EMBEDDING OBSTRUCTIONS AND 4-DIMENSIONAL THICKENINGS OF 2-COMPLEXES

VYACHESLAV S. KRUSHKAL

Abstract. The vanishing of Van Kampen’s obstruction is known to be necessary and sufficient for embeddability of a simplicial n-complex into ℝ^{2n} for n ≠ 2, and it was recently shown to be incomplete for n = 2. We use algebraic-topological invariants of four-manifolds with boundary to introduce a sequence of higher embedding obstructions for a class of 2-complexes in ℝ^{4}.

1. Introduction

By general position any n-dimensional simplicial complex K PL embeds into ℝ^{2n+1}, while the image of a generic map of K into ℝ^{2n} has a finite number of double points. By counting double points of an immersion one gets the cohomological obstruction to embeddability of an n-complex into ℝ^{2n}, introduced by Van Kampen [12]. He also constructed for each n examples which do not admit an embedding. Note that the question of embeddability is not invariant under homotopy equivalence: by a theorem of Stallings [10] every n-dimensional simplicial complex is homotopy equivalent to one which embeds into ℝ^{2n}. An application of Whitney trick shows that this obstruction is complete for n > 2, see [12], [9], [13], [4]. It follows from Kuratowski’s planarity criterion for graphs [6] that this result also holds for n = 1. The remaining case, n = 2, was open until recently when the obstruction was shown in [4] to be incomplete.

This paper is centered around the question of embeddability of 2-complexes in ℝ^{4}, and is motivated by the result of [4]. We define for 2-complexes K with H_{1}(K; ℚ) = 0 a sequence of higher embedding obstructions {o_{m}(K)}, using Massey products on the boundary of a 4-dimensional thickening M^{4} of K. Roughly, Van Kampen’s obstruction corresponds in this setting to the intersection pairing on M, modulo the choice of a thickening M. Since different thickenings may give different Massey products, {o_{m}(K)} in general are subsets of the corresponding cohomology groups; o_{m+1}(K) is defined if o_{m}(K) contains zero. If K embeds into ℝ^{4} then 0 ∈ o_{m}(K) for each m. We prove that these higher obstruction detect non-embeddability of the examples introduced in [4], by showing that o_{m}(K) does not contain zero for some m. Our proof uses the result of Conway - Gordon and Sachs that any embedding of a complete graph on 6 vertices into S^{3} contains two disjoint linking cycles ([1], [8]).

In the simplest relative case, for the disjoint union of 2-disks with a prescribed embedding of their boundaries into S^{3}, by a result of Turaev [11] Massey products correspond to Milnor’s μ̄-invariants of the link in S^{3}, so our obstructions may be thought of as an absolute analogue of μ̄-invariants. As in the case of μ̄-invariants (for example, Whitehead double of the Hopf link is not slice, while all μ̄-invariants vanish), one does not expect that the entire sequence of obstructions defined here is complete, although no examples are known at this time.

The definition of Van Kampen’s obstruction is recalled in section 2. In section 3 we prove its reformulation in the context of thickenings, and we introduce the sequence of higher obstructions {o_{m}(K)}. We review the examples of 2-complexes in [4] in section 4, and we compute the obstructions for these examples. Section 5 gives a reformulation of Van Kampen’s obstruction in terms of configuration spaces, which suggests another approach to defining higher embedding obstructions.

The present study of the embedding problem for 2-complexes in ℝ^{4} is motivated, in part, by the 4-dimensional topological surgery conjecture, via its (A,B)-slice reformulation [3]. More precisely, the surgery conjecture is equivalent to the relative embedding question for a certain family of 4-dimensional handlebodies – “thickenings” of 2-complexes in the sense of section 3. However, many interesting examples of these handlebodies have non-trivial first homology, and for this application the obstructions {o_{i}(K)} need to be extended to the general case.

Acknowledgements. I would like to thank Mike Freedman, Richard Stong and Peter Teichner for many discussions.

2. Van Kampen’s obstruction

In this section we briefly review the definition of Van Kampen’s obstruction, see [4] for more details. In 1933 Van Kampen [12] introduced an obstruction o(K) ∈ H_{ℤ/2}^{2n}(K^{∗}; ℤ) to piecewise-linear embeddability of an n-dimensional simplicial complex K into ℝ^{2n}. The cohomology in question is ℤ/2-equivariant cohomology where ℤ/2 acts on the deleted product K^{∗} = K × K ∖ Δ of a complex K by exchanging the factors of K^{∗} and acts on the coefficients by multiplication with (−1)^{n}. The diagonal Δ consists of all products σ × τ such that simplices σ and τ have at least one vertex in common. Note that for n even (in particular, in the case of main interest in this paper, n = 2) the action of ℤ/2 on the coefficients is trivial, and o(K) is an element of the ordinary cohomology group H^{2n}(K^{∗}/(ℤ/2); ℤ).

Let f be any PL immersion of K into ℝ^{2n}. The obstruction is defined on the cochain level by counting algebraic intersection numbers of the images of disjoint top-dimensional simplices of K: o_{f}(σ^{n} × τ^{n}) = f(σ) · f(τ). Here σ × τ is viewed as an oriented generator of the 2n-th chain group of K × K ∖ Δ. The cohomology class o(K) of o_{f} is independent of the chosen immersion f. Clearly o(K) is trivial if K embeds into ℝ^{2n}. Shapiro [9] and Wu [13] made this definition precise and proved, using the Whitney trick, the converse in dimension n greater than 2.

Theorem 2.1 ([12], [9], [13], [6]). For n ≠ 2 an n-dimensional simplicial complex K embeds into ℝ^{2n} if and only if o(K) vanishes.

2-COMPLEXES IN ℝ^{4} 3

See [4] for a modern exposition of the proof for n > 2. For n = 1 this theorem follows from Kuratowski’s planarity criterion. The obstruction in the remaining case, for n = 2, was shown to be incomplete in [4]. We recall the construction of examples in [4] in section 4.

3. Obstructions via 4-dimensional thickenings of 2-complexes

In this section we give a rational reformulation of Van Kampen’s obstruction o(K) in terms of thickenings of K, and we introduce the sequence of higher embedding obstructions {o_{m}(K)} for 2-complexes whose rational first homology vanishes. Throughout this section all coefficients are ℚ, unless stated otherwise, and K denotes a simplicial 2-complex.

Definition 3.1. A thickening of K is a 4-manifold M with boundary, obtained by replacing each i-simplex of K with a 4-dimensional i-handle, i = 0, 1, 2. The attaching map of each 2-handle is required to be isotopic, within the union of 0- and 1-handles, to the attaching map of the corresponding 2-dimensional simplex.

In general, K has different thickenings depending on the choice of attaching maps of the 2-handles. For example, the boundary-connected sum S^{2} × D^{2}♮S^{2} × D^{2} and S^{2} × S^{2} ∖ 4-cell are both thickenings of S^{2} ∨ S^{2}.

The intersection pairing on M defines an element ι ∈ Hom(H_{2}(M) ⊗ H_{2}(M), ℚ). Let ῑ ∈ H^{4}(K × K ∖ Δ; ℚ) denote its image under the homorphism

Hom(H_{2}(M) ⊗ H_{2}(M), ℚ) ≅ Hom(H_{2}(K) ⊗ H_{2}(K), ℚ) ≅
≅ H^{4}(K × K) ⟶ H^{4}(K × K ∖ Δ)

where the last map is the restriction homomorphism.

Theorem 1. The image of (rational) Van Kampen’s obstruction o(K) under the homomorphism

H_{ℤ/2}^{4}(K × K ∖ Δ; ℚ) ⟶ H^{4}(K × K ∖ Δ; ℚ)

coincides with −ῑ.

Proof. Suppose a thickening M is induced by an immersion f: K ↬ ℝ^{4}, so that f extends to an immersion M ↬ ℝ^{4}. By subdividing the complex K, if necessary, one may assume that f(σ) ∩ f(τ) ≠ ∅ for all 2-simplices σ ≠ τ with σ × τ ∈ Δ, and f|_{σ} is an embedding for each σ. Let ō_{f}: C_{4}(K × K) ⟶ ℚ denote the extension by zero on the diagonal of Van Kampen’s cochain o_{f}. It suffices to prove that [ō_{f}] and −ῑ define identical elements in Hom(H_{2}(K) ⊗ H_{2}(K), ℚ). Let a, b be two classes in H_{2}(K) and let α = Σα_{i}σ_{i}, β = Σβ_{i}σ_{i} be their cycle representatives, where {σ_{i}} is the set of 2-simplices of K. In order to compute a · b in M, we perturb α and β to α̃ and β̃ which intersect each other transversely (in a finite number of double points). Since f is an immersion into ℝ^{4}, the intersection number of two cycles f(α̃) and f(β̃) is trivial. On the other hand, f(α̃)·f(β̃) is the sum of two terms: one is the intersection number of α̃ and β̃ in M, the other is obtained by considering the intersections of f(α̃) and f(β̃), which are singular points of f. This last term is equal to ō_{f}(α × β), and this proves

[ō_{f}] = −ῑ: H_{2}(K) ⊗ H_{2}(K) ⟶ ℚ.

The restriction of [ō_{f}] to H^{4}(K × K ∖ Δ; ℚ) coincides with o(K), thus the result is proved for thickenings induced by immersions.

In general not every thickening of K may be immersed into ℝ^{4}. Let M^{4} be an arbitrary thickening of K and let f: K ↬ ℝ^{4} be any immersion. Again one may assume that f(σ) ∩ f(τ) = ∅ if σ × τ ∈ Δ, σ ≠ τ, and f|_{σ} is an embedding for each simplex σ. The immersion f extends to an embedding of 0- and 1-handles of M. There is an integer obstruction to extending it over each 2-handle, due to a possible difference of the framing of the 2-handle and of the normal bundle of the 2-simplex in ℝ^{4}. However, each 2-handle may be mapped into ℝ^{4} as a bundle over the corresponding 2-simplex, pinched over several points.

The proof, given above in the case of an immersion, carries through, if one extends o_{f} to ō_{f} by setting ō_{f}(σ × σ) to be equal to the difference in framings, discussed above, and setting ō_{f}(σ × τ) = 0 for all σ × τ ∈ Δ, σ ≠ τ.

Remark. In general the intersection pairing varies within the homotopy type of a 4-manifold M. In the example above the intersection pairing on S^{2} × D^{2}♮S^{2} × D^{2} is trivial, while the pairing on S^{2} × S^{2} ∖ 4-cell is non-degenerate. However, theorem 1 shows that the pull-back of the intersection pairing on thickenings to a cohomology class on K×K∖Δ is an invariant of K, which coincides with (negative) Van Kampen’s obstruction.

As a corollary to the proof of theorem 1, we have the following result.

Lemma 3.2. Let K be a 2-complex such that Van Kampen’s obstruction o(K) vanishes. Then there is a 4-dimensional thickening M of K with the trivial intersection pairing ι = 0 ∈ Hom(H_{2}(M) ⊗ H_{2}(M); ℚ).

Proof. Any cochain representative of the obstruction o(K) is given by o_{f} for some immersion f, see [12] or [4]. Since o(K) vanishes, there exists an immersion f: K ↬ ℝ^{4}, giving rise to the trivial Van Kampen’s cochain o_{f} = 0. Let M denote the thickening, induced by f. It follows from the proof of theorem 1 that if one extends o_{f} by zero on the diagonal to a cochain ō_{f} on K ×K, then ῑ = [ō_{f}] = 0 ∈ Hom(H_{2}(M)⊗ H_{2}(M); ℚ). This completes the proof of lemma 3.2.

Before introducing higher embedding obstructions, we review the definition of Massey products. See [5] for proofs and additional properties.

Definition 3.3. Let X be a space, and let α_{1}, …, α_{m} be elements in H^{1}(X). Suppose there is a collection of 1-cochains S = {c_{ij} ∈ C^{1}(X)|1 ≤ i ≤ j ≤ m, (i, j) ≠ (1, m)} satisfying

[c_{ii}] = α_{i} for each i = 1, …, m,

δc_{ik} = ∑_{j=i}^{k−1} c_{ij} ∪ c_{j+1,k} for i < k.

A computation shows that the cochain ∑_{j=1}^{m−1} c_{1j} ∪ c_{j+1,m} is a cocycle, and its cohomology class in H^{2}(X) is called the Massey product of α_{1}, …, α_{m} defined by the system S. The set of Massey products corresponding to all such defining systems is denoted by <α_{1}, …, α_{m} >.

Massey product of two elements is just a cup product. Note that given some classes α_{1}, …, α_{m}, <α_{1}, …, α_{m} > is not necessarily defined. However, if all Massey products of less than m elements vanish, then for any α_{1}, …, α_{m}, <α_{1}, …, α_{m} > is a well-defined element.

The following lemma justifies our definition of higher embedding obstructions.

Lemma 3.4. Let M be a 4-manifold with boundary and with H_{1}(M; ℚ) = 0, and suppose M admits an embedding into ℝ^{4}. Then all Massey products on H^{1}(∂M; ℤ) vanish.

Proof. Let N denote the complement ℝ^{4} ∖ M. By Alexander duality, H_{2}(N) and H^{2}(N) are trivial. The map i^{∗}: H^{1}(N) ⟶ H^{1}(∂M) in the cohomology sequence of the pair (N, ∂M) is an isomorphism, since by assumption and by Poincare duality H^{1}(N, ∂M) ≅ H_{3}(N) and H^{2}(N, ∂M) ≅ H_{2}(N) are trivial. Since Massey products are functorial, for any α_{1}, …, α_{m} ∈ H^{1}(∂M) one has

<α_{1}, …, α_{m} >= i^{∗}<(i^{∗})^{−1}α_{1}, …, (i^{∗})^{−1}α_{m} > ∈ H^{2}(∂M).

However, this is the image of an element in H^{2}(N) = 0, and the result follows.

Let K be a 2-complex with H_{1}(K; ℚ) = 0, and assume o(K) vanishes. Let M be a thickening of K with trivial intersection pairing (its existence is given by lemma 3.2.) Note that the map H_{2}(∂M) ⟶ H_{2}(M) is an isomorphism, since by assumption on K, H_{2}(M, ∂M) ≅ H^{1}(M) = 0, and the map H_{2}(M) ⟶ H_{2}(M, ∂M) is trivial by assumption on the intersection pairing.

We now give the definition of higher embedding obstructions. Let a_{1}, a_{2}, a_{3} be classes in H_{2}(K), and let α_{1}, α_{2}, α_{3} ∈ H^{1}(∂M) denote their images under the isomorphism

H_{2}(K) ≅ H_{2}(M) ≅ H_{2}(∂M) ≅ H^{1}(∂M).

The triple cup product (α_{1}∪α_{2}∪α_{3})[∂M] defines a homomorphism H_{2}(K)⊗H_{2}(K)⊗ H_{2}(K) ⟶ ℚ, and an element o_{3}(K, M) ∈ H^{6}(K × K × K). The cohomology class o_{3}(K, M) depends in general on the choice of a thickening M, thus we define the third obstruction o_{3}(K) to be the subset {o_{3}(K, M)} ⊂ H^{6}(K^{3}; ℚ) where M is to vary over all thickenings of K with trivial intersection pairing.

Definition 3.5. Assume by induction that o_{m−1}(K) is defined and contains zero for some m > 3, so there is a thickening M of K such that all Massey products on H^{1}(∂M) of at most (m−2) elements vanish. Let a_{1}, …, a_{m}, be classes in H_{2}(K), and let α_{1}, …, α_{m} denote the corresponding elements in H^{1}(∂M). The class o_{m}(K, M) ∈ H^{2i}(K^{i}; ℚ) is defined by the homomorphism

H_{2m}(K^{i}) ≅ ⊗_{1}^{m}H_{2}(K) ≅ ⊗_{1}^{m}H^{1}(∂M) ⟶ ℚ

which sends a_{1}⊗ ⋯ ⊗a_{m} to (<α_{1}, …, α_{m−1} >∪ α_{m})[∂M].

Here since o_{m−1}(K, M) vanishes, all Massey products on H^{1}(∂M) of less than (m −1) elements vanish, and <α_{1}, …, α_{m−1} >∈ H^{2}(∂M) is a well-defined element.

Definition 3.6. The obstruction o_{m}(K) is defined to be the subset

{o_{m}(K, M)} ⊂ H^{2m}(K^{m}),

where M is to vary over all thickenings such that o_{m−1}(K, M) = 0.

Lemma 3.4 implies the following corollary.

Corollary 3.7. If K admits an embedding into ℝ^{4} then o_{m}(K) is defined and contains zero for each m.

Some computations are given in section 4. We show that o_{m}(K) does not contain zero for some m for examples in [4], thus giving another proof that they do not embed into ℝ^{4}.

The relative embedding problem

Let K be a 2-complex with H_{1}(K; ℚ) = 0, and let L be a 1-dimensional subcomplex of K with a prescribed embedding ϕ: L ↪ S^{3}. Consider the relative embedding problem: does there exist an embedding K ↪ B^{4} which extends ϕ? Denote B^{4}∪_{ϕ}(thickening of K) by M, where thickening is taken in the sense of definition 3.1. Note that various embeddings ϕ correspond to different thickenings of K∪(cone on L). Let K_{L}^{m} denote the subset in K^{m} consisting of all m-tuples (x_{1}, …, x_{m}) such that x_{i} ∈ L for some i. Assume that π_{0}(L) ⟶ π_{0}(K) is injective to have H_{1}(M) = 0. Analogously to the absolute case, Massey products on ∂M define an element, depending on M, in the relative cohomology group H^{2m}(K^{m}, K_{L}^{m}). Let o_{m}(K, L, ϕ) denote the set of these elements in H^{2m}(K^{m}, K_{L}^{m}), where M is to vary over all thickenings for which the (m −1)-st obstruction is zero. If there is an embedding of K into B^{4}, extending ϕ, clearly there is a 4-dimensional thickening M which embeds into S^{4}, so 0 ∈ o_{m}(K, L, ϕ) for each m.

Consider the simplest relative case, when (K, L) = (D^{2}⨿…⨿D^{2}, S^{1}⨿…⨿S^{1}). By the result of Turaev [11], the first non-trivial obstruction coincides in this case with the first non-trivial Milnor’s μ̄-invariants of the link ϕ(L) in S^{3}. In this this sense the obstructions o_{m}(K) may be thought of as an absolute analogue of μ̄-invariants. However, since 2-complexes in general have a more complicated topology, {o_{m}(K)} have a larger indeterminacy.

4. Examples

First we recall the construction of examples in [4]. Let C denote the 2-skeleton of the 6-simplex with vertices v_{1}, …, v_{7}, with one 2-cell, with vertices v_{1}v_{2}v_{3}, removed. Take another copy C′ of C, with vertices v′_{1}, …, v′_{7}, and denote by \overline{C} the union of C and C′, identified along their last vertices, v_{7} = v′_{7}. This 2-complex is easily seen to admit an embedding into ℝ^{4} (see [12]). Let γ (resp. γ′) denote the loop v_{7}v_{1}v_{2}v_{3}v_{1}v_{7} (resp. v_{7}v′_{1}v′_{2}v′_{3}v′_{1}v_{7}) in \overline{C}.

Denote by F the free group on two generators, and fix a positive integer m. Let α be an element in F^{m}, the m-th term of the lower central series of F. We identify F with π_{1}(γ ∨ γ′), and we associate to each word in F its “standard” representative curve in the wedge of two circles γ ∨ γ′. Finally, we construct the 2-complex K_{α} by attaching a 2-cell to \overline{C} along α.

Theorem 4.1 ([4]). Let α be a non-trivial element in F^{m} for some m ≥ 2. Then Van Kampen’s obstruction o(K_{α}) vanishes, but K_{α} does not admit an embedding into ℝ^{4}.

We now present a computation of the obstructions {o_{i}(K_{α})}. The class m of the commutator α is reflected in non-vanishing of the obstruction o_{m+1}(K_{α}).

Theorem 2. Let α be an element in F^{m} for some m ≥ 2, and assume α ∉ F^{m+1}. Then o_{m+1}(K_{α}) is defined and does not contain zero. In particular, K_{α} does not embed into ℝ^{4}.

Proof. First we construct a thickening M of K_{α} with trivial intersection pairing and such that o_{i}(K, M) = 0 for all i ≤ m. The complex K_{α} is obtained from \overline{C} = C ∨ C′ by attaching a 2-cell along the commutator α ∈ F^{m}. Van Kampen constructed in [12] an immersion of the 2-skeleton of the 6-simplex with vertices v_{1}, …, v_{7} into ℝ^{4} such that the 2-cells with vertices v_{1}v_{2}v_{3} and v_{4}v_{5}v_{6} intersect in one point, and all other simplices are disjoint and embedded. Consider the corresponding embedding of \overline{C}, and let \overline{M} denote its thickening in ℝ^{4}. Clearly the intersection pairing on \overline{M} is trivial, and all Massey products on H^{1}(∂\overline{M}) vanish. Recall that C and C′ have the vertex v_{7} in common. Consider the handle decomposition of \overline{M}, given by thickenings of simplices of K_{α} in ℝ^{4}. The union of the handles in \overline{M} corresponding to all simplices in \overline{C}, containing v_{7}, is a 4-ball B. The remaining 2-handles are attached to B along a link \overline{L} in S^{3} = ∂B. Each attaching curve is isotopic, within the union of 0- and 1-handles, to the boundary curve of the corresponding 2-simplex. There is no 2-cell attached to v_{1}v_{2}v_{3}, however we introduce in S^{3} a circle, isotopic to it. Because of the choice of the embedding of K into ℝ^{4}, \overline{L} is a slice link, and the curves isotopic to v_{1}v_{2}v_{3} and v_{4}v_{5}v_{6} (respectively v′_{1}v′_{2}v′_{3} and v′_{4}v′_{5}v′_{6}) have linking number one. The remaining 2-cell of K_{α} is attached along the commutator of v_{1}v_{2}v_{3} and v′_{1}v′_{2}v′_{3}. Choosing appropriately the corresponding curve l in S^{3}, we get the link L = \overline{L} ∪ l such that all Milnor’s μ̄-invariants of L of length less than m + 1 vanish, and a μ̄-invariant of length m + 1 of the 3-component link (v_{4}v_{5}v_{6}, v′_{4}v′_{5}v′_{6}, l) is nontrivial. It is a result of Turaev [11] that the first non-vanishing μ̄-invariant is equal to the corresponding Massey product on ∂M, where M = \overline{M} ∪_{l} 2-handle. (Note that the framings of the components of \overline{L} are zero, since the intersection pairing on \overline{M} vanishes, and we choose the framing of l also to be zero.) This proves that o_{i}(K, M) = 0 for all i ≤ m, and o_{m+1}(K_{α}, M) ≠ 0.

It remains to show that o_{m+1}(K_{α}) does not contain zero. Let M be any thickening with trivial intersection pairing and with o_{m}(K, M) = 0. As above, the union of the handles in M corresponding to all simplices, containing v_{7}, is a 4-ball B. By a theorem of Conway - Gordon [1] and Sachs [8] any embedding of the complete graph on 6 vertices in S^{3} contains two disjoint linking cycles. Consider the complete graph on vertices v_{1}, …, v_{6} in \overline{C}. According to the definition of M, the attaching curves of the 2-handles in S^{3} = ∂B are isotopic to the attaching maps of simplices of K_{α}. As above, we introduce in S^{3} a curve isotopic to v_{1}v_{2}v_{3}. Now we have in S^{3} a perturbed version of the complete graph on 6 vertices. Since, according to definition 3.1, these perturbations take place in the union of 0- and 1-handles, at least two of the curves must have a non-trivial linking number. Since the intersection pairing on M vanishes, these two circles are the ones isotopic to v_{1}v_{2}v_{3} and to v_{4}v_{5}v_{6}. Similarly we have in S^{3} another, disjoint, copy of a perturbed graph on v′_{1}, …, v′_{6}, and two linking circles isotopic to v′_{1}v′_{2}v′_{3} and to v′_{4}v′_{5}v′_{6}. Recall that there are no 2-handles attached to v_{1}v_{2}v_{3} or v′_{1}v′_{2}v′_{3}, however there is a 2-handle whose attaching curve l is a commutator of these circles. It is easily seen that the link (v_{4}v_{5}v_{6}, v′_{4}v′_{5}v′_{6}, l) has a non-trivial μ̄-invariant of length m+1. As above, this is translated into non-vanishing of o_{m+1}(K, M).

Remark. The idea of the proof of the fact that any embedding of the complete graph on 6 vertices in S^{3} contains two linking cycles ([1], [8]) is conceptually similar to the proof of Van Kampen that the 2-skeleton of the 6-simplex does not embed into ℝ^{4} [12]. (In both cases one shows that a certain number is invariant mod 2 for different maps - in one case, the total linking number, in other case, the total number of singular points of an immersion.) In this sense our proof of theorem 2 is similar to the proof of theorem 4.1 in [4].

5. A note on configuration spaces

In this section we consider an approach to the embedding problem, suggested by obstruction theory and configuration spaces. We give a reformulation of Van Kampen’s obstruction in this context, which suggests another approach to defining higher embedding obstructions. Given a space X, C^{m}(X) will denote its m-fold symmetric product:

C^{m}(X) = {(x_{1}, …, x_{m}) ∈ X^{m} | x_{i} ≠ x_{j} if i ≠ j}.

In the simplicial category, the symmetric product of a complex K is defined to be

C^{m}(K) = {σ_{1}×…×σ_{m} ⊂ K^{m} | simplices σ_{i}, σ_{j} have no vertices in common, i ≠ j}.

The 2-fold symmetric product is sometimes called deleted product and is also denoted by X^{∗}. The symmetric groups are denoted by S_{m}; S_{m} acts freely on C^{m}(K), and on its i-skeleton (C^{m}(K))^{i} for each i, by exchanging the coordinates.

A necessary condition for the existence of an embedding K^{n} ↪ ℝ^{2n} is the existence, for each m, of an S_{m}-equivariant map C^{m}(K) ⟶ C^{m}(ℝ^{2n}). We will now analyze the first embedding obstruction, corresponding to m = 2, that is, the obstruction to existence of a ℤ_{2}-equivariant map

K × K ∖ Δ ⟶ ℝ^{2n} × ℝ^{2n} ∖ Δ ≃ S^{2n−1}.

The ℤ_{2}-equivariant homotopy equivalence above is given by the projection of ℝ^{2n} × ℝ^{2n}∖Δ onto the unit sphere in the antidiagonal {(x, −x)} ⊂ ℝ^{2n}×ℝ^{2n}. The diagonal Δ in K × K is the “simplicial” diagonal, as defined in section 2, while Δ ⊂ ℝ^{2n} × ℝ^{2n} is the usual set-theoretic diagonal. Recall that the spaces above are denoted in short by K^{∗} and (ℝ^{2n})^{∗} respectively.

Theorem 3. The obstruction to existence of a ℤ_{2}-equivariant map K^{∗} ⟶ (ℝ^{2n})^{∗} lies in H_{ℤ/2}^{2n}(K^{∗}; ℤ) and coincides with Van Kampen’s obstruction o(K).

Proof. Since K^{∗} is a (2n)-dimensional CW-complex, the only non-trivial obstruction group in this setting is H_{ℤ_{2}}^{2n}(K^{∗}; π_{2n−1}(S^{2n−1})) ≅ H_{ℤ/2}^{2n}(K^{∗}; ℤ).

Let f: K ↬ ℝ^{2n} be any immersion. Since f(σ) and f(ν) are disjoint for any n-simplex σ and any (n−1)-simplex ν, f ×f restricted to the (2n−1)-skeleton of K^{∗} is a ℤ_{2}-equivariant embedding into (ℝ^{2n})^{∗}. Let σ, τ be two n-dimensional simplices of K and consider σ × τ as an oriented generator of (2n)-dimensional cellular chains on K^{∗}. The obstruction cochain c_{f} assigns to σ × τ the element

c_{f}(σ × τ) = [(f × f)(∂(σ × τ))] ∈ π_{2n−1}(S^{2n−1}).

The map f × f sends σ × τ into ℝ^{2n} × ℝ^{2n}, and one has

o_{f}(σ × τ) = f(σ) · f(τ) = (f × f)(σ × τ) ∩ Δ_{ℝ^{2n}} = (f × f)(∂(σ × τ)) = c_{f}(σ × τ).

This shows that the homotopy-theoretic obstruction coincides with Van Kampen’s obstruction even on the cochain level, when the map of (2n − 1)-skeleton of K^{∗} corresponds to the chosen immersion f. This completes the proof, since the cohomology class [c_{f}] is independent of the choice of a map of (2n − 1)-skeleton of K^{∗}, being the first non-trivial obstruction.

Remarks. (1) Theorem 3 has been independently stated in [7].

(2) It is interesting to note that by theorems 2.1 and 3, the existence of a ℤ_{2} - equivariant map K^{∗} ⟶ (ℝ^{2n})^{∗} is equivalent to existence of an embedding K^{n} ↪ ℝ^{2n} for n ≠ 2.

We conclude by suggesting the following approach to defining higher embedding obstructions. Suppose o(K) vanishes, so there exists a ℤ/2-equivariant map K^{∗} ⟶ (ℝ^{4})^{∗}. One may consider the obstructions to existence of an equivariant, with respect to the free action of the symmetric group S_{m}, map C^{m}(K) ⟶ C^{m}(ℝ^{4}), m = 3, 4, . . .. Note that Fadell - Neuwirth [2] have determined homotopy types of the symmetric products of Euclidean spaces, thus (rationally) explicitely giving the coefficients of obstruction groups.

References

[1] J. H. Conway and C. McA. Gordon, Knots and links in spatial graphs, J. Graph Theory 7 (1983), No. 4, 445-453.

[2] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111-118.

[3] M. H. Freedman, Are the Borromean rings (A, B)-slice?, Topology Appl., 24 (1986), 143-145.

[4] M. H. Freedman, V. S. Krushkal and P. Teichner, Van Kampen’s Embedding Obstruction is Incomplete for 2-Complexes in ℝ^{4}, Math. Res. Lett., 1 (1994), No. 2, 167-176.

[5] D. Kraines, Massey higher products, Trans. Amer. Math. Soc., 124 (1966), 431-449.

[6] C. Kuratowski, Sur le problème des courbes gauches en topologie, Fund. Math. 15(1930), 271-283.

[7] D. Repovš and A. Skopenkov, Embeddability and isotopy of polyhedra in Euclidean spaces, preprint.

[8] H. Sachs, On spatial representations of finite graphs, Finite and infinite sets, Vol. I, II, Colloq. Math. Soc. Janos Bolyai, 37, North-Holland, Amsterdam-New York, 1984.

[9] A. Shapiro, Obstructions to the imbedding of a complex in a euclidean space I, Ann. of Math. 66 (1957) No.2, 256-269.

[10] J. Stallings, The embedding of homotopy types into manifolds, mimeographed notes, Princeton University, 1965.

[11] V. G. Turaev, Milnor invariants and Massey products, Zap. Nauchn. Sem. LOMI 66 (1976), 189-203.

[12] E. R. Van Kampen, Komplexe in euklidischen Räumen, Abh. Math. Sem. Univ. Hamburg, vol.9 (1933), pp. 72-78 and 152-153.

[13] W. T. Wu, A theory of imbedding, immersion, and isotopy of polytopes in a euclidean space, Science Press, Peking 1965.

Michigan State University, East Lansing, MI 48824-1027, USA Current address: Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, D-53225

Bonn, Germany

E-mail address: krushkal@math.msu.edu