Embedding obstructions in {\mathbb R}^d from the Goodwillie-Weiss calculus and Whitney disks
EEMBEDDING OBSTRUCTIONS IN R d FROM THE GOODWILLIE-WEISSCALCULUS AND WHITNEY DISKS
GREGORY ARONE AND VYACHESLAV KRUSHKAL
Abstract.
Given an m -dimensional CW complex K , we use a version of the Goodwillie-Weiss tower to formulate an obstruction theory for embeddings into a Euclidean space R d .For 2-complexes in R , a geometric analogue is also introduced, based on intersections ofWhitney disks and more generally on the intersection theory of Whitney towers developedby Schneiderman and Teichner. The focus in this paper is on the first obstruction beyondthe classical embedding obstruction of van Kampen. In this case we show the two ap-proaches give the same result, and also relate it to the Arnold class in the cohomology ofconfiguration spaces. The obstructions are shown to be realized in a family of examples.Conjectures are formulated, relating higher versions of these homotopy-theoretic, geometricand cohomological theories. Introduction
Let K be a finite CW complex. In this paper we investigate, and compare, two approachesto constructing obstructions to the existence of a topological embedding K (cid:44) → R d , withspecial focus on the case of 2-dimensional complexes in R .Our first approach is inspired by the embedding calculus of Goodwillie and Weiss [35, 14],which provides a systematic framework for studying embedding spaces. The difference be-tween the two settings is that their theory was developed for studying smooth embeddingsof smooth manifolds, while we adapt their ideas to the context of topological (or PL) em-beddings of finite complexes.The embedding calculus of Goodwillie-Weiss works particularly well for the study of smoothembeddings in codimension at least 3; in this case the tower converges to the embeddingspace. In some instances it is known to give rise to highly non-trivial invariants in codi-mension 2 as well; for example in the case of long knots in R this theory is closely relatedto Vassiliev invariants, cf. [33], [6], [16]. We use a weaker version of the Goodwillie-Weisstower to formulate obstructions to embeddability of a complex K into R d . In particular, weapply this theory to 2-complexes in R , another instance of codimension 2 embeddings. The(weak) convergence of this tower to the embedding space is an open problem, see Section 8.For 2-complexes in R we also consider an alternative, geometric approach based on thefailure of the Whitney trick in this dimension. Some instances of this approach are well-known, for example in the study of Milnor’s invariants [21]. More generally, Schneidermanand Teichner [26] developed the intersection theory of Whitney towers in 4-manifolds. Weuse these ideas to formulate embedding obstructions for 2-complexes in R .Focusing on the first new obstruction, we show that these a priori unrelated approachesin fact give the same result (Theorem 5.1). This provides a useful perspective on both of a r X i v : . [ m a t h . A T ] J a n GREGORY ARONE AND VYACHESLAV KRUSHKAL them: the homotopy-theoretic obstruction is manifestly well-defined but lacks an immediategeometric interpretation; the Whitney tower approach has a clear geometric meaning butestablishing its well-definedness directly is a challenging problem.Before outlining our work in more detail, we recall some of the classical results in this subject.When 2 m < d , an m -dimensional simplicial complex K embeds in R d by general position.If 2 dim( K ) ≥ d there is a classical obstruction, proposed by van Kampen [32], based on thefollowing idea. Suppose there exists a topological embedding f : K (cid:44) → R d . Then the map f × f : K × K → R d × R d restricts to a Σ -equivariant map, which we call the deleted square of f :(1.1) f : C ( K, → C ( R d , . Here C ( X,
2) = X × X \ X is the configuration space of ordered pairs of distinct points in X , and Σ n denotes the symmetric group of degree n ; in particular Σ ∼ = Z / Z .Thus the existence of a Σ -equivariant map C ( K, → C ( R d ,
2) is a necessary conditionfor the existence of a topological embedding f : K (cid:44) → R d . The van Kampen obstruction is acohomological obstruction to the existence of such a map. It is an element of the equivariantcohomology group H d Σ (C ( K, Z [( − d ]), where the notation indicates the action of Σ by ( − d on the integers. (The original formulation of van Kampen [32] predated a formaldefinition of cohomology, and it was based on the geometric approach discussed in Section4. Moreover, van Kampen’s formulation focused on the case 2dim( K ) = d .)The van Kampen obstruction is known to be complete when 2 dim( K ) = d (cid:54) = 4. Fordim( K ) >
2, this follows from the validity of the Whitney trick [30, 36]; a modern treatmentmay be found in [12]. For 1-complexes in R this follows from the Kuratowski graph planaritycriterion [18] and the naturality of van Kampen’s obstruction under embeddings. When K is a 2-dimensional complex and d = 4, it was shown in [12] that the existence of a Σ -equivariant map C ( K, → C ( R d ,
2) is insufficient for embeddability, and thus the vanKampen obstruction is incomplete. The underlying geometric reason, the failure of theWhitney trick in 4 dimensions, is well-known. However, as in many other aspects of 4-manifold topology, it is a non-trivial problem to formulate an invariant that captures thisgeometric fact. In this paper, as we discuss below, we formulate such an invariant in thecontext of 2-complexes in R .Building on work of Haefliger [15], Weber [34] extended the embeddability result to the“metastable range” of dimensions. More precisely, it is shown in [34] that given an m -dimensional simplicial complex K and is a Σ -equivariant map f : C ( K, −→ C ( R d , d ≥ m + 1), there exists a PL embedding f : K −→ R m such that the induced map f is Σ -equivariantly homotopic to f .In this paper we produce a new obstruction to the existence of an embedding of a finite com-plex in R d beyond the metastable range, which is defined when the van Kampen obstructionvanishes. Our first new obstruction O ( K ), where the index refers to 3-point configurationspaces used to define it, depends on a choice of a Σ -equivariant map f as in (1.1), and itreally is an obstruction to f being Σ -equivariantly homotopic to the deleted square f ofsome embedding f : K (cid:44) → R d .We will give several topological, geometric and algebraic interpretations of O ( K ). Onthe topological side, we interpret O ( K ) as the fundamental class of the subspace of points MBEDDING OBSTRUCTIONS IN R d ( k , k , k ) ∈ C ( K,
3) for which the vectors f ( k , k ), f ( k , k ) and f ( k , k ) are co-directed(Section 3.1). On the geometric side, we show that it counts intersections of K with theWhitney disks that arise from the vanishing of the van Kampen obstruction (Sections 4, 5).On the algebraic side, we interpret it as the kernel of the Arnold relation (Lemma 6.5). Weuse this algebraic interpretation to verify that O ( K ) is non-zero in a family of examples(Section 6).For concreteness, in this paper we focus on embeddings of 2-dimensional complexes in R , andjust on the first obstruction beyond van Kampen’s. But the general approach underpinningour construction is applicable in other dimensions, and leads to an infinite sequence ofobstructions. Moreover, our methods should lead to a formulation of an obstruction theoryfor embedding complexes into more general manifolds, not just R d . We intend to pursue thiselsewhere. In this paper we will only give an outline of the general approach, and state afew conjectural claims.Work in progress [2], joint with Danica Kosanovi´c, Rob Schneiderman and Peter Teichner,formulates the non-repeating version of the Goodwillie-Weiss tower in a different context,for link maps . In this setting the results of [2], which apply when the ambient manifoldhas an arbitrary fundamental group, establish an equivalence between the homotopy liftingobstructions from the non-repeating version of the Goodwillie-Weiss tower and the highergeometric intersection theory of Whitney towers.We now proceed to outline our construction; see Sections 2, 7 for more details. Let Emb( K, R d )be the space of topological embeddings of K into R d . We are not going to use the topologyof this space (except when K is a finite set), so the reader can think of Emb( K, R d ) as justa set. Our sole concern is whether this set is empty or not. Our obstruction theory is basedon a tower of spaces under Emb( K, R d ), which we denote as follows(1.2) Emb( K, R d ) → · · · → T n Emb( K, R d ) → T n − Emb( K, R d ) → · · · → T Emb( K, R d ) . Note that we make no claim that the induced mapEmb( K, R d ) −→ holim T n Emb( K, R d )is an equivalence. Nevertheless, since there is a map Emb( K, R d ) → T n Emb( K, R d ), anecessary condition for Emb( K, R d ) to be non-empty is that T n Emb( K, R d ) is non-emptyfor all n .It is easy to see that T Emb( K, R d ) (cid:39) map(C ( K, , C ( R d , Σ As we mentioned above, there is a well-known cohomological obstruction O ( K )for this space to be non-empty, and this is the van Kampen obstruction.From this point on, our strategy is to look for an obstruction for a path component of T n Emb( K, R d ) to be in the image of a path component of T n +1 Emb( K, R d ). We will nowdescribe explicitly how to do it in the case n = 2. This case is our main focus in this paper.Suppose K is a finite-dimensional complex for which the van Kampen obstruction vanishes.Then there exists a Σ -equivariant map f : C ( K, −→ C ( R d , GREGORY ARONE AND VYACHESLAV KRUSHKAL
Our goal is to give an effective necessary condition for the existence of an embedding f : K (cid:44) → R d such that f : C ( K, → C ( R d ,
2) is equivariantly homotopic to f . LetC ( X,
3) = (cid:8) ( x , x , x ) ∈ X | x i (cid:54) = x j for i (cid:54) = j (cid:9) be the configuration space of ordered triples of distinct points in X . There is a cubicaldiagram of configuration spaces, where the projection p i omits the i -th coordinate:(1.3) C ( X, X C ( X,
3) C ( X, X {∗} C ( X, X p p p p p p p p p Now suppose we have a topological embedding f : K (cid:44) → R d . Such an embedding inducesa Σ n -equivariant map of configuration spaces C ( K, n ) −→ C ( R d , n ) for each n ; moreoverit induces a map of cubical diagrams (1.3) for K , R d . In the diagram for R d the spaceC ( R d ,
1) = R d is contractible, and (up to homotopy) the map of cubical diagrams may bereplaced by a smaller diagram (1.4) below. Denote by p X the canonical Σ -equivariant map p X : C ( X, −→ C ( X, × C ( X, × C ( X, x , x , x ) (cid:55)→ ( x , x ) , ( x , x ) , ( x , x )Then f induces a commutative diagram(1.4) C ( K,
3) C ( R d , K, × C ( K, × C ( K,
2) C ( R d , × C ( R d , × C ( R d , f p K P R d ( f ) Therefore, given a Σ -equivariant map f : C ( K, −→ C ( R d , R d , K,
3) C ( K, × C ( R d , × p R d p K ( f ) There exists a cohomological obstruction to the existence of a Σ -equivariant dashed arrowthat makes the diagram commute up to homotopy. We denote this obstruction by O ( K ). Itturns out to be an element of the equivariant cohomology group H d − (C ( K, Z [( − d − ]).Details can be found in Section 2. MBEDDING OBSTRUCTIONS IN R d Remark 1.1.
In terms of the tower T n Emb( K, R d ), O ( K ) is precisely the obstruction forthe path component of f in T Emb( K, R d ) to be in the image of the map T Emb( K, R d ) → T Emb( K, R d ). Remark 1.2.
The analogue of O ( K ) in the context of smooth embeddings was studied byMunson [23]. In fact, Munson did not consider the cohomological obstruction, but its lift toequivariant framed cobordism (a.k.a. stable cohomotopy). We will also investigate a lift ofthe cohomology class O ( K ) to a framed cobordism class O fr3 ( K ) in Sections 2 and 3.Focusing on the case of simplicial 2-complexes in R , geometrically (as we recall in Section 4)the vanishing of the van Kampen obstruction implies that a general position map f : K −→ R may be found such that for any two non-adjacent 2-simplices σ i , σ j of K , the algebraicintersection number f ( σ i ) · f ( σ j ) is zero. In higher dimensions in this setup the Whitneytrick enables one to find an actual embedding, cf. [12, Theorem 3]. In dimension 4 one maystill consider Whitney disks W ij pairing up the intersections points f ( σ i ) ∩ f ( σ j ) but theWhitney disks themselves have self-intersections and intersect other 2-cells, see [13, Section1.4] and also Figure 2 in Section 4 below.Our geometric obstruction W ( K ) is an element of the equivariant cohomology group H (C s ( K, Z [( − s ( K, simplicial configuration space, that is K minus the simplicial diagonal consistingof products σ × σ × σ of simplices where two of them have a vertex in common. Theobstruction is defined on the cochain level by sending a 6-cell σ × σ × σ (where σ i is a2-simplex of K ) to the sum of intersection numbers W ij · f ( σ k ) over distinct indices i, j, k ;see Section 4.4 for details. Informally, the obstruction may be thought of as measuring thefailure of the Whitney trick in 4 dimensions. In the special case of disks in the 4-ball with aprescribed boundary – a link in the 3-sphere ∂D – the analogous invariant equals the Milnor¯ µ -invariant of a 3-component link, sometimes referred to as the triple linking number. For knots , a similar expression measuring self-intersections of a disk in D equals the Arf invarint,see Remark 4.6 and references therein.The obstruction W ( K ) depends on the map f : K −→ R and also on Whitney disks W ij . Infact, we show in Lemma 4.3 that a choice of Whitney disks determines a Σ -equivariant mapC s ( K, −→ C ( R , O ( K ), W ( K ) are in fact equal. The proofproceeds by localizing the problem, using subdivisions of the 2-complex K and splittingsof Whitney disks, and identifying the Whitehead product in the homotopy fiber of themap p R : C ( R d , −→ C ( R d , × in the notation of (1.5) using the Pontryagin-Thomconstruction; see Section 5 for details.Finally, a cohomological interpretation in terms of the Arnold relation is given in Section6. We use this interpretation to show that our obstruction is non-trivial for the examplesconstructed in [12]. In that reference 2-complexes were constructed that have trivial vanKampen’s obstruction but which do not admit an embedding into R . The proof of non-embeddability in [12] is group-theoretic in nature (using the Stallings theorem) and is quitedifferent from the methods of this paper. Our work provides a general obstruction theory and GREGORY ARONE AND VYACHESLAV KRUSHKAL gives a conceptual homotopy-theoretic and geometric framework for analyzing the embeddingproblem in this dimension.The following is a brief outline of the structure of the paper. Section 2 starts with the discus-sion of van Kampen’s obstruction and its properties, and proceeds to define the obstruction O ( K ) which is the main focus of this paper. In the case when 3 dim( K ) = 2 d − O ( K ) to aan equivariant framed coborodism class O fr3 ( K ), which is defined in terms of a classifyingmap C ( R d , → (cid:98) Ω Ω ∞ Σ ∞ (cid:98) S d . The class O fr3 ( K ) is a complete obstruction to the liftingproblem (1.5) whenever dim( K ) + 2 ≤ d . An explicit construction of O fr3 ( K ) is given inSection 3; in particular it leads to a topological interpretation of O ( K ) in terms of the setof points satisfying a certain collinearity condition, see Section 3.1. Section 4 starts by recall-ing the geometric definition of van Kampen’s obstruction and basic operations on Whitneydisks in dimension 4. Lemma 4.3 establishes a relation between Whitney disks and mapsof configuration spaces, which illustrates a key connection between geometry and homotopytheory explored in this paper. Section 4.4 defines W ( K ) and analyzes its properties. Theconstruction of higher obstructions W n ( K ), in terms of intersection theory of Whitney towersof Schneiderman-Teichner, is outlined in Section 4.5. The main result of Section 5, Theorem5.1, relates the obstructions O ( K ) and W ( K ). Section 6 recalls the examples of [12] andshows that the obstruction O ( K ) detects their non-embeddability in R . In the process ofdoing this, O ( K ) is related to the Arnold class in Lemma 6.5. In fact, this point of viewprovides a uniform perspective on non-embeddability of 2-complexes in R from [12] and ofthe examples in other dimensions outside the metastable range from [29, 28], see Remark6.6. Section 7 gives the construction of the tower T n Emb( K, R n ), formulates the higher ob-structions O n ( K ), and discusses their properties including a conjectural framed cobordismlift. We conclude by stating a number of questions and conjectures motivated by our resultsin Section 8. Acknolwedgements . We would like to thank Danica Kosanovi´c, Rob Schneiderman andPeter Teichner for many discussions on the embedding calculus and Whitney towers.A substantial part of this project was completed while the authors visited EPFL, Lausanne,as part of the Bernoulli Brainstorm program in July 2019. We are grateful to the BernoulliCenter for warm hospitality and support.GA was supported in part by Swedish Research Council, grant number 2016-05440. VKwas supported in part by the Miller Institute for Basic Research in Science at UC Berkeley,Simons Foundation fellowship 608604, and NSF Grant DMS-1612159.2.
The obstruction
We begin this section by reviewing the classical van Kampen obstruction O ( K ) from ahomotopy-theoretic perspective. Then we introduce our main construction: a higher coho-mological obstruction O ( K ), defined when O ( K ) = 0, and depending on a choice of aΣ -equivariant map f : C ( K, → C ( R d , O fr2 ( K )and O fr3 ( K ) of O ( K ) and O ( K ) respectively into classes that reside in framed cobordismrather than cohomology. We give a homotopy-theoretic description of O ( K ) as a map intoan Eilenberg-Mac Lane space, and a geometric interpretation as a framed cobordism class. MBEDDING OBSTRUCTIONS IN R d Let K continue denoting an m -dimensional CW (or simplicial) complex. We are interestedin the question whether there exists a topological (or PL) embedding of K in R d . As we sawin the introduction, a necessary condition for the existence of an embedding, is the existenceof a Σ -equivariant map f : C ( K, → C ( R , -equivariant mapC ( K, → (cid:101) S d − , where (cid:101) S d − denotes the sphere with the antipodal action of Σ . Recallthat there is a Σ -equivariant homotopy equivalence C ( R d , (cid:39) −→ (cid:101) S d − that sends ( x , x ) to x − x | x − x | . We will occasionally switch back and forth between these spaces.There is a well-known homotopical/cohomological obstruction to the existence of a Σ -equivariant map f : C ( K, → C ( R d , (cid:98) R d be the d -dimensional sign representation of Σ . Let (cid:98) S d be the one-point compactification of (cid:98) R d ,considered as a space with an action of Σ . Equivalently, (cid:98) S d is the unreduced suspension of (cid:101) S d − . Note that (cid:98) S d has two points fixed by Σ , corresponding to 0 and ∞ in the compact-ificaton of (cid:98) R d . By convention, ∞ is the basepoint of (cid:98) S d . The following elementary lemmagives several conditions for the existence of a Σ -map K × K \ K → (cid:101) S d − . Lemma 2.1.
Conditions (1) and (2) below are equivalent(1) There exists a Σ -equivariant map K × K \ K → (cid:101) S d − .(2) The vector bundle (2.1) ( K × K \ K ) × Σ (cid:98) R d → ( K × K \ K ) Σ has a nowhere vanishing section.Furthermore, conditions (1) and (2) above imply conditions (3) and (4) below. Under theassumption d ≥ dim( K ) + 2 , the conditions (1)-(4) are equivalent.(3) The constant map K × K \ K → (cid:98) S d that sends K × K \ K to zero is Σ -equivariantlynull-homotopic.(4) The constant map K × K \ K → Ω ∞ Σ ∞ (cid:98) S d which is the map of part 3 followed bythe suspension map (cid:98) S d → Ω ∞ Σ ∞ (cid:98) S d is Σ -equivariantly null-homotopic.Proof. The vector bundle (2.1) has a nowhere vanishing section if and only if the spherebundle ( K × K \ K ) × Σ (cid:101) S d − → ( K × K \ K ) Σ has a section. It is well-known that sections of this bundle are in bijective correspondencewith Σ -equivariant maps K × K \ K → (cid:101) S d − , which is why (1) and (2) are equivalent.Suppose there is a Σ -equivariant map K × K \ K → (cid:101) S d − . It induces Σ -equivariant maps( K × K \ K ) × I → (cid:101) S d − × I → (cid:98) S d where the latter map is the obvious quotient. This composite map is a null homotopy of theconstant zero map K × K \ K → (cid:98) S d . This is why (1) implies (3). It is obvious that (3)implies (4).For the reverse implication in the last statement of the lemma, let (cid:101) Ω (cid:98) S d be the space of pathsin (cid:98) S d from the basepoint ∞ to 0. There is a canonical Σ -equivariant map (cid:101) S d − → (cid:101) Ω (cid:98) S d . It GREGORY ARONE AND VYACHESLAV KRUSHKAL follows from the Blakers-Massey theorem that this map is 2 d − K × K \ K, (cid:101) S d − ) Σ → map( K × K \ K, (cid:101) Ω (cid:98) S d ) Σ is 2 d − K ) − d − dim( K ) ≥ π . But a Σ -equivariant map K × K \ K → (cid:101) Ω (cid:98) S d is thesame thing as a Σ -equivariant null homotopy of the constant zero map from K × K \ K to (cid:98) S d . Thus, under the assumption d ≥ dim( K ) + 2, condition (3) implies (1).Finally, the map (cid:98) S d → Ω ∞ Σ ∞ (cid:98) S d is 2 d − d ≥ dim( K ) + 1, which is a weaker condition than statedin the lemma. (cid:3) Lemma 2.1 points to several (equivalent) ways to define a cohomological obstruction to theexistence of a Σ -equivariant map K × K \ K → (cid:101) S d − . To begin with, the map given inpart (4) of the lemma can be interpreted as an element of an equivariant stable cohomotopygroup, or equivalently an equivariant framed cobordism group of K × K \ K . We denote thiselement by O fr2 ( K ). Lemma 2.1 says that O fr2 ( K ) is a complete obstruction to the existenceof a Σ -equivariant map K × K → (cid:101) S d − when dim( K ) + 2 ≤ d .The natural map of spectra Σ ∞ S → H Z induces a Σ -equivariant map(2.2) Ω ∞ Σ ∞ (cid:98) S d → Ω ∞ H Z ∧ (cid:98) S d (cid:39) K ( Z [( − d ] , d ) . Here K ( Z [( − d ] , d ) denotes the Eilenberg-Mac Lane space with an action of Σ , that onthe non-trivial homotopy group realizes the representation Z [( − d ], which is the trivialrepresentation if d is even and the sign representation if d is odd. Any two such Eilenberg-Mac Lane spaces are weakly equivariantly equivalent.Composing the maps in Lemma 2.1(4) and (2.2), we obtain a Σ -equivariant map K × K \ K → K ( Z [( − d ] , d ) . This map defines an element in the equivariant cohomology group O ( K ) ∈ H d Σ ( K × K \ K ; Z [( − d ]). This is the classical van Kampen obstruction. It is the same as the Euler classof the vector bundle (2.1). The classical van Kampen obstruction is a complete obstructionto the existence of a Σ -equivariant map K × K \ K → (cid:101) S d − when d = 2 dim( K ). Weare going to focus on the case when 4 = d = 2 dim( K ) = dim( K ) + 2. In this case, thecohomological obstruction is a complete obstruction to the existence of an equivariant map(but not to the existence of an embedding K (cid:44) → R d ), and using the framed cobordism versiondoes not add information. But in other situations O fr2 ( K ) contains more information than O ( K ). Remark 2.2.
The framed cobordism viewpoint points to a geometric interpretation of thevan Kampen obstruction. It is perhaps even more convincing in the context of smoothmanifolds. In that context, the analogue of the van Kampen obstruction is the obstructionfor lifting from the first to the second stage of the Goodwillie-Weiss tower. In other words,it is the first obstruction to an immersion of a smooth manifold M into R d being regularlyhomotopic to an embedding. This obstruction is an element in the relative equivariantcobordism group Ω (cid:98) R d fr ( M × M, M ), and it can be interpreted as the framed cobordism classof the double points manifold of an immersion. This is explained, for example, in the
MBEDDING OBSTRUCTIONS IN R d introduction to [23]. In the case of topological embeddings of a 2-dimensional complex in R , the van Kampen obstruction also can be interepreted as a double points obstruction. Ofcourse this interpretation is well-known, and indeed it was how van Kampen thought aboutit. We review this in Section 4.1.Now let us consider the next step. Suppose we have a finite complex K for which O ( K ) (or O fr2 ( K )) vanishes, and suppose we choose a Σ -equivariant map f : C ( K, → C ( R d , f is Σ -equivariantly homotopic to the deleted square of some embedding f : K (cid:44) → R d .Suppose W is a space with an action of Σ . Then we endow the space W × W × W withan action of Σ via the homeomorphism W × W × W ∼ = map Σ (Σ , W ). In particular, thespaces C ( X, (for any space X ) and ( (cid:101) S d − ) are equipped with a natural action of Σ inthis way.For any space X , a Σ -equivariant map C ( X, → C ( X, is the same thing as a Σ -equivariant map C ( X, → C ( X, ⊂ Σ is identified with the subgroup per-muting 1 ,
2. There is an obvious Σ -equivariant projection map C ( X, → C ( X,
2) whichsends ( x , x , x ) to ( x , x ). This map induces a canonical Σ -equivariant map(2.3) p X : C ( X, → C ( X, × C ( X, × C ( X, x , x , x ) (cid:55)→ ( x , x ) , ( x , x ) , ( x , x )This map is natural with respect to embeddings of X . Therefore, an embedding f : K (cid:44) → R d induces a commutative square as we saw in the introduction (1.4). Conversely, if f : C ( K, → C ( R d ,
2) is a Σ -equivariant map, then a necessary condition for f to beequivarintly homotopic to the deleted square of an embedding is that the lifting problem inthe following diagram has a Σ -equivariant solution(2.4) C ( R d , K,
3) C ( R d , × p R d ( f ) ◦ p K At this point we want to bring obstruction theory into play. For this, we need to examinethe map p R d : C ( R d , → C ( R d , a little more closely.Let us recall some well-known facts about the effect of this map on cohomology. We shallintroduce the following slight refinement of the notation for configurations spaces:(2.5) C ( R d , { i, j } ) := Emb( { i, j } , R d ) . Next, let us give names to some cohomology classes. Let u ∈ H d − (C ( R d , { , } )) bea generator. More generally, for any two points i, j let u ij ∈ H d − (C ( R d , { i, j } )) be thegenerator corresponding to u under homeomorphism induced by the bijection { , } ∼ = { i, j } . Definition 2.3.
The
Arnold class is the following cohomological element. u ⊗ u ⊗ ⊗ u ⊗ u +( − d − u ⊗ ⊗ u ∈ H d − (C ( R d , { , } ) × C ( R d , { , } ) × C ( R d , { , } )) . The following lemma is well-known
Lemma 2.4. p R d : C ( R d , → C ( R d , is surjective in cohomology, and its kernel in cohomology is the ideal generated by the Arnoldclass. We refer to the statement of this lemma as the Arnold relation. The original reference is [1],where it is proved for configuration spaces in R . The general result is proved in [8, Lemma1.3 and Proposition 1.4]). The following corollary is an an easy consequence of the lemma,and is also well-known Corollary 2.5.
The map p R d is d − -connected, and moreover it induces an isomorphismin homology and cohomology in degrees up to an including d − . In degree d − there isan isomorphism of abelian groups H d − (C ( R d , ∼ = Z and an isomorphism of Σ -modules H d − (C ( R d , ) ∼ = Z [Σ ] ⊗ Z [Σ ] Z [( − k − ] . Moreover, the homomorphism in H d − induced by p R d fits in a short exact sequence of Σ -modules → H d − (C ( R d , → H d − (C ( R d , ) → Z [( − d − ] → where the second homomorphism can be identified with the canonical surjection of Σ -modules Z [Σ ] ⊗ Z [Σ ] Z [( − k − ] → Z [( − k − ] . It is worth noticing that the short exact sequence splits, but not Σ -equivariantly.Let F d be the homotopy fiber of the map p R d . It follows from the corollary that the firstnon-trivial homotopy group of F is π d − ( F ) ∼ = Z [( − d − ]. The following result is a straight-forward application of equivariant obstruction theory. Proposition 2.6.
The first obstruction to the lifting problem in figure (2.4) is an elementof the equivariant cohomology group O ( K ) ∈ H d − (C ( K, Z [( − d − ]) . This obstructionis complete if K ) = 2 d − O ( K ) is a complete obstruction to the lifting problem in (2.4)if dim( K ) = 2 and d = 4.One common way to think of cohomological obstruction to lifting a map is via the Postnikovtower. Now that we are looking at spaces with an action of Σ , let K ( Z [( − d ] , d −
2) denotean Eilenberg-Mac Lane space with an action of Σ that acts by Z [( − d ] on the non-trivialhomotopy group. Lemma 2.8 below is, again, an easy consequence of Corollary 2.5. Beforestating the lemma, let us review the definition of a k -(co)cartesian square diagram. Definition 2.7.
Suppose that we have a commutative diagram X X X X One says that the diagram is k -cartesian if the induced map from X to the homotopypullback of X → X ← X MBEDDING OBSTRUCTIONS IN R d is k -connected. Dually, the diagram is k -cocartesian if the induced map from the homotopypushout X ← X → X to X is k -connected.Notice that if, say, X (cid:39) ∗ then k -cartesian means that the map from X to the homotopyfiber of the map X → X is k -connected. Lemma 2.8.
For a suitable model of K ( Z [( − d ] , d − , there is a Σ -equivariant map C ( R d , → K ( Z [( − d − ] , d − , such that the composite map C ( R d , p R d −−→ C ( R d , → K ( Z [( − d − ] , d − is equivariantly null-homotopic, and the following diagram is d − -cartesian (2.6) C ( R d ,
3) C ( R d , ∗ K ( Z [( − d − ] , d − . In terms of this lemma, the obstruction class O ( K ) is represented by the following compo-sition(2.7) C ( K, p K −→ C ( K, f −→ C ( R d , → K ( Z [( − d − ] , d − . We saw earlier that the classical, cohomological van Kampen obstruction has a naturallift to a potentially stronger obstruction that lives in equivariant stable cohomotopy, a.k.aequivariant framed cobordism. The obstruction O ( K ) has a similar lift, which we denote O fr3 ( K ).Till the end of this section, and in the next section, we focus on spaces with an action of Σ and no other symmetric groups. Until the end of next section, let (cid:98) R be the reduced standardrepresentation of Σ , let (cid:98) R d = (cid:98) R ⊗ R d , and let (cid:98) S d be the one-point compactification of (cid:98) R d . As a space, (cid:98) S d is simply the 2 d -dimensional sphere. The ‘hat’ is there to indicate thatit is a space with a specific action of Σ . In the same vein, let (cid:98) Ω (cid:98) S d = map ∗ ( (cid:98) S , (cid:98) S d ) be thedouble loop space Ω S d , on which Σ acts via both S and S d . Similarly define the spacewith Σ -action (cid:98) Ω Ω ∞ Σ ∞ (cid:98) S d . The following proposition is a refinement of Lemma 2.8 Proposition 2.9.
There is a d − -cartesian diagram of spaces with an action of Σ (2.8) C ( R d ,
3) C ( R d , ∗ (cid:98) Ω Ω ∞ Σ ∞ (cid:98) S d . We will prove this proposition in the next section. For the rest of the section, we con-sider some consequences. It follows from the proposition that given a Σ -equivariant map f : C ( K, → C ( R d , f to be equivariantly homotopic to the deleted square of someembedding) is that the following composition is Σ -equivariantly null-homotopic (comparewith (2.7)): C ( K, p K −→ C ( K, f −→ C ( R d , → (cid:98) Ω Ω ∞ Σ ∞ (cid:98) S d . We interpret this composition as an element in the equivariant stable cohomotopy of C ( K, O fr3 ( K ) ∈ Ω (cid:98) R d − fr (C ( K, O fr3 ( K ) is an obstruction to a solution of the lifting problem (2.4). O fr3 ( K ) is a refinementof O ( K ) in the same way as O fr2 ( K ) is a refinement of O ( K ). O fr3 ( K ) is a completeobstruction to the lifting problem if 3 dim( K ) ≤ d −
5, while O ( K ) is a complete obstructionif 3 dim( K ) ≤ d −
2. Of course when dim( K ) = 2 and d = 4 both conditions hold, and O fr3 ( K ) does not really provide more information than O ( K ). Remark 2.10.
The obstruction O fr3 ( M ) in the context of smooth embeddings is the subjectof Munson’s paper [23]. In particular, Proposition 2.9 is proved there. We give a differentproof in the next section. As a result, we will give a geometric interpretation of O fr3 ( K ) thatis hinted at in [op. cit.].3. Construction of a classifying map
In this section we prove Proposition 2.9. That is, we will construct a Σ -equivariant mapC ( R d , → (cid:98) Ω Ω ∞ Σ ∞ (cid:98) S d that makes the square 2.8 3 d − -equivariant map (recall that (cid:101) S d − is Σ -equivariantly equivalent to C ( R d , f : ( (cid:101) S d − ) → (cid:98) Ω (cid:98) S d such that the following composition is Σ -equivariantly null-homotopicC ( R , → ( (cid:101) S d − ) f −→ (cid:98) Ω (cid:98) S d and moreover the following square diagram is 3 d − R d ,
3) ( (cid:101) S d − ) ∗ (cid:98) Ω (cid:98) S df . We call a map f with these properties a classifying map . Since there is a natural map (cid:98) Ω (cid:98) S d → (cid:98) Ω Ω ∞ Σ ∞ (cid:98) S d that is 4 d − d − d − MBEDDING OBSTRUCTIONS IN R d Lemma 3.1.
Suppose that we have a Σ -equivariant map f : ( (cid:101) S d − ) → (cid:98) Ω (cid:98) S d satisfying the following conditions:(1) The composite map (3.2) C ( R d , → ( (cid:101) S d − ) → (cid:98) Ω (cid:98) S d is equivariantly null-homotopic.(2) f induces an epimorphism on H d − (or, equivalently, a monomorphism on H d − ).Then f is a classifying map.Proof. By Lemma 2.4 and Corollary 2.5, the homology of the space C ( R d ,
3) is concentratedin degrees 0 , d − , d − (cid:101) S d − ) is concentrated in degrees i ( d − i ≤
3. The map C ( R d , → ( (cid:101) S d − ) induces an isomorphism on H d − and a monomorphism on H d − . The cokernel of this mapin H d − is isomorphic to Z , which is also isomorphic to H d − (Ω S d ). Our assumptionimplies that the homomorphism from the cokernel of f in H d − to H d − (Ω S d ) is anepimorphism from Z to Z . Therefore it is an isomorphism. Since all the spaces in thediagram 3.1 have trivial homology in dimension above 2( d −
1) and below 3( d − d − R d ,
3) to ( (cid:101) S d − ) and to ∗ are 2 d − d − d − (cid:3) Now we are ready to construct a classifying map. We will use the Thom-Pontryagin collapsemap associated with the diagonal inclusion (cid:101) S d − (cid:44) → ( (cid:101) S d − ) . To get a clean description of theΣ -equivariant properties of this collapse map, let us first consider a more general setting,where M is a manifold with a free action of Σ . The action of Σ can be extended to anaction of Σ via the surjective homomorphism Σ (cid:16) Σ . In this way, we consider M as aspace with an action of Σ .The group Σ acts on M via either one of the identifications M ∼ = map Σ (Σ , M ) ∼ = map(Σ / Σ , M ) . The diagonal inclusion ∆ :
M (cid:44) → M is a Σ -equivariant map (note again that the action ofΣ on M is not trivial). The normal bundle of this inclusion has an induced action of Σ .The normal bundle is Σ -equivariantly isomorphic to the quotient bundle 3 τ / ∆( τ ). Here τ is the tangent bundle of M , 3 τ = τ ⊕ τ ⊕ τ , and ∆( τ ) is the diagonal copy of τ in 3 τ . Wedenote the normal bundle by (cid:99) τ . It is the tensor product of τ with (cid:98) R . Let M (cid:98) τ denote theThom space of the normal bundle. The Thom-Pontryagin collapse map associated with ∆is a Σ -equivariant map M → M (cid:98) τ .Now apply this to the case M = (cid:101) S d − , the ( d − . The Thom-Pontryagin collapse map has the form( (cid:101) S d − ) → ( (cid:101) S d − ) (cid:98) τ Note that this is an unpointed map, as the space ( (cid:101) S d − ) does not have an equivariantbasepoint. Sometimes we like to think of the collapse map as a pointed map( (cid:101) S d − ) → ( (cid:101) S d − ) (cid:98) τ Let us take smash product of this map with (cid:98) S , to obtain the following Σ -equivariant map( (cid:101) S d − ) ∧ (cid:98) S → ( (cid:101) S d − ) (cid:98) τ ∧ (cid:98) S . Now observe that there is a homeomorphism( (cid:101) S d − ) (cid:98) τ ∧ (cid:98) S ∼ = ( (cid:101) S d − ) (cid:98) τ ⊕ R ) . Next, recall that there is an isomorphism τ ⊕ R ∼ = R d . It follows that there is a Σ -equivarianthomeomorphism ( (cid:101) S d − ) (cid:98) τ ⊕ R ) ∼ = ( (cid:101) S d − ) (cid:98) R d ) ∼ = (cid:101) S d − ∧ (cid:98) S d . Next we compose this homeomorphism with the collapse map (cid:101) S d − ∧ (cid:98) S d → (cid:98) S d , and pre-compose with the (suspended) Thom-Pontryagin collapse map above. We obtain the map( (cid:101) S d − ) ∧ (cid:98) S → (cid:98) S d . Taking an adjoint, we obtain an unpointed Σ -equivariant map(3.3) ( (cid:101) S d − ) → (cid:98) Ω (cid:98) S d . This is our model for a classifying map.
Lemma 3.2.
The map (3.3) is a classifying map.Proof.
We need to check that the map satisfies the hypotheses of Lemma 3.1. The firsthypothesis is that the composite mapC ( R d , → ( (cid:101) S d − ) → (cid:98) Ω (cid:98) S d is equivariantly null homotopic. By construction, the second map factors through the Thom-Pontryagin collapse map associated with the inclusion of the thin diagonal of ( (cid:101) S d − ) . Clearlythe space C ( R d , (cid:101) S d − ) , is contained inthe complement of the thin diagonal, and therefore the restriction of the Thom-Pontryagincollapse to C ( R d ,
3) is (equivariantly) null homotopic.The second hypothesis that we need to check is that the following homomorphism is anepimorphism H d − (( (cid:101) S d − ) ) → H d − ( (cid:98) Ω (cid:98) S d )This is equivalent to showing that the adjoint map S ∧ ( S d − × S d − × S d − ) + → S d Induces an epimorphism on H d (till the end of this proof we will omit the ‘tilde’ and ‘hat’decorations, since we are not concerned with the action of Σ at this point). Once again werecall that this map factors through the Thom-Pontryagin collapse as follows S ∧ ( S d − × S d − × S d − ) + → S ∧ ( S d − ) τ ∼ = −→ S d − ∧ S d → S d . We need to prove that this composite map induces an epimorphism on H d . To see this,choose a point ∗ ∈ S d − and consider the inclusion S d − × S d − × {∗} (cid:44) → ( S d − ) . This MBEDDING OBSTRUCTIONS IN R d inclusion intersects the thin diagonal transversely at a single point ( ∗ , ∗ , ∗ ) ∈ ( S d − ) . Itfollows quite easily that the composite map S ∧ S d − × S d − × {∗} + → S ∧ ( S d − × S d − × S d − ) + → S d is the double suspension of the Thom-Pontryagin collapse map associated with the inclusionof a point ( ∗ , ∗ ) (cid:44) → S d − × S d − . In other words, it is the double suspension of the map S d − × S d − → S d − that collapses the complement of a Euclidean neighborhood of ( ∗ , ∗ ).Clearly this map is surjective on H d , and therefore the map S ∧ ( S d − × S d − × S d − ) + → S d is also surjective on H d . (cid:3) A geometric interpretation.
Lemma 3.2 leads to a kind of a geometric interpretationof our obstruction, at least when K is a manifold, or if 2 dim( K ) = d = 4, which is the case weare focusing on. In the manifold case, such an interpretation was hinted at by Munson [23].So suppose K is a 2-dimensional complex and we have a Σ -equivariant map f : K × K \ K → (cid:101) S . Consider the set { ( k , k , k ) ∈ C ( K, | f ( k , k ) = f ( k , k ) = f ( k , k ) } . This set is the preimage of the diagonal under the mapC ( K, f ◦ p K −−−→ (cid:101) S × (cid:101) S × (cid:101) S . Under a transversality assumption, this set defines a an element of H (C ( K, , Z [ − O ( K ) (or even O fr3 ( K )).To see why this is plausible, suppose that f is a normalized deleted square of some embedding f : K (cid:44) → R . I.e., suppose that f ( k , k ) = f ( k ) − f ( k ) | f ( k ) − f ( k ) | Then for all k , k , k , the three vectors f ( k ) − f ( k ) , f ( k ) − f ( k ) , f ( k ) − f ( k ) sum upto zero, while our obstruction consists of triples ( k , k , k ) where these three vectors wouldbe co-directed.(It is interesting to compare this with the interpretation of the second coefficient of theConway polynomial of a knot in terms of collinear triples in [6].)4. Embedding obstructions from Whitney towers
This section starts by reviewing a geometric formulation of van Kampen’s obstruction (Sec-tion 4.1) and operations on Whitney disks (Section 4.2) which are commonly used in 4-manifold topology. These techniques are then used to establish new results: a relationbetween Whitney disks and equivariant maps of configuration spaces (Section 4.3) andhigher embedding obstructions for 2-complexes in R based on intersections of Whitneydisks: W ( K ) in Section 4.4 and W n ( K ) , n > W ( K ) to theobstruction O ( K ) defined above is the subject of Section 5. The van Kampen obstruction.
The discussion in the paper so far concerned thegeneral embedding problem for m -complexes in R d . Here we restrict to the original vanKampen’s context where d = 2 m . Later in this section we will specialize further to m = 2.We start by recalling a geometric description of the van Kampen obstruction(4.1) O s2 ( K ) ∈ H m Σ (C s ( K, Z )to embeddability of an m -complex K into R m . This was the construction outlined by vanKampen in [32]; the details were clarified in [30, 36], see also [12]. As in the introductionthe notation C s ( K,
2) denotes the “simplicial” configuration space K × K (cid:114) ∆ where ∆consisting of all products of simplices σ × σ having a vertex in common. The group Σ acts on the configuration space K × K (cid:114) ∆ by exchanging the factors; it may be seen fromthe description below that the action of Σ on the coefficients is trivial, cf. [30, 22] (notethat the sign was misstated as ( − m in [12].)The superscript in the notation O s2 ( K ) in (4.1) stands for “simplicial”: it is an element of thecohomology group of C s ( K, O ( K ) (considered in the introduction and in Section2) is an element of the cohomology group of the configuration space C ( K,
2) defined usingthe point-set diagonal. The invariant O ( K ) is the “universal” van Kampen obstruction,independent of the simplicial structure, and O s2 ( K ) may be recovered from it: O s2 ( K ) = i ∗ O ( K ), where i is the inclusion map C s ( K, ⊂ C ( K, O s2 ( K ) could be a weakerinvariant since it does not keep track of intersections of adjacent simplices. Nevertheless, itis a complete embedding obstruction for m -complexes in R m for m >
2: intersections ofadjacent simplices may be removed using a version of the Whitney trick, cf. [12, Lemma 5].
Remark 4.1.
The obstruction theory in Section 2 was developed for embeddings of finiteCW complexes. The geometric approach presented here is based on intersection theory andit applies to finite simplicial complexes. We will interchangeably use the terms cells and simplices in the context of simplicial complexes; this should not cause confusion.Consider any general position map f : K −→ R m . Endow the m -cells of K with arbi-trary orientations, and for any two m -cells σ , σ without vertices in common, consider thealgebraic intersection number f ( σ ) · f ( σ ) ∈ Z . This gives a Σ -equivariant cochain(4.2) o f : C m ( K × K (cid:114) ∆) −→ Z . Since this is a top-dimensional cochain, it is a cocycle. Its cohomology class equals the vanKampen obstruction O s2 ( K ).The fact that this cohomology class is independent of a choice of f may be seen geometricallyas follows (see [12, Lemma 1, Section 2.4] for more details). Any two general position maps f , f : K −→ R m are connected by a 1-parameter family of maps f t where at a non-generictime t i an m -cell σ intersects an ( m − ν . Topologically the maps f t i − (cid:15) and f t i + (cid:15) differby a “finger move”, that is tubing σ into a small m -sphere linking ν in R m , Figure 1. Theeffect of this elementary homotopy on the van Kampen cochain is precisely the addition ofthe coboundary δ ( u σ,ν ), where u σ,ν is the Σ -equivaraint “elementary (2 m − m − σ × ν, ν × σ .This argument has the following corollary. MBEDDING OBSTRUCTIONS IN R d νσ Figure 1.
Finger move: homotopy of maps f : K −→ R m Lemma 4.2.
Any cocycle representative of the cohomology class O s2 ( K ) ∈ H m Z / ( K × K (cid:114) ∆; Z ) may be realized as the cocycle o f for some general position map f : K −→ R m . Inparticular, if the van Kampen obstruction O s2 ( K ) vanishes then there exists a general positionmap f : K −→ R m such that the cocycle o f is identically zero. In other words, in this casefor any two non-adjacent -cells σ , τ the algebraic intersection number f ( σ ) · f ( τ ) is zero. Operations on Whitney disks.
The rest of Section 4 concerns 2-complexes in R .Assume the van Kampen class O s2 ( K ) vanishes. By Lemma 4.2, using finger moves on 2-cellsas shown in Figure 1, a map f may be chosen so that f ( σ i ) · f ( σ j ) = 0 for any non-adjacent2-cells σ i , σ j . As usual, one groups intersection points f ( σ i ) ∩ f ( σ j ) into canceling pairs,chooses Whitney arcs connecting them in σ i , σ j , and considers Whitney disks W ij for theseintersections. Note that all Whitney arcs in each 2-cell may be assumed to be pairwisedisjoint. Unlike the situation in higher dimensions where by general position a Whitney diskmay be assumed to be embedded and to have interior disjoint from K , in 4-space generically W ij will have self-intersections and also intersect the 2-cells of K . Moreover, the framing(the relative Euler number of the normal bundle of the Whitney disk) might be non-zero,but it may be corrected by boundary twisting [13, Section 1.3]. A detailed discussion ofWhitney disks in this dimension is given in [13, Section 1.4]. This section summarizes theoperations on Whitney disks and their relation with capped surfaces which will be used inthe proofs in Section 5. σ i σ j σ k W ij σ i σ j σ k C (cid:48) C (cid:48)(cid:48) Figure 2.
A Whitney disk and the associated capped surface
Convention.
To avoid cumbersome notation, we will frequently omit the reference to amap f and keep the notation σ for the image of a cell σ under f .A typical configuration is shown on the left in Figure 2. It is a usual representation in3-space R × { } (the ‘present’) of intersecting surfaces in R = R × R where the R factor isthought of as time. Here σ j is pictured as a surface in R while σ i , σ k are arcs which extendas the product (arc × I ) into the past and the future. The Whitney disk W ij ⊂ R × { } pairs up two generic intersection points σ i ∩ σ j of opposite signs, and W ij in the figure hasa generic intersection point with another 2-cell σ k . The result of the Whitney move in thissetting is shown in Figure 3: the two intersection points σ j ∩ σ i are eliminated, but twonew intersection points σ j ∩ σ k are created instead. In fact, the picture is symmetric withrespect to the three sheets σ i , σ j , σ k : a neighborhood of the Whitney disk W ij in R is a4-ball D , and the intersection of these three sheets with the boundary 3-sphere ∂D formsthe Borromean rings, as shown in Figure 9. Thus any two of the sheets can be arranged tobe disjoint in this 4-ball, but not all three simultaneously. σ i σ j σ k Figure 3.
The result of the Whitney moveIt will be convenient to view these intersections in the context of capped surfaces (or moregenerally capped gropes for higher-order intersections) [13, Chapter 2]. This is shown on theright in Figure 2: a tube is added to one of the two sheets, say σ j as shown in the figure, toeliminate the two intersections σ i ∩ σ j at the cost of adding genus to σ j . The new surface,still denoted σ j , has two caps : disks attached to a symplectic pair of curves on σ j . One ofthe caps, C (cid:48) , is obtained from the Whitney disk W ij . The other cap is a disk normal to σ i and may be thought of as a fiber of the normal bundle to σ i . A general translation betweenWhitney towers and capped gropes is discussed in [24]. An advantage of this point of view isthe symmetry between the original map of σ j (intersecting σ i in two points, as shown on theleft in the figure) and the result of the Whitney move where the two intersections σ i ∩ σ j areeliminated but σ j acquires two intersections with σ k . The first case is obtained by ambientsurgery of the capped surface in the figure on the right along the cap C (cid:48)(cid:48) , and the second caseis the surgery along C (cid:48) . There is an intermediate operation, symmetric surgery (also knownas contraction ) [13, Section 2.3] that uses both caps that will be used in the arguments inthe next section. The disk obtained by surgery on C (cid:48) is isotopic to the surgery on C (cid:48)(cid:48) , andthe symmetric surgery may be thought of as the half point of the isotopy.Consider the following splitting operation on Whitney disks. Suppose a Whitney disk W ij pairing up intersections between σ i , σ j intersects two other 2-cells, σ k , σ l as shown on theleft in Figure 4. Consider an arc in W ij (drawn dashed in the figure) which separates theintersections W ij ∩ σ k , W ij ∩ σ l and whose two endpoints are in the interiors of the twoWhitney arcs forming the boundary of W ij . Then a finger move on one of the sheets, say σ i , along the arc introduces two new points of intersection σ i ∩ σ j and splits W ij into twoWhitney disks W (cid:48) ij , W (cid:48)(cid:48) ij as shown in the figure on the right. The advantage of the result isthat each Whitney disk intersects only one other 2-cell. In general, if W ij had m intersectionpoints with other 2-cells, an iterated application of splitting yields m − MBEDDING OBSTRUCTIONS IN R d σ i σ j σ k W ij σ l σ i σ j σ k σ l W (cid:48)(cid:48) ij W (cid:48) ij Figure 4.
Splitting of a Whitney diskThe discussion above referred to the situation where a Whitney disk W ij for σ i ∩ σ j intersects2-cells which are not adjacent to σ i , σ j . In general, W ij will have self-intersections as wellas intersections with σ i , σ j and with 2-cells adjacent to them. Intersections of these typesare not considered in the formulation of the obstruction in Section 4.4. (An obstructioninvolving these more subtle intersections will be explored in a future work. For example,the Arf invariant of a knot in S may be defined using intersections of this type of the diskbounded by the knot in the 4-ball, see Remark 4.6.)An ingredient in the formulation of higher obstructions in Section 4.4 is a local move onsurfaces which replaces an intersection σ k ∩ W ij in Figure 2 with an intersection σ i ∩ W jk or σ j ∩ W ik .To describe this operation in more detail, start with the model situation in Figure 2 where W ij has a single intersection point with σ k . Perform a finger move on σ k along an arc from σ k ∩ W ij to a point on the Whitney arc in σ j . The result is shown on the left in Figure 5: now σ k is disjoint from W ij but there are two new intersections between σ j and σ k . The fingermove isotopy of σ k gives rise to a Whitney disk for these two points, denoted W (cid:48) jk in thefigure. Note however that the two Whitney disks W ij , W (cid:48) jk cannot be both used for Whitneymoves since their boundary arcs intersect in σ j . Resolving this intersection by an isotopy ofthe Whitney arc in the boundary of W (cid:48) jk yields a Whitney disk W jk on the right in Figure5; this Whitney disk has a single intersection point with σ i . (Note that after this operationthe Whitney disk W ij is embedded and disjoint from other 2-cells; a Whitney move alongthis disk can be used to eliminate the original two intersections σ i ∩ σ j .)Therefore to have a well-defined triple intersection number one has to (1) sum over Whitneydisks over all pairs of indices, and (2) require that Whitney arcs are disjoint, see Section 4.4. σ i σ j σ k W (cid:48) jk W ij σ i σ j σ k W jk Figure 5.
From σ k ∩ W ij to σ i ∩ W jk . From Whitney disks to equivariant maps of configuration spaces.
Let K bea 2-complex and suppose the van Kampen obstruction O s2 ( K ) vanishes. Then by Lemma4.2 there is a map f : K −→ R so that the algebraic intersection number of any two non-adjacent 2-cells in R is zero. As in Section 4.2, pair up the intersections with Whitney disks,so that all Whitney arcs are disjoint in each 2-cell. This condition on the Whitney arcs willbe assumed throughout the rest of the paper. The following lemma shows that f togetherwith a choice of Whitney disks W gives rise to a Σ -equivariant map C s ( K, −→ C ( R d , -equivariant map of the 5-skeleton of C s ( K,
3) to C ( R d , split , so that any Whitney disk has at mostone intersection with a 2-cell of K . Lemma 4.3.
Let K be a -complex and f : K −→ R a general position map such that allintersections of non-adjacent -cells are paired up with split Whitney disks W . This datadetermines a Σ -equivariant map F f,W : C s ( K, −→ C ( R d , .Proof . Given any pair of non-adjacent 2-cells σ i , σ j , by assumption all intersections f ( σ i ) ∩ f ( σ j ) are paired up with Whitney disks W ij , and the Whitney arcs in each 2-cell are disjoint.The self-intersections and intersections of the Whitney disks will not be relevant in thefollowing argument because the simplicial diagonal ∆ is missing in the configuration spaceC s ( K, W ij intersects a single 2-cell σ k as in Figure2. We treat the special case that σ k is either σ i or σ j right away: if W ij intersects σ i , performthe Whitney move along W ij on σ i ; if it intersects σ j then perform the Whitney move of σ j . This results in self-intersections of either f ( σ i ) or f ( σ j ) which are irrelevant since we areworking with the simplicial configuration space C s ( K, F f,W does not needto be defined on σ i × σ i , σ j × σ j . Thus the remaining intersections of W ij are with 2-cells σ k , k (cid:54) = i, j .Next we describe the desired map F f,W : C s ( K, −→ C ( R , K are mapped in disjointly by f , so f × f defines a Σ -equivariantmap on the 3-skeleton of C s ( K, F f,W on each product of two non-adjacent 2-cells σ i × σ j . For each such pair σ i , σ j we pick an order ( i, j ); for the other product σ j × σ i the map F f,W will be defined using Σ equivariance.In each 2-cell σ i consider disjoint disk neighborhoods of the Whitney arcs for the intersectionsof f ( σ i ) with other 2-cells; the disk neighborhoods corresponding to W ij are denoted D ij ,Figure 6. (In general W ij denotes the entire collection of Whitney disks for f ( σ i ) ∩ f ( σ j ), and D ij denotes the collection of corresponding disk neighborhoods; we illustrate the case of asingle component since the argument in general is directly analogous.) If f ( σ i ) ∩ f ( σ j ) = ∅ , D ij is defined to be empty. Now consider the map (cid:101) f ij : K −→ R which coincides with f in the complement of the disk D ij . In this disk (cid:101) f ij is defined to be the result of theWhitney move on f ( σ i ) along the Whitney disk W ij , making (cid:101) f ( σ i ) disjoint from (cid:101) f ( σ j ). If W ij intersected another 2-cell σ k as in Figure 2, as a result of this move (cid:101) f ij ( σ i ) intersects (cid:101) f ij ( σ k ) = f ( σ k ). MBEDDING OBSTRUCTIONS IN R d σ i D ij p q C ji σ j D ji p q Figure 6.
Defining the map σ i × σ j −→ C ( R , f ( p ) = f ( p ) and f ( q ) = f ( q ) are two double points in f ( σ i ) ∩ f ( σ j ).Consider a collar C ji = ∂D ji × I on ∂D ji in σ j (cid:114) int( D ji ), Figure 6. The collars are chosensmall enough so that they are disjoint from each other in σ j for various Whitney arcs. Define(4.3) F f,W | σ i × ( σ j (cid:114) ( D ji ∪ C ji )) := ( f × f ) σ i × ( σ j (cid:114) ( D ji ∪ C ji )) . This defines a map into the configuration space C ( R , f ( σ i ) is disjoint from f ( σ j (cid:114) ( D ji ∪ C ji )). On σ i × D ji the map is defined using the result of the Whitney move:(4.4) F f,W | σ i × D ji := ( (cid:101) f ij × (cid:101) f ij ) | σ i × D ji = ( (cid:101) f ij × f ) | σ i × D ji It remains to define F f,W on σ i × C ji interpolating between the maps (4.3), (4.4). If theWhitney disk W ij was framed and embedded then the original map f and the result of theWhitney move (cid:101) f ij would be isotopic, with the isotopy supported in the interior of D ij . Ingeneral, without these assumptions, these maps are homotopic rather than isotopic. Denoteby f tij : K × I −→ R this homotopy f (cid:39) (cid:101) f ij given by the Whitney move, and supported in D ij .Identify ( x, y, t ) ∈ σ i × ∂D ji × [0 ,
1] with ( x, y t ) ∈ σ i × C ji using the product structure on thecollar C ji . Using this identification, the following map sends a point ( x, y t ) to ( f tij ( x ) , f ( y t )):(4.5) F f,W | σ i × C ji := ( f tij × f tij ) | σ i × C ij = ( f tij × f ) | σ i × C ij . This matches (cid:101) f ij × f on D i × ∂D j and f × f on D i × ∂ ( D j ∪ C ). The result is a continuous map σ i × σ j −→ C ( R , -equivariant map C s ( K, −→ C ( R d , (cid:3) A key point in the above proof is that even though the result of the Whitney move (cid:101) f ij ( σ i )intersects (cid:101) f ij ( σ k ) = f ( σ k ), this does not affect the definition of the map F f,W on σ i × σ k .The assumption of Lemma 4.3 is insufficient for producing a map of 3-point configurationspaces, as we make precise in the next subsection.4.4. An obstruction from intersections of Whitney disks.
We are now in a positionto formulate our geometric embedding obstruction for 2-complexes in R which is definedwhen the van Kampen obstruction vanishes. Under this assumption, following Lemma 4.2consider a map f : K −→ R where the intersection number of any two non-adjacent 2-cells f ( σ i ) ∩ f ( σ j ) in R is zero. As in Section 4.3, consider a collection W = { W ij } ofWhitney disks for f ( K ), where W ij denotes the Whitney disks for f ( σ i ) ∩ f ( σ j ). As above,the Whitney arcs are assumed to be disjoint in each 2-cell σ i . The obstruction W ( K ), defined below, depends on the choice of f and of Whitney disks W .Indeed, in the context of obstruction theory one expects that higher obstructions generallydepend on choices of trivializations of lower order obstructions. Recall from Section 2 thatthe obstruction O ( K ) to lifting to a Σ -equivariant map C ( K, −→ C ( R ,
3) depends onthe choice of a Σ -equivariant map f : C ( K, −→ C ( R , f and W – determine such a map f on the simplicial configuration spaceC s ( K, Definition 4.4 ( The obstruction W ( K ) ) . Let
K, f, W be as above, and endow the 2-cellsof K with arbitrary orientations. The orientation of Whitney disks W ij , where ( i, j ) isan ordered pair, is induced from the orientation on its boundary which is oriented from − intersection to + intersection along f ( σ i ) and from from + to − along f ( σ i ). Consider the6-cochain:(4.6) w : C (C s ( K, −→ Z , defined as follows. Let σ i , σ j , σ k be 2-cells of K which pairwise have no vertices in common,and define(4.7) w ( σ i × σ j × σ k ) = W ij · f ( σ k ) + W jk · f ( σ i ) + W ki · f ( σ j ) , where the algebraic intersection numbers are defined using the orientation convention dis-cussed above. Note that changing the order of i, j reverses the orientation of W ij , so thecochain w in (4.7) is Σ equivariant, where Σ acts on Z according to the sign represen-tation. This 6-cochain is a cocycle since it is a top-dimensional cochain on C s ( K, W ( K, f, W ) ∈ H (C s ( K, Z [( − . When f, W are clear from the context, the notation will be abbreviated to W ( K ).It is worth noting that the local move in Figure 5 shifts the intersection numbers betweenthe terms of (4.7); it is the sum that gives a meaningful invariant (see also Remark 4.6below.) Geometrically (4.7) measures intersection numbers that are an obstruction to findingdisjoint embedded Whitney disks needed to construct an embedding K (cid:44) → R . The definitiondepends on various choices: the pairing of ± intersections of f ( σ i ) ∩ f ( σ j ), and choices ofWhitney arcs and of Whitney disks. By comparing it to the obstruction O ( K ) in thenext section, we show that it really depends only on the homotopy class of the map F f,W constructed in Lemma 4.3, a fact that is not apparent from the geometric framework of theabove definition.In addition to these cell-wise intersection considerations, of course properties of the obstruc-tion W ( K ) depend on the cohomology of the configuration space C s ( K, Remark 4.5.
It is not difficult to see that in the example of [12] there is a map of the 2-complex into R with precisely two 2-cells intersecting in two algebraically canceling points,with a Whitney disk intersecting one other 2-cell as in Figure 2. It follows that the cor-responding cochain (4.6) is non-zero on precisely one 6-cell of C s ( K, MBEDDING OBSTRUCTIONS IN R d Remark 4.6.
Our Definition 4.4 extends to the setting of 2-complexes in R the idea ofusing intersections of Whitney disks with surfaces that has been widely used in 4-manifoldtopology. The construction of this type in the simplest relative case: K = (cid:96) D , thedisjoint union of three disks whose boundary curves form a given three-component link L in S = ∂D , is a reformulation of Milnor’s ¯ µ -invariant [21] ¯ µ ( L ), sometimes referred to as thetriple linking number. Such intersections were used to define an obstruction to representingthree homotopy classes of maps of 2-spheres into a 4-manifold by maps with disjoint imagesin [20, 37], and in the non-simply connected setting in [25]. A version considering self-intersections to define the Arf invariant and the Kervaire-Milnor invariant was given in [13,10.8A], and an extension to non-simply connected 4-manifolds in [25].The definition of W ( K ) shares some of the nice features of the geometric definition (4.2) ofthe van Kampen obstruction. Specifically, we will now describe the higher order analogue(“stabilization”) of the finger move homotopy in Figure 1 and of Lemma 4.2. Definition 4.7 ( Stabilization) . This operation applies to any two 2-cells σ , σ and a 1-cell ν of K which are all pairwise non-adjacent, Figure 7a. Perform a finger move introducingtwo canceling σ - σ intersections, and let W (cid:48) denote the resulting embedded Whitney diskpairing these two intersection, Figure 7b. Also consider S ν , a small 2-sphere linking f ( ν ) in R . The final modification applies to the Whitney disk: W is formed as a connected sumof W and S ν , Figure 7c. f ( σ ) f ( σ ) f ( σ ) f ( σ ) f ( ν )(a) W S ν W (cid:48) (c)(b) Figure 7.
Stabilization (modifying the obstruction cocycle by a coboundary)
Proposition 4.8.
Let ( f , W ) be the result of a stabilization applied to ( f, W ) . Then the Σ -equivariant map F f,W : C s ( K, −→ C ( R , associated to ( f , W ) in Lemma 4.3 is Σ -equivariantly homotopic to F f,W .Proof. The Whitney disk W is used only in the restriction of the map F f,W to σ × σ (and equivariantly to σ × σ ). When f ( ν ) and all 2-cells adjacent to it are omitted from thepicture, the Whitney disks W , W (cid:48) in Figure 7 are isotopic. Thus it is clear from the proofof Lemma 4.3 that the maps of configurations spaces corresponding to these two Whitneydisks are homotopic. (Note that the interior of W is disjoint from f ( σ ) since σ , ν wereassumed to be non-adjacent. Thus the result of the Whitney move on f ( σ ) along W is disjoint from f ( σ ).) Moreover, the map f in Figure 7a is isotopic to the result of theWhitney move applied to f in Figure 7b, so the induced maps on configuration spaces areagain homotopic. (cid:3) We are in a position to formulate the analogue of Lemma 4.2 for the new obstruction.
Lemma 4.9.
Any cocycle representative of the cohomology class W ( K, f, W ) ∈ H (C s ( S, Z [ − may be realized as the cocycle w ( K, f (cid:48) , W (cid:48) ) associated to some map f (cid:48) and Whitney disks W (cid:48) . In particular, if the cohomology class W ( K, f, W ) is trivial then there exist f (cid:48) , W (cid:48) whose associated cocycle is identically zero.Proof. Consider a generator C σ ,σ ,ν of Σ -equivariant 5-cochains on C s ( S, σ , σ and 1-cell ν of K . The stabilization operation ( f, W ) (cid:55)→ ( f , W ),shown in Figure 7, changes the cocycle w ( K, f, W ) by a coboundary ± C σ ,σ ,ν , where the signdepends on the orientation of the sphere S ν . Thus changing w ( K, f, W ) by any coboundarymay be realized by a suitable sequence of stabilizations. (cid:3)
As we explain in the next subsection, the vanishing of the cohomology class W ( K, f, W ) hasa geometric consequence: the existence of another layer of Whitney disks, in turn leading toa higher order obstruction.4.5.
Higher order obstructions from Whitney towers.
The notion of Whitney towersencodes higher order intersections of surfaces in 4-manifolds, where the vanishing of theintersections inductively enables one to find the next layer of Whitney disks. In a senseWhitney towers approximate an embedded disk as the number of layers increases. A closelyrelated notion of capped gropes [13, Chapter 2] is extensively used in the theory of topological4-manifolds: they may be found in the context of surgery and of the s -cobordism conjecturewhere surfaces have duals, cf. Proof of Theorem 5.1A in [13]. We will use the notion ofWhitney towers and their intersection theory developed in [26, 27]. Only a brief summaryof the relevant definitions is given below; the reader is referred to the above references fordetails.In the setting of this paper the ambient 4-manifold is R , and the surfaces are the images ofnon-adjacent 2-cells of a 2-complex K under a general position map f : K −→ R . Moreover,we will use the non-repeating version of Whitney towers considered in [27].Whitney towers have a parameter, order , and are defined inductively. Whitney towers oforder 0 are just surfaces in general position in a 4-manifold. Their intersection numbersmay be used to define the van Kampen obstruction, as discussed in Section 4.1. A Whitneytower of order 1 is a collection of surfaces with trivial intersection numbers, together with acollection of Whitney disks pairing up the intersection points. (As in the preceding sections,all Whitney disks are assumed to be framed, and have disjoint boundaries.) This is thesetting for the obstruction in Definition 4.4. Note that the Whitney tower incorporates boththe map f and the Whitney disks W , so W ( K, f, W ) may be thought of as being definedin terms of a Whitney tower.All surface stages and intersection points between them in a general Whitney tower areinductively assigned an order in Z ≥ as follows. The base of the construction ( order
0) is acollection of the original immersed surfaces in R . All surfaces of higher order are Whitneydisks pairing up intersections of surfaces of lower order. The order of an intersection pointof surfaces of orders n , n is defined to be n + n . A Whitney disk pairing up intersectionpoints of order n is said to have order n + 1. MBEDDING OBSTRUCTIONS IN R d Finally, a Whitney tower W of order n + 1 is defined inductively as a Whitney tower oforder n together with a collection of Whitney disks pairing up all intersections of order n .For example, a tower of order 2 is illustrated on the left in Figure 8, with the surfaces σ oforder 0 and Whitney disks V of order 1 and W of order 2. We say that a map f : K −→ R admits a Whitney tower of order n if this condition holds for the images under f of each n -tuple of pairwise non-adjacent 2-cells. Note that given a 2-complex K , an obstruction tothe existence of a map f admitting a Whitney tower of order n for any n ≥ K (cid:44) → R . σ j σ i σ k σ l V W + = 0 = − + Figure 8.
Left: a Whitney tower of order 2 and the associated tree. Right:the AS relation and the IHX relationWith this terminology at hand, we are ready to formulate a geometric consequence of Lemma4.9.
Corollary 4.10.
Let f : K −→ R be an immersion with double points paired up withWhitney disks W , as in Section 4.4. Suppose the cohomology class W ( K, f, W ) ∈ H (C s ( S, Z [ − is trivial. Then there exists a map (cid:101) f : K −→ R which admits a Whitney tower of order . Indeed, by Lemma 4.9 there exists a map f (cid:48) and Whitney disks W (cid:48) such that for each tripleof (pairwise non-adjacent) 2-cells, the intersection invariant (4.7) is trivial. By [26, Theorem2], the map (cid:101) f is regularly homotopic to f (cid:48) which admits admits a Whitney tower of order 2,as claimed.It follows from Lemma 4.2 that if K has trivial van Kampen’s obstruction, there exists a mapof K into R which admits a Whitney tower of height 1. Corollary 4.10 gives the analogue forthe next obstruction: if the class W ( K, f, W ) = 0, there exists a map admitting a Whitneytower of height 2. To define higher obstruction theory, we will now discuss the intersectioninvariants of Whitney towers.The obstruction cochain in equation (4.7) was defined using an explicit formula with in-tersection numbers between Whitney disks and 2-cells. An elegant way of formulating theintersection invariant [26] for a general Whitney tower is in terms of trees, described next.Each unpaired intersection point p of a Whitney tower determines a trivalent tree t p : thetrivalent vertices correspond to Whitney disks and the leaves are labeled by (distinct) 2-cellsof K . The tree embeds in the Whitney tower, as shown on the left in Figure 8, and it inheritsa cyclic orientation of each trivalent vertex from this embedding. (Recall that Whitney disksare oriented as in Definition 4.4.)The relevant obstruction group in our context will be denoted T n . It is defined as a quotientof the free abelian group generated by trivalent trees with n + 2 leaves (and thus n trivalent vertices). The leaves are labeled by non-repeating labels { , . . . , n + 2 } , and the trivalentvertices are cyclically oriented. The quotient is taken with respect to the AS and IHXrelations, shown on the right in Figure 8. These relations are well-known in the study offinite type invariants; in the context of Whitney towers the AS (anti-symmetry) relationcorresponds to switching orientations of Whitney disks, and the IHX relation reflects choicesof Whitney arcs, see [9].Following [26, Section 2.1], the intersection tree τ n of an order n Whitney tower W is definedto be(4.8) τ n ( W ) := (cid:88) p (cid:15) ( p ) t p ∈ T n , where the sum is taken over all unpaired (order n ) intersections points p , and (cid:15) ( p ) is the signof the intersection. For example, for order 1 Whitney tower the intersection trees are the Y tree with two possible cyclic orderings of the trivalent vertex; the obstruction group T isisomorphic to Z , and the intersection invariant matches the formula (4.7).Let C s ( K, n ) denote K × n minus the simplicial diagonal consisting of all products of simplices σ × . . . × σ n , where at least two of the simplices σ i , σ j have a vertex in common for some i (cid:54) = j . The symmetric group Σ n acts in a natural way on the configuration space C s ( K, n )and also on T n − . The following definition extends Definition 4.4 to all n ≥ Definition 4.11 ( The obstruction W n ( K ) ) . Let n ≥ f : K −→ R admits a Whitney tower W of order n −
2. Endow the 2-cells of K with arbitrary orientations;the orientation of all Whitney disks in W are then determined as in Definition 4.4. Considerthe Σ n -equivariant 2 n -cochain:(4.9) w n : C n (C s ( K, n )) −→ T n − , whose value on the 2 n -cell σ × . . . × σ n is given by the intersection invariant (4.8) of theWhitney tower on the 2-cells f ( σ ) , . . . , f ( s n ). It is a cocycle since it is a top-dimensionalcochain on C s ( K, n ). The resulting cohomology class is denoted W n ( K, W ) ∈ H n Σ n (C s ( K, n ); T n − ) . Thus W n ( K, W ) is an obstruction to increasing the order of a given Whitney tower W to n −
1; in particular it is an obstruction to using the data of the Whitney tower W to findan embedding of K . Remark 4.12.
Note that T n − is isomotrphic to Z ( n − , cf. [27, Lemma 19]; compare thiswith the coefficients of the cohomology group in Theorem 7.11.We note that there is an analogue of stabilization in Definition 4.7 for higher trees gener-ating T n , and an analogue of Corollary 4.10 for higher obstructions W n . Thus there is anobstruction theory for 2-complexes in R formulated entirely within the context of intersec-tions of Whitney towers. As we mentioned previously, the focus of this paper is on the firstnew obstruction, W ; we plan to study higher obstructions in more detail in a future work. O ( K ) and W ( K ) are related in the next section; a conjectural relation between O n ( K )and W n ( K ) for n > MBEDDING OBSTRUCTIONS IN R d The obstructions O ( K ) and W ( K ) are equal Here we will relate the obstruction O ( K ) defined in Section 2 and W ( K ) from Section 4;the main result of this section is Theorem 5.1. Before we state the result, a brief digressionis needed to compare the settings of the two obstructions. As discussed in Section 4.1, thetwo versions of the van Kampen obstruction are related by O s2 ( K ) = i ∗ O ( K ), where i is theinclusion map C s ( K, ⊂ C ( K, O ( K ) istrivial; it follows that O s2 ( K ) vanishes as well, and therefore there exists a map f : K −→ R and a collection of Whitney disks for intersections of non-adjacent simplices. Then Lemma4.3 gives a Σ -equivariant map F f,W : C s ( K, −→ C ( R , O ( K ) is a Σ -equivariant map C ( K, −→ C ( R , s ( K, (cid:44) → C ( K,
2) is a homotopy equivalence. Then F f,W induces a map(well defined up to equivariant homotopy) C ( K, −→ C ( R , O ( K ). Without loss of generality we will assume that the Whitney disks are split as discussed inSection 4.2.
Theorem 5.1.
Given a -complex K with trivial van Kampen’s obstruction O ( K ) , let W be a collection of split Whitney disks for double points of a map f : K −→ R . Let F f,W : C s ( K, −→ C ( R , be the Σ -equivariant map determined by f, W in Lemma 4.3.Then (5.1) W ( K, f, W ) = i ∗ O ( K ) ∈ H (C s ( K, Z [( − , where i : C s ( K, −→ C ( K, is the inclusion map.Proof. The pullback i ∗ O ( K, F f,W ) is the obstruction to the existence of a Σ -equivariantdashed map making the following diagram commute up to homotopy.(5.2) C s ( K,
3) C ( R , s ( K, C ( R , p K p R ( F f,W ) The first step of the proof is to use subdivision to reduce to a model situation where preciselyone of the following holds for the image under f of each 2-cell σ of K :(1) σ is mapped in disjointly from all other non-adjacent 2-cells,(2) σ intersects exactly one other non-adjacent 2-cell in two points, or(3) σ has a single intersection point with one of the Whitney disks.(Moreover, the Whitney disks are already assumed to be split, so each one intersects atmost one 2-cell as in Figure 2.) To begin with, each 2-cell σ of K has a finite number ofdisjoint Whitney arcs, as shown in Figure 6, and a finite number of intersection points withWhitney disks. The conditions (1)-(3) above are achieved by subdividing so that each 2-cell The second author would like to thank Pedro Boavida de Brito for motivating questions. There are also other ways of relating the two settings; for example one may define a “simplicial” versionof O ( K ) as the homotopy-lifting obstruction in (1.5) where C s ( K, −→ C s ( K, × is used instead. contains at most one Whitney arc or intersection point with a Whitney disk. For each pairon intersections of 2-cells σ i , σ j as in case (3) we will choose a particular ordering of i, j thatwill determine which sheet is pushed by the Whitney move.Let K (cid:48) denote the 2-complex obtained as the result of the subdivision and let f (cid:48) : K (cid:48) −→ R be the resulting map. The map F f,W in Lemma 4.3 was defined by local modifications of f indisk neighborhoods of the Whitney arcs; F f (cid:48) ,W (cid:48) may be assumed to be defined with respectto the same disk neighborhoods (which are now located in distinct 2-cells of K (cid:48) ). It followsthat F f,W is the compositionC s ( K, −→ C s ( K (cid:48) , −→ C ( R , F f (cid:48) ,W (cid:48) . Moreover, the cochain (4.6) defining W ( K ) is natural withrespect to subdivisions, so W ( K ) is the pullback of W ( K (cid:48) ) under the inclusion C s ( K, −→ C s ( K (cid:48) , K (cid:48) . For the rest of the proof we willrevert to the notation K for the 2-complex, assuming it is subdivided to satisfy conditions(1)-(3).Since the homotopy fiber of the map p R : C ( R , −→ C ( R , is 4-connected, there is alift in (5.2) on the 5-skeleton Sk C s ( K, Construction 5.2.
The construction described below defines a particular Σ -equivariantmap of the -skeleton, F : Sk C s ( K, −→ C ( R , , lifting up to homotopy the Σ -equivariantmap Sk C s ( K, −→ C ( R , . Its specific geometric form will be used for identifying thepoint preimages of the map to S ∨ S in diagram (5.3). The construction relies on the cappedsurface description of the Whitney move (Figure 2), and is an extension of Lemma 4.3. Consider the map on the 4-skeleton induced by f : given any pairwise non adjacent 2-cell σ and 1-cells ν , τ , by general position f ( σ ) , f ( ν ) and f ( τ ) are pairwise disjoint; F is definedon σ × ν × τ (and its orbit under the Σ action) by the Cartesian product f × .The main part of the proof concerns the extension of this map to the 5-cells. We will define F on the boundary of each 6-cell ∂ ( σ × σ × σ ), where σ i , i = 1 , , K , sothat the definition is consistent on the overlap of the boundaries of 6-cells. The map will bedefined for a particular ordering σ , σ , σ and extended to triple products corresponding toother orderings using Σ equivariance.There are three cases:(i) the images of σ i , i = 1 , , , are pairwise disjoint,(ii) two of them, say σ , σ intersect, and W ∩ σ = ∅ ,(iii) two of them, say σ , σ intersect, and W ∩ σ is a point.In case (i) the map F is defined on ∂ ( σ × σ × σ ) as the Cartesian cube f × . Considercase (ii). The boundary of the product ∂ ( σ × σ × σ ) naturally decomposes as the unionof three parts. The definition of F on two of the parts is again f × . The definition of F on σ × ∂s × σ is an analogue of the proof of Lemma 4.3. It is defined on D × ∂σ × σ as f × f × (cid:101) f , where (cid:101) f is the result of the Whitney move on σ , and D is a disk neighborhoodof the Whitney arc in σ . As in the proof of that lemma an isotopy in a collar C on theboundary of D is used, so that on ∂s × ∂s × σ the map F equals f × . MBEDDING OBSTRUCTIONS IN R d Now consider the most interesting case (iii), shown in Figure 2. As in the previous caseconsider a smaller disk neighborhood D of the Whitney arc in σ . It will be convenientto use the capped torus interpretation of the Whitney move, discussed in Section 4.2. Aneighborhood of the Whitney disk W in R is a 4-ball D , and the intersection of σ i , i =1 , , ∂D is a 3-component link, the Borromean rings, cf. [13, Chapter 12]. Anillustration is given in Figure 9; the disk σ may be converted into a punctured torus as inFigure 2. σ D σ W∂σ ∂D ∂σ Figure 9.
Left: the Borromean rings in ∂D . Right: The Whitney disk W intersects σ in a single pointIt will be convenient to represent disks in D as movies in D × [ − ,
1] with time − ≤ t ≤ t = 0. The remaining figures in this sectionillustrate D × { } . Figure 10 shows the capped torus (referred to above) bounded by ∂σ in this representation. The punctured torus consists of two plumbed bands, with caps C (cid:48) (intersecting σ ) and C (cid:48)(cid:48) (intersecting D ). The intersections of D and σ with the slice D × { } are arcs; they extend as (arc × I ) into the past and the future. σ D ∂σ C (cid:48)(cid:48) C (cid:48) Figure 10.
Left: the capped torus bounded by ∂σ with caps C (cid:48) , C (cid:48)(cid:48) . Right:the map f defining F .The disks bounded by σ in Figures 11, 12 are the surgeries along the two caps and thesymmetric surgery, and they will be entirely in the present. The original map f is recoveredby the surgery along the cap C (cid:48)(cid:48) (Figure 11, left), and the result of the Whitney move (cid:101) f isthe surgery on C (cid:48) (Figure 11, right).We will now proceed to define F on the three parts of the boundary ∂ ( D × σ × σ ). Themap F : D × σ × ∂σ −→ C ( R ,
3) is defined as the Cartesian product f × where f is theoriginal map K −→ R ; it is an embedding when restricted to D (cid:96) σ (cid:96) ∂σ (cid:44) → R , Figure10 (right).The maps F : ∂D × σ × σ −→ C ( R , F : D × ∂σ × σ −→ C ( R ,
3) are definedrespectively as f × , ( (cid:101) f ) × = f × f × (cid:101) f where f is again the original map which restricts to an embedding f : ∂D (cid:96) σ (cid:96) σ (cid:44) → R , and (cid:101) f : D (cid:96) ∂σ (cid:96) σ (cid:44) → R is the result of theWhitney move on σ , Figure 11. ∂σ D σ σ ∂D σ Figure 11.
The map f defining F (left) and (cid:101) f defining F (right).The only part of the definition where the map differs from f × is D × ∂σ × σ , where F is defined as ( (cid:101) f ) × = f × f × (cid:101) f . As in case (ii) and in the proof of Lemma 4.3, consider acollar C on ∂D in σ and extend F to C × ∂σ × σ using an isotopy from (cid:101) f to f . The halfpoint of the isotopy, the symmetric surgery discussed above, is shown in Figure 12. Finally,the map is set to be f × on ( σ (cid:114) ( C ∪ D )) × ∂σ × σ . ∂σ ∂σ σ Figure 12.
The symmetric surgery on the capped torus.The map F is well-defined on the 5-skeleton: consider an overlap ∂σ ∩ ∂σ (cid:48) , where σ intersects W as in case (iii) and σ (cid:48) is disjoint from W , as in case (ii). The definition inthe two cases above assigns the same map to σ × ( ∂σ ∩ ∂s (cid:48) ) × ∂σ .The constructed map F : Sk C s ( K, −→ C ( R ,
3) lifts Sk C s ( K, −→ C ( R , up tohomotopy because the surgeries on the two cap, defining F , are isotopic. This concludes thedescription of the map F in Construction 5.2.In the remainder of the proof of Theorem 5.1 we will show that the cohomology classes W ( K ), i ∗ O ( K ) coincide on the cochain level. Recall that i ∗ O ( K ) is the obstructionto lifting in the diagram (5.2). The value of the obstruction cochain on the 6-cell D := σ × σ × σ is the element represented by F ( ∂D ) in π of the homotopy fiber of themap p R : C ( R , −→ C ( R , . This homotopy group is isomorphic to Z , generatedby the Whitehead product of generators of π linking any two of the three diagonals inC ( R , F ( ∂D ) will be determined as follows. Consider the fibration [10], MBEDDING OBSTRUCTIONS IN R d p : C ( R , −→ C ( R , p ( x , x , x ) = ( x , x ):(5.3) R (cid:114) S ∨ S ∂ ( σ × σ × σ ) C ( R , R , S (cid:39) F p p (cid:39) The composition p ◦ F where the map F : ∂ ( σ × σ × σ ) −→ C ( R ,
3) is the result ofConstruction 5.2, is null-homotopic. In fact, it is clear from Figure 10 that p ◦ F is notsurjective: its image is contained in a ball D ⊂ S . Trivializing the fibration over D ,the map F lifts to the fiber, yielding a map (cid:101) F : S = ∂ ( σ × σ × σ ) −→ S ∨ S . Theremainder of the proof of Theorem 5.1 amounts to checking that the homotopy class of thismap in π ( S ∨ S ) represents the Whitehead product of the two wedge summands.The compositions of the map (cid:101) F with the projections of S ∨ S onto the wedge sum-mands are homotopic to p ◦ F , p ◦ F in the diagram (5.4). In both diagrams, the map p ij : C ( R , −→ S ij is given by p ij ( x , x , x ) = ( x i , x j ) / | x i − x j | , i (cid:54) = j ∈ { , , } .(5.4) S ∂ ( σ × σ × σ ) C ( R , S S F p p p Using the Potryagin construction, determining the homotopy class of (cid:101) F in π ( S ∨ S ) canbe determined by the linking number of point preimages of p ◦ F , p ◦ F . ∂σ σ σ σ σ ∂σ Figure 13.
A transverse point preimage of p ◦ F is shown in Figure 13, where a point in S isrepresented as a vector in R (colored red online). The preimage of p ◦ F (defined onthe left in Figure 11) is empty. The preimage of p ◦ F is shown on the left of Figure 13and consists of two disks. The preimage of p ◦ F is shown on the right of Figure 13 andconsists of an annulus. The entire point preimage of p ◦ F is a 2-sphere. Similarly, the point preimage of p ◦ F is analyzed in Figure 14. The two points preimagesare seen to be the two 2-spheres ∂D × {∗} , {∗} × ∂D ⊂ ∂D × D = ∂ ( σ × σ × σ ). Thisconcludes the proof of Theorem 5.1. (cid:3) σ ∂σ σ σ σ ∂σ Figure 14.
Remark 5.3.
Link-homotopy invariants using Whitehead products in configuration spaceswere defined and studied in [17]. The context of the above proof is similar, but the actualmethod and details of the proof are independent of the results of [17].6.
Cohomological obstructions and Examples
An explicit 2-complex K which does not embed into R , but has a vanishing van Kampenobstruction was constructed in [12]. In this section we reprove the non-embeddability of K by showing that our obstruction is realized in this example.Let us begin by reviewing the construction of the complex K in [12]. Let ∆ (respectively ∆ (cid:48) )be the six-dimensional simplex with vertex set v , . . . , v (respectively v (cid:48) , . . . , v (cid:48) ). Denotethe triangle on vertices v a , v b , v c by ∆ abc and similarly the triangle on vertices v (cid:48) a , v (cid:48) b , v (cid:48) c by∆ (cid:48) abc .Let sk n ∆ denote the n -skeleton of ∆ . Let G (respectively G (cid:48) ) be the 2-skeleton of ∆ minus the 2-cell associated with the triangle ∆ (respectively the analogous subcomplex of∆ (cid:48) ).Let K = G ∨ G (cid:48) be the wedge sum obtained by identifying v and v (cid:48) (in [12] the authors addan edge v v (cid:48) , but this difference does not matter). Finally, let K be the complex obtainedby attaching to K a 2-cell along the commutator of the loops v v v v and v (cid:48) v (cid:48) v (cid:48) v (cid:48) . Theclosure of this 2-cell is a torus embedded in K . We denote this torus simply by ∆ × ∆ (cid:48) . Remark 6.1.
This example admits an immediate generalization to a family of examples,where instead of two copies of G and a basic commutator of two loops as above, one takes n copies of the 2-complex G and an element of the mod 2 commutator subgroup of thefree group F n on n generators. The analysis below also goes through for such commutatorswhich are not in the next (second, in the convention of [12, Lemma 7]) term of the mod 2lower central series of F n ; for simplicity of notation we focus on the basic example describedabove. We expect that the examples corresponding to higher commutators are detected byour higher obstructions O n ( K ) , W n ( K ); see Section 8. MBEDDING OBSTRUCTIONS IN R d As explained in [12], van Kampen showed that sk ∆ can not be embedded in R , but G can. It follows that the complex K can be embedded in R .Let S ⊂ G be the sphere that is the union of the four 2-cells that are disjoint from thetriangle ∆ , namely the cells corresponding to ∆ , ∆ , ∆ and ∆ . S is the dualtetrahedron to the triangle ∆ in the 6-simplex. Dually, let S (cid:48) ⊂ G (cid:48) be the dual sphere tothe triangle ∆ (cid:48) .The following key result about embeddings of K into R is proved in [12] (we do not reproveit). Proposition 6.2 ([12], Lemma 6) . For any PL embedding of K into R , the linking numbersof S, S (cid:48) and ∆ , ∆ (cid:48) satisfy the following link( S, ∆ ) ≡ link( S (cid:48) , ∆ (cid:48) ) ≡ . link( S, ∆ (cid:48) ) = link( S (cid:48) , ∆ ) = 0 . (see figure 15) S ∆ S (cid:48) ∆ (cid:48) Figure 15.
The 2-complex K is obtained by attaching a 2-cell along thecommutator of ∆ and ∆ (cid:48) .It is also shown in [12] that the van Kampen obstruction vanishes on K . Now we can statethe main result of this section. Of course it is also proved in [12], using fundamental groupinstead of cohomology. Proposition 6.3.
Suppose f : C ( K, → C ( R , is a Σ -equivariant map, such that therestriction of f to C ( K , is induced by some embedding f : K (cid:44) → R . Then the followingcomposition map (6.1) C ( K, → C ( K, { , } ) × C ( K, { , } ) × C ( K, { , } ) →→ C ( R , { , } ) × C ( R , { , } ) × C ( R , { , } ) does not lift to a map C ( K, → C ( R , . It follows in particular that no embedding K (cid:44) → R can be extended to an embedding K (cid:44) → R .To prove the proposition, we give a cohomological interpretation of our obstruction O ( K )in terms of the Arnold class, which may be of independent interest. Consider, once again, the problem of constructing a Σ -equivariant lift in a diagram of the following formC ( R , { , , } )C ( K,
3) C ( R , { , } ) × C ( R , { , } ) × C ( R , { , } ) ph Recall that for any two points i, j , u ij ∈ H (C ( R , { i, j } )) denotes a generator that corre-sponds to u under the canonical homeomorphism. Then we have the Arnold class u ⊗ u ⊗ ⊗ u ⊗ u − u ⊗ ⊗ u ∈ H (C ( R , { , } ) × C ( R , { , } ) × C ( R , { , } )) . By Lemma 2.4, this class generates the kernel of p in H . We get the following easy sufficientcondition for our obstruction to be non-zero Lemma 6.4.
Referring to the diagram above, suppose h ∗ ( u ⊗ u ⊗ ⊗ u ⊗ u − u ⊗ ⊗ u ) (cid:54) = 0 . Then a lift does not exists and O ( K ) (cid:54) = 0 . One can make the connection between O ( K ) and the Arnold class a little more precise. Bydefinition, O ( K ) is an element in the Σ -equivariant cohomology group H (C ( K, Z ± ).There is a natural homomorphism H (C ( K, Z ± ) → H (C ( K, Z ± ) Σ ⊂ H (C ( K, Lemma 6.5.
The image of O ( K ) in H (C ( K, under this homomorphism is (the imageof ) the Arnold class under the map p k ◦ f : C ( K, → C ( R , .Proof. We saw in Section 2 that O ( K ) is represented by a mapC ( R n , → K ( Z , n − R n , → C ( R n , → K ( Z , n − H n − and in H n − . It follows that the mapC ( R n , → K ( Z , n −
2) representing our obstruction sends a generator of H n − ( K ( Z , n − H n − (C ( R n , ) → H n − (C ( R n , (cid:3) Now let us prove the main result of this section.
Proof of Proposition 6.3.
The map (6.1) induces a homomorphism in cohomology H (C ( R , { , } ) × C ( R , { , } ) × C ( R , { , } )) → H (C ( K, . By Lemma 6.4, it is enough to show that this homomorphism does not send the element u ⊗ u ⊗ ⊗ u ⊗ u − u ⊗ ⊗ u to zero.Inside K there are three disjoint subspaces: the spheres S and S (cid:48) , and the torus ∆ × ∆ (cid:48) .Since these subspaces are disjoint, the obvious inclusion S × S (cid:48) × (∆ × ∆ (cid:48) ) (cid:44) → K × K × K factors through an inclusion S × S (cid:48) × (∆ × ∆ (cid:48) ) (cid:44) → C ( K, . MBEDDING OBSTRUCTIONS IN R d We will want to give names to elements in the cohomology of S × S (cid:48) × (∆ × ∆ (cid:48) ).For this purpose, let u, u (cid:48) , τ , and τ (cid:48) be generators of H ( S ) , H ( S (cid:48) ) , H (∆ ) , H (∆ (cid:48) )respectively.Consider the composition(6.2) S × S (cid:48) × (∆ × ∆ (cid:48) ) → C ( K, → C ( K, { , } ) × C ( K, { , } ) × C ( K, { , } ) →→ C ( R , { , } ) × C ( R , { , } ) × C ( R , { , } ) . We want to analyze the effect of this map on cohomology. So let us consider the threeprojections of this map. The map S × S (cid:48) × (∆ × ∆ (cid:48) ) → C ( R , { , } ) factors throughthe projection S × S (cid:48) × (∆ × ∆ (cid:48) ) → S × S (cid:48) . The map S × S (cid:48) → C ( R , { , } ) is zero onreduced cohomology for the obvious reason that the target only has non-trivial cohomologyin degree 3 and the source has trivial cohomology in degree 3. It follows that the terms ofthe Arnold class that involve u are sent to zero by this map.It remains to see what happens to the term 1 ⊗ u ⊗ u . Let us consider the map S × S (cid:48) × (∆ × ∆ (cid:48) ) → C ( R , { , } ). This map factors as a composition S × S (cid:48) × (∆ × ∆ (cid:48) ) → S × (∆ × ∆ (cid:48) ) → C ( R , { , } ) . It follows from Proposition 6.2 that this composite map sends the generator u of H (C ( R , { , } )to an odd multiple of u ⊗ τ . Similarly, the map S × S (cid:48) × (∆ × ∆ (cid:48) ) → C ( R , { , } )sends the generator u to an odd multiple of u (cid:48) ⊗ τ (cid:48) .It follows that the map (6.2) in cohomology sends the Arnold class to an odd multiple of u ⊗ u (cid:48) ⊗ τ ⊗ τ (cid:48) . In particular, to a non-zero element of H (C ( K, (cid:3) Remark 6.6.
In the discussion above we focused on the case of 2-complexes in R , buta similar calculation shows that the obstruction O ( K ) detects non-embeddability of ex-amples (with vanishing obstruction O ( K )) in all dimensions outside the metastable range,2 d < m + 1), such that d ≥ max(4 , m ). Such examples of m -dimensional complexes wereconstructed and shown to not admit an embedding in R d in [29, 28]. The construction in-volves the Whitehead product of meridional spheres S l , l = d − m −
1, linking two m -spheres S , S (cid:48) , rather than the commutator of loops in the construction above. Still, there are threedisjoint subspaces in the complex: the spheres S, S (cid:48) , and a 2 l -torus, and the calculation of theArnold class analogous to the above shows that it is non-trivial. This gives a unified proofof non-embeddability of the examples in [12] and in [29, 28], while the arguments in theseoriginal references are quite different, both from each other and from the new perspective inthis paper. 7. The tower
In this section we show how the obstruction O ( K ) and O ( K ) can be extended to a sequenceof obstructions O n ( K ), using a primitive version of the Goodwillie-Weiss tower. We will thengive a conjectural description of a framed cobordism refinement of O n ( K ). Definition 7.1.
Let I be the category of finite sets and injective functions between them,and I n ⊂ I be the full subcategory consisting of sets of cardinality at most n . As before, let Emb( K, R d ) denote the space of topological embeddings of K into R d . In thecase when i is a finite set and X is any space, Emb( i, X ) is the configuration space of ordered i -tuples of pairwise distinct points of X . We also denote this space by C ( X, i ) := Emb( i, X ).Given a small category C and functors F, G : C →
Top, we let Nat C ( F, G ) denote the spaceof natural transformations from F to G , and let hNat C ( F, G ) denote the space of derivednatural transformations from F to G . In other words, hNat C ( F, G ) is the space of naturaltransformations from a cofibrant replacement of F to a fibrant replacement of G . The(co)fibrant replacements can be taken in any Quillen model structure on the functor category[ C , Top], where the weak equivalences are defined levelwise. We will use the projective modelstructure, in which every functor is fibrant.
Remark 7.2.
To save notation, if
F, G are functors C op → Top, we will use the notationhNat C ( F, G ) rather than hNat C op ( F, G ).A topological space K determines a functor C ( K, − ) : I op → Top that sends a set i toC ( K, i ) = Emb( i, K ). A topological embedding f : K (cid:44) → R d gives rise to a natural trans-formation C ( K, − ) → C ( R d , − ), which sends an embedding α : i (cid:44) → K to the embedding f ◦ α : i (cid:44) → R d . This gives rise to natural maps.(7.1) Emb( K, R d ) → Nat I (C ( K, − ) , C ( R d , − )) → hNat I (C ( K, − ) , C ( R d , − )) . One useful feature of the space hNat I (C ( K, − ) , C ( R d , − )) is that it admits a natural towerof approximations. Definition 7.3.
For each n ≥ T n Emb( K, R n ) = hNat I n (C ( K, − ) , C ( R d , − ))The inclusions of categories · · · I n − ⊂ I n ⊂ · · · ⊂ I give rise to a tower whose homotopyinverse limit is equivalent to hNat I (C ( K, − ) , C ( R d , − ))hNat I (C ( K, − ) , C ( R d , − )) → · · · → T n Emb( K, R n ) → T n − Emb( K, R d ) → · · · Remark 7.4.
Readers familiar with the embedding calculus of Goodwillie and Weiss willreadily recognize T n Emb( K, R n ) as a primitive analogue of the n -the Taylor approximationin the Goodwillie tower. Indeed, the Goodwillie-Weiss construction is essentially the sameas the one in Definition 7.3, except that instead of the category I n of sets with at most n elements, they use the category whose objects are manifolds diffeomorphic to the disjointunion of at most n copies of R m , and whose morphisms are smooth embeddings. At leastthis is one way to construct the Goodwillie-Weiss tower. For more information about thisapproach to the Goodwillie-Weiss calculus see the paper of Boavido and Weiss [5].The following lemma is an immediate consequence of the existence of the map (7.1). Lemma 7.5. If hNat I n (C ( K, − ) , C ( R d , − )) is empty for some n , then there does not existsan embedding of K into R d . Our goal is to study obstructions for a path component of T n − Emb( K, R d ) to be in theimage of T n Emb( K, R d ). For this purpose it is useful to have an inductive description of T n Emb( K, R d ). Such a description is given by Proposition 7.8 below. The proposition iselementary and no doubt well-known. But for completeness we will give a proof. We needsome preparation. MBEDDING OBSTRUCTIONS IN R d Definition 7.6.
Let C ( R d , n ) = holim S (cid:40) { ,...,n } C ( R d , S )In words, C ( R d , n ) is the homotopy limit of all the ordered configuration spaces of propersubsets of { , . . . , n } into R d . Remark 7.7.
It is worth noting that C ( R d , (cid:39) C ( R d , -a space that we encountered insections 2 and 3. Everything we are doing in this section is a generalization of what we didin those two sections for n = 2 , ( R d , n ) that will come up. Let I n − ↓ n be the category whose objects are injective maps of sets i (cid:44) → n , where n is shorthandfor { , . . . , n } and i ∈ I n − denotes a set with strictly fewer elements than n . Morphisms in I n − ↓ n are commuting triangles. There is a functor from I n − ↓ n to the category (poset) ofproper subsets of { , . . . , n } which sends an injective map i (cid:44) → n to its image. This functor iseasily seen to be faithful, full and surjective, so it is an equivalence of categories. Thereforeit induces an equivalence(7.3) holim S (cid:40) { ,...,n } C ( R d , S ) (cid:39) −→ holim i(cid:44) → n ∈ I n − ↓ n C ( R d , i ) . Another notion that we will use in the proof of Proposition 7.8 is that of a homotopy rightKan extension. Let us quickly review what this is. Suppose C is a category and C is asubcategory. Let F : C →
Top be a functor. We denote the restriction of F to C by F | C (sometimes we may denote the restriction of F simply by F ). Next, suppose G : C → Topis a functor defined on a subcategory of C . Then let RG : C →
Top denote the homotopyright Kan extension of G from C to C . Recall that RG can be defined on the objects of C by the following formula RG ( x ) = holim x → z ∈ x ↓C G ( z ) . The homotopy right Kan extension is a derived right adjoint to the restriction functor. Thismeans that there is a natural equivalence(7.4) hNat C ( F, RG ) (cid:39) hNat C ( F | C , G )The adjunction also means that there is a natural transformation of functors F → RF | C .If C is a full subcategory of C then this natural transformation is an equivalence whenevaluated on objects of C .Now we are ready to state and prove the inductive description of T n Emb( K, R d ) Proposition 7.8.
There is a homotopy pullback square, where the right vertical map isinduced by the canonical map
C ( R d , n ) → C ( R d , n ) T n Emb( K, R d ) → map(C ( K, n ) , C ( R d , n )) Σ n ↓ ↓ T n − Emb( K, R d ) → map(C ( K, n ) , C ( R d , n )) Σ n Proof.
Since T n Emb( K, R d ) = hNat I n (C ( K, − ) , C ( R d , − )), our task is to prove that thereexists a homotopy pullback diagram of the following form(7.5) hNat I n (C ( K, − ) , C ( R d , − )) map(C ( K, n ) , C ( R d , n )) Σ n hNat I n − (C ( K, − ) , C ( R d , − )) map(C ( K, n ) , C ( R d , n )) Σ n The strategy is to express all four corners of this square as spaces of homotopy naturaltransformations between functors defined on I n using homotopy right Kan extension.Let R nn − C ( R d , − ) be the homotopy right Kan extension of the functor C ( R d , − ) from I n − to I n . By (7.4) we know that restriction from I n to I n − induces an equivalencehNat I n (C ( K, − ) , R nn − C ( R d , − )) (cid:39) −→ hNat I n − (C ( K, − ) , C ( R d , − )) . Now let us analyse the functor R nn − C ( R d , − ). There is a natural transformation of (con-travariant) functors on I n C ( R d , − ) → R nn − C ( R d , − ) . This natural transformation is an equivalence when evaluated on objects of I n − because I n − is a full subcategory of I n . On the other hand, we have the following formula for R nn − C ( R d , n ) R nn − C ( R d , n ) (cid:39) holim i(cid:44) → n ∈ I n − ↓ n C ( R d , i )By (7.3) we have an equivalenceC ( R d , n ) = holim S (cid:40) { ,...,n } C ( R d , S ) (cid:39) −→ holim i(cid:44) → n ∈ I n − ↓ n C ( R d , i ) . Therefore there is an equivalence R nn − C ( R d , n ) (cid:39) C ( R d , n ) . And the map C ( R d , n ) → R nn − C ( R d , n ) is equivalent to the natural map C ( R d , n ) → C ( R d , n ). Now consider the full subcategory of I n consisting of the single object n andits endomorphisms. This category is the symmetric group, and we will denote it by Σ n . Afunctor from Σ n to Top is the same thing as a space with an action of Σ n . Given a space X n with an action of Σ n , we let R I Σ X n ( − ) denote the homotopy right Kan extension of thisfunctor from Σ n to I n . Since there are no morphisms in I n from n to smaller sets, it followsthat R I Σ X n ( n ) = X n and R I Σ X n ( i ) (cid:39) ∗ for i < n .From the discussion above we conclude that there is a homotopy pullback square of functorsfrom I n to Top(7.6) C ( R d , − ) R I Σ C ( R d , n )( − ) R nn − C ( R d , − ) R I Σ C ( R d , n )( − )Indeed, when evaluated at a set i < n , the vertical morphisms in this square are equivalences,and when evaluated at n , the horizontal morphisms are equivalences. So it is a homotopypullback square of functors. MBEDDING OBSTRUCTIONS IN R d Applying hNat I n (C ( K, − ) , − ) to (7.6), we obtain a homotopy pullback squarehNat I n (C ( K, − ) , C ( R d , − )) hNat I n (C ( K, − ) , R I Σ C ( R d , n )( − ))hNat I n (C ( K, − ) , R nn − C ( R d , − )) hNat I n (C ( K, − ) , R I Σ C ( R d , n )( − ))Using the fact that right Kan extension is derived right adjoint to restriction we obtain thatthis square is equivalent to the desired square (7.5) at the beginning of the proof. So wehave proved that a homotopy pullback square of this form exists. (cid:3) Remark 7.9.
One can interpret the homotopy pullback square (7.6) as an inductive de-scription of the coskeletal filtration on a functor defined on a (generalized) Reedy category.See [4, Section 6]Proposition 7.8 leads to an inductive procedure for constructing obstructions to the existenceof an embedding
K (cid:44) → R d . Suppose we have a point g n − : hNat I n − (C ( K, − ) , C ( R d , − )),and we want to know if (the path component of) g n − lies in the image ofhNat I n (C ( K, − ) , C ( R d , − )) . The bottom map in diagram (7.5) sends g n − to a Σ n -equivariant map˜ f n : C ( K, n ) → C ( R d , n ) , which really factors as a compositeC ( K, n ) → C ( K, n ) → C ( R d , n ) . The path component of g n − is in the image of a path component of hNat I n (C ( K, − ) , C ( R d , − ))if and only if ˜ f n lifts up to homotopy to a Σ n -equivariant map f n : C ( K, n ) → C ( R d , n ), asper the following diagram(7.7) C ( R d , n )C ( K, n ) C ( R d , n ) ˜ f n f n At this point obstruction theory kicks in. We will assume that d ≥
3, so that the spacesC ( R d , n ) and C ( R d , n ) are simply connected. The first obstruction to the existence of alift f n lies in the equivariant cohomology of C ( K, n ) with coefficients in the first non-trivialhomotopy group of the homotopy fiber of the map C ( R d , n ) → C ( R d , n ). The followingproposition is known [14, 16]. Proposition 7.10.
The map
C ( R d , n ) → C ( R d , n ) is ( d − n −
1) + 1 -connected. Let F be the homotopy fiber of this map. The first non-trivial homotopy group of F is π ( d − n − ( F ) ∼ = Z ( n − . Since the space F is simply-connected, the action of Σ n on these spaces induces a well-defined action on the first non-trivial homotopy group of F . Thus the group Z ( n − is arepresentation of Σ n . Standard obstruction theory implies the following result. Theorem 7.11.
There is a cohomological obstruction O n ( K ) to the existence of a lift f n asabove. The class O n ( K ) is an element of the equivariant cohomology group. H ( d − n − n (cid:0) C (
K, n ) , Z ( n − (cid:1) . If dim( K ) · n = ( d − n −
1) + 2 then O n ( K ) is a complete obstruction to the existence ofa lift f n . In particular, this holds when dim( K ) = 2 and d = 4 . It is easy to see that for n = 2 , O n ( K ) agrees with the definitionsof O ( K ) and O ( K ) that we saw earlier.We end this section by describing a conjectural refinement of O n ( K ) to an obstruction O frn ( K )living in equivariant stable cobordism, extending the definitions of O fr2 ( K ) and O fr3 ( K ) thatwe saw earlier.The construction of O frn ( K ) uses a geometric realization of the group π ( d − n − ( F ) ∼ = Z ( n − as the cohomology of the space of non 2-connected graphs . Recall that a graph G iscalled 2 -connected if G connected, and for every vertex x , G \ { x } is connected. Definition 7.12.
For n >
1, let ∆ n be the poset of non-trivial non 2-connected graphs withvertex set { , . . . , n } . Let T n be the unreduced suspension of the geometric realization of∆ n .The space T n was initially introduced by Vassiliev, and was studied in the paper [3]. Forexample, T = S , T = S , with the standard (non-trivial) action of Σ .The following is well-known [3] Theorem 7.13.
There is a homotopy equivalence T n (cid:39) (cid:95) ( n − S n − Conjecture 7.14.
There is a natural Σ n -equivariant map C ( R d , n ) → map ∗ ( T n , Ω ∞ Σ ∞ S d ( n − ) So that there is an ( d − n + 1 -cartesian square C ( R d , n ) C ( R d , n ) ∗ map ∗ ( T n , Ω ∞ Σ ∞ S d ( n − )Assuming the conjecture, we have natural maps T n − Emb( K, R d ) → map(C ( K, n ) , C ( R d , n )) Σ n → map ∗ (C ( K, n ) + ∧ T n , Ω ∞ Σ ∞ S d ( n − ) Σ n This map associates to a point in T n − Emb( K, R d ) an element of the equivariant stablecohomotopy group of C ( K, n ) ∧ T n . This element is an obstruction O frn ( K ) to the point of T n − Emb( K, R d ) being in the image of T Emb( K, R d ). The obstruction is complete so longas d ≥ dim( K ) + 2. MBEDDING OBSTRUCTIONS IN R d Remark 7.15.
Our reasons to believe Conjecture 7.14 come from Orhogonal calculus. Thefunctor that sends R d to the spectrummap ∗ ( T n , Ω ∞ Σ ∞ S d ( n − )is the bottom non-trivial layer of the difference between C ( R d , n ) and C ( R d , n ). In fact,the conjecture is almost a formal consequence of the existence of Orthogonal Calculus andwhat we know about the derivatives of functors related to C ( R d , n ). However, it would beinteresting to have an explicit mapC ( R d , n ) → map ∗ ( T n , Ω ∞ Σ ∞ S d ( n − )with some sort of geometric interpretation. The map that we defined for the case n = 3 inSection 3 does not seem to generalize easily to higher values of n .8. Questions and conjectures
In conclusion we will mention several problems motivated by the results of this paper.8.1.
Equivalence of higher obstructions.
Given a 2-complex K with trivial van Kam-pen’s obstruction O ( K ), according to Lemma 4.3 a map f : K −→ R together withWhitney disks W for intersections of non-adjacent 2-cells determine a Σ -equivariant map F f,W : C s ( K, −→ C ( R d , W ( K ) equalsthe pullback of O ( K ) to H (C s ( K, Z [( − f : K −→ R which admits a Whitney tower of order n − n ≥ T n − Emb( K, R ), and the obstruction W n ( K ) fromDefinition 4.11 equals the pullback of O n ( K ) to H n Σ n (cid:0) C s ( K, n ) , Z ( n − (cid:1) .8.2. Conjectural higher cohomological obstructions.
Recall the discussion of the Arnoldclass (Definition 2.3) and its relation with the obstruction O ( K ) (Lemma 6.5). Here weformulate a certain version of Massey products, defined when the Arnold class vanishes.For convenience of choosing signs below, we will focus on the case d = 4; analogous coho-mological classes can be constructed for any d .We will use the notation C ( R , { i, j } ) defined in (2.5) and the generators u ij ∈ H (C ( R d , { i, j } ))defined in the paragraph following (2.5). Denote by U ij some cocycle representatives of u ij .Assume that there exists ˜ f : C ( K, −→ C ( R d ,
4) as in diagram (7.7). Denote by V ij the3-cocycles on C ( K,
4) obtained as the pull-backs of U ij , i, j ∈ { , . . . , } . Consider V i,j,k := V ij V jk + V jk V ki + V ki V ij . It follows from the existence of the map ˜ f : C ( K, −→ C ( R d ,
4) and the resulting van-ishing of the Arnold class in Lemma 6.5 that for each subset { i, j, k } ⊂ { , . . . , } thecohomology class of V i,j,k in H (C ( K, X ijk on C ( K, δX ijk = V ijk . Consider the 8-cochain on C ( K, Y (12)(34) := X ( U − U ) + X ( U − U ) + X ( U − U ) + X ( U − U ) One checks that this is a cocycle; in fact there are two additional cocycles which we denote Y (13)(24) , Y (14)(23) ; for example(8.2) Y (13)(24) := X ( U − U ) + X ( U − U ) + X ( U − U ) + X ( U − U )The sum of these three cocycles is zero. We conjecture that these cohomology classes are ob-structions to lifting ˜ f in diagram (7.7) to a map f : C ( K, −→ C ( R d , O ( K ) , W ( K ) ∈ H (C ( K, , Z ) analogously to Lemma 6.5.Moreover, formulas (8.1), (8.2) suggest that these classes admit a systematic generalizationto C ( K, n ) for larger n as well.8.3. 4 -complexes in R . Recall that the validity of the Whitney trick in higher dimensionsimplies that an m -complex K admits an embedding into R m if and only if van Kampen’sobstruction O ( K ) vanishes, m >
2. (And more generally, by Weber’s theorem, this holdsfor m -complexes in R d in the metastable range, 2 d ≥ m + 1).)It is an interesting question whether outside of the metastable range, when the dimensionand codimension are both sufficiently large, the vanishing of just one additional obstructionis sufficient for embeddability. To be specific, suppose K is a 4-complex such that both O ( K ) and O ( K ) are trivial. Does this imply that K admits a PL embedding into R ?It seems reasonable to conjecture that the answer is affirmative. A central role in Sections4, 5 of this paper is played by the study of 2-complexes in R where the Whitney trick fails.The analysis of 4-complexes in R is back in the dimensions where the (suitably generalized)Whitney trick works both for primary and secondary intersections, so one may expect thatthe vanishing of algebraic obstructions should lead to an embedding.8.4. (Weak) convergence of the tower. To our knowledge the problem of convergenceto the tower (7.2) is open in dimensions 2 d < m + 1). (Section 8.3 discussed a particularcase, m = 4 , d = 7.)In particular, the case of 2-complexes in R is interesting. As mentioned in Remark 6.1, theexamples in Section 6 admit a generalization where a 2-cell is attached to an ( n − n − n >
3; we expect that the obstructions W n ( K ) , O n ( K ) detecttheir non-embeddability. Question . Let K be a -complex such that all obstruction O n ( K ) are trivial, n ≥ . Does K necessarily admit an embedding in R , in either PL or topologically flat category? As mentioned in the introduction, in the special relative case where K is the disjoint unionof disks D i and the embedding problem in the 4-ball has a prescribed boundary condition– a link L formed by the boundaried of the disks ∂D i in S = ∂D – our obstructionscorrespond to the Milnor invariants of L . There are well-known examples (boundary links)which have trivial Milnor’s invariants but are not slice. (Further, there are examples [7] oflinks with vanishing Milnor invaraints which are not concordant to boundary links.) Howeverin our context there is no boundary condition present, and there is considerable flexibility inre-embedding; we are not aware of an example contradicting the possibility of an affirmativeanswer to the question above. MBEDDING OBSTRUCTIONS IN R d Intrinsic characterization of the obstructions.
Given a 2-complex K with trivial O ( K ), is there an intrinsic characterization of classes in H (C s ( K, H (C ( R , R as in Lemma 4.3? The proof ofnon-embeddability of examples in Section 6 relies on Proposition 6.2. A characterizationof such classes H (C s ( K, R .8.6. Complexity of embeddings.
There have been recent advances in the subject of com-plexity of embeddings of complexes into Euclidean spaces, both from algorithmic and geo-metric perspectives, cf. [19, 11]. In higher dimensions there is an upper bound O (exp( N (cid:15) ))on the refinement complexity (r.c.), i.e. the number of subdivisions needed to PL embed asimplicial m -complex (with trivial O ( K )) into R m , m >
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Gregory AroneDepartment of Mathematics, Stockholm UniversityVyacheslav KrushkalDepartment of Mathematics, University of Virginia, Charlottesville, VA 22904-4137
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