Let $Q$ be a Riemannian manifold such that the Betti numbers of its free loop space with respect to some coefficient field are unbounded. We show that every contact form on its unit cotangent bundle supporting the natural contact structure has infinitely many simple Reeb orbits. This is an extension of a theorem by Gromoll and Meyer. We also show that if a compact manifold admits a Stein fillable contact structure then there is a possibly different such structure which also has infinitely many simple Reeb orbits for every supporting contact form. We use local Floer homology along with symplectic homology to prove these facts.

A basic result in equivariant $K$–theory, the Atiyah–Segal completion theorem relates the $G$–equivariant $K$–theory of a finite $G$–CW complex to the non-equivariant $K$–theory of its Borel construction. We prove the analogous result for twisted equivariant $K$–theory.

We prove that among four-dimensional ideal right-angled hyperbolic polytopes the $24$–cell is of minimal volume and of minimal facet number. As a corollary, a dimension bound for ideal right-angled hyperbolic polytopes is obtained.

Cannon, Swenson and others have proved numerous theorems about subdivision rules associated to hyperbolic groups with a $2$–sphere at infinity. However, few explicit examples are known. We construct an explicit finite subdivision rule for many $3$–manifolds obtained from polyhedral gluings. The manifolds that satisfy the conditions include all manifolds created from compact right angled hyperbolic polyhedra, as well as many $3$–manifolds with toral or hyperbolic boundary.

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1{\ast }_{Q_1\cap Q_2}{Q}_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, and results for discrete subgroups of isometries of hyperbolic spaces. An application on separability of double cosets of quasiconvex subgroups is included.

We show the ${\mathbb{T}}^2$–cobordism group of the category of 4–dimensional quasitoric manifolds is generated by the ${\mathbb{T}}^2$–cobordism classes of ${ℂ\mathbb{P}}^2$. We construct nice oriented ${\mathbb{T}}^2$ manifolds with boundary whose boundaries are the Hirzebruch surfaces. The main tool is the theory of quasitoric manifolds.

Generalized Baumslag–Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester [Geom. Topol. 6 (2002) 219-267] proved that in most cases the defining graphs are cyclic JSJ decompositions, in the sense of Rips and Sela. Here we extend Forester’s results to graphs of groups with vertex groups that can be either infinite cyclic or quadratically hanging surface groups.

Given a hyperbolic knot, we prove that the Reidemeister torsion of any lift of the holonomy to $SL\left(2,ℂ\right)$ is invariant under mutation along a surface of genus 2, hence also under mutation along a Conway sphere.

We define a notion of concordance based on Euler characteristic, and show that it gives rise to a concordance group $\mathcal{ℒ}$ of links in $S^3$, which has the concordance group of knots as a direct summand with infinitely generated complement. We consider variants of this using oriented and unoriented surfaces as well as smooth and locally flat embeddings.

We investigate the behavior of the $\mathit{SL}\left(2,ℂ\right)$ Casson invariant for $3$–manifolds obtained by Dehn surgery along two-bridge knots. Using the results of Hatcher and Thurston, and also results of Ohtsuki, we outline how to compute the Culler–Shalen seminorms, and we illustrate this approach by providing explicit computations for double twist knots. We then apply the surgery formula of Curtis [Topology 40 (2001), 773–787] to deduce the $\mathit{SL}\left(2,ℂ\right)$ Casson invariant for the $3$–manifolds obtained by $\left(p∕q\right)$–Dehn surgery on such knots. These results are applied to prove nontriviality of the $\mathit{SL}\left(2,ℂ\right)$ Casson invariant for nearly all $3$–manifolds obtained by nontrivial Dehn surgery on a hyperbolic two-bridge knot. We relate the formulas derived to degrees of $A$–polynomials and use this information to identify factors of higher multiplicity in the $Â$–polynomial, which is the $A$–polynomial with multiplicities as defined by Boyer–Zhang.

Given a knot $K$ in $S^3$, let $\Sigma \left(K\right)$ be the double branched cover of $S^3$ over $K$. We show there is a spectral sequence whose $E^1$ page is $\left(\widehat{\mathit{HFK}}\left(\Sigma \left(K\right),K\right)\otimes V^{\otimes \left(n-1\right)}\right)\otimes {\mathbb{Z}}_2\left(\left(q\right)\right)$, for $V$ a ${\mathbb{Z}}_2$–vector space of dimension two, and whose $E^{\infty }$ page is isomorphic to $\left(\widehat{\mathit{HFK}}\left(S^3,K\right)\otimes V^{\otimes \left(n-1\right)}\right)\otimes {\mathbb{Z}}_2\left(\left(q\right)\right)$, as ${\mathbb{Z}}_2\left(\left(q\right)\right)$–modules. As a consequence, we deduce a rank inequality between the knot Floer homologies $\widehat{\mathit{HFK}}\left(\Sigma \left(K\right),K\right)$ and $\widehat{\mathit{HFK}}\left(S^3,K\right)$.

Given an ideal triangulation of a connected $3$–manifold with nonempty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject to Thurston’s gluing equations. From this, one can recover a representation of the fundamental group of the manifold into the isometries of $3$–dimensional hyperbolic space. However, the deformation variety depends crucially on the triangulation: there may be entire components of the representation variety which can be obtained from the deformation variety with one triangulation but not another. We introduce a generalisation of the deformation variety, which again consists of assignments of complex variables to certain dihedral angles subject to polynomial equations, but together with some extra combinatorial data concerning degenerate tetrahedra. This “extended deformation variety” deals with many situations that the deformation variety cannot. In particular we show that for any ideal triangulation of a small orientable $3$–manifold with a single torus boundary component, we can recover all of the irreducible nondihedral representations from the associated extended deformation variety. More generally, we give an algorithm to produce a triangulation of a given orientable $3$–manifold with torus boundary components for which the same result holds. As an application, we show that this extended deformation variety detects all factors of the $PSL\left(2,ℂ\right)$ A–polynomial associated to the components consisting of the representations it recovers.

Let $G={\mathbb{Z}}_{p^k}$ be a cyclic group of prime power order and let $V$ and $W$ be orthogonal representations of $G$ with $V^G=W^G=\left\{0\right\}$. Let $S\left(V\right)$ be the sphere of $V$ and suppose $f:S\left(V\right)\to W$ is a $G$–equivariant mapping. We give an estimate for the dimension of the set $f^{-1}\left\{0\right\}$ in terms of $V$ and $W$. This extends the Bourgin–Yang version of the Borsuk–Ulam theorem to this class of groups. Using this estimate, we also estimate the size of the $G$–coincidences set of a continuous map from $S\left(V\right)$ into a real vector space $W^{\prime }$.

We construct a relation among right-handed Dehn twists in the mapping class group of a compact oriented surface of genus $g$ with $4g+4$ boundary components. This relation gives an explicit topological description of $4g+4$ disjoint $\left(-1\right)$–sections of a hyperelliptic Lefschetz fibration of genus $g$ on the manifold ${ℂ\mathbb{P}}^2\#\left(4g+5\right){\overline{ℂ\mathbb{P}}}^2$.

We give a full solution in terms of $k$–invariants of the $D\left(2\right)$–problem for $D_{4n}$, assuming that $\mathbf{Z}\left[D_{\mathrm{4n}}\right]$ satisfies torsion-free cancellation.

We define and study an equivariant version of Farber’s topological complexity for spaces with a given compact group action. This is a special case of the equivariant sectional category of an equivariant map, also defined in this paper. The relationship of these invariants with the equivariant Lusternik–Schnirelmann category is given. Several examples and computations serve to highlight the similarities and differences with the nonequivariant case. We also indicate how the equivariant topological complexity can be used to give estimates of the nonequivariant topological complexity.

We give here some extensions of Gromov’s and Polterovich’s theorems on $k$–area of ${ℂ\mathbb{P}}^n$, particularly in the symplectic and Hamiltonian context. Our main methods involve Gromov–Witten theory, and some connections with Bott periodicity and the theory of loop groups. The argument is closely connected with the study of jumping curves in ${ℂ\mathbb{P}}^n$, and as an upshot we prove a new symplectic-geometric theorem on these jumping curves.

Using methods developed by Franke in [K-theory Preprint Archives 139 (1996)], we obtain algebraic classification results for modules over certain symmetric ring spectra (S-algebras). In particular, for any symmetric ring spectrum $R$ whose graded homotopy ring ${\pi }_{\ast }R$ has graded global homological dimension $2$ and is concentrated in degrees divisible by some natural number $N\ge 4$, we prove that the homotopy category of $R$–modules is equivalent to the derived category of the homotopy ring ${\pi }_{\ast }R$. This improves the Bousfield-Wolbert algebraic classification of isomorphism classes of objects of the homotopy category of R-modules. The main examples of ring spectra to which our result applies are the $p$–local real connective $K$–theory spectrum $k{o}_{\left(p\right)}$, the Johnson–Wilson spectrum $E\left(2\right)$, and the truncated Brown–Peterson spectrum $BP\langle 1\rangle$, all for an odd prime $p$. We also show that the equivalences for all these examples are exotic in the sense that they do not come from a zigzag of Quillen equivalences.

We consider the rational vector space generated by all rational homology spheres up to orientation-preserving homeomorphism, and the filtration defined on this space by Lagrangian-preserving rational homology handlebody replacements. We identify the graded space associated with this filtration with a graded space of augmented Jacobi diagrams.

We show how to construct homology bases for certain CW complexes in terms of discrete Morse theory and cellular homology. We apply this technique to study certain subcomplexes of the half cube polytope studied in previous works. This involves constructing explicit complete acyclic Morse matchings on the face lattice of the half cube; this procedure may be of independent interest for other highly symmetric polytopes.

The category $D_s$–$U$ of unstable modules over the Steenrod algebra equipped with a compatible module structure over the Dickson algebra $D_s$ is studied at the prime $2$, with applications to the Singer functor $R_s$, considered as a functor from unstable modules $U$ to $D_s$–$U$. An explicit copresentation of $R_{s}M$ is given using Lannes’ $T$–functor when $M$ is a reduced unstable module; applying Lannes’ functor ${Fix}_s$, this is used to show that $R_s$ gives a fully-faithful embedding of $U$ in $D_s$–$U$. In addition, the right adjoint ${\mathfrak{ℨ}}_s$ to $R_s$ is introduced and is related to the indecomposables functor and the functor ${Fix}_s$.

We give estimates on the length of paths defined in the sphere model of outer space using a surgery process, and show that they make definite progress in some sense when they remain in some thick part of outer space. To do so, we relate the Lipschitz metric on outer space to a notion of intersection numbers.