We study the topology of configuration spaces $F_r\left(\Gamma ,2\right)$ of two thick particles (robots) of radius $r>0$ moving on a metric graph $\Gamma$. As the size of the robots increases, the topology of $F_r\left(\Gamma ,2\right)$ varies. Given $\Gamma$ and $r$, we provide an algorithm for computing the number of path components of $F_r\left(\Gamma ,2\right)$. Using our main tool of PL Morse–Bott theory, we show that there are finitely many critical values of $r$ where the homotopy type of $F_r\left(\Gamma ,2\right)$ changes. We study the transition across a critical value $R\in \left(a,b\right)$ by computing the ranks of the relative homology groups of $\left(F_a\left(\Gamma ,2\right),F_b\left(\Gamma ,2\right)\right)$.

This paper investigates the relation between Toda brackets and congruences of modular forms. It determines the $f$–invariant of Toda brackets and thereby generalizes the formulas of J F Adams for the classical $e$–invariant to the chromatic second filtration.

For a closed topological manifold $M$ with $dim\left(M\right)\ge 5$ the topological structure set $\mathcal{S}\left(M\right)$ admits an abelian group structure which may be identified with the algebraic structure group of $M$ as defined by Ranicki. If $dim\left(M\right)=2d-1$, $M$ is oriented and $M$ is equipped with a map to the classifying space of a finite group $G$, then the reduced $\rho$–invariant defines a function,

$$\tilde{\rho }:\mathcal{S}\left(M\right)\to \mathbb{Q}{R}_{Ĝ}^{{\left(-1\right)}^d},$$

to a certain subquotient of the complex representation ring of $G$. We show that the function $\tilde{\rho }$ is a homomorphism when $2d-1\ge 5$.

Along the way we give a detailed proof that a geometrically defined map due to Cappell and Weinberger realises the 8–fold Siebenmann periodicity map in topological surgery.

We construct periodic families of Poincaré complexes, partially solving a question of Hodgson, and infinite families of Poincaré complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.

We show that if $K$ is a knot in $S^3$ and $\Sigma$ is a bridge sphere for $K$ with high distance and $2n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $\Sigma$ via an isotopy is $n$. We also construct a knot with two different bridge spheres with $2n$ and $2n-1$ bridges respectively for which any common perturbation has at least $3n-4$ bridges. We generalize both of these results to bridge surfaces for knots in any $3$–manifold.

This paper contains several results about the Chabauty space of a general locally compact abelian group. Notably, we determine its topological dimension, we characterize when it is totally disconnected or connected; we characterize isolated points.

We generalize the works of Lee [Adv. Math. 197 (2005) 554–586] and Gornik [arXiv math.QA/0402266] to construct a basis for generic deformations of the colored $\mathfrak{s}\mathfrak{l}\left(N\right)$–homology defined in [arXiv 1002.2662v2]. As applications, we construct nondegenerate pairings and co-pairings which lead to dualities of generic deformations of the colored $\mathfrak{s}\mathfrak{l}\left(N\right)$–homology. We also define and study colored $\mathfrak{s}\mathfrak{l}\left(N\right)$–Rasmussen invariants. Among other things, we observe that these invariants vanish on amphicheiral knots and discuss some implications of this observation.

We find a simple algebraic model for rational ${\mathbb{Z}}_p$–equivariant spectra, via a series of Quillen equivalences. This model, along with an Adams short exact sequence, will allow us to easily perform constructions and calculations.

The $SO\left(3\right)$ Kauffman polynomial and the chromatic polynomial of planar graphs are categorified by a unique extension of the Khovanov homology framework. Many structural observations and computations of homologies of knots and spin networks are included.

Cho and McCullough gave a numerical parameterization of the collection of all tunnels of all tunnel number 1 knots and links in the $3$–sphere. Here we give an algorithm for finding the parameter of a given tunnel by using its Heegaard diagram.

We consider the $SO\left(3\right)$ Witten–Reshetikhin–Turaev quantum invariants of random 3–manifolds. When the level $r$ is prime, we show that the asymptotic distribution of the absolute value of these invariants is given by a Rayleigh distribution which is independent of the choice of level. Hence the probability that the quantum invariant certifies the Heegaard genus of a random 3–manifold of a fixed Heegaard genus $g$ is positive but very small, less than $10^{-7}$ except when $g\le 3$. We also examine random surface bundles over the circle and find the same distribution for quantum invariants there.

We show that there exist flat surface bundles with closed leaves having nontrivial normal bundles. This leads us to compute the abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant of certain Hamiltonian diffeomorphisms to surfaces of higher genus and derive a similar formula for the first MMM–class of surface bundles with punctured fibre.

Let $M$ be a hyperbolic $n$–manifold whose cusps have torus cross-sections. In an earlier paper, the authors constructed a variety of nonpositively and negatively curved spaces as “$2\pi$–fillings” of $M$ by replacing the cusps of $M$ with compact “partial cones” of their boundaries. These $2\pi$–fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such $2\pi$–filling is positive, and bounded above by $\frac{Vol\left(M\right)}{v_n}$, where $v_n$ is the volume of a regular ideal hyperbolic $n$–simplex. This result generalizes the fact that hyperbolic Dehn filling of a $3$–manifold does not increase hyperbolic volume.

In particular, we obtain information about the simplicial volumes of some $4$–dimensional homology spheres described by Ratcliffe and Tschantz, answering a question of Belegradek and establishing the existence of $4$–dimensional homology spheres with positive simplicial volume.

We study the maximal entropy per unit generator of point-push mapping classes on the punctured disk. Our work is motivated by fluid mixing by rods in a planar domain. If a single rod moves among $N$ fixed obstacles, the resulting fluid diffeomorphism is in the point-push mapping class associated with the loop in ${\pi }_1\left(D^2-\left\{N\text{ points}\right\}\right)$ traversed by the single stirrer. The collection of motions where each stirrer goes around a single obstacle generate the group of point-push mapping classes, and the entropy efficiency with respect to these generators gives a topological measure of the mixing per unit energy expenditure of the mapping class. We give lower and upper bounds for $Eff\left(N\right)$, the maximal efficiency in the presence of $N$ obstacles, and prove that $Eff\left(N\right)\to log\left(3\right)$ as $N\to \infty$. For the lower bound we compute the entropy efficiency of a specific point-push protocol, ${HSP}_N$, which we conjecture achieves the maximum. The entropy computation uses the action on chains in a $\mathbb{Z}$–covering space of the punctured disk which is designed for point-push protocols. For the upper bound we estimate the exponential growth rate of the action of the point-push mapping classes on the fundamental group of the punctured disk using a collection of incidence matrices and then computing the generalized spectral radius of the collection.

We consider selfmaps of hyperbolic surfaces and graphs, and give some bounds involving the rank and the index of fixed point classes. One consequence is a rank bound for fixed subgroups of surface group endomorphisms, similar to the Bestvina–Handel bound (originally known as the Scott conjecture) for free group automorphisms.

When the selfmap is homotopic to a homeomorphism, we rely on Thurston’s classification of surface automorphisms. When the surface has boundary, we work with its spine, and Bestvina–Handel’s theory of train track maps on graphs plays an essential role.

It turns out that the control of empty fixed point classes (for surface automorphisms) presents a special challenge. For this purpose, an alternative definition of fixed point class is introduced, which avoids covering spaces hence is more convenient for geometric discussions.

We describe families of monotone symplectic manifolds constructed via the symplectic cutting procedure of Lerman [Math. Res. Lett. 2 (1995) 247–258] from the cotangent bundle of manifolds endowed with a free circle action. We also give obstructions to the monotone Lagrangian embedding of some compact manifolds in these symplectic manifolds.

Let $X$ be a Poincaré duality space, $Y$ a space and $f:X\to Y$ a based map. We show that the rational homotopy group of the connected component of the space of maps from $X$ to $Y$ containing $f$ is contained in the rational homology group of a space $L_{f}Y$ which is the pullback of $f$ and the evaluation map from the free loop space $LY$ to the space $Y$. As an application of the result, when $X$ is a closed oriented manifold, we give a condition of a noncommutativity for the rational loop homology algebra $H_{\ast }\left(L_{f}Y;\mathbb{Q}\right)$ defined by Gruher and Salvatore which is the extension of the Chas–Sullivan loop homology algebra.

Two types of Nil-groups arise in the codimension $1$ splitting obstruction theory for homotopy equivalences of finite CW–complexes: the Farrell–Bass Nil-groups in the nonseparating case when the fundamental group is an HNN extension and the Waldhausen Nil-groups in the separating case when the fundamental group is an amalgamated free product. We obtain a general Nil-Nil theorem in algebraic $K$–theory relating the two types of Nil-groups.

The infinite dihedral group is a free product and has an index $2$ subgroup which is an HNN extension, so both cases arise if the fundamental group surjects onto the infinite dihedral group. The Nil-Nil theorem implies that the two types of the reduced $\widetilde{Nil}$–groups arising from such a fundamental group are isomorphic. There is also a topological application: in the finite-index case of an amalgamated free product, a homotopy equivalence of finite CW–complexes is semisplit along a separating subcomplex.

For a compact surface $M$, the free degree $\mathfrak{f}\mathfrak{r}\left(M\right)$ of homeomorphisms on $M$ is the minimum positive integer $n$ with property that for any self homeomorphism $\xi$ of $M$, at least one of the iterates $\xi ,{\xi }^2,\dots ,{\xi }^n$ has a fixed point. This is to say $\mathfrak{f}\mathfrak{r}\left(M\right)$ is the maximum of least periods among all periodic points of self homeomorphisms on $M$. We prove that $\mathfrak{f}\mathfrak{r}\left(F_{g,b}\right)\le 24g-24$ for $g\ge 2$ and $\mathfrak{f}\mathfrak{r}\left(N_{g,b}\right)\le 12g-24$ for $g\ge 3$.

We analyze the homological behavior of the attaching maps in the $2$–local Goodwillie tower of the identity evaluated at $S^1$. We show that they exhibit the same homological behavior as the James–Hopf maps used by N Kuhn to prove the $2$–primary Whitehead conjecture. We use this to prove a calculus form of the Whitehead conjecture: the Whitehead sequence is a contracting homotopy for the Goodwillie tower of $S^1$ at the prime $2$.