On the deleted product criterion for embeddability of manifolds in ${\mathbb{R}}^m$

Abstract

For a space N let $\tilde{N}=\{(x,y)\in N\times N|x\ne y\}$ . Let ${\mathbb{Z}}_2$ act on Ñ and on $S^{m-1}$ by exchanging factors and antipodes, respectively. For an embedding $f:N\to {\mathbb{R}}^m$ define the map $\tilde{f}:\tilde{N}\to S^{m-1}$ by $\tilde{f}(x,y)=\frac{fx-fy}{|fx-fy|}$ .¶Theorem. Let $d=3n-2m+2$ and N be a closed PL n-manifold.¶a) If $d\in \{0,1,2\}$ , N is d-connected and there exists an equivariant map $F:\tilde{N}\to S^{m-1}$ , then N is PL-embeddable in ${\mathbb{R}}^m$ .¶b) If $d\in \{-1,0,1\}$ , $m-n\ge 3$ , N is (d + 1)-connected and $f,g:N\to {\mathbb{R}}^m$ are PL-embeddings such that $\tilde{f},\tilde{g}$ are equivariantly homotopic, then f, g are PL-isotopic.¶Corollary. a) Every closed 6-manifold N such that H1(N) = 0 PL embeds in ${\mathbb{R}}^{10}$ ;¶b) Every closed PL 2-connected 7-manifold PL embeds in ${\mathbb{R}}^{11}$ ;¶c) There are exactly four PL embeddings $S^{2l+1}\times S^{2l+1}\subset {\mathbb{R}}^{6l+4}$ up to PL isotopy (l > 0).