Low Dimensional Topology (Google eBook)
Tomasz Mrowka, Peter Steven Ozsváth
American Mathematical Soc., Jan 1, 2009 - Mathematics - 315 pages
Low-dimensional topology has long been a fertile area for the interaction of many different disciplines of mathematics, including differential geometry, hyperbolic geometry, combinatorics, representation theory, global analysis, classical mechanics, and theoretical physics. The Park City Mathematics Institute summer school in 2006 explored in depth the most exciting recent aspects of this interaction, aimed at a broad audience of both graduate students and researchers. The present volume is based on lectures presented at the summer school on low-dimensional topology. These notes give fresh, concise, and high-level introductions to these developments, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field of low-dimensional topology and to senior researchers wishing to keep up with current developments. The volume begins with notes based on a special lecture by John Milnor about the history of the topology of manifolds. It also contains notes from lectures by Cameron Gordon on the basics of three-manifold topology and surgery problems, Mikhail Khovanov on his homological invariants for knots, John Etnyre on contact geometry, Ron Fintushel and Ron Stern on constructions of exotic four-manifolds, David Gabai on the hyperbolic geometry and the ending lamination theorem, Zoltan Szabo on Heegaard Floer homology for knots and three manifolds, and John Morgan on Hamilton's and Perelman's work on Ricci flow and geometrization.
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2-sphere abelian groups algebra bimodules boundary component bounded braid group classiﬁcation cobordisms compact complex conﬁguration Conjecture construction contact structure corresponding critical points curvature deﬁne Deﬁnition Dehn ﬁllings Dehn surgery denote diﬀeomorphic diﬀerent diﬀerential dimension disjoint disk elliptic embedded Euler characteristic example Exercise ﬁber ﬁbration ﬁeld Figure ﬁnd ﬁnite ﬁrst ﬂat Floer homology foliation fundamental group genus geodesic Geom geometric graded handlebody Heegaard diagram holonomy homeomorphic homology class homotopy hyperbolic 3-manifold hyperbolic knots incompressible inﬁnitely intersection irreducible isomorphic isotopy Jones polynomial Khovanov knot surgery Lecture Lemma lens space link homology Math metric non-trivial oriented Ozsv´ath polynomial proof result Ricci flow Riemannian satisﬁes Seiberg-Witten invariants Seifert self-intersection sequence Show shrinkwrapping simple closed curve simplicial hyperbolic surface simply connected smooth solid torus sphere symplectic manifold Szab´o tangent tensor Theorem theory three-manifolds topology tori toroidal unknot vector