On embeddability of joins and their `factors'

DOI10.1016/J.TOPOL.2023.108409zbMATH Open1520.57028arXiv2003.12285MaRDI QIDQ6337507

Soran Parsa, A. B. Skopenkov

Publication date: 27 March 2020

Abstract: We present a short and clear proof of the following particular case of a 2006 result of Melikhov-Schepin: Let $K$ be a $k$-dimensional simplicial complex and $K*[3]$ the union of three cones over $K$ along their common bases. If $2dge3k+3$ and $K*[3]$ embeds into $mathbb{R}^{d+2}$, then $K$ embeds into $mathbb{R}^d$. We also present a generalization of this theorem. The proofs are based on the Haefliger-Weber `configuration spaces' embeddability criterion, equivariant suspension theorem and simple properties of joins and cones.

Mathematics Subject Classification ID

Simplicial sets and complexes in algebraic topology (55U10) Embeddings and immersions in PL-topology (57Q35) Homology with local coefficients, equivariant cohomology (55N25)

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