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In the 80's V. Poenaru studied a non-compact version of the
geometric connectivity previously defined by Wall,
introducing also the idea of killing 1-handles stably, in order to
understand the topology at infinity of open 3-manifolds. In
particular, he proved that if the product of an open simply
connected 3-manifold with a ball is gsc (geometrically
simply connected), then the manifold is simply connected at
infinity ( sci).
There are several generalizations of the gsc and of the
sci, that assure the tameness of the end of manifolds, such as
the Tucker property, the weak geometric simple connectivity
( wgsc), the missing boundary condition and the
quasi-simple filtration ( qsf), that are closely related to
each other.
In the present talk, I will review all these topological tameness
conditions together with their mutual relationships. Then, I will
also show how to extend Poenaru’s result for manifolds that are not
simply connected.
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