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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A controlled plus construction for crumpled laminations

Authors: R. J. Daverman and F. C. Tinsley
Journal: Trans. Amer. Math. Soc. 342 (1994), 807-826
MSC: Primary 57N70; Secondary 54B15, 57M20, 57N35
MathSciNet review: 1182981
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Abstract: Given a closed n-manifold M $ (n > 4)$ and a finitely generated perfect subgroup P of $ {\pi _1}(M)$, we previously developed a controlled version of Quillen's plus construction, namely a cobordism (W, M, N) with the inclusion $ j:N \mapsto W$ a homotopy equivalence and kernel of $ {i_\char93 }:{\pi _1}(M) \mapsto {\pi _1}(W)$ equalling the smallest normal subgroup of $ {\pi _1}(M)$ containing P together with a closed map $ p:W \mapsto [0,1]$ such that $ {p^{ - 1}}(t)$ is a closed n-manifold for every $ t \in [0,1]$ and, in particular, $ M = {p^{ - 1}}(0)$ and $ N = {p^{ - 1}}(1)$. We accomplished this by constructing an acyclic map of manifolds $ f:M \mapsto N$ having the right fundamental groups, and W arose as the mapping cylinder of f with a collar attached along N. The main result here presents a condition under which the desired controlled plus construction can still be accomplished in many cases even when $ {\pi _1}(M)$ contains no finitely generated perfect subgroups. By-products of these results include a new method for constructing wild embeddings of codimension one manifolds and a better understanding of perfect subgroups of finitely presented groups.

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Keywords: Crumpled lamination, degree one map, almost acyclic, perfect normal subgroup
Article copyright: © Copyright 1994 American Mathematical Society

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