Acyclicity in Three-manifolds
نویسنده
چکیده
An acyclic compactum in an orientable, open 3-manifold has arbitrarily close, polyhedral neighborhoods whose components are compact 3-manifolds with a special structure. Frequently, these 3-manifolds have free fundamental groups. These observations and some results from combinatorial group theory are exploited to deduce facts about the homomorphism of fundamental groups induced by an acyclic mapping. The techniques are applied to relate local homotopy properties of quotient spaces of acyclic upper semicontinuous decompositions, to "UV" (or "shape") properties of the elements in the decomposition. It is shown that a "O-dimensional" monotone decomposition of Euclidean &-space is acyclic if the quotient space is an open ^-manifold. (For &=3, such a decomposition is shown to be cellular.) Some conditions are given under which acyclic decompositions are cellular.
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