CyberSec Research

It is known that, for all n, there exist compact differentiable orientable n-manifolds with dual Stiefel-Whitney class wbar_{n-ahat(n)} nonzero, and this is best possible, but the proof is nonconstructive. Here ahat(n) equals the number of 1's in the binary expansion of n if n equiv 1 mod 4 and exceeds this by 1 otherwise. We find, for all n nonzero mod 4, examples of real Bott manifolds with this property.

In this paper, we study the relationship between the bridge index and the braid index for twist positive knots. A knot is said to be twist positive if it admits a positive braid representative that contains at least one full twist. Motivated by a conjecture of Krishna and Morton, we investigate this relationship via the knot Floer torsion order. As a consequence, we show that the bridge index and the braid index coincide for all twist positive knots.

We prove that every 3-sphere of positive Ricci curvature contains some embedded minimal surface of genus 2. We also establish a theorem for more general 3-manifolds that relates the existence of genus $g$ minimal surfaces to topological properties regarding the set of all embedded singular surfaces of genus $\leq g$.

This paper constructs numerous examples of highly connected Poincar\'{e} complexes, each homotopy equivalent to a topological manifold yet not homotopy equivalent to any smooth manifold. Furthermore, we determine the homotopy type of any closed $2k$-connected framed $(4k+2)$-manifold with Kervaire invariant one for $k=7,15,31$.

It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory.

We construct universal geometric spaces over the real spectrum compactification $\Xi^{\mathrm{RSp}}$ of the character variety $\Xi$ of a finitely generated group $\Gamma$ in $\mathrm{SL}_n$, providing geometric interpretations of boundary points. For an algebraic set $Y(\mathbb{R})$ on which $\mathrm{SL}_n(\mathbb{R})$ acts by algebraic automorphisms (such as $\mathbb{P}^{n-1}(\mathbb{R})$ or an algebraic cover of the symmetric space of $\mathrm{SL}_n(\mathbb{R})$), the projection map $\Xi \times Y \rightarrow \Xi$ extends to a $\Gamma$-equivariant continuous surjection $(\Xi \times Y)^{\mathrm{RSp}} \rightarrow \Xi^{\mathrm{RSp}}$. The fibers of this extended map are homeomorphic to the Archimedean spectrum of $Y(\mathbb{F})$ for some real closed field $\mathbb{F}$, which is a locally compact subset of $Y^{\mathrm{RSp}}$. The Archimedean spectrum is naturally homeomorphic to the real analytification, and we use this identification to compute the image of the fibers in their Berkovich analytification. For $Y=\mathbb{P}^1$, the image is a real subtree.

We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions $n \equiv 3$ mod $4$, there exist finite $n$-complexes which are homotopy equivalent after stabilising with multiple copies of $S^n$, but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with $k$-periodic cohomology which does not have free period $k$. In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction.

We study Milnor fibers and symplectic fillings of links of sandwiched singularities, with the goal of contrasting their algebro-geometric deformation theory and symplectic topology. In the algebro-geometric setting, smoothings of sandwiched singularities are described by de Jong--van Straten's theory: all Milnor fibers are generated from deformations of a singular plane curve germ associated to the surface singularity. We develop an analog of this theory in the symplectic setting, showing that all minimal symplectic fillings of the links are generated by certain immersed disk arrangements resembling de Jong--van Straten's picture deformations. This paper continues our previous work for a special subclass of singularities; the general case has additional difficulties and new features. The key new ingredient in the present paper is given by spinal open books and nearly Lefschetz fibrations: we use recent work of Roy--Min--Wang to understand symplectic fillings and encode them via multisections of certain Lefschetz fibrations. As an application, we discuss arrangements that generate unexpected Stein fillings that are different from all Milnor fibers, showing that the links of a large class of sandwiched singularities admit unexpected fillings.

A key question for $4$-manifolds $M$ admitting symplectic structures is to determine which cohomology classes $\alpha\in H^2(M,\mathbb R)$ admit a symplectic representative. The collection of all such classes, the symplectic cone $\mathcal C_M$, is a basic smooth invariant of $M$. This paper describes the symplectic cone for elliptic surfaces without multiple fibers.

In this paper, we discuss function theory on Teichm\"uller space through Thurston's theory, as well as the dynamics of subgroups of the mapping class group of a surface, with reference to Sullivan's theory on the ergodic actions of discrete subgroups of the isometry group of hyperbolic space at infinity.

Knot and link energies can be computed from sets of closed curves in three dimensional space, and each type of knot or link has a minimum energy associated with it. Here, we consider embeddings of links that locally or globally minimize the M\"obius and Minimum Distance energies. By describing these energies as functions of a small number of free parameters, we can find configurations that minimize the energies with respect to these parameters. It has previous been demonstrated that such minimizers exist, but the specific embeddings have not necessarily been found. We find the geometries leading to minimal configurations of Hopf links and Borromean rings, as well as more complex structures such as chain links and chainmails. We find that scale-invariant properties of these energies can lead to ``non-physical'' minimizers, e.g. that a linear chain of Hopf links will subtend a finite length as its crossing number diverges. This incidentally allows us to derive a conjectural improved universal lower bound for the ropelength of knots and links. We also show that Japanese-style square chainmail networks are more efficient, in terms of excess energy, than square lattice ``4-in-1'' chainmail networks.

We study sequences of modular representations of the symplectic and special linear groups over finite fields obtained from the first homology of congruence subgroups of mapping class groups and automorphism groups of free groups, and the module of coinvariants for the abelianization of the Torelli group. In all cases we compute the composition factors and multiplicities for these representations, and obtain periodic representation stability results in the sense of Church--Farb.

We give an upper bound on the distance between a degeneracy slope for a very full essential lamination and a boundary slope of an essential surface embedded in a compact, orientable, irreducible, atoroidal 3-manifold with incompressible torus boundary. There are three applications: (i) We show that a degeneracy slope for a very full essential lamination in the exterior of a prime alternating knot is meridional. This gives an affirmative answer to part of a conjecture posed by Gabai and Kazez. (ii) We obtain two bounds on boundary slopes for a hyperbolic knot in an integral homology sphere, at least one of which always holds: one concerning the denominators of boundary slopes, and the other concerning the differences between boundary slopes. This generalizes a result on Montesinos knots obtained by the author and Mizushima. (iii) We obtain two bounds on exceptional surgery slopes for a hyperbolic knot in an integral homology sphere, at least one of which always holds: one concerning the denominators of such slopes, and the other concerning their range in terms of the genera of the knots. Both are actually conjectured by Gordon and Teragaito to always hold for hyperbolic knots in the 3-sphere.

Let $\Sigma_g^b$ be a compact oriented surface of genus $g$ with $b$ boundary components, where $b\in\{0,1\}$. The Johnson kernel $\mathcal{K}_g^b$ is the subgroup of the mapping class group $\mathrm{Mod}(\Sigma_g^b)$ generated by Dehn twists about separating simple closed curves. Let $F_n$ be a free group with $n$ generators. The Torelli group for $\mathrm{Out}(F_n)$ is the subgroup $\mathrm{IO}_n\subset\mathrm{Out}(F_n)$ consisting of all outer automorphisms that act trivially on the abelianization of $F_n$. Long standing questions are whether the groups $\mathcal{K}_g^b$ and $[\mathrm{IO}_n,\mathrm{IO}_n]$ or their abelianizations $(\mathcal{K}_g^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_n,\mathrm{IO}_n]^{\mathrm{ab}}$ are finitely generated for $g\ge3$ (respectively, $n\ge3$). During the last 15 years, these questions were answered positively for $g\ge4$ and $n\ge4$, respectively. Nevertheless, the cases of $g=3$ and $n=3$ remained completely unsettled. In this paper, we prove that the abelianizations $(\mathcal{K}_3^b)^{\mathrm{ab}}$ and $[\mathrm{IO}_3,\mathrm{IO}_3]^{\mathrm{ab}}$ are finitely generated. Our approach is based on a new general sufficient condition for a module over a Laurent polynomial ring to be finitely generated as an abelian group.

We define a knot to be $\gamma_0$-sharp if its Seifert genus is detected by the concordance invariant $\gamma_0$, which arises from the immersed curve formalism in bordered Heegaard Floer homology. We show that a connected sum of $\gamma_0$-sharp fibered knots is ribbon exactly when it is of the form $K \mathbin{\#} -K$. Consequently, either iterated cables of tight fibered knots are linearly independent in the smooth concordance group, or the slice--ribbon conjecture fails.

We consider a biquandle-cohomological framework for invariants of oriented immersed surface-links in the four-space. After reviewing projections and Roseman moves for immersed surfaces, we prove that the move types (a,b,c,e,f,g,h) form a minimal generating set, showing in particular that the singular move (h) is independent of the embedded-case set (a,b,c,e,f,g). We extend biquandle colorings to broken surface diagrams with singular points and establish that coloring sets are in bijection for diagrams related by these moves, yielding a coloring number invariant for immersed surface-links. We introduce singular biquandle 3-cocycles: biquandle 3-cocycles satisfying an additional antisymmetry when the singular relations hold. Using such cocycles, we define a triple-point state-sum with Boltzmann weights and prove its invariance under all generating moves, including (h), thereby obtaining a state-sum invariant for immersed surface-links. The theory is illustrated on the Fenn-Rolfsen link example, where a computation yields non-trivial value, demonstrating nontriviality of the invariant in the immersed setting. These results unify and extend biquandle cocycle invariants from embedded to immersed surface-links.

We prove that for strictly negatively curved manifolds of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes if and only if the Cheeger isoperimetric constant is positive. This gives a partial affirmative answer to a conjecture of Kim and Kim. Furthermore, we show that for all strictly negatively curved manifolds of infinite volume, the positivity of the Cheeger constant implies the vanishing of the bounded volume class, solving one direction of the conjecture in full generality.

In this paper, we establish the Poisson integral formula for bounded pluriharmonic functions on the Teichm\"uller space of analytically finite Riemann surfaces of type $(g,m)$ with $2g-2+m>0$. We also discuss a version of the F. and M. Riesz theorem concerning the value distribution of plurisubharmonic functions on the Teichm\"uller space, as well as a Teichm\"uller-theoretic interpretation of the mean value theorem for pluriharmonic functions.