An orientation is defined on a family of curve graphs on which the Torelli group acts. It is shown that the resulting signed stable length of an element of the Torelli group is a cohomology class. This cohomology class is half the dual of the contraction of the Johnson homomorphism, the so-called “Chillingworth class”.

We prove that in any hyperbolic orbifold with one boundary component, the product of any hyperbolic fundamental group element with a sufficiently large multiple of the boundary is represented by a geodesic loop that virtually bounds an immersed surface. In the case that the orbifold is a disk, there are some conditions. Our results generalize work of Calegari–Louwsma and resolve a conjecture of Calegari.

In this paper we will study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with nonempty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman’s identity (in the surface case), and the zeroth moment recovers Basmajian’s identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function.

We prove a rectification theorem for enriched $\infty$–categories: if $V$ is a nice monoidal model category, we show that the homotopy theory of $\infty$–categories enriched in $V$ is equivalent to the familiar homotopy theory of categories strictly enriched in $V$. It follows, for example, that $\infty$–categories enriched in spectra or chain complexes are equivalent to spectral categories and dg–categories. A similar method gives a comparison result for enriched Segal categories, which implies that the homotopy theories of $n$–categories and $\left(\infty ,n\right)$–categories defined by iterated $\infty$–categorical enrichment are equivalent to those of more familiar versions of these objects. In the latter case we also include a direct comparison with complete $n$–fold Segal spaces. Along the way we prove a comparison result for fiberwise simplicial localizations potentially of independent use.

We organize the quantum hyperbolic invariants (QHI) of $3$–manifolds into sequences of rational functions indexed by the odd integers $N\ge 3$ and defined on moduli spaces of geometric structures refining the character varieties. In the case of one-cusped hyperbolic $3$–manifolds $M$ we generalize the QHI and get rational functions ${\mathcal{\mathscr{H}}}_N^{h_f,h_c,k_c}$ depending on a finite set of cohomological data $\left(h_f,h_c,k_c\right)$ called weights. These functions are regular on a determined Abelian covering of degree $N^2$ of a Zariski open subset, canonically associated to $M$, of the geometric component of the variety of augmented $PSL\left(2,ℂ\right)$–characters of $M$. New combinatorial ingredients are a weak version of branchings which exists on every triangulation, and state sums over weakly branched triangulations, including a sign correction which eventually fixes the sign ambiguity of the QHI. We describe in detail the invariants of three cusped manifolds, and present the results of numerical computations showing that the functions ${\mathcal{\mathscr{H}}}_N^{h_f,h_c,k_c}$ depend on the weights as $N\to \infty$, and recover the volume for some specific choices of the weights.

We construct and study an algebraic theory which closely approximates the theory of power operations for Morava $E$–theory, extending previous work of Charles Rezk in a way that takes completions into account. These algebraic structures are made explicit in the case of $K$–theory. Methodologically, we emphasize the utility of flat modules in this context, and prove a general version of Lazard’s flatness criterion for module spectra over associative ring spectra.

We give a necessary, and in some cases sufficient, condition for sliceness inside the family of pretzel knots $P\left(p_1,\dots ,p_n\right)$ with one $p_i$ even. The $3$–stranded case yields two interesting families of examples: The first consists of knots for which the nonsliceness is detected by the Alexander polynomial while several modern obstructions to sliceness vanish. The second family has the property that the correction terms from Heegaard–Floer homology of the double branched covers of these knots do not obstruct the existence of a rational homology ball; however, the Casson–Gordon invariants show that the double branched covers do not bound rational homology balls.

We try to give a cluster-algebraic interpretation of the complex volume of knots. We construct the $R$–operator from cluster mutations, and show that it can be regarded as a hyperbolic octahedron. The cluster variables are interpreted as the edge parameters used by Zickert for computing complex volume.

We consider certain point patterns of a Euclidean space and calculate the Ellis enveloping semigroup of their associated dynamical systems. The algebraic structure and the topology of the Ellis semigroup, as well as its action on the underlying space, are explicitly described. As an example, we treat the vertex pattern of the Amman–Beenker tiling of the plane.

We formulate a theory of punctured affine formal schemes, suitable for describing certain phenomena within algebraic topology. As a proof-of-concept we show that the Morava $K$–theoretic localizations of Morava $E$–theory, which arise in transchromatic homotopy theory, corepresent a Lubin–Tate-type moduli problem in this framework.

Simply connected manifolds of positive sectional curvature are speculated to have a rigid topological structure. In particular, they are conjectured to be rationally elliptic, ie to have only finitely many non-zero rational homotopy groups. In this article, we combine positive curvature with rational ellipticity to obtain several topological properties of the underlying manifold. These results include an upper bound on the Euler characteristic and new evidence for a couple of well-known conjectures due to Hopf and Halperin. We also prove a conjecture of Wilhelm for even-dimensional manifolds whose rational type is one of the known examples of positive curvature.

Inspired by Lurie’s theory of quasi-unital algebras we prove an analogous result for $\infty$–categories. By constructing a suitable model category of non-unital complete Segal spaces, we show that the unital structure of an $\infty$–category can be uniquely recovered from the underlying non-unital structure once suitable candidates for units have been identified. The main result of this paper can be used to produce a proof of the $1$–dimensional cobordism hypothesis, as described in a forthcoming paper of the author.

Biringer, Johnson, and Minsky proved that any pseudo-Anosov whose stable lamination is the limit of disks in a compression body has a power which extends over some non-trivial minimal compression body. This paper presents an alternative proof of their theorem. The key ingredient is the existence of a certain collection of disks whose boundaries are formed from an arc of the stable lamination and an arc of the unstable lamination. The proof here also shows that there are only finitely many minimal compression bodies over which a power of a pseudo-Anosov can extend.

Toric origami manifolds are characterized by origami templates, which are combinatorial models built by gluing polytopes together along facets. In this paper, we examine the topology of orientable toric origami manifolds with coorientable folding hypersurface. We determine the fundamental group. In our previous paper, we studied the ordinary and equivariant cohomology rings of simply connected toric origami manifolds. We conclude this paper by computing some Betti numbers and cohomology rings in the non-simply connected case.

By a Cantor group we mean a topological group homeomorphic to the Cantor set. We show that a compact metric space of rational cohomological dimension $n$ can be obtained as the orbit space of a Cantor group action on a metric compact space of covering dimension $n$. Moreover, the action can be assumed to be free if $n=1$.

Embedded Lagrangian cobordisms between Legendrian submanifolds are produced by isotopy, spinning, and handle-attachment constructions that employ the technique of generating families. Moreover, any Legendrian with a generating family has an immersed Lagrangian filling with a compatible generating family. These constructions are applied in several directions, in particular to a nonclassical geography question: any graded group satisfying a duality condition can be realized as the generating family homology of a connected Legendrian submanifold in ${ℝ}^{2n+1}$ or in the $1$–jet space of any compact $n$–manifold with $n\ge 2$.