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

Abstract

For a space N let $\tilde N = \{(x,y) \in N \times N|x \neq y \}$ . Let $\Bbb {Z}_2$ act on Ñ and on $S^{m-1}$ by exchanging factors and antipodes, respectively. For an embedding $f : N \to \Bbb {R}^m$ define the map $\tilde f : \tilde N \to S^{m-1}$ by $\tilde f(x,y) = {fx-fy \over|fx-fy|}$ .¶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 $\Bbb {R}^m$ .¶b) If $d \in \{-1,0,1\}$ , $m-n \ge 3$ , N is (d + 1)-connected and $f, g : N \to \Bbb {R}^m$ 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 $\Bbb {R}^{10}$ ;¶b) Every closed PL 2-connected 7-manifold PL embeds in $\Bbb {R}^{11}$ ;¶c) There are exactly four PL embeddings $S^{2l+1} \times S^{2l+1} \subset \Bbb {R}^{6l+4}$ up to PL isotopy (l > 0).