# On the deleted product criterion for embeddability of manifolds in $R_{m}$

### A. Skopenkov

Moscow State University, Russian Federation

## Abstract

For a space N let $N~={(x,y)∈N×N∣x=y}$ . Let $Z_{2}$ act on Ñ and on $S_{m−1}$ by exchanging factors and antipodes, respectively. For an embedding $f:N→R_{m}$ define the map $f~ :N~→S_{m−1}$ by $f~ (x,y)=∣fx−fy∣fx−fy $ .¶Theorem. Let $d=3n−2m+2$ and N be a closed PL n-manifold.¶a) If $d∈{0,1,2}$ , N is d-connected and there exists an equivariant map $F:N~→S_{m−1}$ , then N is PL-embeddable in $R_{m}$ .¶b) If $d∈{−1,0,1}$ , $m−n≥3$ , N is (d + 1)-connected and $f,g:N→R_{m}$ are PL-embeddings such that $f~ ,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 $R_{10}$ ;¶b) Every closed PL 2-connected 7-manifold PL embeds in $R_{11}$ ;¶c) There are exactly four PL embeddings $S_{2l+1}×S_{2l+1}⊂R_{6l+4}$ up to PL isotopy (l > 0).

## Cite this article

A. Skopenkov, On the deleted product criterion for embeddability of manifolds in $R_{m}$. Comment. Math. Helv. 72 (1997), no. 4, pp. 543–555

DOI 10.1007/S000140050033