JournalscmhVol. 72 , No. 4DOI 10.1007/s000140050033

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

  • A. Skopenkov

    Moscow State University, Russian Federation
On the deleted product criterion for embeddability of manifolds in $ \Bbb {R}^m $ cover

Abstract

For a space N let N~={(x,y)N×Nxy}\tilde N = \{(x,y) \in N \times N|x \neq y \} . Let Z2\Bbb {Z}_2 act on Ñ and on Sm1S^{m-1} by exchanging factors and antipodes, respectively. For an embedding f:NRmf : N \to \Bbb {R}^m define the map f~:N~Sm1\tilde f : \tilde N \to S^{m-1} by f~(x,y)=fxfyfxfy\tilde f(x,y) = {fx-fy \over|fx-fy|} .¶Theorem. Let d=3n2m+2d = 3n - 2m + 2 and N be a closed PL n-manifold.¶a) If d{0,1,2}d \in \{0,1,2 \} , N is d-connected and there exists an equivariant map F:N~Sm1F : \tilde N \to S^{m-1} , then N is PL-embeddable in Rm\Bbb {R}^m .¶b) If d{1,0,1}d \in \{-1,0,1\} , mn3m-n \ge 3 , N is (d + 1)-connected and f,g:NRmf, g : N \to \Bbb {R}^m are PL-embeddings such that f~,g~\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 R10\Bbb {R}^{10} ;¶b) Every closed PL 2-connected 7-manifold PL embeds in R11\Bbb {R}^{11} ;¶c) There are exactly four PL embeddings S2l+1×S2l+1R6l+4S^{2l+1} \times S^{2l+1} \subset \Bbb {R}^{6l+4} up to PL isotopy (l > 0).