We propose definitions of complex manifolds ${\mathcal{𝒫}}_M\left(X,m,n\right)$ that could potentially be used to construct the symplectic Khovanov homology of $n$–stranded links in lens spaces. The manifolds ${\mathcal{𝒫}}_M\left(X,m,n\right)$ are defined as moduli spaces of Hecke modifications of rank $2$ parabolic bundles over an elliptic curve $X$. To characterize these spaces, we describe all possible Hecke modifications of all possible rank $2$ vector bundles over $X$, and we use these results to define a canonical open embedding of ${\mathcal{𝒫}}_M\left(X,m,n\right)$ into $M^s\left(X,m+n\right)$, the moduli space of stable rank $2$ parabolic bundles over $X$ with trivial determinant bundle and $m+n$ marked points. We explicitly compute ${\mathcal{𝒫}}_M\left(X,1,n\right)$ for $n=0,1,2$. For comparison, we present analogous results for the case of rational curves, for which a corresponding complex manifold ${\mathcal{𝒫}}_M\left({ℂ\mathbb{P}}^1,3,n\right)$ is isomorphic for $n$ even to a space $\mathcal{𝒴}\left(S^2,n\right)$ defined by Seidel and Smith that can be used to compute the symplectic Khovanov homology of $n$–stranded links in $S^3$.

Using methods inspired from algebraic $K$–theory, we give a new proof of the Genauer fibration sequence, relating the cobordism categories of closed manifolds with cobordism categories of manifolds with boundaries, and of the Bökstedt–Madsen delooping of the cobordism category. Unlike the existing proofs, this approach generalizes to other cobordism-like categories of interest. Indeed we argue that the Genauer fibration sequence is an analogue, in the setting of cobordism categories, of Waldhausen’s additivity theorem in algebraic $K$–theory.

We study the magnitude homology of geodesic metric spaces of curvature $\le \kappa$, especially $CAT\left(\kappa \right)$ spaces. We will show that the magnitude homology ${MH}_n^l\left(X\right)$ of such a metric space $X$ vanishes for small $l$ and all $n>0$. Consequently, we can compute magnitude homology in small length gradings for spheres ${\mathbb{𝕊}}^n$, the Euclidean spaces ${\mathbb{𝔼}}^n$, the hyperbolic spaces ${ℍ}^n$ and real projective spaces ${ℝ\mathbb{P}}^n$ with the standard metric. We also show that the existence of a closed geodesic in a metric space guarantees the nontriviality of magnitude homology.

The Legendrian product of two Legendrian knots, as defined by Lambert-Cole, is a Legendrian torus. We show that this Legendrian torus is a twist spun whenever one of the Legendrian knot components is sufficiently large. We then study examples of Legendrian products which are not Legendrian isotopic to twist spuns. In order to do this, we prove a few structural results on the bilinearised Legendrian contact homology and augmentation variety of a twist spun. In addition, we show that the threefold Bohr–Sommerfeld covers of the Clifford torus and Chekanov torus are not twist spuns.

Let $\mathrm{\Sigma }$ be a compact oriented surface. The Dehn twist along every simple closed curve $\gamma \subset \mathrm{\Sigma }$ induces an automorphism of the fundamental group $\pi$ of $\mathrm{\Sigma }$. There are two possible ways to generalize such automorphisms if the curve $\gamma$ is allowed to have self-intersections. One way is to consider the “generalized Dehn twist” along $\gamma$: an automorphism of the Maltsev completion of $\pi$ whose definition involves intersection operations and only depends on the homotopy class $\left[\gamma \right]\in \pi$ of $\gamma$. Another way is to choose in the usual cylinder $U:=\mathrm{\Sigma }\times \left[-1,+1\right]$ a knot $L$ projecting onto $\gamma$, to perform a surgery along $L$ so as to get a homology cylinder $U_L$, and let $U_L$ act on every nilpotent quotient $\pi ∕{\mathrm{\Gamma }}_j\pi$ of $\pi$ (where ${\mathrm{\Gamma }}_j\pi$ denotes the subgroup of $\pi$ generated by commutators of length $j$). In this paper, assuming that $\left[\gamma \right]$ is in ${\mathrm{\Gamma }}_k\pi$ for some $k\ge 2$, we prove that (whatever the choice of $L$ is) the automorphism of $\pi ∕{\mathrm{\Gamma }}_{2k+1}\pi$ induced by $U_L$ agrees with the generalized Dehn twist along $\gamma$ and we explicitly compute this automorphism in terms of $\left[\gamma \right]$ modulo ${\mathrm{\Gamma }}_{k+2}\pi$. As applications, we obtain new formulas for certain evaluations of the Johnson homomorphisms showing, in particular, how to realize any element of their targets by some explicit homology cylinders and/or generalized Dehn twists.

We study $CAT\left(0\right)$ groups and their splittings as graphs of groups. For one-ended $CAT\left(0\right)$ groups with isolated flats we prove a theorem characterizing exactly when the visual boundary is locally connected. This characterization depends on whether the group has a certain type of splitting over a virtually abelian subgroup. In the locally connected case, we describe the boundary as a tree of metric spaces in the sense of Świątkowski.

A significant tool used in the proofs of the above results is a general convex splitting theorem for arbitrary $CAT\left(0\right)$ groups. If a $CAT\left(0\right)$ group splits as a graph of groups with convex edge groups, then the vertex groups are also $CAT\left(0\right)$ groups.

We obtain combinatorial model categories of parametrised spectra, together with systems of base change Quillen adjunctions associated to maps of parameter spaces. We work with simplicial objects and use Hovey’s sequential and symmetric stabilisation machines. By means of a Grothendieck construction for model categories, we produce combinatorial model categories controlling the totality of parametrised stable homotopy theory. The global model category of parametrised symmetric spectra is equipped with a symmetric monoidal model structure (the external smash product) inducing pairings in twisted cohomology groups.

As an application of our results we prove a tangent prolongation of Simpson’s theorem, characterising tangent $\infty$–categories of presentable $\infty$–categories as accessible localisations of $\infty$–categories of presheaves of parametrised spectra. Applying these results to the homotopy theory of smooth $\infty$–stacks produces well-behaved (symmetric monoidal) model categories of smooth parametrised spectra. These models, which subsume previous work of Bunke and Nikolaus, provide a concrete foundation for studying twisted differential cohomology.

For any positive integer $g$, we completely determine the minimal genus function for ${\mathrm{\Sigma }}_g\times T^2$. We show that the lower bound given by the adjunction inequality is not sharp for some class in $H_2\left({\mathrm{\Sigma }}_g\times T^2\right)$. However, we construct a suitable embedded surface for each class and we have exact values of minimal genus functions.

Building on Quillen’s rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply connected space $X$ there is a dg Lie algebra ${\mathrm{\Lambda }}_X$ and a (coassociative cocommutative) dg coalgebra $C_X$ that model the rational homotopy type. We prove that the rational homotopy type of an $X$–parametrised spectrum is completely encoded by a ${\mathrm{\Lambda }}_X$–representation and also by a $C_X$–comodule. The correspondence between rational parametrised spectra and algebraic data is obtained by means of symmetric monoidal equivalences of homotopy categories that vary pseudofunctorially in the parameter space $X$.

Our results establish a comprehensive dictionary enabling the translation of topological constructions into homological algebra using Lie representations and comodules, and conversely. For example, the fibrewise smash product of parametrised spectra is encoded by the derived tensor product of dg Lie representations and also by the derived cotensor product of dg comodules. As an application, we obtain novel algebraic descriptions of rational homotopy classes of fibrewise stable maps, providing new tools for the study of section spaces.

We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik–Schnirelmann category and provide lower bounds for the numbers of critical orbits of $SO\left(n\right)$–invariant functions on spaces of $n$–spheres in a manifold. Lower bounds on these invariants are derived using weights of cohomology classes. As an application, we prove new existence results for closed geodesics on Finsler manifolds of positive flag curvature satisfying a pinching condition.