Let F be a singular Riemannian foliation on a compact connected Riemannian manifold M. We demonstrate that global foliated vector fields generate a distribution tangent to the strata defined by the closures of leaves of F and which, in each stratum, is transverse to these closures of leaves.

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...

Nous caractérisons, en terme de dimension (topologique et de Hausdorff) des fibres des espaces de limites de tangents et du cône de Whitney, les conditions de régularité $b_{\mathrm{cod}q}$ et $b^{*}$ sur une stratification $C^1$. Nous précisons ces résultats lorsque les espaces qui interviennent ne sont pas fractals, en particulier lorsque la stratification est sous-analytique.

We study the cohomology properties of the singular foliation ℱ determined by an action Φ: G × M → M where the abelian Lie group G preserves a riemannian metric on the compact manifold M. More precisely, we prove that the basic intersection cohomology $ℍ{*}_{p̅}(M/ℱ)$ is finite-dimensional and satisfies the Poincaré duality. This duality includes two well known situations: ∙ Poincaré duality for basic cohomology (the action Φ is almost free). ∙ Poincaré duality for intersection cohomology (the group G is compact...

We define suitable Sobolev topologies on the space ${𝒞}_P(B_k,f)$ of connections of bounded geometry and finite Yang-Mills action and the gauge group and show that the corresponding configuration space is a stratified space. The underlying open manifold is assumed to have bounded geometry.

It is known that, for a regular riemannian foliation on a compact manifold, the properties of its basic cohomology (non-vanishing of the top-dimensional group and Poincaré duality) and the tautness of the foliation are closely related. If we consider singular riemannian foliations, there is little or no relation between these properties. We present an example of a singular isometric flow for which the top-dimensional basic cohomology group is non-trivial, but the basic cohomology does not satisfy...

Given adjacent subanalytic strata $(X,Y)$ in ${\mathbf{R}}^n$ verifying Kuo’s ratio test $\left(r\right)$ (resp. Verdier’s $\left(w\right)$-regularity) we find an open dense subset of the codimension $k$$C^1$ submanifolds $W$ (wings) containing $Y$ such that $(X\cap W,Y)$ is generically Whitney $\left(b^{\pi }\right)$-regular is exactly one more than the dimension...