For a space N let . Let act on Ñ and on by exchanging factors and antipodes, respectively. For an embedding define the map by .¶Theorem. Let and N be a closed PL n-manifold.¶a) If , N is d-connected and there exists an equivariant map , then N is PL-embeddable in .¶b) If , , N is (d + 1)-connected and are PL-embeddings such that are equivariantly homotopic, then f, g are PL-isotopic.¶Corollary. a) Every closed 6-manifold N such that H1(N) = 0 PL embeds in ;¶b) Every closed PL 2-connected 7-manifold PL embeds in ;¶c) There are exactly four PL embeddings up to PL isotopy (l > 0).