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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)
Keywords:
acyclic 2-complex; embedding