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Algebraic & Geometric Topology
Algebraic & Geometric Topology
In a previous paper [Homology cylinders: an enlargement of the mapping class group, Algebr. Geom. Topol. 1 (2001) 243–270], a group [math] of homology cylinders over the oriented surface of genus [math] is defined. A filtration of [math] is defined, using the Goussarov-Habiro notion of finite-type. It is erroneously claimed that this filtration essentially coincides with the relative weight filtration. The present note corrects this error and studies the actual relation between the two filtrations.
We show that in the mapping class group of a surface any relation between Dehn twists of the form [math] ( [math] a multitwist) is the lantern relation, and any relation of the form [math] (where [math] commutes with [math] ) is the 2–chain relation.
Let [math] be a group generated by two positive multi-twists. We give some sufficient conditions for [math] to be free or have no “unexpectedly reducible” elements. For a group [math] generated by two Dehn twists, we classify the elements in [math] which are multi-twists. As a consequence we are able to list all the lantern-like relations in the mapping class groups. We classify groups generated by powers of two Dehn twists which are free, or have no “unexpectedly reducible” elements. In the end we pose similar problems for groups generated by powers of [math] twists and give a partial result.
We utilize the obstruction theory of Galewski–Matumoto–Stern to derive equivalent formulations of the Triangulation Conjecture. For example, every closed topological manifold [math] with [math] can be simplicially triangulated if and only if the two distinct combinatorial triangulations of [math] are simplicially concordant.
Given a topological space [math] denote by [math] the space of non-empty subsets of [math] of size at most [math] , topologised as a quotient of [math] . This space may be regarded as a union over [math] of configuration spaces of [math] distinct unordered points in [math] . In the special case [math] we show that: (1) [math] has the homotopy type of an odd dimensional sphere of dimension [math] or [math] ; (2) the natural inclusion of [math] into [math] is multiplication by two on homology; (3) the complement [math] of the codimension two strata in [math] has...
Strickland, N P
Let [math] and [math] be complex vector bundles over a space [math] . We use the theory of divisors on formal groups to give obstructions in generalised cohomology that vanish when [math] and [math] can be embedded in a bundle [math] in such a way that [math] has dimension at least [math] everywhere. We study various algebraic universal examples related to this question, and show that they arise from the generalised cohomology of corresponding topological universal examples. This extends and reinterprets earlier work on degeneracy classes in ordinary cohomology or intersection theory.
An open question asks if every knot of 4–genus [math] can be changed into a slice knot by [math] crossing changes. A counterexample is given.
The perturbative expression of Chern–Simons theory for links in Euclidean 3–space is a linear combination of integrals on configuration spaces. This has successively been studied by Guadagnini, Martellini and Mintchev, Bar-Natan, Kontsevich, Bott and Taubes, D. Thurston, Altschuler and Freidel, Yang and others. We give a self-contained version of this study with a new choice of compactification, and we formulate a rationality result.
Cattaneo, Alberto S; Cotta-Ramusino, Paolo; Longoni, Riccardo
The real cohomology of the space of imbeddings of [math] into [math] , [math] , is studied by using configuration space integrals. Nontrivial classes are explicitly constructed. As a by-product, we prove the nontriviality of certain cycles of imbeddings obtained by blowing up transversal double points in immersions. These cohomology classes generalize in a nontrivial way the Vassiliev knot invariants. Other nontrivial classes are constructed by considering the restriction of classes defined on the corresponding spaces of immersions.
Answering a question of W S Wilson, I introduce a [math] –equivariant Atiyah–Real analogue of Johnson–Wilson cohomology theory [math] , whose coefficient ring is the [math] –chromatic part of Landweber’s Real cobordism ring.
Let [math] be a 2–dimensional finite flag complex. We study the CAT(0) dimension of the ‘Bestvina–Brady group’, or ‘Artin kernel’, [math] . We show that [math] has CAT(0) dimension 3 unless [math] admits a piecewise Euclidean metric of non-positive curvature. We give an example to show that this implication cannot be reversed. Different choices of [math] lead to examples where the CAT(0) dimension is 3, and either (i) the geometric dimension is 2, or (ii) the cohomological dimension is 2 and the geometric dimension is not known.
In this paper we define a quantity called the rank of an outer automorphism of a free group which is the same as the index introduced in [D Gaboriau, A Jaeger, G Levitt and M Lustig, An index for counting fixed points for automorphisms of free groups, Duke Math. J. 93 (1998) 425–452] without the count of fixed points on the boundary. We proceed to analyze outer automorphisms of maximal rank and obtain results analogous to those in [D J Collins and E Turner, An automorphism of a free group of finite rank with maximal rank fixed point subgroup fixes a primitive element, J. Pure and...
Pawałowski, Krzysztof; Solomon, Ronald
In 1960, Paul A. Smith asked the following question. If a finite group [math] acts smoothly on a sphere with exactly two fixed points, is it true that the tangent [math] –modules at the two points are always isomorphic? We focus on the case [math] is an Oliver group and we present a classification of finite Oliver groups [math] with Laitinen number [math] or [math] . Then we show that the Smith Isomorphism Question has a negative answer and [math] for any finite Oliver group [math] of odd order, and for any finite Oliver group [math] with a cyclic quotient...
Gilmer, Patrick M; Kania-Bartoszynska, Joanna; Przytycki, Jozef H
We show how the periodicity of a homology sphere is reflected in the Reshetikhin–Turaev–Witten invariants of the manifold. These yield a criterion for the periodicity of a homology sphere.
A surface in the 4–sphere is trivially embedded, if it bounds a 3–dimensional handle body in the 4–sphere. For a surface trivially embedded in the 4–sphere, a diffeomorphism over this surface is extensible if and only if this preserves the Rokhlin quadratic form of this embedded surface.
We compute the Ozsváth–Szabó Floer homologies [math] and [math] for three-manifolds obtained by integer surgery on a two-bridge knot.
Niblo, Graham A; Williams, Ben T
We examine residual properties of word-hyperbolic groups, adapting a method introduced by Darren Long to study the residual properties of Kleinian groups.
We construct a family of rings. To a plane diagram of a tangle we associate a complex of bimodules over these rings. Chain homotopy equivalence class of this complex is an invariant of the tangle. On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle. When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.
A “total Chern class” invariant of knots is defined. This is a universal Vassiliev invariant which is integral “on the level of Lie algebras” but it is not expressible as an integer sum of diagrams. The construction is motivated by similarities between the Kontsevich integral and the topological Chern character.
Ahearn, Stephen T; Kuhn, Nicholas J
Let [math] denote the mapping space of continuous based functions between two based spaces [math] and [math] . If [math] is a fixed finite complex, Greg Arone has recently given an explicit model for the Goodwillie tower of the functor sending a space [math] to the suspension spectrum [math] . ¶ Applying a generalized homology theory [math] to this tower yields a spectral sequence, and this will converge strongly to [math] under suitable conditions, eg if [math] is connective and [math] is at least [math] connected. Even when the convergence is more problematic, it appears the spectral sequence can still shed considerable...