In this paper, we give a proof of a conjecture which says that $w\left(K\right)\ge n^{2}w\left(J\right)$, where $w\left(\cdot \right)$ is the width of a knot, $K$ is a satellite knot with $J$ as its companion, and $n$ is the winding number of the pattern. We also show that equality holds if $K$ is a satellite knot with braid pattern.

Given a monotone Lagrangian submanifold invariant under a loop of Hamiltonian diffeomorphisms, we compute a piece of the closed-open string map into the Hochschild cohomology of the Lagrangian which captures the homology class of the loop’s orbit.

Our applications include split-generation and nonformality results for real Lagrangians in projective spaces and other toric varieties; a particularly basic example is that the equatorial circle on the $2$–sphere carries a nonformal Fukaya $A_{\infty }$ algebra in characteristic $2$.

We investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the homology sphere the knot is in. By finding a different bound on the number of slopes, we show that non-null-homologous knots in certain homology $ℝP^3$ are determined by their complements. We also prove the surgery characterisation of the unknot for null-homologous knots in $L$–spaces. This leads to showing that all knots in some lens spaces are determined by their complements. Finally, we establish that knots of genus greater than $1$ in the Brieskorn sphere $\mathrm{\Sigma }\left(2,3,7\right)$ are also determined by their complements.

We identify tight contact structures on small Seifert fibered $L$–spaces as exactly the structures having nonvanishing contact invariant, and classify them by their induced ${Spin}^c$ structures. The result (in the new case of $M\left(-1;r_1,r_2,r_3\right)$) is based on the translation between convex surface theory and the tightness criterion of Lisca and Stipsicz.

A closed orientable splitting surface in an oriented 3–manifold is a topologically minimal surface of index $n$ if its associated disk complex is $\left(n-2\right)$–connected but not $\left(n-1\right)$–connected. A critical surface is a topologically minimal surface of index 2. In this paper, we use an equivalent combinatorial definition of critical surfaces to construct the first known critical bridge spheres for nontrivial links.

A second cohomology class of the group cohomology of the symplectomorphism group is defined for a symplectic manifold with first Chern class proportional to the class of symplectic form and with trivial first real cohomology. Some properties of it are studied. In particular, it is characterized in terms of cohomology classes of the universal symplectic fiber bundle over the classifying space of the symplectomorphism group with the discrete topology.

We show that Dranishnikov’s asymptotic property C is preserved by direct products and the free product of discrete metric spaces. In particular, if $G$ and $H$ are groups with asymptotic property C, then both $G\times H$ and $G\ast H$ have asymptotic property C. We also prove that a group $G$ has asymptotic property C if $1\to K\to G\to H\to 1$ is exact, $asdimK<\infty$ and $H$ has asymptotic property C. The groups are assumed to have left-invariant proper metrics and need not be finitely generated. These results settle questions of Dydak and Virk (2016), of Bell and Moran (2015) and an open problem in topology.

Let $R$ be either ${\mathbb{F}}_p$ or a field of characteristic $0$. For each $R$–good topological space $Y$, we define a collection of higher cohomology operations which, together with the cohomology algebra $H^{\ast }\left(Y;R\right)$, suffice to determine $Y$ up to $R$–completion. We also provide a similar collection of higher cohomology operations which determine when two maps $f_0,f_1:Z\to Y$ between $R$–good spaces (inducing the same algebraic homomorphism $H^{\ast }\left(Y;R\right)\to H^{\ast }\left(Z;R\right)$) are $R$–equivalent.

Given a hyperbolic knot $K$ and any $n\ge 2$ the abelian representations and the holonomy representation each give rise to an $\left(n-1\right)$–dimensional component in the $SL\left(n,ℂ\right)$–character variety. A component of the $SL\left(n,ℂ\right)$–character variety of dimension $\ge n$ is called high-dimensional.

It was proved by D Cooper and D Long that there exist hyperbolic knots with high-dimensional components in the $SL\left(2,ℂ\right)$–character variety. We show that given any nontrivial knot $K$ and sufficiently large $n$ the $SL\left(n,ℂ\right)$–character variety of $K$ admits high-dimensional components.

We suggest a relatively simple and totally geometric conjectural description of uncolored daha superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov–Rozansky polynomials) via the flagged Jacobian factors (new objects) of the corresponding unibranch plane curve singularities. This generalizes the Cherednik–Danilenko conjecture on the Betti numbers of Jacobian factors, the Gorsky combinatorial conjectural interpretation of superpolynomials of torus knots and that by Gorsky and Mazin for their constant term. The paper mainly focuses on nontorus algebraic knots. A connection with the conjecture due to Oblomkov, Rasmussen and Shende is possible, but our approach is different. A motivic version of our conjecture is related to $p$–adic orbital $A$–type integrals for anisotropic centralizers.

For a complex projective space the inertia group, the homotopy inertia group and the concordance inertia group are isomorphic. In complex dimension $4n+1$, these groups are related to computations in stable cohomotopy. Using stable homotopy theory, we make explicit computations to show that the inertia group is nontrivial in many cases. In complex dimension $9$, we deduce some results on geometric structures on homotopy complex projective spaces and complex hyperbolic manifolds.

We provide negative answers to questions posed by Durham, Hagen and Sisto on the existence of boundary maps for some hierarchically hyperbolic spaces, namely maps from right-angled Artin groups to mapping class groups. We also prove results on existence of boundary maps for free subgroups of mapping class groups.

We show that on a hyperbolic knot $K$ in $S^3$, the distance between any two finite surgery slopes is at most $2$, and consequently, there are at most three nontrivial finite surgeries. Moreover, in the case where $K$ admits three nontrivial finite surgeries, $K$ must be the pretzel knot $P\left(-2,3,7\right)$. In the case where $K$ admits two noncyclic finite surgeries or two finite surgeries at distance $2$, the two surgery slopes must be one of ten or seventeen specific pairs, respectively. For $D$–type finite surgeries, we improve a finiteness theorem due to Doig by giving an explicit bound on the possible resulting prism manifolds, and also prove that $4m$ and $4m+4$ are characterizing slopes for the torus knot $T\left(2m+1,2\right)$ for each $m\ge 1$.

We give a characterization for asymptotic dimension growth. We apply it to $CAT\left(0\right)$ cube complexes of finite dimension, giving an alternative proof of Wright’s result on their finite asymptotic dimension. We also apply our new characterization to geodesic coarse median spaces of finite rank and establish that they have subexponential asymptotic dimension growth. This strengthens a recent result of S̆pakula and Wright.

A $1$–truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of ${Map}_{\ast }\left(BG,BH\right)$, $Map\left(BG,BH\right)$, and $Map\left(EG,B_{G}H\right)$ for compact Lie groups $G$ and $H$ with $H$ $1$–truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case when $H$ is compact abelian due to Lashof, May, and Segal.

In this paper, we compute the second mod $2$ homology of an arbitrary Artin group, without assuming the $K\left(\pi ,1\right)$ conjecture. The key ingredients are (A) Hopf’s formula for the second integral homology of a group and (B) Howlett’s result on the second integral homology of Coxeter groups.

K Orr defined a Milnor-type invariant of links that lies in the third homotopy group of a certain space $K_{\omega }$. The problem of nontriviality of this third homotopy group has been open. We show that it is an infinitely generated group. The question of realization of its elements as links remains open.

We introduce a new approach to phantom maps which largely extends the rational-ization-completion approach developed by Meier and Zabrodsky. Our approach enables us to deal with the set $Ph\left(X,Y\right)$ of homotopy classes of phantom maps and the subset $SPh\left(X,Y\right)$ of homotopy classes of special phantom maps simultaneously. We give a sufficient condition for $Ph\left(X,Y\right)$ and $SPh\left(X,Y\right)$ to have natural group structures, which is much weaker than the conditions obtained by Meier and McGibbon. Previous calculations of $Ph\left(X,Y\right)$ have generally assumed that $\left[X,\Omega Ŷ\right]$ is trivial, in which case generalizations of Miller’s theorem are directly applicable, and calculations of $SPh\left(X,Y\right)$ have rarely been reported. Here, we calculate not only $Ph\left(X,Y\right)$ but also $SPh\left(X,Y\right)$ in many important cases of nontrivial $\left[X,\Omega Ŷ\right]$.

We determine the ring structure of the loop homology of some global quotient orbifolds. We can compute by our theorem the loop homology ring with suitable coefficients of the global quotient orbifolds of the form $\left[M∕G\right]$ for $M$ being some kinds of homogeneous manifolds, and $G$ being a finite subgroup of a path-connected topological group $\mathcal{G}$ acting on $M$. It is shown that these homology rings split into the tensor product of the loop homology ring ${ℍ}_{\ast }\left(LM\right)$ of the manifold $M$ and that of the classifying space of the finite group, which coincides with the center of the group ring $Z\left(k\left[G\right]\right)$.