Journal of Topology and Analysis

Lazarev and Lieb showed that finitely many integrable functions from the unit interval to $ℂ$ can be simultaneously annihilated in the $L^2$ inner product by a smooth function to the unit circle. Here, we answer a question of Lazarev and Lieb proving a generalization of their result by lower bounding the equivariant topology of the space of smooth circle-valued functions with a certain $W^{1,1}$-norm bound. Our proof uses a variety of motion planning algorithms that instead of contractibility yield a lower bound for the $\mathbb{Z}/2$-coindex of a space.

Let G be a group acting properly and by isometries on a metric space X; it follows that the quotient or orbit space X/G is also a metric space. We study the Vietoris–Rips and Čech complexes of X/G. Whereas (co) homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.

Let G be a semisimple Lie group with finite center, $K\subset G$ a maximal compact subgroup, and $P\subset G$ a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on $G/K\times G/P$ to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on G/P to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense.

The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that $G=SU(n+1,1)$. Thus, G/P is $S^{2n+1}$ with its natural CR structure and the relevant BGG complex is the Rumin complex, while G/K is complex hyperbolic space of complex dimension $n+1$. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

We study smooth isotopy classes of complex curves in complex surfaces from the perspective of the theory of bridge trisections, with a special focus on curves in ${ℂ\mathbb{P}}^2$ and ${ℂ\mathbb{P}}^1\times {ℂ\mathbb{P}}^1$. We are especially interested in bridge trisections and trisections that are as simple as possible, which we call efficient. We show that any curve in ${ℂ\mathbb{P}}^2$ or ${ℂ\mathbb{P}}^1\times {ℂ\mathbb{P}}^1$ admits an efficient bridge trisection. Because bridge trisections and trisections are nicely related via branched covering operations, we are able to give many examples of complex surfaces that admit efficient trisections. Among these are hypersurfaces in ${ℂ\mathbb{P}}^3$, the elliptic surfaces $E(n)$, the Horikawa surfaces $H(n)$, and complete intersections of hypersurfaces in ${ℂ\mathbb{P}}^N$. As a corollary, we observe that, in many cases, manifolds that are homeomorphic but not diffeomorphic have the same trisection genus, which is consistent with the conjecture that trisection genus is additive under connected sum. We give many trisection diagrams to illustrate our examples.

The tail of a quantum spin network in the two-sphere is a q-series associated to the network. We study the existence of the head and tail functions of quantum spin networks colored by 2n. We compute the q-series for an infinite family of quantum spin networks and give the relation between the tail of these networks and the tail of the colored Jones polynomial. Finally, we show that the family of quantum spin networks under study satisfies a natural product structure.

Let ${SL}_n(\mathbb{Z})$ be the special linear group over integers and $M^r=S^{r_1}\times S^{r_2},$ $T^{r_1}\times S^{r_2}$, or $T^{r_0}\times S^{r_1}\times S^{r_2},$ products of spheres and tori. We prove that any group action of ${SL}_n(\mathbb{Z})$ on $M^r$ by diffeomorphims or piecewise linear homeomorphisms is trivial if $r<n-1.$ This confirms a conjecture on Zimmer’s program for these manifolds.

We show that closed arithmetic hyperbolic 3-dimensional orbifolds with larger and larger volumes give rise to triangulations of the underlying spaces whose 1-skeletons are harder and harder to embed nicely in Euclidean space. To show this we generalize an inequality of Gromov and Guth to hyperbolic n-orbifolds and find nearly optimal geodesic triangulations of arithmetic hyperbolic 3-orbifolds.

We initiate the study of perturbation of von Neumann algebras relatively to the Banach–Mazur distance. We first prove that the type decomposition is continuous, i.e. if two von Neumann algebras are close, then their respective summands of each type are close. We then prove that, under some vanishing conditions on its Hochschild cohomology groups, a von Neumann algebra is Banach–Mazur stable, i.e. any von Neumann algebra which is close enough is actually Jordan ∗-isomorphic. These vanishing conditions are possibly empty.