We show that the proper homotopy type of any properly c-connected locally finite n-dimensional CW-complex is represented by a closed polyhedron in ${ℝ}^{2n-c}$ (Theorem I). The case n - c ≥ 3 is a special case of a general proper homotopy embedding theorem (Theorem II). For n - c ≤ 2 we need some basic properties of “proper” algebraic topology which are summarized in Appendices A and B. The results of this paper are the proper analogues of classical results by Stallings [17] and Wall [20] for finite CW-complexes;...

We construct an epsilon coincidence theory which generalizes, in some aspect, the epsilon fixed point theory proposed by Robert Brown in 2006. Given two maps f, g: X → Y from a well-behaved topological space into a metric space, we define µ ∈(f, g) to be the minimum number of coincidence points of any maps f 1 and g 1 such that f 1 is ∈ 1-homotopic to f, g 1 is ∈ 2-homotopic to g and ∈ 1 + ∈ 2 < ∈. We prove that if Y is a closed Riemannian manifold, then it is possible to attain µ ∈(f, g) moving...

If f:G → H is a group homomorphism and p,q are the projections from the free product G*H onto its factors G and H respectively, let the group ${}_f\subseteq G*H$ be the equalizer of fp and q:G*H → H. Then p restricts to an epimorphism $p_f{=p|}_f{:}_f\to G$. A right inverse (section) $G{\to }_f$ of $p_f$ is called a coaction on G. In this paper we study ${}_f$ and the sections of $p_f$. We consider the following topics: the structure of ${}_f$ as a free product, the restrictions on G resulting from the existence of a coaction, maps of coactions and the resulting...

We survey results related to the problem of the existence of equilibria in some classes of infinitely repeated two-person games of incomplete information on one side, first considered by Aumann, Maschler and Stearns. We generalize this setting to a broader one of principal-agent problems. We also discuss topological results needed, presenting them dually (using cohomology in place of homology) and more systematically than in our earlier papers.

The non-commutative torus $C^{*}({\mathbb{Z}}^n,\omega )$ is realized as the $C^{*}$-algebra of sections of a locally trivial $C^{*}$-algebra bundle over $\widehat{S_{\omega }}$ with fibres isomorphic to $C^{*}({\mathbb{Z}}^n/S_{\omega },{\omega }_1)$ for a totally skew multiplier ${\omega }_1$ on ${\mathbb{Z}}^n/S_{\omega }$. D. Poguntke [9] proved that $A_{\omega }$ is stably isomorphic to $C\left(\widehat{S_{\omega }}\right)\otimes C^{*}({\mathbb{Z}}^n/S_{\omega },{\omega }_1)\cong C\left(\widehat{S_{\omega }}\right)\otimes A_{\varphi }\otimes M_{kl}\left(ℂ\right)$ for a simple non-commutative torus $A_{\varphi }$ and an integer $kl$. It is well-known that a stable isomorphism of two separable $C^{*}$-algebras is equivalent to the existence of equivalence bimodule between them. We construct an $A_{\omega }$-$C\left(\widehat{S_{\omega }}\right)\otimes A_{\varphi }$-equivalence bimodule.