We determine the Zariski closure of the representations of the braid groups that factor through the Birman–Wenzl–Murakami algebra, for generic values of the parameters $\alpha ,s$. For $\alpha ,s$ of modulus 1 and close to 1, we prove that these representations are unitarizable, thus deducing the topological closure of the image when in addition $\alpha ,s$ are algebraically independent.

Let $M={\mathbb{S}}^2$ or $ℝP^2$. Let $P{B}_n\left(M\right)$ and $B_n\left(M\right)$ be the pure and the full braid groups of $M$ respectively. If $\Gamma$ is any of these groups, we show that $\Gamma$ satisfies the Farrell–Jones Fibered Isomorphism Conjecture and use this fact to compute the lower algebraic $K$–theory of the integral group ring $\mathbb{Z}\Gamma$, for $\Gamma =P{B}_n\left(M\right)$. The main results are that for $\Gamma =P{B}_n\left({\mathbb{S}}^2\right)$, the Whitehead group of $\Gamma$, ${\tilde{K}}_0\left(\mathbb{Z}\Gamma \right)$ and $K_i\left(\mathbb{Z}\Gamma \right)$ vanish for $i\le -1$ and $n>0$. For $\Gamma =P{B}_n\left(ℝP^2\right)$, the Whitehead group of $\Gamma$ vanishes for all $n>0$, ${\tilde{K}}_0\left(\mathbb{Z}\Gamma \right)$ vanishes for all $n>0$ except for the cases $n=2,3$ and $K_i\left(\mathbb{Z}\Gamma \right)$ vanishes for all $i\le -1$.

In this article, we classify all involutions on $S^6$ with $3$–dimensional fixed point set. In particular, we discuss the relation between the classification of involutions with fixed point set a knotted $3$–sphere and the classification of free involutions on homotopy $ℂP^3$’s.

We construct a cofibrantly generated Quillen model structure on the category of small $n$–fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An $n$–fold functor is a weak equivalence if and only if the diagonal of its $n$–fold nerve is a weak equivalence of simplicial sets. This is an $n$–fold analogue to Thomason’s Quillen model structure on $Cat$. We introduce an $n$–fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the $n$–fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and $n$–fold categories are natural weak equivalences.

In [arXiv:0706.0741], Lawrence Roberts, extending the work of Ozsváth and Szabó in [Adv. Math 194 (2005) 1-33], showed how to associate to a link $\mathbb{L}$ in the complement of a fixed unknot$B\subset S^3$ a spectral sequence whose $E^2$ term is the Khovanov homology of a link in a thickened annulus defined by Asaeda, Przytycki and Sikora in [Algebr. Geom. Topol. 4 (2004) 1177-1210], and whose $E^{\infty }$ term is the knot Floer homology of the preimage of $B$ inside the double-branched cover of $\mathbb{L}$.

In [Adv. Math. 223 (2010) 2114-2165], we extended the aforementioned Ozsváth–Szabó paper in a different direction, constructing for each knot $K\subset S^3$ and each $n\in {\mathbb{Z}}_{+}$, a spectral sequence from Khovanov’s categorification of the reduced, $n$–colored Jones polynomial to the sutured Floer homology of a reduced $n$–cable of $K$. In the present work, we reinterpret Roberts’ result in the language of Juhasz’s sutured Floer homology [Algebr. Geom. Topol. 6 (2006) 1429–1457] and show that the spectral sequence of [Adv. Math. 223 (2010) 2114-2165] is a direct summand of the spectral sequence of Roberts’ paper.

In this paper we study the small dilatation pseudo-Anosov mapping classes arising from fibrations over the circle of a single 3–manifold, the mapping torus for the “simplest hyperbolic braid”. The dilatations that occur include the minimum dilatations for orientable pseudo-Anosov mapping classes for genus $g=2,3,4,5$ and $8$. We obtain the “Lehmer example” in genus $g=5$, and Lanneau and Thiffeault’s conjectural minima in the orientable case for all genus $g$ satisfying $g=2$ or $4\left(mod6\right)$. Our examples show that the minimum dilatation for orientable mapping classes is strictly greater than the minimum dilatation for non-orientable ones when $g=4,6$ or $8$. We also prove that if ${\delta }_g$ is the minimum dilatation of pseudo-Anosov mapping classes on a genus $g$ surface, then

$$\underset{g\to \infty }{limsup}{\left({\delta }_g\right)}^g\le \frac{3+\sqrt{5}}{2}.$$

We introduce the category $\mathcal{P}stem\left[n\right]$ of $n$–stems, with a functor $\mathcal{P}\left[n\right]$ from spaces to $\mathcal{P}stem\left[n\right]$. This can be thought of as the $n$–th order homotopy groups of a space. We show how to associate to each simplicial $n$–stem ${\mathcal{Q}}_{\bullet }$ an $\left(n+1\right)$–truncated spectral sequence. Moreover, if ${\mathcal{Q}}_{\bullet }=\mathcal{P}\left[n\right]X_{\bullet }$ is the Postnikov $n$–stem of a simplicial space $X_{\bullet }$, the truncated spectral sequence for ${\mathcal{Q}}_{\bullet }$ is the truncation of the usual homotopy spectral sequence of $X_{\bullet }$. Similar results are also proven for cosimplicial $n$–stems. They are helpful for computations, since $n$–stems in low degrees have good algebraic models.

In In [Ann. Math. (2) 106 (1977) 469–516], Miller, Ravenel and Wilson defined generalized beta elements in the $E_2$–term of the Adams–Novikov spectral sequence converging to the stable homotopy groups of spheres, and in [Hiroshima Math. J. 7 (1977) 427–447], Oka showed that the beta elements of the form ${\beta }_{t{p}^2∕r}$ for positive integers $t$ and $r$ survive to the homotopy of spheres at a prime $p>3$, when $r\le 2p-2$ and $r\le 2p$ if $t>1$. In this paper, for $p>5$, we expand the condition so that ${\beta }_{t{p}^2∕r}$ for $t\ge 1$ and $r\le p^2-2$ survives to the stable homotopy groups.

We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the $\left(1+1\right)$–dimensional topological quantum field theory defined by Honda, Kazez and Matić in [arXiv:0807.2431]. The ${\mathbb{Z}}_2$ SFH of these solid tori forms a “categorification of Pascal’s triangle”, and contact structures correspond bijectively to chord diagrams, or sets of disjoint properly embedded arcs in the disc. Their contact elements are distinct and form distinguished subsets of SFH of order given by the Narayana numbers. We find natural “creation and annihilation operators” which allow us to define a QFT–type basis of each SFH vector space, consisting of contact elements. Sutured Floer homology in this case reduces to the combinatorics of chord diagrams. We prove that contact elements are in bijective correspondence with comparable pairs of basis elements with respect to a certain partial order, and in a natural and explicit way. The algebraic and combinatorial structures in this description have intrinsic contact-topological meaning.

In particular, the QFT–basis of SFH and its partial order have a natural interpretation in pure contact topology, related to the contact category of a disc: the partial order enables us to tell when the sutured solid cylinder obtained by “stacking” two chord diagrams has a tight contact structure. This leads us to extend Honda’s notion of contact category to a “bounded” contact category, containing chord diagrams and contact structures which occur within a given contact solid cylinder. We compute this bounded contact category in certain cases. Moreover, the decomposition of a contact element into basis elements naturally gives a triple of contact structures on solid cylinders which we regard as a type of “distinguished triangle” in the contact category. We also use the algebraic structures arising among contact elements to extend the notion of contact category to a $2$–category.

We define the stable $4$–genus of a knot $K\subset S^3$, $g_{st}\left(K\right)$, to be the limiting value of $g_4\left(nK\right)∕n$, where $g_4$ denotes the $4$–genus and $n$ goes to infinity. This induces a seminorm on the rationalized knot concordance group, ${\mathcal{C}}_Q=\mathcal{C}\otimes Q$. Basic properties of $g_{st}$ are developed, as are examples focused on understanding the unit ball for $g_{st}$ on specified subspaces of ${\mathcal{C}}_Q$. Subspaces spanned by torus knots are used to illustrate the distinction between the smooth and topological categories. A final example is given in which Casson–Gordon invariants are used to demonstrate that $g_{st}\left(K\right)$ can be a noninteger.

This paper continues the investigation of the configuration space of two distinct points on a graph. We analyze the process of adding an additional edge to the graph and the resulting changes in the topology of the configuration space. We introduce a linking bilinear form on the homology group of the graph with values in the cokernel of the intersection form (introduced in Part I of this work). For a large class of graphs, which we call mature graphs, we give explicit expressions for the homology groups of the configuration space. We show that under a simple condition, adding an edge to a mature graph yields another mature graph.

If $\Gamma$ is a group acting properly by semisimple isometries on a proper $CAT\left(0\right)$ space $X$, then we build models for the classifying spaces $E_{\mathcal{V}\mathcal{C}}\Gamma$ and $E_{\mathcal{ℱ}\mathcal{ℬ}\mathcal{C}}\Gamma$ under the additional assumption that the action of $\Gamma$ has a well-behaved collection of axes in $X$. We verify that the latter assumption is satisfied in two cases: (i) when $X$ has isolated flats, and (ii) when $X$ is a simply connected real analytic manifold of nonpositive sectional curvature. We conjecture that $\Gamma$ has a well-behaved collection of axes in the great majority of cases.

Our classifying spaces are natural variations of the constructions due to Connolly, Fehrman and Hartglass [arXiv:math.AT/0610387] of $E_{\mathcal{V}\mathcal{C}}\Gamma$ for crystallographic groups $\Gamma$.

We show that subsurfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.

In this article we study the curvature properties of the order complex of a bounded graded poset under a metric that we call the “orthoscheme metric”. In addition to other results, we characterize which rank $4$ posets have $CAT\left(0\right)$ orthoscheme complexes and by applying this theorem to standard posets and complexes associated with four-generator Artin groups, we are able to show that the $5$–string braid group is the fundamental group of a compact nonpositively curved space.

Since the set of volumes of hyperbolic $3$–manifolds is well ordered, for each fixed $g$ there is a genus–$g$ surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus–$g$ bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumption, we prove that this is indeed the case when $g$ is large. The proof combines a soft geometric limit argument with a detailed Neumann–Zagier asymptotic formula for the volumes of Dehn fillings.

Our examples are all Dehn fillings on the sibling of the Whitehead manifold, and we also analyze the dilatations of all closed surface bundles obtained in this way, identifying those with minimal dilatation. This gives new families of pseudo-Anosovs with low dilatation, including a genus 7 example which minimizes dilatation among all those with orientable invariant foliations.

The symmetric homology of a unital algebra $A$ over a commutative ground ring $k$ is defined using derived functors and the symmetric bar construction of Fiedorowicz. For a group ring $A=k\left[\Gamma \right]$, the symmetric homology is related to stable homotopy theory via $H{S}_{\ast }\left(k\left[\Gamma \right]\right)\cong H_{\ast }\left(\Omega {\Omega }^{\infty }{S}^{\infty }\left(B\Gamma \right);k\right)$. Two chain complexes that compute $H{S}_{\ast }\left(A\right)$ are constructed, both making use of a symmetric monoidal category $\Delta S_{+}$ containing $\Delta S$. Two spectral sequences are found that aid in computing symmetric homology. The second spectral sequence is defined in terms of a family of complexes, ${Sym}_{\ast }^{\left(p\right)}$. ${Sym}^{\left(p\right)}$ is isomorphic to the suspension of the cycle-free chessboard complex ${\Omega }_{p+1}$ of Vrećica and Živaljević, and so recent results on the connectivity of ${\Omega }_n$ imply finite-dimensionality of the symmetric homology groups of finite-dimensional algebras. Some results about the $k{\Sigma }_{p+1}$–module structure of ${Sym}^{\left(p\right)}$ are devloped. A partial resolution is found that allows computation of $H{S}_1\left(A\right)$ for finite-dimensional $A$ and some concrete computations are included.

We prove that the free group ${\mathbb{F}}_2$ admits a faithful discrete representation into ${Diff}_{+}^1\left[0,1\right]$. We also prove that ${\mathbb{F}}_2$ admits a faithful discrete representation of bi-Lipschitz class into ${Homeo}_{+}\left[0,1\right]$. Some properties of these representations are studied.

In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive – a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an associated coarse assembly map from the topologically excisive functor to the coarsely excisive functor.

We conjecture that this coarse assembly map is an isomorphism for uniformly contractible spaces with bounded geometry, and show that the coarse isomorphism conjecture, along with some mild technical conditions, implies that a corresponding equivariant assembly map is injective. Particular instances of this equivariant assembly map are the maps in the Farrell–Jones conjecture, and in the Baum–Connes conjecture.

Une ligne d’étirement cylindrique est une ligne d’étirement au sens de Thurston dont la lamination horocyclique est une multicourbe pondérée. Nous montrons ici que deux lignes cylindriques correctement paramétrées sont parallèles si et seulement si ces lignes convergent vers le même point du bord de Thurston de l’espace de Teichmüller.

A cylindrical stretch line is a stretch line, in the sense of Thurston, whose horocyclic lamination is a weighted multicurve. In this paper, we show that two correctly parameterized cylindrical lines are parallel if and only if these lines converge towards the same point in Thurston’s boundary of Teichmüller space.