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Generalizing Kuratowski’s theorem from \(R^ 2\) to \(R^ 4\). (English) Zbl 0617.05026

This paper concerns 2-complexes that do not embed in \(R^ 4\). The main results are phrased in terms of the intrinsic 1-skeleton, which is derived from the given 2-complex by deletion of those points with locally planar neighborhoods and of the intrinsic 0-skeleton, which arises from the 1-skeleton by removal of vertices of degree two. Two theorems are given:
Theorem 1. Any finite 2-dimensional simplicial complex whose intrinsic 1- skeleton is a proper subset of \(K_ 7\) embeds in \(R^ 4.\)
Theorem 2. Any finite 2-dimensional simplicial complex whose intrinsic 0- skeleton consists of six or fewer points embeds in \(R^ 4\).
Reviewer: M.Marx

MSC:

05C10 Planar graphs; geometric and topological aspects of graph theory
57M15 Relations of low-dimensional topology with graph theory
57M20 Two-dimensional complexes (manifolds) (MSC2010)
57Q35 Embeddings and immersions in PL-topology
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