The disjoint curve property

Detecting codimension one manifold factors with the piecewise disjoint arc-disk property and related properties

We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note that in every example presently known to the authors of a codimension one manifold factor of dimension $n\geq 4$ determined by general position properties, the piecewise disjoint arc-disk property is satisfied.

A classification of pairs of disjoint nonparallel primitives in the boundary of a genus two handlebody

Embeddings of pairs of disjoint nonparallel primitive simple closed curves in the boundary of a genus two handlebody are classified. Briefly, two disjoint primitives either lie on opposite ends of a product $F \boldsymbol{\times} I$, or they lie on opposite ends of a kind of "twisted" product $F \widetilde{\boldsymbol{\times}} I$, where $F$ is a once-punctured torus. If one of the curves is a proper power of a primitive, the situation is simpler. Either the curves lie on opposite sides of a separating disk in the handlebody, or they bound a nonseparating essential annulus in the handlebody.

Lines and circles joining components of a link

We estimate from below the number of lines meeting each of given 4 disjoint smooth closed curves in a given cyclic order in the real projective 3-space and in a given linear order in the Euclidean 3-space. Similarly, we estimate the number of circles meeting in a given cyclic order given 6 disjoint smooth closed curves in Euclidean 3-space. The estimations are formulated in terms of linking numbers of the curves and obtained by orienting of the corresponding configuration spaces and evaluating of their signatures. This involves a study of a surface swept by lines meeting 3 given disjoint smoot…

Tunnel-number-one knot exteriors in $S^3$ disjoint from proper power curves

As one of the background papers of the classification project of hyperbolic primitive/Seifert knots in $S^3$ whose complete list is given in [BK20], this paper classifies all possible R-R diagrams of two disjoint simple closed curves $R$ and $β$ lying in the boundary of a genus two handlebody $H$ up to equivalence such that $β$ is a proper power curve and a 2-handle addition $H[R]$ along $R$ embeds in $S^3$ so that $H[R]$ is the exterior of a tunnel-number-one knot. As a consequence, if $R$ is a nonseparating simple closed curve on the boundary of a genus two handlebody such that $H[R]$ embeds…

Busemann $G$-空间的不相交圆盘性质

The disjoint disks property for Busemann $G$-spaces

本文证明维数至少为五的有限维 Busemann $G$-空间具有不相交圆盘性质，通过度量球面的同伦 $Z_2$-集性质及 Daverman 乘积定理实现，将 Busemann 猜想归结为分解问题。

Path concordances as detectors of codimension-one manifold factors

We present a new property, the Disjoint Path Concordances Property, of an ENR homology manifold X which precisely characterizes when X times R has the Disjoint Disks Property. As a consequence, X times R is a manifold if and only if X is resolvable and it possesses this Disjoint Path Concordances Property.