Coperfectly Hopfian Groups and Shape Fibrator's Properties

TL;DR

This paper explores shape fibrator properties of manifolds with coperfectly Hopfian fundamental groups, establishing new conditions under which products of such manifolds are shape m$_{ m simpl}$-fibrators.

Contribution

Introduces the class of coperfectly Hopfian groups and demonstrates their role in determining shape m$_{ m simpl}$-fibrators among product manifolds.

Findings

Coperfectly Hopfian groups are a new class of Hopfian groups.

Product manifolds with coperfectly Hopfian fundamental groups are shape m$_{ m simpl}$-fibrators under certain conditions.

Manifolds homotopically determined by $ ext{pi}_1$ and with coperfectly Hopfian groups are shape m$_{ m simpl}$-fibrators if they are codimension-2 fibrators.

Abstract

This paper provides further investigation of the concept of shape m$_{\rm simpl}$-fibrators (previously introduced by the author). The main results identify shape m$_{\rm simpl}$-fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined by $\pi_1$ with coperfectly Hopfian group (a new class of Hopfian groups that are introduced here) is a shape m$_{\rm simpl}$o-fibrator if it is a codimension-2 fibrator (Theorem 5.4). The main result (Theorem 6.2) states that the direct product of two closed orientable manifolds (of different dimension) homotopically determined by $\pi_1$ and with coperfectly Hopfian fundamental groups (one normally incommensurable with the other one) is a shape m$_{\rm simpl}$o-fibrator, if it is a Hopfian manifold and a codimension-2 fibrator.