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C. Gordon :
“Dehn surgery and 3-manifolds ,”
pp. 21–71
in
Low dimensional topology .
Edited by T. S. Mrowka and P. S. Ozsváth .
IAS/Park City Mathematics Series 15 .
American Mathematical Society (Providence, RI ),
2009 .
MR
2503492 Zbl
1194.57003 incollection
Abstract
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These notes are somewhat expanded versions of the six lectures given at the 2006 Park City Mathematics Institute Graduate Summer School. The main focus of the lectures was exceptional Dehn surgeries on knots, and, more generally, exceptional Dehn fillings on hyperbolic 3-manifolds.
In Lecture 1 we describe the crude classification of 3-manifolds that comes from cutting them along essential surfaces of non-negative Euler characteristic, and say what this means for exteriors of knots. In Lecture 2 we discuss Dehn surgery on knots, and in particular describe a construction, framed surgery on knots on surfaces, which is the source of many examples of exceptional Dehn surgeries. Lecture 3 summarizes some facts and conjectures about exceptional Dehn surgeries on knots. In Lecture 4 we introduce rational tangle fillings on tangles; these induce Dehn fillings on the double branched cover of the tangle. Tangle fillings have the advantage that they are easy to visualize, and although they impose a symmetry on the manifold in question, nevertheless it turns out that many examples of exceptional Dehn fillings arise in this way. Lecture 5 gives more examples of exceptional Dehn fillings derived from tangles. In Lecture 6 we discuss some classification results about exceptional Dehn fillings; many of these take the form that a hyperbolic 3-manifold has a pair of non-hyperbolic Dehn fillings of a particular kind if and only if it is the double branched cover of one of a certain explicit family of tangles. We conclude with a sketch of the proof of one of these classification results, describing in particular how one shows that the fillings under consideration arise from tangle fillings.
@incollection {key2503492m,
AUTHOR = {Gordon, Cameron},
TITLE = {Dehn surgery and 3-manifolds},
BOOKTITLE = {Low dimensional topology},
EDITOR = {Mrowka, Tomasz S. and Ozsv\'ath, Peter
S.},
SERIES = {IAS/Park City Mathematics Series},
NUMBER = {15},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {2009},
PAGES = {21--71},
NOTE = {MR:2503492. Zbl:1194.57003.},
ISSN = {1079-5634},
ISBN = {9780821847664},
}
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