We introduce a skeletal structure $(M,U)$ in ${ℝ}^{n+1}$, which is an $n$- dimensional Whitney stratified set $M$ on which is defined a multivalued “radial vector field” $U$. This is an extension of notion of the Blum medial axis of a region in ${ℝ}^{n+1}$ with generic smooth boundary. For such a skeletal structure there is defined an “associated boundary” $ℬ$. We introduce geometric invariants of the radial vector field $U$ on $M$ and a “radial flow” from $M$ to $ℬ$. Together these allow us to provide sufficient numerical conditions for...

Recently, T. Fukui and L. Paunescu introduced a weighted version of the $\left(w\right)$-regularity condition and Kuo’s ratio test condition. In this approach, we consider the $\left(w\right)$- regularity condition and $\left(c\right)$-regularity related to a Newton filtration.

Teardrops are generalizations of open mapping cylinders. We prove that the teardrop of a stratified approximate fibration X → Y × ℝ with X and Y homotopically stratified spaces is itself a homotopically stratified space (under mild hypothesis). This is applied to manifold stratified approximate fibrations between manifold stratified spaces in order to establish the realization part of a previously announced tubular neighborhood theory.

To apply surgery theory to the problem of classifying pairs of closed manifolds, it is necessary to know the subgroup of the group $L{P}_{*}$ generated by those elements which are realized by normal maps to a pair of closed manifolds. This closely relates to the surgery problem for a closed manifold and to the computation of the assembly map. In this paper we completely determine such subgroups for many cases of Browder-Livesay pairs of closed manifolds. Moreover, very explicit results are obtained in the...