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Let and be orientable irreducible –manifolds with connected boundary and suppose . Let be a closed –manifold obtained by gluing to along the boundary. We show that if the gluing homeomorphism is sufficiently complicated, then is not homeomorphic to and all small-genus Heegaard splittings of are standard in a certain sense. In particular, , where denotes the Heegaard genus of . This theorem is also true for certain manifolds with multiple boundary components.
We show that there are at most finitely many one cusped orientable hyperbolic –manifolds which have more than eight nonhyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable.
The Maskit embedding of a surface is the space of geometrically finite groups on the boundary of quasifuchsian space for which the “top” end is homeomorphic to , while the “bottom” end consists of triply punctured spheres, the remains of when a set of pants curves have been pinched. As such representations vary in the character variety, the conformal structure on the top side varies over the Teichmüller space .
We investigate when is a twice punctured torus, using the method of pleating rays. Fix a projective measure class supported on closed curves on . The pleating ray consists of those groups in for which the bending measure of the top component of the convex hull boundary of the associated –manifold is in . It is known that is a real –submanifold of . Our main result is a formula for the asymptotic direction of in as the bending measure tends to zero, in terms of natural parameters for the complex –dimensional representation space and the Dehn–Thurston coordinates of the support curves to relative to the pinched curves on the bottom side. This leads to a method of locating in .
It is known that perturbative invariants of rational homology 3–spheres can be constructed by using arithmetic perturbative expansion of quantum invariants of them. However, we could not make arithmetic perturbative expansion of quantum invariants for 3–manifolds with positive Betti numbers by the same method.
In this paper, we explain how to make arithmetic perturbative expansion of quantum invariants of 3–manifolds with the first Betti number . Further, motivated by this expansion, we construct perturbative invariants of such 3–manifolds. We show some properties of the perturbative invariants, which imply that their coefficients are independent invariants.
KEYWORDS: metrics of positive scalar curvature, moduli space of positive scalar curvature metrics, classifying space of a diffeomorphism group, Gromov–Lawson surgery parametrized by a Morse function, rational homotopy type, Hatcher map, 53-02, 55-02
We show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov–Lawson to certain nonlinear smooth sphere bundles constructed by Hatcher.
We study fillings of contact structures supported by planar open books by analyzing positive factorizations of their monodromy. Our method is based on Wendl’s theorem on symplectic fillings of planar open books. We prove that every virtually overtwisted contact structure on has a unique filling, and describe fillable and nonfillable tight contact structures on certain Seifert fibered spaces.
Cet article est la suite de l’article [arXiv :0902.3143] dans lequel l’auteur caractérisait le fait d’être de volume fini pour une surface projective convexe. On montre ici que l’espace des modules des structures projectives convexes de volume fini sur la surface de genre à pointes est homéomorphe à .
Enfin, on montre que s’identifie à une composante connexe de l’espace des représentations du groupe fondamental de dans qui conservent les paraboliques à conjugaison près.
This article follows the article [arXiv :0902.3143] in which the author characterizes the fact of being of finite volume for a convex projective surface. We show here that the moduli space of convex projective structures on the surface of genus with punctures is homeomorphic to .
Finally, we show that can be identified with a connected component of the space of representations of the fundamental group of in which keep the parabolics modulo conjugation.
We determine the rational homology of the space of long knots in for . Our main result is that the Vassiliev spectral sequence computing this rational homology collapses at the page. As a corollary we get that the homology of long knots (modulo immersions) is the Hochschild homology of the Poisson algebras operad with bracket of degree , which can be obtained as the homology of an explicit graph complex and is in theory completely computable.
Our proof is a combination of a relative version of Kontsevich’s formality of the little –disks operad and of Sinha’s cosimplicial model for the space of long knots arising from Goodwillie–Weiss embedding calculus. As another ingredient in our proof, we introduce a generalization of a construction that associates a cosimplicial object to a multiplicative operad. Along the way we also establish some results about the Bousfield–Kan spectral sequences of a truncated cosimplicial space.
We show that the introduction of polar coordinates in toric geometry smoothes a wide class of equivariant mappings, rendering them locally trivial in the topological category. As a consequence, we show that the Betti realization of a smooth proper and exact mapping of log analytic spaces is a topological fibration, whose fibers are orientable manifolds (possibly with boundary). This turns out to be true even for certain noncoherent log structures, including some families familiar from mirror symmetry. The moment mapping plays a key role in our proof.
We prove that two angle-compatible Coxeter generating sets of a given finitely generated Coxeter group are conjugate provided one of them does not admit any elementary twist. This confirms a basic case of a general conjecture which describes a potential solution to the isomorphism problem for Coxeter groups.
We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones , and . This gives a new proof of the Hamilton–Ivey–Perelman classification of –dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of , , , or .
If there are any –component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions.
The simplest plausible counterexample to the Generalized Property R Conjecture could be a –component link containing the square knot. We characterize all two-component links that contain the square knot and which surger to . We exhibit a family of such links that are probably counterexamples to Generalized Property R. These links can be used to generate slice knots that are not known to be ribbon.
We show that the Adams operation , , in complex –theory lifts to an operation in smooth –theory. If is a –oriented vector bundle with Thom isomorphism , then there is a characteristic class such that in for all . We lift this class to a –valued characteristic class for real vector bundles with geometric –structures.
If is a –oriented proper submersion, then for all we have in , where is the stable –oriented normal bundle of . To a smooth –orientation of we associate a class refining . Our main theorem states that if is compact, then in for all . We apply this result to the –invariant of bundles of framed manifolds and –invariants of flat vector bundles.
In this article we prove that for a smooth fiberwise convex Hamiltonian, the asymptotic Hofer distance from the identity gives a strict upper bound to the value at 0 of Mather’s function, thus providing a negative answer to a question asked by Siburg [Duke Math. J. 92 (1998) 295-319]. However, we show that equality holds if one considers the asymptotic distance defined in Viterbo [Math. Ann. 292 (1992) 685-710].
Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into does not contain parabolics, then it is also the sequence’s geometric limit. We construct examples that demonstrate the failure of this theorem for certain sequences of unfaithful representations, and offer a suitable replacement.
Suppose that is the Gromov–Hausdorff limit of a sequence of Riemannian manifolds with a uniform lower bound on Ricci curvature. In a previous paper the authors showed that when is compact the universal cover is a quotient of the Gromov–Hausdorff limit of the universal covers . This is not true when is noncompact. In this paper we introduce the notion of pseudo-nullhomotopic loops and give a description of the universal cover of a noncompact limit space in terms of the covering spaces of balls of increasing size in the sequence.