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We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q := H_q(N; \mathbb Z)$. Our main result is a readily calculable classification of embeddings $N\to\mathbb R^7$ up to isotopy, with an indeterminancy. Such a classification was only known before for $H_1=0$ by our earlier work from 2008. Our classification is complete when $H_2=0$ or when the signature of $N$ is divisible neither by 64 nor by 9. The group of knots $S^4\to\mathbb R^7$ acts on the set of embeddings $N\to\mathbb R^7$ up to isotopy by embedded connected sum. In Part I we classified the quotient of this action. The main novelty of this paper is the description of this action for $H_1\ne0$, with an indeterminancy. Besides the invariants of Part I, detecting the action of knots involves a refinement of the Kreck invariant from our work of 2008. For $N=S^1\times S^3$ we give a geometrically defined 1--1 correspondence between the set of isotopy classes of embeddings and a certain explicitly defined quotient of the set $\mathbb Z\oplus\mathbb Z\oplus\mathbb Z_{12}$., Comment: 19 pages, exposition improved

This paper is expository and is accessible to students. We define simple invariants of knots or links (linking number, Arf-Casson invariants and Alexander-Conway polynomials) motivated by interesting results whose statements are accessible to a non-specialist or a student. The simplest invariants naturally appear in an attempt to unknot a knot or unlink a link. Then we present certain `skein' recursive relations for the simplest invariants, which allow to introduce stronger invariants. We state the Vassiliev-Kontsevich theorem in a way convenient for calculating the invariants themselves, not only the dimension of the space of the invariants. No prerequisites are required; we give rigorous definitions of the main notions in a way not obstructing intuitive understanding.

A map $f\colon K\to \mathbb R^d$ of a simplicial complex is an almost embedding if $f(\sigma)\cap f(\tau)=\emptyset$ whenever $\sigma,\tau$ are disjoint simplices of $K$. Theorem. Fix integers $d,k\ge2$ such that $d=\frac{3k}2+1$. (a) Assume that $P\ne NP$. Then there exists a finite $k$-dimensional complex $K$ that does not admit an almost embedding in $\mathbb R^d$ but for which there exists an equivariant map $\tilde K\to S^{d-1}$. (b) The algorithmic problem of recognition almost embeddability of finite $k$-dimensional complexes in $\mathbb R^d$ is NP hard. The proof is based on the technique from the Matou\v{s}ek-Tancer-Wagner paper (proving an analogous result for embeddings), and on singular versions of the higher-dimensional Borromean rings lemma and a generalized van Kampen--Flores theorem., Comment: 14 pages

A map $\varphi:K\to R^2$ of a graph $K$ is approximable by embeddings, if for each $\varepsilon>0$ there is an $\varepsilon$-close to $\varphi$ embedding $f:K\to R^2$. Analogous notions were studied in computer science under the names of cluster planarity and weak simplicity. This short survey is intended not only for specialists in the area, but also for mathematicians from other areas. We present criteria for approximability by embeddings (P. Minc, 1997, M. Skopenkov, 2003) and their algorithmic corollaries. We introduce the van Kampen (or Hanani-Tutte) obstruction for approximability by embeddings and discuss its completeness. We discuss analogous problems of moving graphs in the plane apart (cf. S. Spiez and H. Torunczyk, 1991) and finding closest embeddings (H. Edelsbrunner). We present higher dimensional van Kampen obstruction, its completeness result and algorithmic corollary (D. Repovs and A. Skopenkov, 1998)., Comment: 11 pages, 6 figures, minor corrections

The topological Tverberg conjecture was considered a central unsolved problem of topological combinatorics. The conjecture asserts that for any integers $r,d>1$ and any continuous map $f:\Delta\to\mathbb R^d$ of the $(d+1)(r-1)$-dimensional simplex there are pairwise disjoint faces $\sigma_1,\ldots,\sigma_r\subset\Delta$ such that $f(\sigma_1)\cap \ldots \cap f(\sigma_r)\ne\emptyset$. The conjecture was proved for a prime power $r$. Recently counterexamples for other $r$ were found. Analogously, the $r$-fold van Kampen-Flores conjecture holds for a prime power $r$ but does not hold for other $r$. The arguments form a beautiful and fruitful interplay between combinatorics, algebra and topology. We present a simplified exposition accessible to non-specialists in the area. We also mention some recent developments and open problems., Comment: 42 pages, 7 figures, exposition significantly improved and updated

We work in the smooth category. Let $N$ be a closed connected orientable 4-manifold with torsion free $H_1$, where $H_q:=H_q(N;Z)$. Our main result is a complete readily calculable classification of embeddings $N\to R^7$, up to the equivalence relation generated by isotopy and embedded connected sum with embeddings $S^4\to R^7$. Such a classification was already known only for $H_1=0$ by the work of Bo\'echat, Haefliger and Hudson from 1970. Our classification involves the Bo\'echat-Haefliger invariant $\varkappa(f)\in H_2$, Seifert bilinear form $\lambda(f):H_3\times H_3\to Z$ and $\beta$-invariant $\beta(f)$ which assumes values in a quotient of $H_1$ defined by values of $\varkappa(f)$ and $\lambda(f)$. In particular, for $N=S^1\times S^3$ we give a geometrically defined 1-1 correspondence between the set of equivalence classes of embeddings and an explicit quotient of the set $Z\oplus Z$. Our proof is based on development of Kreck modified surgery approach, involving some simpler reformulations, and also uses parametric connected sum., Comment: 52 pages, exposition improved

Let N be a closed connected smooth four-manifold with H1(N; ℤ) = 0. Our main result is the following classification of the set E7(N) of smooth embeddings N → ℝ7 up to smooth isotopy. Haefliger proved that E7(S4) together with the connected sum operation is a group isomorphic to ℤ12. This group acts on E7(N) by an embedded connected sum. Boéchat and Haefliger constructed an invariant ℵ: E7(N) → H2(N;ℤ) which is injective on the orbit space of this action; they also described im (ℵ). We determine the orbits of the action: for u ∈ im (ℵ) the number of elements in ℵ-1(u) is GCD (u/2, 12) if u is divisible by 2, or is GCD(u, 3) if u is not divisible by 2. The proof is based on Kreck's modified formulation of surgery.

We work in the smooth category. If there are knotted embeddings S^n\to R^m, which often happens for 2m, 32 pages, a link to http://www.springerlink.com added, to appear in Math. Zeit

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., 32 pages, 3 figures, some sections are rewritten

We prove a theorem on equivariant maps implying the following two corollaries: (1) Let N and M be compact orientable n-manifolds with boundaries such that M\subset N, the inclusion M\to N induces an isomorphism in integral cohomology, both M and N have (n-d-1)-dimensional spines and m > max {n+2, (3n+1-d)/2} . Then the restriction-induced map E^m(N)\to E^m(M) is bijective. Here E^m(X) is the set of embeddings X\to R^m up to isotopy (in the PL or smooth category). (2) For a 3-manifold N with boundary whose integral homology groups are trivial and such that N\not\cong D^3 (or for its special 2-spine N) there exists an equivariant map from the deleted product of N to S^2, although N does not embed into R^3. The second corollary completes the answer to the following question: for which pairs (m,n) for each n-polyhedron N the existence of an equivariant map from the deleted product of N to S^{m-1} implies the embeddability of N into R^m? An answer was known for each pair (m,n) except (3,3) and (3,2)., 7 pages, no figures

We work in the smooth category. The following problem was suggested by E. Rees in 2002: describe the precomposition action of self-diffeomorphisms of S^p x S^q on the set of isotopy classes of embeddings S^p x S^q -> R^m. Let g : S^p x S^q -> R^m be an embedding such that g |_{a x S^q} : a x S^q -> R^m - g (b x S^q) is null-homotopic for some pair of different points a,b in S^p. Theorem. If h is an autodiffeomorphism of S^p x S^q identical on a neighborhood of a x S^q for some a\in S^p and p R^m, g : S^p x S^q -> R^m isotopy classes of embeddings. As a corollary we obtain that under certain conditions for orientation-preserving embeddings s : S^p x D^q -> N the S^p-parametric embedded connected sum f#_sg depends only on f,g and the homology class of s|_{S^p x 0}., Comment: 12 pages, no figures, exposition improved

For K = K1t . . .tKs and a link map f : K → R letK = ⊔ i

A very short proof of the following smooth homogeneity theorem of D. Repovs, E. V. Scepin and the author is presented. Let N be a locally compact subset of a smooth manifold M. Assume that for each two points x,y in N there exist their neighborhoods Ux and Uy in M and a diffeomorphism h : Ux \to Uy such that h(x)=y and h (Ux \cap N) = Uy \cap N. Then N is a smooth submanifold of M., Comment: 4 pages, meaning-distorting typos corrected

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 Mathematicae