We compute the fundamental group of each connected component of the space of formal Legendrian embeddings in ${ℝ}^3$. We use it to show that previous examples in the literature of nontrivial loops of Legendrian embeddings are already nontrivial at the formal level. Likewise, we compute the fundamental group of the different connected components of the space of formal horizontal embeddings into the standard Engel ${ℝ}^4$. We check that the natural inclusion of the space of horizontal embeddings into the space of formal horizontal embeddings induces an isomorphism at ${\pi }_1$–level.

We show that the number of double points of smoothly immersed $2$–spheres representing certain homology classes of an oriented, smooth, closed, simply connected $4$–manifold $X$ must increase with the complexity of corresponding $h$–cobordisms from $X$ to $X$. As an application, we give results restricting the minimal number of double points of immersed spheres in manifolds homeomorphic to rational surfaces.

Let $P_k$ be the subgroup generated by $k^{\text{ th}}$ powers of primitive elements in $F_r$, the free group of rank $r$. We show that $F_2∕P_k$ is finite if and only if $k$ is $1$, $2$ or $3$. We also fully characterize $F_2∕P_k$ for $k=2,3,4$. In particular, we give a faithful $9$–dimensional representation of $F_2∕P_4$ with infinite image.

We show that unrolled quantum groups at odd roots of unity give rise to relative modular categories. These are the main building blocks for the construction of $\left(1+1+1\right)$–TQFTs extending CGP invariants, which are nonsemisimple quantum invariants of closed $3$–manifolds decorated with ribbon graphs and cohomology classes. When we consider the zero cohomology class, these quantum invariants are shown to coincide with the renormalized Hennings invariants coming from the corresponding small quantum groups.

We compute the Picard group of the category of $K\left(2\right)$–local module spectra over the ring spectrum $E^{h{C}_4}$, where $E$ is a height $2$ Morava $E$–theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K\left(2\right)$–local $E$–modules in genuine $C_4$–spectra. We show that in addition to a cyclic subgroup of order $32$ generated by $E\wedge S^1$, the Picard group contains a subgroup of order $2$ generated by $E\wedge S^{7+\sigma }$, where $\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO\left(C_4\right)$–graded Mackey functor homotopy fixed point spectral sequence for the $C_4$–spectrum $E“$.

The $n$–strand braid group can be defined as the fundamental group of the configuration space of $n$ unlabeled points in a closed disk based at a configuration where all $n$ points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points remain pointwise fixed forms a subgroup isomorphic to a braid group with fewer strands. We generalize this phenomenon by introducing the notion of boundary braids. A boundary braid is a braid that has a representative where some specified subset of the points remains in the boundary cycle of the disk. Although boundary braids merely form a subgroupoid rather than a subgroup, they play an interesting geometric role in the piecewise Euclidean dual braid complex defined by Tom Brady and the second author. We prove several theorems in this setting, including the fact that the subcomplex of the dual braid complex determined by a specified set of boundary braids metrically splits as the direct metric product of a Euclidean polyhedron and a dual braid complex of smaller rank.

Let $\bar{\gamma }$ be a link in a Seifert-fibered space $M$ over a hyperbolic $2$–orbifold $\mathcal{𝒪}$ that projects injectively to a filling multicurve of closed geodesics $\gamma$ in $\mathcal{𝒪}$. We prove that the complement $M_{\bar{\gamma }}$ of $\bar{\gamma }$ in $M$ admits a hyperbolic structure of finite volume, and we give combinatorial bounds of its volume.

We show that infinitely many of the simply connected $4$–manifolds constructed by Levine and Lidman that do not admit PL spines actually admit topological spines.

We define new differential graded algebras $\mathcal{𝒜}\left(n,k,\mathcal{𝒮}\right)$ in the framework of Lipshitz, Ozsváth and Thurston’s and Zarev’s strands algebras from bordered Floer homology. The algebras $\mathcal{𝒜}\left(n,k,\mathcal{𝒮}\right)$ are meant to be strands models for Ozsváth and Szabó’s algebras $\mathcal{ℬ}\left(n,k,\mathcal{𝒮}\right)$; indeed, we exhibit a quasi-isomorphism from $\mathcal{ℬ}\left(n,k,\mathcal{𝒮}\right)$ to $\mathcal{𝒜}\left(n,k,\mathcal{𝒮}\right)$. We also show how Ozsváth and Szabó’s gradings on $\mathcal{ℬ}\left(n,k,\mathcal{𝒮}\right)$ arise naturally from the general framework of group-valued gradings on strands algebras.

In terms of Turaev’s shadows, we provide a sufficient condition for a compact, smooth, acyclic $4$–manifold with boundary the $3$–sphere to be diffeomorphic to the standard $4$–ball. As a consequence, we prove that if a compact, smooth, acyclic $4$–manifold with boundary the $3$–sphere has shadow-complexity at most $2$, then it is diffeomorphic to the standard $4$–ball.

We explain a direct topological proof for the multiplicativity of the Duflo isomorphism for arbitrary finite-dimensional Lie algebras, and derive the explicit formula for the Duflo map. The proof follows a series of implications, starting with “the calculation $1+1=2$ on a 4D abacus”, using the study of homomorphic expansions (aka universal finite-type invariants) for ribbon $2$–knots, and the relationship between the corresponding associated graded space of arrow diagrams and universal enveloping algebras. This complements the results of the first author, Le and Thurston, where similar arguments using a “3D abacus” and the Kontsevich integral were used to deduce Duflo’s theorem for metrized Lie algebras; and results of the first two authors on finite-type invariants of w–knotted objects, which also imply a relation of $2$–knots with the Duflo theorem in full generality, though via a lengthier path.