On the deleted product criterion for embeddability in ℝ^{𝕞}

Skopenkov, A.

doi:10.1090/S0002-9939-98-04142-2

On the deleted product criterion for embeddability in $\mathbb R^m$
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Proc. Amer. Math. Soc. 126 (1998), 2467-2476 Request permission

Abstract:

For a space $K$ let $\tilde K=\{(x,y)\in K\times K| x\not =y\}$. Let $\mathbb {Z}_{2}$ act on $\tilde K$ and on $S^{m-1}$ by exchanging factors and antipodes respectively. We present a new short proof of the following theorem by Weber: For an $n$-polyhedron $K$ and $m\geqslant \frac {3(n+1)}{2}$, if there exists an equivariant map $F: \tilde K\rightarrow S^{m-1}$, then $K$ is embeddable in $\mathbb {R}^{m}$. We also prove this theorem for a peanian continuum $K$ and $m=2$. We prove that the theorem is not true for the 3-adic solenoid $K$ and $m=2$.

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