TOPOLOGICAL 4-MANIFOLDS WITH 4-DIMENSIONAL FUNDAMENTAL GROUP

Let $\pi$ be a group satisfying the Farrell–Jones conjecture and assume that $B\pi$ is a 4-dimensional Poincaré duality space. We consider topological, closed, connected manifolds with fundamental group $\pi$ whose canonical map to $B\pi$ has degree 1, and show that two such manifolds are s-cobordant if and only if their equivariant intersection forms are isometric and they have the same Kirby–Siebenmann invariant. If $\pi$ is good in the sense of Freedman, it follows that two such manifolds are homeomorphic if and only if they are homotopy equivalent and have the same Kirby–Siebenmann invariant. This shows rigidity in many cases that lie between aspherical 4-manifolds, where rigidity is expected by Borel’s conjecture, and simply connected manifolds where rigidity is a consequence of Freedman’s classification results.

EMBEDDINGS, NORMAL INVARIANTS AND FUNCTOR CALCULUS

This paper investigates the space of codimension zero embeddings of a Poincaré duality space in a disk. One of our main results exhibits a tower that interpolates from the space of Poincaré immersions to a certain space of “unlinked” Poincaré embeddings. The layers of this tower are described in terms of the coefficient spectra of the identity appearing in Goodwillie’s homotopy functor calculus. We also answer a question posed to us by Sylvain Cappell. The appendix proposes a conjectural relationship between our tower and the manifold calculus tower for the smooth embedding space.

Virtual Religion and Duality of Religious Spaces

To present a relatively apparent communication picture from the virtual religion and its features in “the current virtualized world” or in the ground of “dual-globalized” processes requires the discussion of four basic questions in relation to the virtual religion. Although it is not possible to provide detailed and analytical answers to these questions, aspects of the questions and facets of the answers will be analyzed. As Martin Heidegger (AP 1373:1)* has suggested, a question itself possesses leading features and opens ways of thought: 1) What is the virtual religion? 2) Which ontology can offer a presence of religion in virtual-real space — vireal space? 3) Is there any fundamental difference between a virtual and the real religion? And eventually, it is important to answer the question: 4) What are the outcomes of duality space and what is the place of religion's existence? To answer these questions we will look at ongoing literature on “the virtual religion” and on religious typology in virtual space from an “ontological” view and we will analyze structural differences between these two spaces. And finally we will address the representation of religion in two spaces through a “dual globalization paradigm” and “communication paradigm of dual spacization”.

Actions of finite p-groups on homology manifolds

Let a cyclic group of odd prime order p act on a ℤ(p)-Poincaré duality space X. We prove a relation between the Witt classes associated to the [ ]p-cohomology rings of the fixed point set of this action and of X. This is applied to show a similar result for actions of finite p-groups on ℤ(p)-homology manifolds.