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A controlled plus construction for crumpled laminations. (English) Zbl 0799.57013

Let \(\text{Wild}(G)\) denote the unique maximal perfect subgroup of any group, \(G\). If \(S\) is any subset of \(G\), then let \(\text{ncl}(S,G)\) denote the normal closure of \(S\) in \(G\). A finite 2-complex \(K\) is almost acyclic if \(H_ 2(K;Z) \cong 0\) and \(H_ 1(K;Z)\) is free. The main result is the paper is as follows.
Theorem. If \(K\) is an almost acyclic 2-complex and \(f: K\to M^ n\), \(n>4\), is a locally tame embedding such that \(f_ \# (\pi_ 1(K))< \text{ncl} (f_ \# (\text{Wild} (\pi_ 1(K))), \pi_ 1(M)),\) then there exists a crumpled lamination \((W,M,N,p)\) with \(p^{-1} ([0, {3\over 4})) \cong M\times [0,1), p^{-1} [{3\over 4}, 1])\cong N\times [0,1]\), and \(\pi_ 1(N)\cong \pi_ 1(M)/ \text{ncl} (\text{image}(f_ \#), \pi_ 1(M)).\) This result provides a controlled version of Quillen’s plus construction, where it is even possible \(\pi_ 1(M)\) will contain no finitely generated perfect subgroup. Geometric consequences of this result include a new method for constructing wild embeddings of codimension one manifolds.

MSC:

57N45 Flatness and tameness of topological manifolds
57N15 Topology of the Euclidean \(n\)-space, \(n\)-manifolds (\(4 \leq n \leq \infty\)) (MSC2010)
54B15 Quotient spaces, decompositions in general topology
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57N65 Algebraic topology of manifolds
57N70 Cobordism and concordance in topological manifolds
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