A smooth curve $\gamma :\left[0,1\right]\to {\mathbb{S}}^2$ is locally convex if its geodesic curvature is positive at every point. J A Little showed that the space of all locally convex curves $\gamma$ with $\gamma \left(0\right)=\gamma \left(1\right)=e_1$ and ${\gamma }^{\prime }\left(0\right)={\gamma }^{\prime }\left(1\right)=e_2$ has three connected components ${\mathcal{ℒ}}_{-1,c}$, ${\mathcal{ℒ}}_{+1}$, ${\mathcal{ℒ}}_{-1,n}$. The space ${\mathcal{ℒ}}_{-1,c}$ is known to be contractible. We prove that ${\mathcal{ℒ}}_{+1}$ and ${\mathcal{ℒ}}_{-1,n}$ are homotopy equivalent to $\left(\Omega {\mathbb{S}}^3\right)\vee {\mathbb{S}}^2\vee {\mathbb{S}}^6\vee {\mathbb{S}}^{10}\vee \cdots$ and $\left(\Omega {\mathbb{S}}^3\right)\vee {\mathbb{S}}^4\vee {\mathbb{S}}^8\vee {\mathbb{S}}^{12}\vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $Free\left({\mathbb{S}}^1,{\mathbb{S}}^2\right)$ of free curves $\gamma :{\mathbb{S}}^1\to {\mathbb{S}}^2$ (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $Free\left(\left[0,1\right],{\mathbb{S}}^2\right)$ with fixed initial and final frames.

The $\Theta$–invariant is the simplest $3$–manifold invariant defined by configuration space integrals. It is actually an invariant of rational homology spheres equipped with a combing over the complement of a point. It can be computed as the algebraic intersection of three propagators associated to a given combing $X$ in the $2$–point configuration space of a $\mathbb{Q}$–sphere $M$. These propagators represent the linking form of $M$ so that $\Theta \left(M,X\right)$ can be thought of as the cube of the linking form of $M$ with respect to the combing $X$. The invariant $\Theta$ is the sum of $6\lambda \left(M\right)$ and $p_1\left(X\right)∕4$, where $\lambda$ denotes the Casson–Walker invariant, and $p_1$ is an invariant of combings, which is an extension of a first relative Pontrjagin class. In this article, we present explicit propagators associated with Heegaard diagrams of a manifold, and we use these “Morse propagators”, constructed with Greg Kuperberg, to prove a combinatorial formula for the $\Theta$–invariant in terms of Heegaard diagrams.

Given a closed hyperbolic $3$–manifold $M$ with a quasigeodesic flow, we construct a ${\pi }_1$–equivariant sphere-filling curve in the boundary of hyperbolic space. Specifically, we show that any complete transversal $P$ to the lifted flow on ${ℍ}^3$ has a natural compactification to a closed disc that inherits a ${\pi }_1$–action. The embedding $P\hookrightarrow {ℍ}^3$ extends continuously to the compactification, and restricts to a surjective ${\pi }_1$–equivariant map $\partial P\to \partial {ℍ}^3$ on the boundary. This generalizes the Cannon–Thurston theorem, which produces such group-invariant space-filling curves for fibered hyperbolic $3$–manifolds.

We introduce a method to construct $G_2$–instantons over compact $G_2$–manifolds arising as the twisted connected sum of a matching pair of building blocks. Our construction is based on gluing $G_2$–instantons obtained from holomorphic vector bundles over the building blocks via the first author’s work. We require natural compatibility and transversality conditions which can be interpreted in terms of certain Lagrangian subspaces of a moduli space of stable bundles on a $K3$ surface.

This is the fifth in a series of papers on the ‘$k$–shifted symplectic derived algebraic geometry’ of Pantev, Toën, Vaquié and Vezzosi. We extend our earlier results from (derived) schemes to (derived) Artin stacks. We prove four main results:

(a) If $\left(X,{\omega }_X\right)$ is a $k$–shifted symplectic derived Artin stack for $k<0$, then near each $x\in X$ we can find a ‘minimal’ smooth atlas $\phi :U\to X$, such that $\left(U,{\phi }^{\ast }\left({\omega }_X\right)\right)$ may be written explicitly in coordinates in a standard ‘Darboux form’.

(b) If $\left(X,{\omega }_X\right)$ is a $\left(-1\right)$–shifted symplectic derived Artin stack and $X=t_0\left(X\right)$ the classical Artin stack, then $X$ extends to a ‘d–critical stack’ $\left(X,s\right)$, as by Joyce.

(c) If $\left(X,s\right)$ is an oriented d–critical stack, we define a natural perverse sheaf ${\check{P}}_{X,s}^{\bullet }$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, $T$ is locally modelled on a critical locus $Crit\left(f:U\to {\mathbb{A}}^1\right)$, and $t^{\ast }\left({\check{P}}_{X,s}^{\bullet }\right)\left[n\right]$ is modelled on the perverse sheaf of vanishing cycles ${\mathcal{P}\mathcal{V}}_{U,f}^{\bullet }$ of $f$.

(d) If $\left(X,s\right)$ is a finite-type oriented d–critical stack, we can define a natural motive ${MF}_{X,s}$ in a ring of motives ${\overline{\mathcal{ℳ}}}_X^{st,\widehat{\mu }}$ on $X$, such that if $T$ is a scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is modelled on a critical locus $Crit\left(f:U\to {\mathbb{A}}^1\right)$, and ${\mathbb{L}}^{-n∕2}\odot t^{\ast }\left({MF}_{X,s}\right)$ is modelled on the motivic vanishing cycle ${MF}_{U,f}^{mot,\varphi }$ of $f$.

Our results have applications to categorified and motivic extensions of Donaldson–Thomas theory of Calabi–Yau $3$–folds.

We show that the span of the variable $q$ in the Lawrence–Krammer–Bigelow representation matrix of a braid is equal to twice the dual Garside length of the braid, as was conjectured by Krammer. Our proof is close in spirit to Bigelow’s geometric approach. The key observation is that the dual Garside length of a braid can be read off a certain labeling of its curve diagram.

Let $G$ be a complex reductive linear algebraic group and let $K\subset G$ be a maximal compact subgroup. Given a nilpotent group $\Gamma$ generated by $r$ elements, we consider the representation spaces $Hom\left(\Gamma ,G\right)$ and $Hom\left(\Gamma ,K\right)$ with the natural topology induced from an embedding into $G^r$ and $K^r$ respectively. The goal of this paper is to prove that there is a strong deformation retraction of $Hom\left(\Gamma ,G\right)$ onto $Hom\left(\Gamma ,K\right)$. We also obtain a strong deformation retraction of the geometric invariant theory quotient $Hom\left(\Gamma ,G\right)∕∕G$ onto the ordinary quotient $Hom\left(\Gamma ,K\right)∕K$.

We show how to lift positive Ricci and almost nonnegative curvatures from an orbit space $M∕G$ to the corresponding $G“$–manifold, $M$. We apply the results to get new examples of Riemannian manifolds that satisfy both curvature conditions simultaneously.

We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous completeness). Coupled with earlier work, this completes the well-posedness theory for instantaneously complete Ricci flows on surfaces.

We show that any $n$–dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of ${ℝ}^n$ by an action of a crystallographic group. We describe all such actions.

We characterize the rate of convergence of a converging volume-normalized Yamabe flow in terms of Morse-theoretic properties of the limiting metric. If the limiting metric is an integrable critical point for the Yamabe functional (for example, this holds when the critical point is nondegenerate), then we show that the flow converges exponentially fast. In general, we make use of a suitable Łojasiewicz–Simon inequality to prove that the slowest the flow will converge is polynomially. When the limit metric satisfies an Adams–Simon-type condition we prove that there exist flows converging to it exactly at a polynomial rate. We conclude by constructing explicit examples of this phenomenon. These seem to be the first examples of a slowly converging solution to a geometric flow.

We study properties of the space of horizontal paths joining the origin with a vertical point on a generic two-step Carnot group. The energy is a Morse-Bott functional on paths and its critical points (sub-Riemannian geodesics) appear in families (compact critical manifolds) with controlled topology. We study the asymptotic of the number of critical manifolds as the energy grows. The topology of the horizontal-path space is also investigated, and we find asymptotic results for the total Betti number of the sublevels of the energy as it goes to infinity. We interpret these results as local invariants of the sub-Riemannian structure.

In this paper, we study Manolescu’s construction of the relative Bauer–Furuta invariants arising from the Seiberg–Witten equations on $4$–manifolds with boundary. The main goal of this paper is to introduce a new gauge fixing condition in order to apply the finite-dimensional approximation technique. We also hope to provide a framework to extend Manolescu’s construction to general $4$–manifolds.

A link in the $3$–sphere is called (smoothly) slice if its components bound disjoint smoothly embedded disks in the $4$–ball. More generally, given a $4$–manifold $M$ with a distinguished circle in its boundary, a link in the $3$–sphere is called $M$–slice if its components bound in the $4$–ball disjoint embedded copies of $M$. A $4$–manifold $M$ is constructed such that the Borromean rings are not $M$–slice but the Hopf link is. This contrasts the classical link-slice setting where the Hopf link may be thought of as “the most nonslice” link. Further examples and an obstruction for a family of decompositions of the $4$–ball are discussed in the context of the A-B slice problem.

Let $X$ be a compact complex manifold, $D_c^b\left(X\right)$ be the bounded derived category of constructible sheaves on $X$, and $Fuk\left(T^{\ast }X\right)$ be the Fukaya category of $T^{\ast }X$. A Lagrangian brane in $Fuk\left(T^{\ast }X\right)$ is holomorphic if the underlying Lagrangian submanifold is complex analytic in $T^{\ast }{X}_{ℂ}$, the holomorphic cotangent bundle of $X$. We prove that under the quasiequivalence between $D_c^b\left(X\right)$ and $DFuk\left(T^{\ast }X\right)$ established by Nadler and Zaslow, holomorphic Lagrangian branes with appropriate grading correspond to perverse sheaves.

This article investigates the complex symplectic geometry of the deformation space of complex projective structures on a closed oriented surface of genus at least 2. The cotangent symplectic structure given by the Schwarzian parametrization is studied carefully and compared to the Goldman symplectic structure on the character variety, clarifying and generalizing a theorem of S Kawai. Generalizations of results of C McMullen are derived, notably quasifuchsian reciprocity. The symplectic geometry is also described in a Hamiltonian setting with the complex Fenchel–Nielsen coordinates on quasifuchsian space, recovering results of I Platis.