Journal of Topology and Analysis

In this paper, we study symplectic embedding questions for the ${\ell }_p$-sum of two discs in ${ℝ}^4$, when $1\le p\le \infty$. In particular, we compute the symplectic inner and outer radii in these cases, and show how different kinds of embedding rigidity and flexibility phenomena appear as a function of the parameter $p$.

For a nonorientable surface, the twist subgroup is an index $2$ subgroup of the mapping class group generated by Dehn twists about two-sided simple closed curves. In this paper, we consider involution generators of the twist subgroup and give generating sets of involutions with smaller number of generators than the ones known in the literature using new techniques for finding involution generators.

An $n$-dimensional knot $S^n\subset S^{n+2}$ is called doubly slice if it occurs as the cross section of some unknotted $(n+1)$-dimensional knot. For every $n$ it is unknown which knots are doubly slice, and this remains one of the biggest unsolved problems in high-dimensional knot theory. For $\ell >1$, we use signatures coming from $L^{(2)}$-cohomology to develop new obstructions for $(4\ell -3)$-dimensional knots with metabelian knot groups to be doubly slice. For each $\ell >1$, we construct an infinite family of knots on which our obstructions are nonzero, but for which double sliceness is not obstructed by any previously known invariant.

We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

We consider the problem whether for a group $G$ there exists a constant $\mathrm{\Lambda }(G)>1$ such that for any $(r,s)$-matrix $A$ over the integral group ring $\mathbb{Z}G$ the Fuglede–Kadison determinant of the $G$-equivariant bounded operator $L^2{(G)}^r\to L^2{(G)}^s$ given by right multiplication with $A$ is either one or greater or equal to $\mathrm{\Lambda }(G)$. If $G$ is the infinite cyclic group and we consider only $r=s=1$, this is precisely Lehmer’s problem.

We define a system of groups, a duality system of groups and a $\mathrm{\text{PD}}(n)$-system of groups, generalizing the corresponding concepts of pairs of groups. We give several characterizations of duality systems of groups and apply these results to derive an important characterization of $\mathrm{\text{PD}}(n)$-systems of groups.

Any knot $K$ in genus-1 1-bridge position can be moved by isotopy to lie in a union of $n$ parallel tori tubed by $n-1$ tubes so that $K$ intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal $n$ for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the (1, 1)-length. We show that the (1, 1)-length equals the level number. We then find braid descriptions for (1,1)-positions of all 2-bridge knots providing upper bounds for their level numbers and also show that the (-2, 3, 7)-pretzel knot has level number two.

We construct a geodesic net in the plane with four unbalanced (boundary) vertices that has sixteen balanced vertices and does not contain nontrivial subnets. This is the first example of an irreducible geodesic net in the Euclidean plane with four boundary vertices that contain cycles of balanced vertices.

We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the perturbation is outward, then the next mean curvature flow singularity is cylindrical and collapsing from outside; if the perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.

An estimate for the modulus of convexity and characteristic of convexity of a general direct sum of Banach spaces is established. Using direct sums, we construct a space with given characteristic of convexity and the value of the modulus of convexity at 2. A result on uniform convexity of spaces obtained with the general discrete interpolation method is proved.