Some minimal nonembeddable complexes
The author presents a method for constructing finite simplicial complexes which cannot be embedded into \(\mathbb{R}^ n\) and are minimal with respect to this property (in the sense that any proper subcomplex can be embedded into \(\mathbb{R}^ n\)). The method also covers many of the previously known examples of such complexes, notably those of \textit{E. R. van Kampen} [Abh. Math. Semin. Univ. Hamb. 9, 72-78, 152-153 (1932; Zbl 0005.02604)], \textit{Flores} [Erg. Math. Kolloq. 5, 17-24 (1933; Zbl 0007.36803)], \textit{B. Grünbaum} [Comment. Math. Helv. 44, 502-513 (1969; Zbl 0188.286)], \textit{J. Zaks} [Proc. Am. Math. Soc. 20, 439-444 (1969; Zbl 0176.532)] and \textit{K. S. Sarkaria} [Topology 30, 67-76 (1991; Zbl 0721.57012)]. We briefly describe the method. The author calls a finite simplicial complex \({\mathcal K}\) nice if it is a simplex (more precisely, the complex of all faces of a simplex \(\sigma\), including \(\sigma\)) or if it is a subcomplex of a simplex such that for every subset \(S\) of the vertex set \(\text{vert }{\mathcal K}\), exactly one of the subsets \(S\) or \(\text{vert } {\mathcal K}\setminus S\) spans a simplex of \({\mathcal K}\). It is then shown that if \({\mathcal K}\) is the join of nice complexes \({\mathcal K}_ i\) \((i=1, \dots, r)\) having \(n_ i+2\) vertices if \({\mathcal K}_ i\) is a simplex and \(n_ i+3\) vertices else, then \({\mathcal K}\) cannot be embedded into \(\mathbb{R}^ n\) where \(n= 2r-2+ \sum_{i=1}^ r n_ i\). Moreover, \({\mathcal K}\) is minimal except if all \({\mathcal K}_ i\) are simplexes, of it \(r=1\) and \({\mathcal K}_ 1\) consists of the proper faces of a simplex and an isolated vertex. (It is asserted without proof that if non-embeddability into the sphere \(\mathbb{S}^ n\) instead of \(\mathbb{R}^ n\) is considered, then all these complexes are minimal.) The proof of non-embeddability uses a result of Borsuk-Ulam type by \textit{C.-T. Yang} [Ann. Math., II. Ser. 60, 262-282 (1954; Zbl 0057.391)]. In order to apply this result to a complex \({\mathcal K}\) of the type given above, the author considers the cell complex \({\mathcal K}^*\) whose cells are the products \(\sigma\times \tau\) where \(\sigma\) and \(\tau\) are disjoint simplices of \({\mathcal K}\), and proves that the sum of all maximal cells is a cycle in \(C_ n ({\mathcal K}^*; \mathbb{Z}_ 2)\) of index \(\neq 0\).
scientific article; zbMATH DE number 500707
scientific article