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The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie’s classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we use the abstract theory of filtered and graded Lie algebras. We also describe all differential and integral invariants of new Lie algebras found in the paper and discuss the infinite-dimensional case.
A classical –symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of . This abstract association is traditionally used simply to express the symmetry of the –symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the –symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-off is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.
Let and be smooth manifolds without boundary. Immersion theory suggests that an understanding of the space of smooth embeddings should come from an analysis of the cofunctor from the poset of open subsets of to spaces. We therefore abstract some of the properties of this cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie style, with Taylor series and so on. The terms of the Taylor series for the cofunctor are explicitly determined. In a sequel to this paper, we introduce the concept of an analytic cofunctor from to spaces, and show that the Taylor series of an analytic cofunctor converges to . Deep excision theorems due to Goodwillie and Goodwillie–Klein imply that the cofunctor is analytic when .
Let and be smooth manifolds. For an open let be the space of embeddings from to . By the results of Goodwillie and Goodwillie–Klein, the cofunctor is analytic if . We deduce that its Taylor series converges to it. For details about the Taylor series, see Part I
The Mahler volume of a centrally symmetric convex body is defined as . Mahler conjectured that this volume is minimized when is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body has least volume when is an ellipsoid. If true, the bottleneck conjecture would strengthen the best current lower bound on the Mahler volume due to Bourgain and Milman. We also generalize the bottleneck conjecture in the context of indefinite orthogonal geometry and prove some special cases of the generalization.
We produce examples of taut foliations of hyperbolic 3–manifolds which are –covered but not uniform — ie the leaf space of the universal cover is , but pairs of leaves are not contained in bounded neighborhoods of each other. This answers in the negative a conjecture of Thurston. We further show that these foliations can be chosen to be close to foliations by closed surfaces. Our construction underscores the importance of the existence of transverse regulating vector fields and cone fields for –covered foliations. Finally, we discuss the effect of perturbing arbitrary –covered foliations.
A smooth, compact 4–manifold with a Riemannian metric and has a non-trivial, closed, self-dual 2–form. If the metric is generic, then the zero set of this form is a disjoint union of circles. On the complement of this zero set, the symplectic form and the metric define an almost complex structure; and the latter can be used to define pseudo-holomorphic submanifolds and subvarieties. The main theorem in this paper asserts that if the 4–manifold has a non zero Seiberg–Witten invariant, then the zero set of any given self-dual harmonic 2–form is the boundary of a pseudo-holomorphic subvariety in its complement.
Integral symplectic 4–manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive “symplectic volume”; it evaluates positively with the Kähler class. Some other applications of the signature formula and some more general results for genus two fibrations are discussed.
Artin groups of finite type are not as well understood as braid groups. This is due to the additional geometric properties of braid groups coming from their close connection to mapping class groups. For each Artin group of finite type, we construct a space (simplicial complex) analogous to Teichmüller space that satisfies a weak nonpositive curvature condition and also a space “at infinity” analogous to the space of projective measured laminations. Using these constructs, we deduce several group-theoretic properties of Artin groups of finite type that are well-known in the case of braid groups.
In this article, we generalize Eberlein’s Rigidity Theorem to the singular case, namely, one of the spaces is only assumed to be a CAT(0) topological manifold. As a corollary, we get that any compact irreducible but locally reducible locally symmetric space of noncompact type does not admit a nonpositively curved (in the Aleksandrov sense) piecewise Euclidean structure. Any hyperbolic manifold, on the other hand, does admit such a structure.
We show that the unit tangent bundle of and a real cohomology admit Riemannian metrics with positive sectional curvature almost everywhere. These are the only examples so far with positive curvature almost everywhere that are not also known to admit positive curvature.
Let be a closed manifold with , and let be a circle-valued Morse function. We define an invariant which counts closed orbits of the gradient of , together with flow lines between the critical points. We show that our invariant equals a form of topological Reidemeister torsion defined by Turaev [Math. Res. Lett. 4 (1997) 679–695].
We proved a similar result in our previous paper [Topology 38 (1999) 861–888], but the present paper refines this by separating closed orbits and flow lines according to their homology classes. (Previously we only considered their intersection numbers with a fixed level set.) The proof here is independent of the previous proof, and also simpler.
Aside from its Morse-theoretic interest, this work is motivated by the fact that when is three-dimensional and , the invariant equals a counting invariant which was conjectured in our previous paper to equal the Seiberg–Witten invariant of . Our result, together with this conjecture, implies that the Seiberg–Witten invariant equals the Turaev torsion. This was conjectured by Turaev and refines the theorem of Meng and Taubes [Math. Res. Lett 3 (1996) 661–674].
The Burau representation is a natural action of the braid group on the free –module of rank . It is a longstanding open problem to determine for which values of this representation is faithful. It is known to be faithful for . Moody has shown that it is not faithful for and Long and Paton improved on Moody’s techniques to bring this down to . Their construction uses a simple closed curve on the –punctured disc with certain homological properties. In this paper we give such a curve on the –punctured disc, thus proving that the Burau representation is not faithful for .
We consider an oriented surface and a cellular complex of curves on , defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of , following the ideas of Hatcher–Thurston and Harer.