On the Deleted Product Criterion For Embeddability in $\mathbb{R}^{m}$

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 $F:\nobreak \tilde K\rightarrow S^{m-1}$, 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 $m=2$.