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Embeddings of acyclic 2-complexes in \(S^ 4\) with contractible complement. (English) Zbl 0765.57004
Topology and combinatorial group theory, Proc. Fall Foliage Topology Semin., New Hampshire/UK 1986-88, Lect. Notes Math. 1440, 122-129 (1990).
[For the entire collection see Zbl 0701.00019.]
The author is interested in the following question: Can each acyclic 2- complex be embedded in \(S^ 4\) with simply-connected complement? He proves that if \(P\) is a finite presentation of a group with the same number of generators as relations and such that the relators abelianize to the generators, then the standard 2-complex \(K_ P\) modeled on \(P\) has an embedding in \(S^ 4\) with contractible complement. This is done by showing that if \(P\) is any finite presentation and if one chooses the “simplest” embedding of \(K_ P\) in \(S^ 4\), then the closed complement of a regular neighborhood of \(K_ P\) in \(S^ 4\) collapses to a 2-complex \(K_{P^*}\), where \(P^*\) can easily be described in terms of \(P\).

MSC:
57M20 Two-dimensional complexes (manifolds) (MSC2010)