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Van Kampen’s embedding obstruction is incomplete for 2-complexes in \(\mathbb{R}^ 4\). (English) Zbl 0847.57005

E. R. van Kampen [Abh. Math. Semin. Univ. Hamb. 9, 72-78, 152-153 (1932; Zbl 0005.02604)] gave a rough description of an obstruction \(o(K) \in H^{2n}_{\mathbb{Z}/2} (K^*, \mathbb{Z})\) which vanishes if and only if an \(n\)-dimensional simplicial complex \(K\) admits a piecewise-linear embedding into \(\mathbb{R}^{2n}\), for \(n \geq 3\) or \(n = 1\). The authors exhibit a 2-complex \(K\) with 14 vertices, 43 1-cells and 69 2-cells for which \(o (K)\) is trivial but which does not admit an embedding into \(\mathbb{R}^4\).

MSC:

57M20 Two-dimensional complexes (manifolds) (MSC2010)
57Q35 Embeddings and immersions in PL-topology

Citations:

Zbl 0005.02604
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