A conjecture of Colin and Honda states that the number of periodic Reeb orbits of universally tight contact structures on hyperbolic manifolds grows exponentially with the period, and they speculate further that the growth rate of contact homology is polynomial on nonhyperbolic geometries. Along the line of the conjecture, for manifolds with a hyperbolic component that fibers on the circle, we prove that there are infinitely many nonisomorphic contact structures for which the number of periodic Reeb orbits of any nondegenerate Reeb vector field grows exponentially. Our result hinges on the exponential growth of contact homology, which we derive as well. We also compute contact homology in some nonhyperbolic cases that exhibit polynomial growth, namely those of universally tight contact structures on a circle bundle nontransverse to the fibers.

It is well known that any $3$–manifold can be obtained by Dehn surgery on a link, but not which ones can be obtained from a knot or which knots can produce them. We investigate these two questions for elliptic Seifert fibered spaces (other than lens spaces) using the Heegaard Floer correction terms or $d$–invariants associated to a $3$–manifold $Y$ and its torsion ${Spin}^c$ structures. For ${\pi }_1\left(Y\right)$ finite and $\left|H_1\left(Y\right)\right|\le 4$, we classify the manifolds which are knot surgery and the knot surgeries which give them; for $\left|H_1\left(Y\right)\right|\le 32$, we classify the manifolds which are surgery and place restrictions on the surgeries which may give them.

We show that the sutured Floer homology of a sutured $3$–manifold of the form $\left(D^2\times S^1,F\times S^1\right)$ can be expressed as the homology of a string-type complex, generated by certain sets of curves on $\left(D^2,F\right)$ and with a differential given by resolving crossings. We also give some generalisations of this isomorphism, computing “hat” and “infinity” versions of this string homology. In addition to giving interesting elementary facts about the algebra of curves on surfaces, these isomorphisms are inspired by, and establish further, connections between invariants from Floer homology and string topology.

Given an integer homology class of a finitely presentable group, the systolic volume quantifies how tight a geometric realization of this class could be. In this paper, we study various aspects of this numerical invariant showing that it is a complex and powerful tool for investigating topological properties of homology classes of finitely presentable groups.

A complex projective tower, or simply $ℂP$ tower, is an iterated complex projective fibration starting from a point. In this paper, we classify a certain class of $8$–dimensional $ℂP$ towers up to diffeomorphism. As a consequence, we show that cohomological rigidity is not satisfied by the collection of $8$–dimensional $ℂP$ towers: there are two distinct $8$–dimensional $ℂP$ towers that have the same cohomology rings.

We consider moduli spaces of quilted strips with markings. By identifying each compactified moduli space with the nonnegative real part of a projective toric variety, we conclude that it is homeomorphic under the moment map to the moment polytope. The moment polytopes in these cases belong to a certain class of graph associahedra, which include the associahedra and permutahedra as special cases. In fact, these graph associahedra are precisely the polytopes whose facet combinatorics encode the $A_{\infty }$ equations of $A_{\infty }$ $n$–modules.

We study the cokernel of the Johnson homomorphism for the mapping class group of a surface with one boundary component. A graphical trace map simultaneously generalizing trace maps of Enomoto and Satoh and Conant, Kassabov and Vogtmann is given, and using technology from the author’s work with Kassabov and Vogtmann, this is is shown to detect a large family of representations which vastly generalizes series due to Morita and Enomoto and Satoh. The Enomoto–Satoh trace is the rank-$1$ part of the new trace, and it is here that the new series of representations is found. The rank-$2$ part is also investigated, though a fuller investigation of the higher-rank case is deferred to another paper.

We study the semitopologization functor of Friedlander and Walker from the perspective of motivic homotopy theory. We construct a triangulated endofunctor on the stable motivic homotopy category $\mathcal{S}\mathcal{\mathscr{H}}\left(ℂ\right)$, which we call homotopy semitopologization. As applications, we discuss the representability of several semitopological cohomology theories in $\mathcal{S}\mathcal{\mathscr{H}}\left(ℂ\right)$, a construction of a semitopological analogue of algebraic cobordism and a construction of Atiyah–Hirzebruch type spectral sequences for this theory.

Using the Seiberg–Witten Floer spectrum and $Pin\left(2\right)$–equivariant $KO$–theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$–spheres. We then give explicit constrains on the intersection forms of spin $4$–manifolds bounded by Brieskorn spheres $\pm \Sigma \left(2,3,6k\pm 1\right)$. Along the way, we also give an alternative proof of Furuta’s improvement of $\frac{10}{8}$–theorem for closed spin $4$–manifolds.

Let $S_g$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_g$ and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group. Next, we demonstrate that minimally intersecting filling pairs are combinatorially optimal, in the sense that there are many simple closed curves intersecting the pair exactly once. We conclude by initiating the study of a topological Morse function ${\mathcal{ℱ}}_g$ over the moduli space of Riemann surfaces of genus $g$, which, given a hyperbolic metric $\sigma$, outputs the length of the shortest minimally intersecting filling pair for the metric $\sigma$. We completely characterize the global minima of ${\mathcal{ℱ}}_g$ and, using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper, we show that the number of such minima grows at least exponentially in $g$.

Algebraic homology and cohomology theories for quandles have been studied extensively in recent years. With a given quandle $2$–cocycle ($3$–cocycle) one can define a state-sum invariant for knotted curves (surfaces). In this paper we introduce another version of quandle (co)homology theory, called positive quandle (co)homology. Some properties of positive quandle (co)homology groups are given and some applications of positive quandle cohomology in knot theory are discussed.

A Bott manifold is the total space of some iterated $ℂ{\mathbb{P}}^1$–bundles over a point. We prove that any graded ring isomorphism between the cohomology rings of two Bott manifolds preserves their Pontrjagin classes. Moreover, we prove that such an isomorphism is induced from a diffeomorphism if the Bott manifolds are $\mathbb{Z}∕2$–trivial, where a Bott manifold is called $\mathbb{Z}∕2$–trivial if its cohomology ring with $\mathbb{Z}∕2$–coefficients is isomorphic to that of a product of copies of $ℂ{\mathbb{P}}^1$.

For a finitely generated group $G$, we introduce an asymmetric pseudometric on projectivized deformation spaces of $G$–trees, using stretching factors of $G$–equivariant Lipschitz maps, that generalizes the Lipschitz metric on Outer space and is an analogue of the Thurston metric on Teichmüller space. We show that in the case of irreducible $G$–trees distances are always realized by minimal stretch maps, can be computed in terms of hyperbolic translation lengths and geodesics exist. We then study displacement functions on projectivized deformation spaces of $G$–trees and classify automorphisms of $G$. As an application, we prove the existence of train track representatives for irreducible automorphisms of virtually free groups and nonelementary generalized Baumslag–Solitar groups that contain no solvable Baumslag–Solitar group $BS\left(1,n\right)$ with $n\ge 2$.

A Legendrian or transverse knot in an overtwisted contact $3$–manifold is nonloose if its complement is tight and loose if its complement is overtwisted. We define three measures of the extent of nonlooseness of a nonloose knot and show they are distinct.

In this series of papers, we study the correspondence between the following: (1) the large scale structure of the metric space ${\bigsqcup }_{m}Cay\left({\mathbb{G}}^{\left(m\right)}\right)$ consisting of Cayley graphs of finite groups with $k$ generators; (2) the structure of groups that appear in the boundary of the set $\left\{{\mathbb{G}}^{\left(m\right)}\right\}$ in the space of $k$–marked groups. In this third part of the series, we show the correspondence among the metric properties “geometric property $(T)$”, “cohomological property $(T)$” and the group property “Kazhdan’s property $(T)$”. Geometric property $(T)$ of Willett–Yu is stronger than being expander graphs. Cohomological property $(T)$ is stronger than geometric property $(T)$ for general coarse spaces.

We reprove and expand results of Bonahon and Wong on central elements of the Kauffman bracket skein modules at roots of $1$ and on the existence of the Chebyshev homomorphism, using elementary skein methods.

The $n$–solvable filtration ${\left\{{\mathcal{ℱ}}_n\right\}}_{n=0}^{\infty }$ of the smooth knot concordance group (denoted by $\mathcal{C}$) due to Cochran, Orr and Teichner has been instrumental in the study of knot concordance in recent years. Part of its significance is due to the fact that certain geometric attributes of a knot imply membership in various levels of the filtration. We show the counterpart of this fact for two new filtrations of $\mathcal{C}$ due to Cochran, Harvey and Horn; the positive and negative filtrations, denoted by ${\left\{{\mathcal{P}}_n\right\}}_{n=0}^{\infty }$ and ${\left\{{\mathcal{N}}_n\right\}}_{n=0}^{\infty }$ respectively. In particular, we show that if a knot $K$ bounds a Casson tower of height $n+2$ in $B^4$ with only positive (resp. negative) kinks in the base-level kinky disk, then $K\in {\mathcal{P}}_n$ (resp. ${\mathcal{N}}_n$). En route to this result we show that if a knot $K$ bounds a Casson tower of height $n+2$ in $B^4$, it bounds an embedded (symmetric) grope of height $n+2$ and is therefore $n$–solvable. We also define a variant of Casson towers and show that if $K$ bounds a tower of type $\left(2,n\right)$ in $B^4$, it is $n$–solvable. If $K$ bounds such a tower with only positive (resp. negative) kinks in the base-level kinky disk then $K\in {\mathcal{P}}_n$ (resp. $K\in {\mathcal{N}}_n$). Our results show that either every knot which bounds a Casson tower of height three is topologically slice or there exists a knot in $\bigcap {\mathcal{ℱ}}_n$ which is not topologically slice. We also give a $3$–dimensional characterization, up to concordance, of knots which bound kinky disks in $B^4$ with only positive (resp. negative) kinks; such knots form a subset of ${\mathcal{P}}_0$ (resp. ${\mathcal{N}}_0$).

An embedding of a graph into ${ℝ}^3$ is said to be linear if any edge of the graph is sent to a line segment. And we say that an embedding $f$ of a graph $G$ into ${ℝ}^3$ is free if ${\pi }_1\left({ℝ}^3-f\left(G\right)\right)$ is a free group. It is known that the linear embedding of any complete graph is always free.

In this paper we investigate the freeness of linear embeddings by considering the number of vertices. It is shown that the linear embedding of any simple connected graph with at most 6 vertices whose minimal valency is at least 3 is always free. On the contrary, when the number of vertices is much larger than the minimal valency or connectivity, the freeness may not be an intrinsic property of the graph. In fact we show that for any $n\ge 1$ there are infinitely many connected graphs with minimal valency $n$ which have nonfree linear embeddings and furthermore that there are infinitely many $n$–connected graphs which have nonfree linear embeddings.

A finite-volume hyperbolic $3$–manifold geometrically bounds if it is the geodesic boundary of a finite-volume hyperbolic $4$–manifold. We construct here an example of a noncompact, finite-volume hyperbolic $3$–manifold that geometrically bounds. The $3$–manifold is the complement of a link with eight components, and its volume is roughly equal to $29.311$.

The paper contains a new proof that a complete, non-compact hyperbolic $3$–manifold with finite volume contains an immersed, closed, quasi-Fuchsian surface.

Dmitri Pavlov and Jakob Scholbach have pointed out that part of Proposition 6.3, and hence Proposition 4.28(a), of Harper [Algebr. Geom. Topol. 9 (2009) 1637–1680] are incorrect as stated. While all of the main results of that paper remain unchanged, this necessitates modifications to the statements and proofs of a few technical propositions.