Embeddings of $k$-connected $n$-manifolds into $\mathbb {R}^{2n-k-1}$

Abstract

Abstract. We obtain estimations for isotopy classes of embeddings of closed k-connected n-manifolds into R 2n−k−1 for n ≥ 2k + 6 and k ≥ 0. This is done in terms of an exact sequence involving the Whitney invariants and an explicitly constructed action of H k+1 (N ; Z 2 ) on the set of embeddings. The proof involves a reduction to the classification of embeddings of a punctured manifold and uses the parametric connected sum of embeddings.Corollary. Suppose that N is a closed almost parallelizable k-connected n-manifold and n ≥ 2k + 6 ≥ 8. Then the set of isotopy classes of embeddings N → R 2n−k−1 is in 1-1 correspondence with H k+2 (N ; Z 2 ) for n − k = 4s + 1.

Classification of knotted tori

“…The sum operation on E m (S p × S q ) is 'S p -parametric connected sum', cf. [19,26,31], [33, theorem 8]. See accurate definitions in § 2.1; cf.…”

“…Classification of knotted tori is a natural next step after the Haefliger link theory [12] and the classification of embeddings of highly-connected manifolds [27, § 2], [15]. Such a step gives some insight or even precise information concerning embeddings of arbitrary manifolds [26,31,33], and reveals new interesting relations to algebraic topology.…”

How do autodiffeomorphisms act on embeddings?

“…There is an isotopy R t i ψ R t between i ψ and i ψ. By the standardization lemma of [29,33] there is a standardized representativeḡ of g. Then a representativeh of g +[i]ψ is defined by ( * * ) forf ,ḡ, s replaced byḡ, i ψ , i, respectively. We haveh =ḡ =ḡψ on In this subsection we denote by the same letter a symmetric autodiffeomorphism of T p,q and its restriction S p × D q ± → S p × D q ± .…”

“…Proof of theorem 1.16. By the standardization lemma of [29,33] there are s-standardized and i-standardized representativesf of f andḡ of g, respectively. Then a representativeh of f # s g is defined by ( * * ).…”