Significant information about the topology of a bounded domain $\Omega$ of a Riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial \Omega }$, from the boundary of $\Omega$. We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of $d_{\partial \Omega }$, as well as applications to homotopy equivalence.

Let A, $X_1$ and $X_2$ be topological spaces and let $i_1:A\to X_1$, $i_2:A\to X_2$ be continuous maps. For all self-maps $f_A:A\to A$, $f_1:X_1\to X_1$ and $f_2:X_2\to X_2$ such that $f_1{i}_1=i_1{f}_A$ and $f_2{i}_2=i_2{f}_A$ there exists a pushout map f defined on the pushout space $X_1{\bigsqcup }_A{X}_2$. In [F] we proved a formula relating the generalized Lefschetz numbers of f, $f_A$, $f_1$ and $f_2$. We had to assume all the spaces involved were connected because in the original definition of the generalized Lefschetz number given by Husseini in [H] the space was assumed to be connected. So, to extend the result of [F] to the not...

We prove that any compact simply connected manifold carrying a structure of Riemannian 3- or 4-symmetric space is formal in the sense of Sullivan. This result generalizes Sullivan's classical theorem on the formality of symmetric spaces, but the proof is of a different nature, since for generalized symmetric spaces techniques based on the Hodge theory do not work. We use the Thomas theory of minimal models of fibrations and the classification of 3- and 4-symmetric spaces.

If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism φ: π₁(X) → π̌₁(X) from the fundamental group to its first shape homotopy group is an isomorphism. We present a partial converse to this result: a path-connected topological space X admits a generalized universal covering space if φ: π₁(X) → π̌₁(X) is injective. This generalized notion of universal covering p: X̃ → X enjoys most of the usual properties, with the possible exception of evenly...

For G = SU(n), Sp(n) or Spin(n), let $C_G\left(SU\left(2\right)\right)$ be the centralizer of a certain SU(2) in G. We have a natural map $J:G/C_G\left(SU\left(2\right)\right)\to \Omega ₀³G$. For a generator α of $H⁎(G/C_G\left(SU\left(2\right)\right);\mathbb{Z}/2)$, we describe J⁎(α). In particular, it is proved that $J⁎:H⁎(G/C_G\left(SU\left(2\right)\right);\mathbb{Z}/2)\to H⁎(\Omega ₀³G;\mathbb{Z}/2)$ is injective.

We study the genus and SNT sets of connective covering spaces of familiar finite CW-complexes, both of rationally elliptic type (e.g. quaternionic projective spaces) and of rationally hyperbolic type (e.g. one-point union of a pair of spheres). In connection with the latter situation, we are led to an independently interesting question in group theory: if f is a homomorphism from Gl(ν,A) to Gl(n,A), ν < n, A = ℤ, resp. ${\mathbb{Z}}_p$, does the image of f have infinite, resp. uncountably infinite, index in...