1,720,985 research outputs found
Amenable L-2-Theoretic Methods and Knot Concordance
No full textWe reveal new structures in the topological knot concordance group. As a key ingredient, we develop obstructions using L-2-theoretic methods for amenable groups in Strebel's class recently introduced by Orr and the author. Concerning (h)-solvable knots, which are defined in terms of certain Whitney towers of height h in bounding 4-manifolds, we show the following: for any n>1, there are (n)-solvable but non-(n. 5)-solvable (and therefore nonslice) knots, which are not detected by prior methods using Cochran-Orr-Teichner L-2-signature obstructions as well as Levine algebraic obstructions and Casson-Gordon invariants.X1197sciescopu
Complexity of surgery manifolds and Cheeger-Gromov invariants
No full textWe present new lower bounds on the complexity of Dehn surgery manifolds of knots, using our recent result on the Cheeger-Gromov. invariants and triangulations. As an application, we give explicit examples of closed hyperbolic 3-manifolds with fixed first homology for which the gap between the Gromov norm and the complexity is arbitrarily large.1110sciescopu
A Topological Approach to Cheeger-Gromov Universal Bounds for von Neumann rho-Invariants
No full textUsing deep analytic methods, Cheeger and Gromov showed that for any smooth (4k-1)-manifold there is a universal bound for the von Neumann L-2-invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological (4k-1)-manifolds, using L-2-signatures of bounding 4k-manifolds. We give explicit linear universal bounds for 3-manifolds in terms of triangulations, Heegaard splittings, and surgery descriptions. We show that our explicit bounds are asymptotically optimal. As an application, we give new lower bounds of the complexity of 3-manifolds that can be arbitrarily larger than previously known lower bounds. As ingredients of the proofs that seem interesting on their own, we develop a geometric construction of efficient 4-dimensional bordisms of 3-manifolds over a group and develop an algebraic topological notion of uniformly controlled chain homotopies.(c) 2016 Wiley Periodicals, Inc.1141sciescopu
THE EFFECT OF MUTATION ON LINK CONCORDANCE, 3-MANIFOLDS, AND THE MILNOR INVARIANTS
No full textWe study the effect of mutation on link concordance and 3-manifolds. We show that the set of links concordant to sublinks of homology boundary links is not closed under positive mutation. We also show that mutation does not preserve homology cobordism classes of 3-manifolds. A significant consequence is that there exist 3-manifolds which have the same quantum SU(2) -invariants but are not homology cobordant. These results are obtained by investigating the effect of mutation on the Milnor (mu) over bar -invariants, or equivalently the Massey products.X112sciescopu
FIBRED KNOTS AND TWISTED ALEXANDER INVARIANTS
No full textWe study the twisted Alexander invariants of fibred knots. We establish necessary conditions on the twisted Alexander invariants for a knot to be fibred, and develop a practical method to compute the twisted Alexander invariants from the homotopy type of a monodromy. It is illustrated that the twisted Alexander invariants carry more information on fibredness than the classical Alexander invariants, even for knots with trivial Alexander polynomials.open1135sciescopu
THE STRUCTURE OF THE RATIONAL CONCORDANCE GROUP OF KNOTS
No full textWe study the group of rational concordance classes of codimension two knots in rational homology spheres. We give a full calculation of its algebraic theory by developing a complete set of new invariants. For computation, we relate these invariants with limiting behaviour of the Artin reciprocity over an infinite tower of number fields and analyze it using tools from algebraic number theory. In higher dimensions it classifies the rational concordance group of knots whose ambient space satisfies a certain cobordism theoretic condition. In particular, we construct infinitely many torsion elements. We show that the structure of the rational concordance group is much more complicated than the integral concordance group from a topological viewpoint. We also investigate the structure peculiar to knots in rational homology 3-spheres. To obtain further nontrivial obstructions in this dimension, we develop a technique of controlling a certain limit of the von Neumann L-2-signature invariants.open1123sciescopu
SYMMETRIC WHITNEY TOWER COBORDISM FOR BORDERED 3-MANIFOLDS AND LINKS
Full text linkWe introduce the notion of a symmetric Whitney tower cobordism between bordered 3-manifolds, aiming at the study of homology cobordism and link concordance. It is motivated by the symmetric Whitney tower approach to slicing knots and links initiated by T. Cochran, K. Orr, and P. Teichner. We give amenable Cheeger-Gromov ρ-invariant obstructions to bordered 3-manifolds being Whitney tower cobordant. Our obstruction is related to and generalizes several prior known results, and also gives new interesting cases. As an application, our method applied to link exteriors reveals new structures on (Whitney tower and grope) concordance between links with nonzero linking number, including the Hopf link. © 2014 American Mathematical Society.open119Nsciescopu
Topological minimal genus and L-2-signatures
No full textWe obtain new lower bounds for the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov rho-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples with arbitrary slice genus for which our lower bound is optimal but all previously known bounds vanish.open111614sciescopu
Link concordance, homology cobordism, and Hirzebruch-type defects from iterated p-covers
No full textWe obtain new invariants of topological link concordance and homology cobordism of 3-manifolds from Hirzebruch-type intersection form defects of towers of iterated p-covers. Our invariants can extract geometric information from an arbitrary depth of the derived series of the fundamental group, and can detect torsion which is invisible via signature invariants. Applications illustrating these features include the following: (1) There are infinitely many homology equivalent rational 3-spheres which are indistinguishable via multisignatures, eta-invariants, and L-2-signatures but have distinct homology cobordism types. (2) There is an infinite family of 2-torsion (amphichiral) knots with non-slice iterated Bing doubles; as a special case, we give the first proof of the conjecture that the Bing double of the figure eight knot is not slice. (3) There exist infinitely many torsion elements at any depth of the Cochran-Orr-Teichner filtration of link concordance.X111918sciescopu
Injectivity theorems and algebraic closures of groups with coefficients
No full textRecently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain homology localization of groups. In order to give a concrete combinatorial description and existence proof of the necessary homology localization, we introduce a new version of algebraic closures of groups with coefficients by considering certain types of equations.X1199sciescopu
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