On embeddability of joins and their `factors'

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 $K\ast [3]$ the union of three cones over $K$ along their common bases. If $2d\ge 3k+3$ and $K\ast [3]$ 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.

4 pages, exposition improved, references updated

References in corpus (8)

• Embedding and knotting of manifolds in Euclidean spaces
• Combinatorial Lefschetz theorems beyond positivity
• The telescope approach to embeddability of compacta
• On the metastable Mabillard-Wagner conjecture
• Invariants of graph drawings in the plane
• A short exposition of S. Parsa's theorems on intrinsic linking and non-realizability
• On the Smith classes, the van Kampen obstruction and embeddability of $[3]\ast K$
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Cited by in corpus (2)

• On different reliability standards in current mathematical research
• Geometric Embeddability of Complexes is $\exists \mathbb{R}$-complete