Classification of knotted tori

For a smooth manifold N denote by E^m(N) the set of smooth isotopy classes of smooth embeddings N -> R^m. A description of the set E^m (S^p x S^q) was known only for p=q=0 or for p=0, m\ne q+2 or for 2m > 2(p+q)+\max\{p,q\}+3 (in terms of homotopy groups of spheres and Stiefel manifolds). For m > 2p+q+2 an abelian group structure on E^m (S^p x S^q) is introduced. This group is described up to an extension problem: this group and E^m (D^{p+1} x S^q) + ker l + E^m (S^{p+q}) are associated to the same group for some filtrations of length four. Here l : E -> pi_q(S^{m-p-q-1}) is the linking coefficient defined on the subset E of E^m (S^q U S^{p+q}) formed by isotopy classes of embeddings whose restriction to each component is unknotted. This result and its proof have corollaries which, under stronger dimension restrictions, more explicitly describe E^m (S^p x S^q) in terms of homotopy groups of spheres and Stiefel manifolds. The proof is based on relations between different sets E^m (N), in particular, on a recent exact sequence of M. Skopenkov.Comment: 32 pages, 3 figures, some sections are rewritte

Eliminating higher-multiplicity intersections in the metastable dimension range

The $r$-fold analogues of Whitney trick were `in the air' since 1960s. However, only in this century they were stated, proved and applied to obtain interesting results. Here we prove and apply a version of the $r$-fold Whitney trick when general position $r$-tuple intersections have positive dimension. Theorem. Assume that $D=D_1\sqcup\ldots\sqcup D_r$ is disjoint union of $k$-dimensional disks, $rd\ge (r+1)k+3$, and $f:D\to B^d$ a proper PL (smooth) map such that $f\partial D_1\cap\ldots\cap f\partial D_r=\emptyset$. If the map $$f^r:\partial(D_1\times\ldots\times D_r)\to (B^d)^r-\{(x,x,\ldots,x)\in(B^d)^r\ |\ x\in B^d\}$$ extends to $D_1\times\ldots\times D_r$, then there is a proper PL (smooth) map $\overline f:D\to B^d$ such that $\overline f=f$ on $\partial D$ and $\overline fD_1\cap\ldots\cap \overline fD_r=\emptyset$.Comment: 13 pages, 2 figures, exposition improve

A new invariant and parametric connected sum of embeddings

We define an isotopy invariant of embeddings N -> R^m of manifolds into Euclidean space. This invariant together with the \alpha-invariant of Haefliger-Wu is complete in the dimension range where the \alpha-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows to obtain new completeness results for the \alpha-invariant and the following estimation of isotopy classes of embeddings. For the piecewise-linear category, a (3n-2m+2)-connected n-manifold N and (4n+4)/3 < m < (3n+3)/2 each preimage of \alpha-invariant injects into a quotient of H_{3n-2m+3}(N), where the coefficients are Z for m-n odd and Z_2 for m-n even.Comment: 13 pages, to appear in Fundamenta Mathematica

Homotopy type of the complement of an immersion and classification of embeddings of tori

This paper is devoted to the classification of embeddings of higher dimensional manifolds. We study the case of embeddings $S^p\times S^q\to S^m$, which we call knotted tori. The set of knotted tori in the the space of sufficiently high dimension, namely in the metastable range $m\ge p+3q/2+2$, $p\le q$, which is a natural limit for the classical methods of embedding theory, has been explicitely described earlier. The aim of this note is to present an approach which allows for results in lower dimension