Given a 4-manifold X and an imbedding of T^{2} x B^2 into X, we describe an
algorithm X --> X_{p,q} for drawing the handlebody of the 4-manifold obtained
from X by (p,q)-logarithmic transforms along the parallel tori. By using this
algorithm, we obtain a simple handle picture of the Dolgachev surface
E(1)_{p,q}, from that we deduce that the exotic copy E(1)_{p,q} # 5(-CP^2) of
E(1) # 5(-CP^2) differs from the original one by a codimension zero simply
connected Stein submanifold M_{p,q}, which are therefore examples of infinitely
many Stein manifolds that are exotic copies of each other (rel boundaries).
Furthermore, by a similar method we produce infinitely many simply connected
Stein submanifolds Z_{p} of E(1)_{p,2} # 2(-CP^2)$ with the same boundary and
the second Betti number 2, which are (absolutely) exotic copies of each other;
this provides an alternative proof of a recent theorem of the author and Yasui
[AY4]. Also, by using the description of S^2 x S^2 as a union of two cusps
glued along their boundaries, and by using this algorithm, we show that
multiple log transforms along the tori in these cusps do not change smooth
structure of S^2 x S^2.Comment: Updated, with 17 pages 21 figure
More Cappell-Shaneson spheres are standard
Akbulut has recently shown that an infinite family of Cappell-Shaneson
homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present
paper, a strictly larger family is shown to be standard by a simpler method.
This new approach uses no Kirby calculus except through the relatively simple
1979 paper of Akbulut and Kirby showing that the simplest example with
untwisted framing is standard. Instead, hidden symmetries of the original
Cappell-Shaneson construction are exploited. In the course of the proof, we
give an example showing that Gluck twists can sometimes be undone using
symmetries of fishtail neighborhoods.Comment: 11 pages, 2 figures. This (v2) is essentially the published version,
with minor mathematical improvements over v
Twisting 4-manifolds along RP^2
We prove that the Dolgachev surface E(1)_{2,3} (which is an exotic copy of
the elliptic surface E(1)=CP^2 # 9(-CP^2)) can be obtained from E(1) by
twisting along a simple "plug", in particular it can be obtained from E(1) by
twisting along an RP^2.Comment: 5 papes, 5 figures. Appeared in Proceedings of GGT
Cork twisting Schoenflies problem
The stable Andrews-Curtis conjecture in combinatorial group theory is the
statement that every balanced presentation of the trivial group can be
simplified to the trivial form by elementary moves corresponding to
"handle-slides" together with "stabilization" moves. Schoenflies conjecture is
the statement that the complement of any smooth embedding S^3 into S^4 are pair
of smooth balls. Here we suggest an approach to these problems by certain cork
twisting operation on contractible manifolds, and demonstrate it on the example
of the first Cappell-Shaneson homotopy sphere.Comment: 10 pages, 17 figure
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