Knotted tori and the deleted product criterion
by
Arkady Skopenkov
Kolmogorov College

The description of isotopy classes of embeddings S^p×S^q --> R^m (originated in [Zee63, Hud63]) is an important problem because
1) it is a generalization of an important classical theory of 2-componented links;
2) just as the link theory, it provides interesting examples (e.g. [Hud63] and below);
3) it is a natural next step (from the link theory) towards understanding isotopy classes of an arbitrary manifold in R^m (by the Handle Decomposition Theorem).

Our first main result is

Theorem 1 (for m=2q+p+1 [Hud63]) The set of PL (smooth) embeddings S^p×S^q --> R^m up to PL (smooth) isotopy is in 1-1 correspondence with \pi_q(V_{m-q, p+1})\oplus\pi_p(V_{m-p, q+1}), provided p <= q and m >= \frac3q2+p+2 (either m >= \frac3(q+p)2+2 or [m=\frac3(p+q)2+1, 2 <= p < q and p+q=6\mod8]). 

Note that \pi_p(V_{m-p, q+1})=0 for 2p+q <= m-2 (this is so for the smooth case of Theorem 1). For calculations of \pi_q(V_ab) see [Pae56, Sko]. Theorem 1 is a corollary of the Haefliger-Weber Theorem on the deleted product criterion for embeddability and isotopy of manifolds in R^m, its improvement by the author and of [Boe71]. Note that these criteria are versions of the Gromov h-principle [Gro86, 2.1.E]. To state our next results, let us state the Haefliger-Weber Theorem, restricted to the case of closed PL (smooth) manifolds. Let [N\tilde]={(x, y) in N×N | x =/= y} be the deleted product of N and let \Z₂ act on [N\tilde] and on S^m-1 by exchanging factors and antipodes, respectively. For an embedding f:N --> R^m the map [f\tilde]:[N\tilde] --> S^m-1 is defined by [f\tilde](x, y)=\fracfx-fy|fx-fy|. Consider the following assertions for a PL (smooth) manifold N:

A) If \Phi:[N\tilde] --> S^m-1 is an equivariant map, then there is a PL (smooth) embedding f:N --> R^m such that [f\tilde] =~ _eq\Phi.

B) If f₀, f₁:N --> R^m are PL (smooth) embeddings and [(f₀)\tilde] =~ _eq[(f₁)\tilde], then f₀ and f₁ are PL (smoothly) isotopic.

Date received: May 21, 1999