Results 1  10
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213
Special Lagrangian fibrations. I. Topology,” Integrable systems and algebraic geometry (Kobe/Kyoto
 156–193, World Sci. Publ., River Edge, NJ
, 1997
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Simplicial manifolds, bistellar flips and a 16vertex triangulation of the Poincaré homology 3sphere
 Math
, 2000
"... We present an algorithm based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16vertex triangulation of the Poincaré homology 3sphere; we construct an infinite series of nonPL ddimensional spheres with ..."
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Cited by 32 (11 self)
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We present an algorithm based on bistellar operations that provides a useful tool for the construction of simplicial manifolds with few vertices. As an example, we obtain a 16vertex triangulation of the Poincaré homology 3sphere; we construct an infinite series of nonPL ddimensional spheres with d+13 vertices for d 5; and we show that if a dmanifold admits any triangulation on n vertices, then it admits a noncombinatorial triangulation on n + 12 vertices (d 5).
4manifolds with inequivalent symplectic forms and 3manifolds with inequivalent fibrations
, 2003
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Ricci curvature, minimal volumes, and SeibergWitten
, 2001
"... We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a nontrivial SeibergWitten invariant. These allow one, for example, to exactly compute the infimum of the L 2norm of Ricci curvature for all complex surfaces ..."
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Cited by 26 (2 self)
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We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4manifold with a nontrivial SeibergWitten invariant. These allow one, for example, to exactly compute the infimum of the L 2norm of Ricci curvature for all complex surfaces of general type. We are also able to show that the standard metric on any complex hyperbolic 4manifold minimizes volume among all metrics satisfying a pointwise lower bound on sectional curvature plus suitable multiples of the scalar curvature. These estimates also imply new nonexistence results for Einstein metrics. 1
Grope cobordism of classical knots
, 2000
"... We explain the notion of a grope cobordism between two knots in a 3manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the GoussarovHabiro approach to finite type invariants of knots is closely relat ..."
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Cited by 24 (8 self)
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We explain the notion of a grope cobordism between two knots in a 3manifold. Each grope cobordism has a type that can be described by a rooted unitrivalent tree. By filtering these trees in different ways, we show how the GoussarovHabiro approach to finite type invariants of knots is closely related to our notion of grope cobordism. Thus our results can be viewed as a geometric interpretation of finite type invariants. An interesting refinement we study are knots modulo symmetric grope cobordism in 3space. On one hand this theory maps onto the usual Vassiliev theory and on the other hand it maps onto the CochranOrrTeichner filtration of the knot concordance group, via symmetric grope cobordism in 4space. In particular, the graded theory contains information on finite type invariants (with degree h terms mapping to Vassiliev degree 2 h), Blanchfield forms or Sequivalence at h = 2, CassonGordon invariants at h = 3, and for h = 4 one has the new von Neumann signatures of a knot. 1
Subexponential groups in 4manifold topology
 Geom. Topol
"... Abstract. We present a new, more elementary proof of the FreedmanTeichner result that the geometric classification techniques (surgery, scobordism, and pseudoisotopy) hold for topological 4manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to th ..."
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Cited by 23 (13 self)
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Abstract. We present a new, more elementary proof of the FreedmanTeichner result that the geometric classification techniques (surgery, scobordism, and pseudoisotopy) hold for topological 4manifolds with groups of subexponential growth. In an appendix Freedman and Teichner give a correction to their original proof, and reformulate the growth estimates in terms of course geometry. The disk embedding theorem for 4manifolds with “good ” fundamental group is the key ingredient of the classification theory: it is used in the proof of the 4dimensional surgery theorem, and the 5dimensional scobordism theorem and pseudoisotopy theorems. The homotopy hypotheses of the theorem always allow one to find a 2stage immersed capped grope. If one can find such a grope so that loops in the image are nullhomotopic in the ambient manifold, then Freedman’s theorem [F, FQ] shows there is a topologically flat embedded disk. The current focus, therefore, is on obtaining this π1nullity condition. Freedman [F1] showed this is possible if the fundamental group of the manifold is poly(finite or cyclic). This was extended to groups of polynomial growth in [S]. The current best result
Knot concordance and von Neumann ρinvariants
"... Abstract. We prove the nontriviality at all levels of the filtration of the classical topological knot concordance group C · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0 ⊆ C. defined in [COT]. This filtration is significant because not only is it strongly connected to Whitney tower constructions of Casson and ..."
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Cited by 20 (13 self)
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Abstract. We prove the nontriviality at all levels of the filtration of the classical topological knot concordance group C · · · ⊆ Fn ⊆ · · · ⊆ F1 ⊆ F0 ⊆ C. defined in [COT]. This filtration is significant because not only is it strongly connected to Whitney tower constructions of Casson and Freedman, but all previouslyknown concordance invariants are related to the first few terms in the filtration. In [COT] we proved nontriviality at the first new level n = 3 by using von Neumann ρinvariants of the 3manifolds obtained by surgery on the knots. For larger n we use the CheegerGromov estimate for such ρinvariants, as well as some rather involved algebraic arguments using our noncommutative Blanchfield forms. In addition, we consider a closely related filtration, {Gn}, of C defined in terms of Gropes in the 4ball. We show that this filtration is also nontrivial for all n> 2. 1.