EMBEDDINGS

. We obtain estimations for isotopy classes of embeddings of closed k -connected n -manifolds into R 2 n − k − 1 for n ≥ 2 k + 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 ≥ 2 k +6 ≥ 8 . Then the set of isotopy classes of embeddings N → R 2 n − k − 1 is in 1–1 correspondence with H k +2 ( N ; Z 2 ) for n − k = 4 s +1 .


Introduction
This paper is on the classical Knotting Problem: for an n-manifold N and a number m describe the set E m (N ) of isotopy classes of embeddings N → R m . For recent surveys, see [RS99], [Sk08], [HCEC]; whenever possible we refer to these surveys, not to original papers.
Denote CAT = DIFF (smooth) or PL (piecewise linear). If the category is omitted, then a statement is correct (or a definition is given) for both categories.
By Z (k) we denote Z for k even and Z 2 for k odd. The Haefliger-Zeeman Unknotting Theorem states that for a closed k-connected orientable n-manifold N , each two embeddings N → R m are isotopic for m ≥ 2n − k + 1 and n ≥ 2k + 2 [Sk08, Theorem 2.8.b].
The classification of embeddings of N into R 2n−k−1 which was known for N = S k+1 × S n−k−1 : E 2n−k−1 (S k+1 × S n−k−1 ) is in 1-1 corespondence with 2 • Z ⊕ Z 2 for n even and to Z 2 for n odd, provided k = 0 and n ≥ 6; 3378 A. SKOPENKOV • Z 4 , 0, Z 2 ⊕ Z 2 , Z 2 according to n − k ≡ 0, 1, 2, 3 mod 4, provided n ≥ 2k + 6 ≥ 8. Now we state the main result, then describe which parts of it are new. After that we state an open problem and define maps used in the statement.
An n-manifold N is called p-parallelizable if each embedding S p → N extends to an embedding S p × D n−p → N . 3 If the coefficients of a homology group are omitted, then they are Z. For a group G, denote by G * Z 2 the set of elements of order at most 2 in G.
Main Theorem 1.1. (a) Let N be a closed k-connected n-manifold. Suppose that k ≥ 1, n ≥ 2k + 6 and N embeds 4 into R 2n−k−1 . For n − k odd, assume that N is (k + 2)-parallelizable. Then there is an exact sequence of sets 5 with an action b: (b) Under the assumptions of (a) for n−k = 4s+1 there is a 1-1 correspondence In this paper N is a closed connected n-manifold. Denote N 0 := N − Int B n , where B n ⊂ N is a codimension 0 ball. Consider the coefficient exact sequence Here 2 is the multiplication by 2, ρ 2 is the reduction modulo 2 and β is the Bockstein homomorphism.
Main Theorem 1.1. (c) Let N be a closed connected orientable n-manifold. 6 If n is odd, assume that N is spin and the Hurewicz homomorphism π 2 (N ) → H 2 (N ) is epimorphic. For n ≥ 6 (and for n = 5 in the PL category) there is an exact sequence of sets with an action b: Here r is the restriction-induced map and a(x, f ) : In Main Theorem 1.1(c) the right-hand exactness implies that im(W × r) is in 1-1 correspondence with im ρ 2 × (2W 0 ) −1 (0).
For N = S k+1 × S n−k−1 Main Theorem 1.1 is covered by the known result cited before the formulation (this result does not follow from Main Theorem 1.1). Main Theorem 1.1(c) is new. Main Theorem 1.1(a,b) is new for k = 1. For k ≥ 2 the new part of Main Theorem 1.1(a,b) is a direct geometric description of maps b, W, W ; the exact sequences could apparently be obtained using homotopy classification of 3 Note that 1-parallelizability is equivalent to orientability and 2-parallelizability is equivalent to being a spin manifold. A reader who is bothered by new terms can replace in this paper the p-parallelizability by the almost parallelizability. 4 The embeddability into R 2n−k−1 is equivalent to W n−k−1 (N ) = 0, where W n−k−1 (N ) is the normal Stiefel-Whitney class [Sk08, §2, Pr07, 11.3].
5 The right-hand term can be represented by a formula valid for both odd and even n − k: H k+2 (N ) ⊗ Z (n−k) × H k+1 (N ; Z (n−k−1) ) * Z 2 . The validity for n − k even is obvious and for n − k odd follows by the Universal Coefficients Formula.
maps from an (n − k − 1)-polyhedron to an (n − k − 2)-connected space and the following result [BG71, Corollary 1.3]: if N is a closed k-connected orientable nmanifold embeddable into R m , m ≥ 2n − 2k + 1 and 2m ≥ 3n + 4, then there is There exists a reduction of the classification of embeddings N → R 2n−k−1 to an equivariant homotopy problem [Sk08,§5]. However, an explicit solution of that problem is hard to obtain. Our result is explicit enough e.g. to yield the following corollary: under the assumptions of Main Theorem 1.1(a) the set E 2n−k−1 (N ) is infinite if and only if n − k is even and H k+2 (N ) is infinite.
Our proof is not a generalization of the classical arguments as in [BG71, Corollary 1.3] or [Sk08,§8]. Our proof is direct geometric and is a generalization of the Haefliger-Hirsch-Hudson-Vrabec argument for the proof of the bijectivity of W 2n−k [Sk08, Theorem 2.13]. Our classification involves the explicit construction of all embeddings from a given embedding; see Remark 2.10.
A classification of E 2n−1 (N ) is announced in [Ya83] (it was probably meant for n ≥ 6). Although no details are available via Google Scholar, in [Ya83] important preliminaries were set. Main Theorem 1.1(c) could be useful because the set E 2n−1 (N 0 ) is apparently easier to describe explicitly than E 2n−1 (N ) (e.g. using methods of [Ya83] or [Sa99], cf. The self-intersection set of a map H : Take a general position homotopy H : N × I → R 2n−k−1 × I between f 0 and f . Since n ≥ 2k +5, by general position, Σ(H) is a (k +2)-submanifold (not necessarily compact). The closure Cl Σ(H) is a closed (k + 2)-submanifold. For n − k is even it has a natural orientation. 8 Define the Whitney invariant In Main Theorem 1.1(a,b), k ≥ 1, so N is orientable. For an equivalent definition see the Difference Lemma 2.3 below or [Sk081,§1]. 8 Definition of the orientation is analogous to [Sk08, §2.3, p. 263]. Take smooth triangulations T and T of the domain and the range of H such that H is simplicial. Then Cl Σ(H) is a subcomplex of T . Take any oriented simplex σ ⊂ Cl Σ(H). Let us show how to decide whether the orientation of σ is right or to be changed. By general position there is a unique simplex τ of T such that f σ = f τ . The orientation on σ induces an orientation on f σ and then on τ . The orientations on σ and τ induce orientations on normal spaces in N × I to these simplices. These two orientations (in this order) together with the orientation on f σ induce an orientation on R 2n−k−1 × I. If this orientation agrees with the fixed orientation of R 2n−k−1 × I, then the orientation of σ is right, otherwise it should be changed. Since n − k is even, these orientations agree for adjacent simplices [Hu69, Lemma 11.4]. So they define an orientation of Cl Σ(H). by By [Vr89, Theorem 3.1] the map W equals (up to sign for n − k odd) the composition Here r is the restriction map and W 0 is defined as follows.
The singular set of a smooth map H : Take a general position homotopy H : For an embedding f :

the composition of Alexander and Poincaré isomorphisms,
Definition 1.4 of the action b = b N . 11 We give a definition for n ≥ 2k + 6 and H k (N ) = 0. Take an embedding f : By general position and Alexander duality, C is (n−2)-connected. Hence h n−1 is an isomorphism. Consider Definition of the orientation. Recall the notation from the previous footnote. Take a (k + 1)simplex α ⊂ S(H). By general position there are (k + 2)-simplices σ, τ ⊂ Cl Σ(H) such that f σ = f τ and σ ∩ τ = α. Define the 'right' orientation of α to be the orientation induced by the 'right' orientation of σ. This is well-defined because the 'right' orientation of τ induces the same orientation of α. (Indeed, since n − k is odd, normal spaces of σ and of τ in N × I are even-dimensional, so the 'right' orientations on σ and on τ induce the same orientation on f σ.) 10 We use W 0 r, not W in the proof. Although we do not need this, note that W 0 r is a regular homotopy invariant; if k = 0 and n is even, 11 A reader who is not interested in explicit constructions can omit this definition and set b( 12 Since N is k-connected and n ≥ 2k + 3, we can represent x by an embedding x : S k+1 → N . If the restriction to x (S k+1 ) of the normal bundle ν f : ∂C → N is trivial, then the spheroid h −1 n−1 ADx can be constructed directly as follows; cf.
Let us show how to make an embedded surgery of S k+1 × * ⊂ X to obtain an (n − 1)-sphere S n−1 ∼ = Σ ⊂ C whose inclusion into C represents h −1 n−1 ADx. Take a vector field on S k+1 × * normal to X in R 2n−k−1 . Extend S k+1 × * along this vector field to a smooth map x : D k+2 → S 2n−k−1 . Since 2n − k − 1 > 2k + 4 and n + k + 2 < 2n − k − 1, by general position we may assume that x is a smooth embedding and x(Int D k+2 ) misses f (N ) ∪ X. Denote l := 2n − 2k − 3. Since n − k − 1 > k+ 1, we have π k+1 (V l,n−k−2 ) = 0. Hence the standard framing of S k+1 × * in X extends to an l-framing on x(D k+2 ) in R 2n−k−1 . Thus x extends to The connected sum in C of this composition with f | B n is homotopic (relative to the boundary) to an embedding x : B n → C by Theorem 2.5. Define b(x)f to be f on N 0 and x on B n . This is well-defined (i.e. is independent of the choices of x and of x ) for n ≥ 2k+6 and is an action by the equivalent definition given after the Construction 2.6 of ψ below.

Proof of Main Theorem 1.1
Main tools. The proof is based on the construction and application of the following commutative diagram: Here • N is a closed homologically k-connected orientable n-manifold, f : N → R 2n−k−1 is an embedding, n ≥ 2k + 6 and N 0 := N − Int B n , where B n ⊂ N is a codimension 0 ball, • C, AD, h are defined at the end of §1, • W and W 0 are defined above in Definitions 1.3 of the Whitney invariants, • r is the restriction-induced map, • β is the Bockstein homomorphism defined only for n − k odd, • ψ f is defined below in Construction 2.6 of ψ, • ρ (n−k) is the identity for n − k even and the reduction modulo 2 for n − k odd, • b is defined in §1 (and can be alternatively defined as b : of the Alexander duality, the coefficient isomorphism, tensor product of the Hurewicz and the Pontryagin isomorphisms, and the composition map. 13 The proof of Main Theorem 1.1 in the next subsection shows how to apply this diagram. That proof uses statements of lemmas below, not their proofs. Proof. By general position and Alexander duality, C is (n − 2)-connected. Since n ≥ 4, by [Wh50] there is an exact sequence forming the first line of the following diagram: Now the lemma follows by Alexander duality.
The Whitney Invariant Lemma 2.2. Let N be a closed k-connected orientable n-manifold embeddable into R 2n−k−1 .
(W 0 ) The map W 0 is a 1-1 correspondence for k ≥ 1; the map W 0 is surjective for k = 0 and n even; im W 0 ⊃ H 1 (N ) * Z 2 for k = 0 and n odd.
(r) Assume that n ≥ 2k + 6. The restriction-induced map r : Part (W 0 ) for k = 0 follows by [Ya83, Main Theorem (i) and (iii)]. Part (W 0 ) for k ≥ 1 and part (r) are proved in the PL category in [Vr89, Theorem 2.1, Theorem 2.4 and Corollary 3.2] and in the smooth category in [Ri70]. For the reader's convenience, the proofs of (W 0 ) and (r) are sketched below.
Sketch of the proof of (W 0 ). Let Y be the set of regular homotopy classes of immersions N 0 → R 2n−k−1 . Since N is k-connected, N 0 collapses to an (n − k − 1)polyhedron. So by general position the forgetful map E 2n−k−1 (N 0 ) → Y is surjective and, for k ≥ 1, injective (see details e.g. in [Vr89, proof of Theorem 2.1] on p. 167). The map W 0 is a composition of the forgetful map and a certain map Y → H k+1 (N ; Z (n−k−1) ) that is a 1-1 correspondence for k ≥ 1 by the Smale-Hirsch (in the smooth category) or the Haefliger-Poenaru (in the PL category) classification of immersions. Now assume that k = 0. Then im W 0 ⊃ im W . By [Ya83, Main Theorem (i) and (iii)] W is surjective for n even and im W = im β = H 1 (N ) * Z 2 for n odd. This implies the required result on im W 0 . 15 Sketch of the proof of (r). Let f : N 0 → R 2n−k−1 be an embedding. If n − k is even, then the homology class of f (∂N 0 ) in H n−1 (C) is trivial [Vr89, proof of Theorem 2.4 and Addendum 2.2]. By general position and Alexander duality, C is (n − 2)-connected. Hence h n−1 is an isomorphism. Therefore the homotopy class of f (∂N 0 ) in π n−1 (C) is trivial. Then by Theorem 2.5(a) f extends to an embedding N → R 2n−k−1 . Thus r is surjective.
If n − k is odd, then the homology class of f (∂N 0 ) equals 2AD(W 0 (f )) [Vr89, proof of Theorem 2.4 and Addendum 2.2]. Thus im r = ker(2W 0 ) analogously to the case when n − k is even. 16 Proof of (β). Take Proof. Take a map F : B n+1 → R 2n−k−1 in general position with f (N 0 ) and such that F  F (Int B n+1 ). There is a general position homotopy H between f and f such that pr N Cl Σ(H) = f −1 F (Int B n+1 ). For n − k even observe that in this formula the signs of corrresponding simplices (in a certain smooth or PL triangulation of N ) are the same. So the lemma follows.

Construction Lemma 2.4.
Let N be a closed homologically k-connected orientable n-manifold, f : N → R 2n−k−1 an embedding and n ≥ 2k + 6. Then there is  16 For n − k odd the inclusion im r ⊂ ker(2W 0 ) of part (r) also follows by part (β) or by an analogue of the Boechat-Haefliger Lemma [Sk081,§2]. 17 In this formula B n is B n with reversed orientation; we have is an invariant of an isotopy (of f and f ) relative to N 0 . Construction 2.6 of ψ (analogous to [Sk081], proof of the surjectivity of W in §5). Take x ∈ π n (C) represented by a map x : S n → C. The connected sum x #f | B n in C of x with f | B n is homotopic rel ∂B n to a proper embedding x : B n → C coinciding with f on ∂B n , and x is uniquely defined by x up to isotopy rel ∂B n . 18 Define ψ(x) to be f on N 0 and x on B n . For

Proof of the Construction Lemma
Part (a) and the Difference Lemma 2.3 imply (b). Part (c) follows analogously to the uniqueness of x in the Construction 2.6 of ψ.
If r(f 1 ) = r(f ) for an embedding f 1 : N → R 2n−k−1 , then f 1 is isotopic to an embedding f such that f = f on N 0 . Then by (c) ψ[f | B n ∪f | B n ] is isotopic rel N 0 to f and hence to f 1 . This implies (d).
Let us prove part (e). Take x, y, x+y ∈ π n (C) represented by maps x , y , (x+y) : S n → C. We have that x #(y #f | B n ) is homotopic rel ∂B n to (x + y )#f | B n . Hence ψ f (x + y) is isotopic rel N 0 to ψ ψ f (y) (x) analogously to the uniqueness of x in the Construction 2.6 of ψ. Theorem 1.1(a,c).

Proof of Main
Proof of Main Theorem 1.1(a) for n − k even. The map W = W 0 r is surjective by the Whitney Invariant Lemma 2.2(r),(W 0 ). Since ρ (n−k) = id and h n is epimorphic, by the Construction Lemma 2.4(b,d), W × W is surjective.
By the Complement Lemma 2.1, h n b = 0. Hence by the Difference Lemma 2.3, W (f ) = W (ψ f b (x)). Since r is a factor of W and r(f ) = r(ψ f b (x)), we have Suppose that W (f ) = W (g) and W (f ) = W (g). Then r(f ) = r(g) by the Whitney Invariant Lemma 2.2 (W 0 ) because W = W 0 r. Thus g = ψ f (y) for some y ∈ π n (C) by the Construction Lemma 2.4(d). By the Construction Lemma 2.4(b), h n (y) = 0. Hence by the Complement Lemma 2.1, Main Theorem 1.1(c) for n even. By the Whitney Invariant Lemma 2.2(r) and the Construction Lemma 2.4(b,d) the map r × W is surjective.

Proof of
Clearly, r(f ) = r (ψ f b (x)). Analogously to the previous proof, Now we turn to the case when n − k is odd. The proof of the following result is postponed.

Proof of
Analogously to the case of n − k even W (f ) = W (ψ f b (x)) for each x.
If W (f ) = W (g), then W 0 r(f ) = W 0 r(g) by the Whitney Invariant Lemma 2.2(β). Hence by the Whitney Invariant Lemma 2.2(W 0 ) we have r(f ) = r(g), i.e. g is isotopic to an embedding g 1 such that g 1 = f on N 0 . Since W (g 1 ) = W (g) = W (f ), by the Difference Lemma 2.3, d(g 1 , f) is even. Hence by the Twisting Lemma 2.7, g 1 is isotopic to an embedding g 2 such that g 2 = f on N 0 and d(g 2 , f) = 0. By the Construction Lemma 2.4(d) there is y ∈ π n (C) such that g 2 is isotopic to ψ f (y) relative to N 0 . By the Construction Lemma 2.4(a), Proof of Main Theorem 1.1(c) for n odd. If W (f ) = W (g) and r(f ) = r(g), then analogously to the proof of (a) for n − k odd, g = b(z)f for some z ∈ H 1 (N ; Z 2 ).
Let us prove that im(W × r) ⊃ ker a. Take x ∈ H 2 (N ; Z 2 ) and f : N 0 → R 2n−1 such that β(x) = W 0 (f ). Then 2W 0 (f ) = 2β(x) = 0. Hence by the Whitney Invariant Lemma 2.2(r), f extends to an embedding f 1 : N → R 2n−1 . By the Whitney Invariant Lemma 2.2(β) we have βW (f 1 ) = W 0 (f ) = β(x). Hence W (f 1 )−x = ρ 2 y for some y ∈ H 2 (N ). Since h f 1 ,n is surjective, there is y ∈ π n (C f 1 ) such that AD f 1 h f 1 ,n (y) = y . Then by the Construction Lemma 2.4(b), Parametric connected sum of embeddings. In this subsection we recall, with only minor modifications, some results of [Sk07], [PCS]. Denote For m ≥ n + 2 denote by t m p,n−p the CAT standard embedding that is the composition Take an embedding s : − is the restriction of the standard embedding.

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A. SKOPENKOV • each embedding N → S m is isotopic to an s-standardized embedding, and • each concordance between s-standardized embeddings is isotopic relative to the ends to an s-standardized concordance.
Summation Lemma 2.9. Assume that m ≥ n + p + 3, N is a closed connected n-manifold and f : N → S m , g : T p,n−p → S m are embeddings.
(a) By the Standardization Lemma 2.8 we can make concordances and assume that f and g are s-standardized and i-standardized, respectively. Then an embedding where R k is the symmetry of S k with respect to the hyperplane where p < n/2 and d(g, t m p,n−p ) ∈ H p (T p,n−p ) is considered as an integer.
Proof. The argument for (a) is easy and similar to [Sk06,Sk07]. In order to prove that f # s g is well-defined we need to show that the concordance class of f # s g depends only on concordance classes of f and g but not on the chosen standardizations of f and g. Take concordances F : N × I → S m × I and G : T p,n−p × I → S m × I between different standardizations of f and of g. By the 'concordance' part of the Standardization Lemma 2.8 we can take concordances relative to the ends and assume that F and G are s-standardized and i-standardized, respectively. Define a concordance If F is a concordance from f 0 to f 1 and G is a concordance from g 0 to g 1 , then F # s G is a concordance from f 0 # s g 0 to f 1 # s g 1 .
Parts (b) and (c) are clear.
Since 2n − k − 1 ≥ n + k + 2 + 3, embeddings f = f # x t and f := f # x t 1 are welldefined and are isotopic by the Summation Lemma 2.9(a). We have d(f # x t 1 , f) = d(t 1 , t)x = 2x by the Construction Lemma 2.4(a) and the Summation Lemma 2.9(c). 1.1(b). By Main Theorem 1.1(a) it remains to prove that b N = 0 for n − k = 4s + 1.

Proof of Main Theorem
Since N is k-connected, the composition π k+1 (N ) → H k+1 (N ) ρ 2 → H k+1 (N ; Z 2 ) of the Hurewicz isomorphism and the reduction modulo 2 is an epimorphism. Hence by Theorem 2.5(a) for each x ∈ H k+1 (N ; Z 2 ) there is an embedding S k+1 → N realizing x (because n ≥ 2k + 3). Since N is (k + 1)-parallelizable, this embedding extends to an embedding x : S k+1 × D n−k−1 → N . Denote T := S k+1 × S n−k−1 , t := t 2n−k−1 k+1,n−k−1 and γ := b T (1)t, where 1 ∈ H k+1 (T ; Z 2 ) is the generator. Since n − k ≡ 1 mod 4, by [Sk08, Theorem 3.9 and tables] γ is isotopic to t. Therefore Here the parametric connected sums are well-defined because 2n − k − 1 ≥ n + k + 1 + 3. In order to prove the first equality we assume in the construction of γ that S n , S n−1 ⊂ R 2n−k−1 + . Then we may assume that γ is standardized. So f # x γ is obtained from f by linked connected summation along x with a composition S n → S n−1 × D n−k → S 2n−k−1 − f (N ) of two embeddings, the one representing Σ n−3 η and the other representing AD(x). Hence b N (x)f = f # x γ by definition of b.