Graph invariants from the topology of rigid isotopy classes
aa r X i v : . [ m a t h . A T ] A ug GRAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPYCLASSES
MARA BELOTTI, ANTONIO LERARIO, AND ANDREW NEWMAN
Abstract.
We define a new family of graph invariants, studying the topology of the modulispace of their geometric realizations in Euclidean spaces, using a limiting procedure reminis-cent of Floer homology.Given a labeled graph G on n vertices and d ≥ , W G,d ⊆ R d × n denotes the space ofnondegenerate realizations of G in R d . For example if G is the empty graph then W G,d ishomotopy equivalent to the configuration space of n points in R d . Questions about when acertain graph G exists as a geometric in R d have been considered in the literature and inour notation have to do with deciding when W G,d is nonempty. However W G,d need not beconnected, even when it is nonempty, and we refer to the connected components of W G,d as rigid isotopy classes of G in R d . We study the topology of these rigid isotopy classes.First, regarding the connectivity of W G,d , we generalize a result of Maehara that W G,d isnonempty for d ≥ n to show that W G,d is k -connected for d ≥ n + k + 1 , and so W G, ∞ isalways contractible.While π k ( W G,d ) = 0 for G , k fixed and d large enough, we also prove that, in spite of this,when d ! ∞ the structure of the nonvanishing homology of W G,d exhibits a stabilizationphenomenon. The nonzero part of its homology is concentrated in at most ( n − -manyequally spaced clusters in degrees between d − n and ( n − d − , and whose structuredoes not depend on d , for d large enough. This leads to the definition of a family of graphinvariants, capturing the asymptotic structure of the homology of the rigid isotopy class. Forinstance, the sum of the Betti numbers of W G,d does not depend on d , for d large enough; wecall this number the Floer number of the graph G . This terminology comes by analogy withFloer theory, because of the shifting phenomenon in the degrees of positive Betti numbers of W G,d as d tends to infinity.Finally, we give asymptotic estimates on the number of rigid isotopy classes of R d –geometric graphs on n vertices for d fixed and n tending to infinity. When d = 1 we showthat asymptotically as n ! ∞ each isomorphism class corresponds to a constant number ofrigid isotopy classes, on average. For d > we prove a similar statement at the logarithmicscale. Introduction
Let P = ( p , . . . , p n ) be a point in R d × n . The geometric graph associated to P is the labeledgraph G ( P ) whose vertices and edges are, respectively: V ( G ( P )) = { (1 , p ) , . . . , ( n, p n ) } and E ( G ( P )) = { (( i, p i ) , ( j, p j )) | i < j, k p i − p j k < } . If a graph G on n vertices is isomorphic to a geometric graph G ( P ) , as a labeled graph,for some P ∈ R d × n we say it is realizable as an R d –geometric graph on n vertices. It wasproved by Maehara in [Mae84] that when d ≥ n every graph on n vertices is realizable as an R d -geometric graph. In particular, if we denote by d,n the number of isomorphims classes of A.N. was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)Graduiertenkolleg 2434 "Facets of Complexity". From now on, unless differently specified, the word “graph” stands for “labeled graph”. labeled R d –geometric graphs on n vertices, then for d ≥ n we have(1.1) d,n = 2( n ) . This statement can be rephrased using the theory of discriminants from real algebraicgeometry. To explain this idea let us first introduce the notion of nondegenerate geometricgraph: the R d –geometric graph G ( P ) is called nondegenerate if there is no pair of indices ≤ i < j ≤ n such that k p i − p j k = 1 . Studying nondegenerate graphs is not an actualrestriction, since the set of isomorphism classes of labeled nondegenerate R d –geometric graphscoincides with the set of all possible isomorphism classes of labeled R d –geometric graphs (seeLemma 15 below). Moreover, nondegenerate geometric graphs are simpler to study, becauseof their stability under small perturbations of the defining points.In this setting the discriminant consists of the set of degenerate R d –geometric graphs: ∆ d,n = { P ∈ R d × n | there exist ≤ i < j ≤ n such that k p i − p j k = 1 } ⊂ R d × n . This discriminant partitions R d × n \ ∆ d,n into many disjoint, connected open sets, which wewill call chambers . If two points P and P belong to the same chamber in R d × n \ ∆ d,n thenclearly G ( P ) and G ( P ) are isomorphic, but the reverse implication does not hold in general,leading to the following definition. Definition 1.
If two points P , P ∈ R d × n \ ∆ d,n belong to the same chamber, that is if thereis a continuous curve P : [0 , ! R d × n \ ∆ d,n with P (0) = P and P (1) = P , we will say thatthe geometric graphs G ( P ) and G ( P ) are rigidly isotopic .As an example of R d -geometric graphs which are isomorphic but not rigidly isotopic, con-sider P = ( − , and P = (0 , − : the R –geometric graphs G ( P ) and G ( P ) are isomorphic;they are both the graph on vertices with no edges, but they are not rigid isotopic since anycurve P ( t ) ∈ R × with P (0) = P and P (1) = P must intersect the discriminant. Anotherexample is depicted in Figure 1.For n and d the number of rigid isotopy classes of geometric graphs on n vertices in R d isexactly given by b ( R d × n \ ∆ d,n ) , and we always clearly have b ( R d × n \ ∆ d,n ) ≥ d,n . One natural question therefore is for what values of n and d do the two notions coincide.Moreover, one could consider higher-dimensional notions of connectivity of R d × n \ ∆ d,n andstudy the higher homology and the homotopy groups of the space of its connected components.As we will see, this study will lead us to a definition of a new graph invariant, which remindsof Floer homology, as well as precise asymptotics for the enumeration of rigid isotopy andisomorphism classes of geometric graphs.1.1. The case d ! ∞ . As we will prove in Corollary 45 below, for d ≥ n + 1 , the two notionsof isomorphic and rigidly isotopic coincide and b ( R d × n \ ∆ d,n ) = d,n . Therefore, adoptingthis language we can reformulate the identity in (1) as: b ( R d × n \ ∆ d,n ) = 2( n ) , which is true for d ≥ n + 1 . The realizability result of Maehara [Mae84] and the fact thatfor large d “rigid isotopy” and “isomorphism” are the same notion, seem to settle all relevant Here and below, for a topological space X we will denote by b k ( X ) = dim Z ( H k ( X ; Z )) its k –th Bettinumber and by b ( X ) = P ∞ k =0 b k ( X ) , its total Betti number, whenever these numbers are defined. This willhappen for all the spaces that we will consider in this paper: they will all be homotopy equivalent to finiteCW–complexes. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 3 p p ′ Figure 1.
Here we are drawing points in R together with the circles cen-tered at those points and with radius √ . Now, let us define P and P ′ pointsin R × in such a way that p is the point inside the big circle and p ′ isthe point outside the big circle while p i = p ′ i for i > and they are thepoints on the big circle. Then, the two geometric graphs G ( P ) and G ( P ′ ) are isomorphic but not rigidly isotopic.questions related to the study of the asymptotics for the number of chambers of b ( R d × n \ ∆ d,n ) for fixed n and large d . However, as we will see, the topology of the chambers of the complementof the discriminant is extremely rich and some unexpected structure emerges as d ! ∞ . In order to explain this phenomenon, let us label the chambers of R d × n \ ∆ d,n with thecorresponding isomorphism class of labeled geometric graphs: given a graph G on n verticeswe define W G,d = { P ∈ R d × n \ ∆ d,n | G ( P ) ∼ = G } ⊂ R d × n . In other words, W G,d consists of all the points P ∈ R d × n not on the discriminant whoseassociated geometric graph is isomorphic to G . For small d this set could be a union of severalchambers, but for large d it is an actual chamber, that is a connected open set. This can berephrased by saying that for every graph G on n vertices and for large enough d , the homotopygroup π ( W G,d ) consists of a single element. In fact, as we will show, the same statement istrue for all the homotopy groups, once the group is fixed and d becomes large enough. Theorem 2.
For every k ≥ and for d ≥ k + n + 1 we have π k ( W G,d ) = 0 . Theorem 2 in fact generalizes the result of Maehara [Mae84]. Taking the standard conven-tion that a topological space is said to be ( − -connected provided it is nonempty, Theorem2 for k = − is Maehara’s result that every graph on n vertices can be realized as a geometricgraph in R d , for d ≥ n .Notice that there is a natural sequence of inclusions:(1.2) · · · ֒ −! W G,d ֒ −! W G,d +1 ֒ −! · · · obtained by simply including R d × n into R ( d +1) × n by appending a list of zeroes to the coor-dinates of P . The proof of the Theorem 2 goes through two intermediate steps, which are ofindependent interest: we first prove that for every k ≥ and for d ≥ k + n + 1 the inclusion MARA BELOTTI, ANTONIO LERARIO, AND ANDREW NEWMAN W G,d ֒ −! W G,d +1 induces an injection on the homotopy classes of maps, and then we provethat the inclusion W G,d ֒ −! W G,d + n is homotopic to a constant map. Example 3 (Homotopy groups of the configuration space of n points in R d ) . Let us considerthe graph G consisting of n vertices and no edges. It is easy to see that the correspondingchamber W G,d is homotopy equivalent to the configuration space of n distinct points in R d : W G,d ∼ Conf n ( R d ) . In this case one can compute exactly the homotopy groups of W G,d : for every k ≥ and for d ≥ we have (see [FH01, Chapter 2, Theorem 1.1]) π k ( W G,d ) ≃ π k (Conf n ( R d )) ≃ n − M j =1 π k (cid:18) S d − ∨ · · · ∨ S d − | {z } bouquet of j spheres (cid:19) . Since π k ( S d − ∨ · · · ∨ S d − ) = 0 for d ≥ k + 2 , in this case we immediately see that also π k ( W G,d ) = 0 for d ≥ k + 2 . It is natural at this point to put the sequence of inclusions (1.1) into the infinite dimensionalspace R ∞× n of n –tuples of sequences ( p , . . . , p n ) such that for every j = 1 , . . . , n all but finitelymany elements in the sequence p j are zero: R ∞× n = lim −! R d × n . The definition of geometric graph and discriminant also makes sense in this infinite dimensionalspace, see Section 4.1. The chambers, are now defined as follows: for a given graph G on n vertices, we set W G, ∞ = (cid:8) P = ( p , . . . , p n ) ∈ R ∞× n \ ∆ ∞ ,n (cid:12)(cid:12) G ( P ) ∼ = G (cid:9) . From Theorem 2 we deduce the following.
Theorem 4.
For every graph G , the set W G, ∞ = lim −! W G,d is contractible.
Floer homology of a graph.
Summarizing the picture so far: as d ! ∞ each W G,d becomes eventually k –connected and its direct limit W G, ∞ has no homotopy. Moreover, byHurewicz Theorem, each fixed reduced Betti number of W G,d vanishes for d large enough:more precisely, for every k > there exists d ( k ) > such that b k ( W G,d ) = 0 ∀ d ≥ d ( k ) . But this is not the whole story. Before we continue let us discuss one more, likely familiar,example.
Example 5 (The infinite dimensional sphere) . Let G be the graph consisting of two disjointpoints. Then W G,d ≃ R d × S d − ∼ S d − and W G, ∞ ≃ S ∞ × R d × ( n − ∼ S ∞ . In this case (1.1)become: · · · ֒ −! S d − ֒ −! S d ֒ −! · · · ֒ ! S ∞ = lim −! S d . If now we look at the Betti numbers of S d we see that there is a hole in dimension d that movesto infinity as d ! ∞ and it disappears when d = ∞ . The sphere S ∞ has no cohomology exceptin dimension zero, but still it has cohomology every time we cut it with a finite–dimensionalspace. Here and below, for two topological spaces X and Y we use the symbol X ≃ Y to denote that they arehomeomorphic and X ∼ Y to denote that they are homotopy equivalent. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 5 i · d ( n − · dd ! ∞ d ! ∞ b k ( W G,d ) k Figure 2.
A plot of the Betti numbers of W G,d . The width of each non zerocluster of holes is (cid:0) n (cid:1) + 1 , which is a constant. Each of these clusters is placedat a multiple of d and as d ! ∞ they shift to infinity. The total Betti numberof W G,d , i.e. the blue area, becomes constant for d large enough.The phenomenon described in Example 5 can be interpreted using some extraordinarycohomology theory, in the context of the Leray–Schauder degree and, more generally, of Floerhomology theories (see [Szu92, GG73, Abb97]). This behaviour has also been observed fornon–holonomic loop spaces in Carnot groups, see [AGL15]. In all these examples we aredealing with a sequence of spaces X d whose direct limit X ∞ is contractible, but for every d large enough each space carries the same amount of cohomology, just shifted in its dimension.A more general family of examples where this occurs is the iterated suspension. Example 6 (The iterated suspension) . Let X be a CW-complex and define X d = SX d − ,where SX is the suspension of X : SX = ( X × I ) / ∼ , and the equivalence relation “ ∼ ” is given by: ( x , ∼ ( x , and ( x , ∼ ( x , for all x , x ∈ X . We have a natural sequence of inclusions(1.3) · · · ֒ −! X d ֒ −! X d +1 ֒ −! · · · given by mapping X d ! SX d homeomorphically to X d × { } . We denote by X ∞ = lim −! X d thedirect limit of the sequence of inclusions (6). If X = { x , x } , then X d = S d and X ∞ = S ∞ . For every k the space X d becomes eventually k –connected when d is large enough, and X ∞ is contractible. If one looks at the homology of X d , this is made by a cluster of holes thatshifts to infinity as d ! ∞ . This can be expressed, for example, by looking at the Poincarépolynomial of X d : P X d ( t ) = 1 + t d ( P X ( t ) − . These holes are not present when d = ∞ , but the sum of the Betti numbers of X d is constant: b ( X d ) = P X d (1) ≡ P X (1) = b ( X ) . We will prove that a similar phenomenon happens, for all the spaces W G,d : their reducedcohomology is made of “clusters of holes” that “shift” to infinity as d ! ∞ , see Figure 2. Infact we show also that for G on n vertices there are at most n − such clusters. More precisely,we have the following result. Theorem 7.
For every graph G on n vertices there exist polynomials Q G, , . . . , Q G,n − eachof degree at most (cid:0) n (cid:1) + 1 such that for d ≥ (cid:0) n (cid:1) + 2 the Poincaré polynomial of W G,d is P W G,d ( t ) = 1 + t d − ( n ) − Q G, ( t ) + · · · + t md − ( n ) − Q G,m ( t ) + · · · + t ( n − d − ( n ) − Q G,n − ( t ) . These polynomials only depend on G and not on d . MARA BELOTTI, ANTONIO LERARIO, AND ANDREW NEWMAN
In particular, there exists β ( G ) > such that b ( W G,d ) = P W G,d (1) ≡ n − X ℓ =1 Q G,ℓ (1) = β ( G ) for d large enough (i.e. the sum of the Betti numbers of W G,d becomes a constant, whichdepends on G only). Each polynomial Q G,m corresponds to one of the clusters above and keeps track of the Bettinumbers b k ( W G,d ) with ≤ dm − k ≤ (cid:0) n (cid:1) , i.e. with index k located “near” dm (rememberthat (cid:0) n (cid:1) is a constant in this asymptotic regime). These clusters are the “Floer homologies”of the graph.The proof of Theorem 7 uses a spectral sequence argument: each W G,d can be described asa system of quadratic inequalities and one can use the technique developed in [Agr88a, AL12]for the study of its Betti numbers. In this case, as d ! ∞ , the spectral sequence that weneed to consider converges at the second step (i.e. E = E ∞ ). One can prove that both E and the second differential d have an asymptotic stable shape: as d ! ∞ the nonzero part ofthe spectral sequence looks like a skinny table where there is no room for higher differentialsand the nonzero elements of this table are located near the rows which are labeled by indiceswhich are multiples of d (see Figure 3 below).An interesting question arising from Theorem 7 is, for a given graph G on n vertices, tocompute the number β ( G ) = lim d ! ∞ b ( W G,d ) . We call this number the
Floer number of the graph G . This number is a graph invariant, aswell as the polynomials from Theorem 7. Table 1 shows the value of this number for all thepossible graphs on vertices, but the general case is still mysterious. There is more discussionabout some of the details of Table 1 in Example 65.We can make some observations in a few cases that suggest conjectures about β ( G ) forgeneral graphs. In the case of graphs on four or fewer vertices, we see that the Poincarépolynomial of W G,d has a general form with exponents given in terms of d . Once d is largeenough that G has a realization as a geometric graph in R d we see that the Poincaré polynomialis determined by the general form. From this we conjecture that for every graph G there isa general form of the Poincaré polynomial of W G,d with exponents given in terms of d thatis valid as long as d is large enough that W G,d is nonempty. This would imply that as soonas G can be realized as a geometric graph in R d , b ( W G,d ) = β ( G ) . In the case of graphsrealizable in R , we would have that β ( G ) counts the number of chambers of W G, . To see thisrecall that in the d = 1 case R × n \ ∆ ,n is a disjoint union of polyhedra so every chamber iscontractible; therefore homology can only exists in degree zero. A first step toward the proofof this conjecture would be to prove that all the differentials of the spectral sequence that weuse in the proof of Theorem 7 are zero for all d ≥ n . Example 8 (Betti numbers of the configuration space of n points in R d ) . The Poincarépolynomial of
Conf n ( R d ) for d ≥ is given by (see [FH01, Chapter V, Corollary 1.4]) P Conf n ( R d ) ( t ) = n − Y j =1 (1 + jt d − ) . Consequently b (Conf n ( R d )) = P Conf n ( R d ) ( t )(1) ≡ n ! . In other words, for the graph G consistingof n disjoint points we have β ( G ) = n ! . RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 7
Graph Poincaré polynomial β ( G ) Labeled copies t d − + 11 t d − + 6 t d −
24 1 t d − + 2 t d − t d − t d − + t d − t d − t d − + t d − + t d − t d − t d − t d − + t d − + t d − t d − Table 1.
Poincaré polynomials and Floer numbers for W G,d for G on 4 ver-tices. For every graph “labeled copies” refers to the number of isomorphismclasses of labeled graphs with the same unlabeled graph.1.3. The case n ! ∞ . Concerning the other asymptotic regime, the first case of interest iswhen d = 1 : here ∆ ,n is a hyperplane arrangement, since each quadric {| p i − p j | = 1 } ⊂ R × n is the union of the two hyperplanes { p i − p j = 1 } and { p i − p j = − } . It turns out that thenumber of chambers of the complement of such a hyperplane arrangement, that is the numberof rigid isotopy classes of R –geometric graphs on n vertices, equals the number of labeledsemiorders of [ n ] . Using techniques from analytic combinatorics, we will prove the followingtheorem. Theorem 9.
The number of rigid isotopy classes of R –geometric graphs on n vertices equals: b (cid:0) R × n \ ∆ ,n (cid:1) = 1 n · r log · ne log ! n (cid:16) O ( n − ) (cid:17) . It is in fact possible also to compute the asymptotics of ,n as n ! ∞ , we do this inTheorem 50. It is remarkable that the two numbers b ( R × n \ ∆ ,n ) and ,n have the sameasymptotic, up to a multiplicative constant :(1.4) b ( R × n \ ∆ ,n ) = 8 e · ,n (cid:16) O ( n − ) (cid:17) . The case when d ≥ is more delicate to handle. This is in large part because the dis-criminant in higher dimensions is an arrangement of quadrics rather than an arrangment ofhyperplanes. For this general case we will prove the following upper and lower bounds for thenumber of rigid isotopy classes and isomorphism classes. Theorem 10.
For d ≥ fixed and for n ≥ d + 1 one has the following bounds: The constant e is approximatively 7.36. MARA BELOTTI, ANTONIO LERARIO, AND ANDREW NEWMAN (cid:18) d + 1) e (cid:19) dn n dn ≤ d,n ≤ b (cid:16) R d × n \ ∆ d,n (cid:17) ≤ dn (cid:18) e d (cid:19) dn n dn Note that these bounds imply that d,n and b (cid:0) R d × n \ ∆ d,n (cid:1) become equivalent at thelogarithmic scale, giving the following analogue of (1.3): log b ( R d × n \ ∆ d,n ) = (log d,n ) (1 + o (1)) as n ! ∞ . For the proof of the upper bound we will use the fact that ∆ d,n is a real algebraic set: usingAlexander–Pontryagin duality, we bound the topology of the complement of the discriminant R d × n \ ∆ d,n by studying the topology the one-point compactification of ∆ d,n which we denoteby ˆ∆ d,n . This one-point compactification can also be described in an algebraic way and studiedusing [Mil64].For the lower bound, our proof is an higher dimensional version of [MM14], where theauthors prove that in the case d = 2 there exists a constant α > such that ,n ≥ α n n n .Finally we observe that one of our central questions here has been to establish bounds onthe number of connected components of R d × n \ ∆ d,n and more generally on b k ( R d × n \ ∆ d,n ) for values of k that are fixed but not necessarily zero. Moving from these low-degree Bettinumbers, it is also interesting to look at the Betti numbers of R d × n \ ∆ d,n in degrees close to nd . Because of Alexander duality to determine the ( dn − k ) –th Betti number of R d × n \ ∆ d,n it suffices to determine the ( k − st Betti number of ˆ∆ d,n . The next result gives informationon some of the top Betti numbers; this result holds in general with no restriction that d and n be sufficiently large. Theorem 11.
For every d , n , ˆ∆ d,n is ( n + d − -connected, but not ( n + d − -connected.By Alexander–Pontryagin duality this implies that H nd − n − d +1 ( R d × n \ ∆ d,n ) = 0 , but all highercohomology groups of R d × n \ ∆ d,n vanish. The description of ∆ d,n as a union of quadrics allows us to use a Mayer-Vietoris typeargument, and the theorem follows by an application of a generalized nerve lemma.2. Preliminaries
Geometric graphs.
There are several different notions of geometric graphs in the lit-erature. The type of geometric graph we consider here are also sometimes called intersectiongraphs or space graphs. Formally the definition for geometric graph we use here is the follow-ing.
Definition 12 (Geometric graph) . Given a point P ∈ R d × n we denote by G ( P ) the labeledgraph whose vertices and edges are, respectively: V ( G ( P )) = { (1 , p ) , . . . , ( n, p n ) } and E ( G ( P )) = { (( i, p i ) , ( j, p j )) | i < j, k p i − p j k < } . We say that G ( P ) is an R d –geometric graph. If a labeled graph G is isomorphic to G ( P ) forsome P ∈ R d × n , we say that G is realizable as an R d –geometric graph.We say that the geometric graph G ( P ) is nondegenerate if P / ∈ ∆ d,n , where ∆ d,n = { P ∈ R d × n | there exist ≤ i < j ≤ n such that k p i − p j k = 1 } ⊂ R d × n . RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 9
Remark . The reason for considering in our definition the list of pairs { (1 , p ) , . . . , ( n, p n ) } as the set of vertices of G ( P ) , instead of the list { p , . . . , p n } , is just formal. In other settingsit may be more natural to take p , ..., p n to be distinct points in R d and then to define agraph with vertex set { p , ..., p n } and edges ( p i , p j ) provided that k p i − p j k < . This isthe approach taken by Maehara [Mae84] who studies the sphericity of graphs, the minimumdimension d in which a graph may be realized as a geometric graph in R d with vertices given bydistinct points. For us it makes sense to associate graphs on n vertices in R d to points of R d × n therefore the actual points in R d may not all be unique from one another. The two-coordinateapproach to describe the vertices allows for such repetition and naturally associates each pointin R d \ ∆ d,n to unique labeled graph.We will consider the following notions of equivalence of geometric graphs. Definition 14.
Let G ( P ) , G ( P ) be two R d –geometric graphs on n vertices, with P , P ∈ R d × n . We will say that they are isomorphic if they are isomorphic as labeled geometric graphs.Moreover, if they are both nondegenerate, we will say that they are rigidly isotopic if thereexists a continuous curve P : [0 , ! R d × n \ ∆ d,n such that P (0) = P and P (1) = P . Notice that, since ∆ d,n is an algebraic set, its complement is a semialgebraic set and itspath-components are the same as its connected components. Therefore two nondegenerategeometric graphs G ( P ) , G ( P ) are rigidly isotopic if and only if P and P belong to the sameconnected component of R d × n \ ∆ d,n .Let us introduce the following notation:(2.1) d,n := { isomorphism classes of geometric graphs on n vertices in R d } . Notice that in the definition of d,n we did not assume the nondegeneracy of the graphs.However, the following lemma proves that isomorphism classes of nondegenerate graphs arethe same as all isomorphism classes as in (2.1) (see also [ALL20, Lemma 2.2] for an analogousstatement in the more general context of geometric complexes).
Lemma 15.
For every P ∈ ∆ d,n there exists e P ∈ R d × n \ ∆ d,n such that G ( P ) and G ( e P ) areisomorphic as labeled graphs.Proof. Take P = ( p , . . . , p n ) ∈ ∆ d,n and for ǫ > small enough consider: e P := (1 + ǫ ) P. Then for ǫ small enough we have: (( i, p i ) , ( j, p j )) ∈ E ( G ( P )) ⇐⇒ (( i, (1 + ǫ ) p i ) , ( j, (1 + ǫ ) p j )) ∈ E ( G ( e P )) , which proves that G ( P ) and G ( e P ) are isomorphic as labeled graphs. Moreover, again for ǫ small enough, we have that e P / ∈ ∆ d,n . (cid:3) By definition, for nondegenerate graphs we haverigidly isotopic = ⇒ isomorphic , this means that isomorphism classes are union of rigid isotopy classes. The number of rigidisotopy classes of geometric graphs on n vertices in R d is given b ( R d × n \ ∆ d,n ) and Lemma 15implies we can compare the number of rigid isotopy classes with the number of isomorphismclasses: d,n ≤ b ( R d × n \ ∆ d,n ) . Below we will prove, as Corollary 45, that for d ≥ n + 1 , two R d –geometric graphs on n verticesare isomorphic if and only if they are rigid isotopic, i.e. that d,n = b ( R d × n \ ∆ d,n ) for d ≥ n + 1 . We will deal with the asymptotic of d,n and b ( R d × n \ ∆ d,n ) in the case d fixed and n ! ∞ in Section 5.2.2. Alexander duality and the discriminant.
As we are interested in the topology of R d × n \ ∆ d,n , the topology of ∆ d,n should play an important role as well and in some case itwill be easier to study. The key tool to connect the topology of the two is Alexander duality.Given a compact, locally contractible, nonempty and proper subspace X of the N -dimensionalsphere S N , Alexander duality [Hat02, Corollary 3.45] provides a way to study the topologyof X from the topology of S N \ X . Namely for every k we have the following isomorphismsbetween the homology of X and the cohomology of S N \ X ˜ H k ( X ) ∼ = ˜ H N − k − ( S N \ X ) . Remark . If we are working with Z coefficients, as we will throughout, and with a space X ⊂ S N with finitely generated homology, we can relate the Betti numbers of X with thoseof its complement in the sphere (i.e. we can freely identify homology and cohomology). Whenworking with compact semialgebraic sets in the sphere, this last requirement will be satisfiedthanks to [BCR98, Theorem 9.4.1].In order to use this duality in the present setting we work in the one-point compactificationof ∆ d,n ⊂ R d × n which we will denote by ˆ∆ d,n ⊂ S d × n . Now ˆ∆ d,n contains the point at infinityso S d × n \ ˆ∆ d,n = R d × n \ ∆ d,n .The discriminant itself is a union of quadratic hypersurfaces of the form(2.2) ∆ i,jd,n = { ( x , ..., x n ) ∈ R d × n | k x i − x j k = 1 } . Each of these quadrics is topologically S d − × R d × ( n − and establishing bounds on the topBetti number of ˆ∆ d,n establishes bounds on the number of rigid isotopy classes of R d -geometricgraphs on n vertices. We take such an approach in Section 5.3.The discriminant itself is also interesting from the perspective of graph theory. One definesthe unit-distance graph on R d to be the graph whose vertex set is all of R d and two vertices x and y are connected by an edge if and only if k x − y k = 1 . As a remark, the unit distancegraph on R d , especially in the case d = 2 , is studied in the case of the well known Hadwiger–Nelson problem to establish the chromatic number of the plane; that is the chromatic numberof the unit distance graph on R .In our situation ∆ d,n is connected to homomorphisms from graphs on n vertices to the unitdistance graph on R d . Indeed each quadric ∆ i,jd,n is the space of images of homomorphismsfrom the graph on [ n ] with an edge between vertex i and vertex j . More generally, we denotean intersection of quadrics by ∆ Gd,n for a graph G = ([ n ] , E ) by(2.3) ∆ Gd,n := \ ( i,j ) ∈ E ∆ ( i,j ) d,n . Then ∆ Gd,n is the space of images of homomorphisms from the graph G to the unit distancegraph on R d . Putting all of this together we have that ∆ d,n itself is the set of all points in R d × n that are the image of a graph homomorphism for a nonempty graph on n vertices. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 11
Semialgebraic triviality.
A most useful technical tool that we will use in the paper isTheorem 18, which relates the structure of semialgebraic families and their homotopy.
Definition 17.
Let
S, T and T ′ be semialgebraic sets, T ′ ⊂ T , and let f : S ! T be acontinuous semialgebraic mapping. A semialgebraic trivialization of f over T ′ , with fibre F ,is a semi-algebraic homeomorphism θ : T ′ × F ! f − ( T ′ ) , such that f ◦ θ is the projectionmapping π : T ′ × F ! T ′ . We say that the semialgebraic trivialization θ is compatible with asubset S ′ of S if there is a subset F ′ of F such that θ ( T ′ × F ′ ) = S ′ ∩ f − ( T ′ ) . Theorem 18 (Semialgebraic triviality) . Let S and T be two semialgebraic sets, f : S ! T asemialgebraic mapping, ( S j ) j =1 ,...,q a finite family of semialgebraic subsets of S . There exista finite partition of T into semialgebraic sets T = S rl =1 T l and, for each l , a semialgebraictrivialization θ l : T l × F l ! f − ( T l ) of f over T l compatible with S j , for j = 1 , . . . , q , i.e,there exists F jl ⊂ F l such that θ l ( T l × F jl ) = S j ∩ f − ( T l ) .Proof. This is [BCR98, Theorem 9.3.2]. (cid:3)
Corollary 19.
Let S be a semialgebraic set and f : S ! R be a continuous semialgebraicfunction. Then for ǫ > small enough the inclusion { f ≥ ǫ } ֒ − ! { f > } is an homotopy equivalence.Proof. Thanks to semialgebraic triviality, we know that for ǫ sufficiently small there exists T l such that (0 , ǫ ] ⊂ T l . Then, we define a map H : { f > } × [0 , ! { f > } H ( x, t ) = ( x if f ( x ) (0 , ǫ ) θ l ((1 − t ) · ( π ◦ θ − l ) + tǫ, π ◦ θ − l ) if f ( x ) ∈ (0 , ǫ ] and this is a continuous function because the two expressions agree on f − ( ǫ ) × [0 , and bothof them are continuous on closed subsets. The map H is therefore a deformation retraction of { f > } onto f ≥ ǫ . (cid:3) Systems of quadratic inequalities.
In this section we recall a general constructionfrom [Agr88a, AL12] for computing the Betti numbers of the set of solutions of a system ofquadratic inequalities.To start with, let h : R N +1 ! R k +1 be a quadratic map , i.e. a map whose components h = ( h , . . . , h k ) are homogeneous quadratic forms. Let also K ⊆ R k +1 be a closed convexpolyhedral cone (centered at the origin). We are interested in the Betti numbers of(2.4) V = h − ( K ) ∩ S N ⊂ R N +1 Such a set V can be seen as the set of solutions of a system of homogeneous quadratic inequal-ities on the sphere S N : in fact, since K is polyhedral, we have K = { η ≤ , . . . , η ℓ ≤ } for some linear forms η , . . . , η ℓ ∈ ( R k +1 ) ∗ and V = { η h ≤ , . . . , η ℓ h ≤ } ∩ S N , which is a system of quadratic inequalities. (Here given a linear form η ∈ ( R k +1 ) ∗ and aquadratic map h : R N +1 ! R k +1 we simply denote by ηh the composition of the two). Everyhomogeneous system can be written in this way. We denote by K ◦ the polar of K , i.e. K ◦ = { η ∈ ( R k +1 ) ∗ | η ( y ) ≤ ∀ y ∈ K } , and we set: Ω = K ◦ ∩ S k , where S k denotes the unit sphere in ( R k +1 ) ∗ , with respect to a fixed scalar product. The scalarproduct on R k +1 plays no role, but we will also use a scalar product on R N +1 , by choosinga positive definite quadratic form g on R N +1 . This scalar product will play a role, and wedenote it by h· , ·i g ; it is defined by h x, x i g = g ( x ) for all x ∈ R N +1 ; for practical purposes wewill omit the “ g ” subscripts when not needed.Once the scalar product on R N +1 has been fixed, we can associate to a quadratic form q : R N +1 ! R a real symmetric matrix, via the equation:(2.5) q ( x ) = h x, Qx i g ∀ x ∈ R N +1 . We will often use small letters for the quadratic form and capital letters for the associatedmatrices.Accordingly we can define the eigenvalues of q (with respect to g ) as those of Q : λ ( q ) ≥ · · · ≥ λ N +1 ( q ) . The eigenvalues (and the eigenvectors) of q depend therefore on the chosen scalar product,but again we will omit this dependence in the notation if not needed.2.4.1. The index function.
Using the above notation, we will denote by ind + ( q ) = ind + ( Q ) the positive inertia index, i.e. the number of positive eigenvalues of the symmetric matrix Q .Note that the index of a quadratic form does not depend on the chosen scalar product.When we are in the situation as above, i.e. when we are given a homogeneous quadraticmap h : R N +1 ! R k +1 , for every covector η ∈ ( R k +1 ) ∗ we can consider the composition ηh ,which is a quadratic form. For every natural number j ≥ we define the sets: Ω j = { ω ∈ Ω | ind + ( ωh ) ≥ j } . These sets are open and semialgebraic, as it is easily verified. Moreover, these sets do not depend on the choice of the scalar product g .Observe now that over each set Ω j \ Ω j +1 the function ind + ≡ j is constant, i.e. the numberof positive eigenvalues of the corresponding matrices is j and there exists a natural vectorbundle P j ⊆ Ω j \ Ω j +1 × R N +1 R j P j Ω j \ Ω j +1 whose fiber over a point ω is the positive eigenspace of ωH = ω H + · · · + ω k H k . In fact thisbundle is the restriction of a more general bundle over the set D j = { ω | λ j ( ωH ) = λ j +1 ( ωH ) } i.e. the set where the j -th eigenvalue of ωH is distinct from the ( j + 1) -th. We still denotethis bundle by P j ⊂ D j × R N +1 :(2.6) R j P j D j RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 13
Here the fiber over a point ω ∈ D j consists of the eigenspace of ωH associated to the first j eigenvalues (this is well defined); however note that the bundle over D j depends on the choiceof the scalar product (since D j itself depends on this choice).We denote by ν j ∈ H ( D j ) the first Stiefel-Whitney class of this bundle. The following Lemma will be useful for us: Lemma 20.
The cup-product with the class ν j defines a map: (2.7) ( · ) ` ν j : H ∗ (Ω j , Ω j +1 ) ! H ∗ +1 (Ω j , Ω j +1 ) . Proof.
To see that the previous cup product is well defined observe that we can write: Ω j = A ∪ B, A = Ω j +1 , B = Ω j ∩ D j . In fact, if a point ω belongs to Ω j then, either ω ∈ Ω j +1 or ind + ( ωH ) = j and consequently ω ∈ Ω j ∩ D j . Since both A and B are open, by excision we get: H ∗ (Ω j , Ω j +1 ) ≃ H ∗ (Ω j ∩ D j , Ω j +1 ∩ D j ) . In particular, in order to see that (20) is well defined, it is enough to see that ( · ) ` ν j : H ∗ (Ω j ∩ D j , Ω j +1 ∩ D j ) ! H ∗ +1 (Ω j ∩ D j , Ω j +1 ∩ D j ) is well defined. Let ψ ∈ C k (Ω j ) a singular cochain representing a cohomology class in H k (Ω j ∩ D j , Ω j +1 ∩ D j ) and φ j ∈ C ( D j ) a cochain representing ν j . The cup product of ψ and φ j isdefined on a singular chain σ : [ v , . . . , v k +1 ] ! Ω j ∩ D j in the usual way: ( ψ ` φ j )( σ ) = ψ ( σ | [ v ,...,v k ] ) φ j ( σ | [ v k ,v k +1 ] ) , from which we see that ψ ` φ j vanishes on C k +1 (Ω j +1 ∩ D j ) and defines an element of H k +1 (Ω j ∩ D j , Ω j +1 ∩ D j ) . (cid:3) Since we have inclusions Ω j ⊇ Ω j +1 ⊇ Ω j +2 , we also consider the connecting homomophisms ∂ : H ∗ (Ω j +1 , Ω j +2 ) ! H ∗ +1 (Ω j , Ω j +1 ) of the long exact sequence for the triple (Ω j , Ω j +1 , Ω j +2 ) . We summarize the directions of these homomorphism in the following non-commutative diagram of maps: H i (Ω j +1 , Ω j +2 ) H i +1 (Ω j , Ω j +1 ) H i +1 (Ω j +1 , Ω j +2 ) H i +2 (Ω j , Ω j +1 ) ∂ ( · ) ` ν j +1 ( · ) ` ν j ∂ Remark . Let ∂ : H i ( X, Y ) ! H i +1 ( Z, X ) be the boundary operator in the exact sequenceof the triple ( Z, X, Y ) , where all spaces are open. Following [Hat02, pag. 201], thanks tothe fact that we are working with Z coefficients, we have that ∂ ([ φ ]) = [ φ ◦ π ◦ δ ] where δ : C i +1 ( Z, X ) ! C i ( X ) is the boundary operator and π : C i ( X ) ! C i ( X, Y ) is the projectionoperator. Let us also consider ˜ X ⊂ A both open and such that ( X ∩ ˜ X ) ∪ Y is open. If wetake the relative cup product H i ( X, Y ) × H ( ˜ X ) ! H ( X, Y ) as defined in Lemma 20, then this coincides with the cup product H i ( X, Y ) × H ( A ) ! H ( X, Y ) as defined in Lemma 20, meaning that given a ∈ H i ( X, Y ) and b ∈ H ( A ) then a ` b = a ` r ∗ ( b ) where r ∗ is just the restriction. In the same way, if we suppose that ˜ Z ⊂ A , we canrepeat a similar reasoning for the cup product H i ( Z, X ) × H ( ˜ Z ) ! H i +1 ( Z, X ) . Now, given [ γ ] ∈ H ( A ) , we claim that(2.8) ∂ ([ a ] ` [ γ ]) = ∂ [ a ] ` [ γ ] . At the level of cochains if we take c i +1 ∈ C i +1 ( Z, X ) a singular chain we have ∂ ( a ` γ )( c i +1 ) =( a ` γ )( π ◦ δc i +1 ) . Reasoning as in the proof of Lemma 3.6 in [Hat02, pag. 206], we see that ( a ` γ )( π ◦ δc i +1 ) = ∂a ` γ + ( − i a ` δ ∗ ( γ ) and (21) follows from δ ∗ ( γ ) = 0 .2.4.2. The spectral sequence.
For the computation of the Betti numbers of V , defined in (4)we will need the following result, which is an adaptation from [Agr88a, AL12]. Clearly thecomputation of the cohomology of V is equivalent to that of S N \ V , by Alexander duality, andin [Agr88a, AL12] it is introduced a spectral sequence for computing the latter.The delicate part here is that in [Agr88a] the spectral sequence is defined for nondegenerate systems of quadrics, i.e. for systems such that the map h is transversal to the cone K , in thesense of [AL12]; in [Ler11] the spectral sequence is defined also for degenerate systems, but thesecond differential is not computed explicitly, and in [AL12] it is defined also for degeneratesystems, and the second differential is explicitly computed, but the solutions are studied in theprojective space rather than the sphere. Since in our case the system of quadratic inequalitiesmight be degenerate, we will need to prove the existence of such a spectral sequence and tocompute its second differential. Theorem 22.
Let V = h − ( K ) ∩ S N be defined by a system of quadratic inequalities, asabove. There exists a cohomology spectral sequence ( E r , d r ) r ≥ converging to H ∗ ( S N \ V ; Z ) such that: (1) the second page of the spectral sequence is given, for j > , by E i,j = H i (Ω j +1 , Ω j +2 ; Z ) . For j = 0 , the elements of the second page of the spectral sequence fit into a long exactsequence: (2.9) · · · ! H i (Ω ; Z ) ! E i, ! H i (Ω , Ω ; Z ) ( · ) ⌣ν −! H i +1 (Ω ; Z ) ! · · · . (2) for j ≥ the second differential d i,j : H i (Ω j +1 , Ω j +2 ) ! H i +2 (Ω j , Ω j +1 ) is given by d i,j ξ = ∂ ( ξ ` ν j +1 ) + ∂ξ ` ν j . Proof.
The proof proceeds similar to [AL12, Theorem 25 and Theorem 28], using a regular-ization process and taking the limit over the regularizing parameter. More precisely, let q bea positive definite quadratic form, chosen as in [AL12, Lemma 13], and for t > consider theset: B ( t ) = { ( ω, x ) ∈ Ω × S N | ωh ( x ) − tq ( x ) ≥ } . The choice of q as in [AL12, Lemma 13] makes the map ω ωh − tq nondegenerate withrespect to K and will allow to compute the second differential of our spectral sequence. Bysemialgebraic trivilality, for t > small enough the set B ( t ) is homotopy equivalent to B = { ( ω, x ) ∈ Ω × S N | ωh ( x ) > } . RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 15
Moreover the projection on the second factor (i.e. p : Ω × S n ! S N ) restricts to a homotopyequivalence B ∼ p ( B ) = S N \ V , see [Ler11, Section 3.2]. Therefore, for t > small enough H ∗ ( S N \ V ) ≃ H ∗ ( B ( t )) . We consider now the Leray spectral sequence ( E r [ t ] , d r [ t ]) r ≥ of the map p t := p | B ( t ) : B ( t ) ! Ω . This spectral sequence converges to the cohomology of B ( t ) , For the first part of the statement, the structure of E i,j in the case j > is proved in[Ler11, Section 3.2], as follows. If t < t then B ( t ) ֒ − ! B ( t ) is an homotopy equivalence and p t | B ( t ) = p t . For < t < t < δ the inclusion defines a morphism of filtered differentialgraded modules φ ( t , t ) : ( E [ t ] , d [ t ]) ! ( E [ t ] , d [ t ]) turning { E [ t ] } t into an inverse system and thus { ( E r [ t ] , d r [ t ]) } t into an inverse system ofspectral sequences. Then, we can define a new spectral sequence ( E r , d r ) := lim − t { ( E r [ t ] , d r [ t ]) } . The proof shows that for j > we have E i,j [ t ] = H i (Ω n − j [ t ] , Ω n − j − [ t ]; Z ) where the sets Ω k [ t ] are defined by: Ω k [ t ] := { w ∈ Ω | i − ( w · h − tg ) ≤ k } . Moreover we also have that for j > the isomorphism φ ( t , t ) is just the homomorphisminduced in cohomology by the inclusion Ω j [ t ] ⊆ Ω j [ t ] and that E i,j = lim − t E i,j [ t ] = H i (Ω j , Ω j +1 ; Z ) . Thanks to this, by semialgebraic triviality φ ( t , t ) is an isomorphism for < t < t < δ with δ sufficiently small and therefore also φ ∞ ( t , t ) is an isomorphism, assuring the convergenceof ( E r , d r ) to B ( t ) (see also the proof of [AL12, Theorem 25] for more details on this point).Let us call e ∗ t : H ∗ (Ω j , Ω j +1 ; Z ) ! H ∗ (Ω n − j ( t ) , Ω n − j − ( t ); Z ) the isomorphism induced by the inclusion. From now on we choose our scalar product on R N +1 to be g = q , in such a way that the matrix associated to q through the polarizationidentity (2.4) is the identity matrix.For the case j = 0 we know thanks to [Agr88b] that there exist a long exact sequence · · · ! H i (Ω n − [ t ]; Z ) ! E i, [ t ] !! H i (Ω n − [ t ] , Ω n − [ t ]; Z ) ( · ) ⌣ν −! H i +1 (Ω n − [ t ]; Z ) ! · · · . We can pass to the inverse limit of these long exact sequences respect to t in the obvious wayobtaining a long exact sequence · · · ! H i (Ω ; Z ) ! E i, ! H i (Ω , Ω ; Z ) ( · ) ⌣ν −! H i +1 (Ω ; Z ) ! · · · , where we have used the fact that ( e ∗ ) − ◦ (( · ) ` ν j ) ◦ e ∗ = ( · ) ` ν j (we will get back to thispoint later). In this long exact sequence we are still using ν because we chose our scalar product to be g . Same forthe definition of d ( t ) , where we used ν j . This proves point (1) of the statement. For what concerns the differential, thanks to [Agr88b,Theorem 3] we know that the second differential d [ t ] of the spectral sequence ( E r [ t ] , d r [ t ]) with d i,j [ t ] : H i (Ω n − j − [ t ] , Ω n − j − [ t ]; Z ) ! H i +2 (Ω n − j [ t ] , Ω n − j − [ t ]; Z ) has the form d i,j [ t ] ξ = ∂ t ( i ∗ t ) − ( i ∗ t ξ ` ν j +1 ) + ( i ∗ t ) − ( i ∗ t ∂ t ξ ` ν j ) , where ∂ t : H i (Ω n − j − [ t ] , Ω n − j − [ t ]; Z ) ! H i +1 (Ω n − j [ t ] , Ω n − j − [ t ]; Z ) is the connecting homomorphism in the exact sequence of the triple (Ω n − j [ t ] , Ω n − j − [ t ] , Ω n − j − [ t ]) and the map i ∗ t : H i (Ω n − j [ t ] , Ω n − j − [ t ]; Z ) ! H i (Ω n − j [ t ] ∩ D j , Ω n − j − ( t ) ∩ D j ; Z ) is the map induced by the inclusion; this map is an isomorphism by excision.The second differential for j > of our new spectral sequence ( E i,jr , d r ) is given by d i,j :=( e ∗ t ) − ◦ d i,j ( t ) ◦ e ∗ t . More explicitly, d i,j = ∂ ( i ∗ t ◦ e ∗ t ) − (( i ∗ t ◦ e ∗ t ) ξ ` ν j +1 ) + ( i ∗ t ◦ e ∗ t ) − (( i ∗ t ◦ e ∗ t ) ∂ξ ` ν j ) thanks to the naturality of the connecting homomorphism.Let us now consider the following diagram where all the maps are inclusions: (Ω j , Ω j +1 )(Ω n − j − [ t ] ∩ D j , Ω n − j − [ t ] ∩ D j ) (Ω j ∩ D j , Ω j +1 ∩ D j ) j t e t ◦ i t i and all the induced homomorphisms in cohomology are isomorphisms. We can write d i,j = ∂ ( j ∗ t ◦ i ∗ ) − (( j ∗ t ◦ i ∗ ) ξ ` ν j +1 ) + ( j ∗ t ◦ i ∗ ) − (( j ∗ t ◦ i ∗ ) ∂ξ ` ν j ) == ∂ ( i ∗ ) − ( j ∗ ) − ( j ∗ t ( i ∗ ξ ) ` j ∗ t ◦ ν j +1 ) + ( i ∗ ) − ( j ∗ t ) − ( j ∗ t ( i ∗ ∂ξ ) ` j ∗ t ◦ ν j ) == ∂ ( i ∗ ) − ( i ∗ ξ ` ν j +1 ) + ( i ∗ ) − ( i ∗ ∂ξ ` ν j ) where the pull-back property of the pullback in the third equality holds true because in thatcase it is just the standard cup-product. Because of how we defined the cup product in Lemma20, we have the claim. (cid:3) Analytic Combinatorics.
In order to study the asymptotic of the number of isotopyclasses of geometric graphs on the real line we will need some tools from analytic combinatorics.For a full introduction to the topic see [FS09]. Given a generating function G ( x ) = P ∞ n =0 a n x n of a sequence a n we want to study the asymptotic of such a sequence. There are varioustechniques to do this. Definition 23.
We say that a sequence { a n } is of exponential order K n which we abbreviateas a n ⊲⊳ K n if and only if lim sup | a n | n = K .If we have a n ⊲⊳ K n then a n = K n θ ( n ) with lim sup | θ ( n ) | n = 1 . The term θ ( n ) is calledsubexponential factor. In order to study the subexponential factor θ ( n ) we should look at thesingularities of the generating function. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 17
Definition 24.
Given two numbers φ , R with R > and < φ < π , the open domain D ( φ, R ) is defined as D ( φ, R ) := { z | | z | < R, z = 1 , | arg ( z − | > φ } . A domain of this type is called D − domain.Denoting by S the set of all meromorphic functions of the form S := { (1 − z ) − α | α ∈ R } , we recall the next result [FS09, Theorem VI.4], which we will need in the sequel. Theorem 25.
Let G ( z ) be an analytic function at with a singularity at ζ , such that G ( z ) can be continued to a domain of the form ζ · D , for a D − domain D , where ζ · D is theimage of D by the mapping z ! ζz . Assume there exists two functions σ, τ, where σ is afinite linear combination of elements in S and τ ∈ S such that G ( z ) = σ (cid:18) zζ (cid:19) + O ( τ (cid:18) zζ (cid:19) ) as z ! ζ in ζ · D . Then the coefficients of G ( z ) satisfy the asymptotic estimate a n = ζ − n σ n + O ( ζ − n τ ∗ n ) where σ ( z ) = P ∞ n =0 σ n z n and τ ∗ n = n α − , if τ ( z ) = (1 − z ) − α .Remark . For a later use, we record the following. The Newton Binomial Series is definedby (1 − z ) − α = ∞ X n =0 b n z n where b n = (cid:18) n + α − n (cid:19) . Using (cid:18) n + α − n (cid:19) = Γ( n + α )Γ( α )Γ( n + 1) , we get the following asymptotic for its coefficients: b n = n α − Γ( α ) (cid:18) O (cid:18) n (cid:19)(cid:19) . Homology of the chambers and the Floer number
Graphs and sign conditions.
Recall that given a graph G on n vertices we have defined W G,d = { P ∈ R d × n \ ∆ d,n | G ( P ) ∼ = G } ⊂ R d × n . In other words, W G,d consists of all the points P ∈ R d × n not on the discriminant whosecorresponding graph is isomorphic to G . For small d this set could be a union of severalchambers, but for large d it is an actual chamber (a connected open set).Now we introduce an alternative notation for labelling the sets W G,d . For every ≤ i For every G graph on n vertices there exists a sign condition σ = σ ( G ) such that W G,d = W σ,d . Viceversa, for every σ there exists G ( σ ) such that W σ,d = W G,σ . In other words,the signs of the family of quadrics { q ij } ≤ i In this section we study the asymptotic distributionof the Betti numbers of the chambers. Before giving the main result, we will need someintermediate steps.3.2.1. Some preliminary reductions. Using Lemma 28 we can immediately switch from thegraph labelling to the sign condition one and given G there exists σ such that W G,d = W σ,d .In this way we describe the chamber we are interested in with a system of quadratic inequalities,and we will take advantage of this description.Our first step is to replace W σ,d with another set which has the same homology and which iscompact. To start with, we have W σ,d = { sign ( q ij ) = σ i,j , ∀ ≤ i < j ≤ n } , with the quadrics q ij : R d × n ! R defined above. Since σ is fixed, it will be convenient for us to define the newquadrics: s ij = σ ij q ij and h ij ( x, z ) = σ ij ( k x i − x j k − z ) . We set N = nd and k = (cid:0) n (cid:1) and for every ǫ > consider the set: W σ,d ( ǫ ) = { [ x ] ∈ R P N | h ij ( x, z ) ≥ ǫz ∀ ≤ i < j ≤ n, k x k ≤ ǫ − z } ⊆ R P N . Notice that W σ,d ( ǫ ) ∩ { z = 0 } = ∅ , because if z = 0 then the last inequality defining W σ,d ( ǫ ) forces x = 0 . Therefore, in the affine chart { z = 1 } the set W σ,d ( ǫ ) can be described as W σ,d ( ǫ ) = { s ij ( x ) ≥ ǫ ∀ ≤ i < j ≤ n, k x k ≤ ǫ − } and can be identified with a subset of W σ,d . Proposition 29. For every d > there exists ǫ ( d ) such that for all ǫ < ǫ ( d ) the inclusion W σ,d ( ǫ ) ֒ −! W σ,d induces an isomorphism on the homology groups. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 19 Proof. Observe that { W σ,d ( ǫ ) } ǫ> is an increasing family of compact sets such that: [ ǫ> W σ,d ( ǫ ) = W σ,d . Therefore:(3.3) H ∗ ( W σ,d ) = lim −! { H ∗ ( W σ,d ( ǫ )) } . On the other hand, consider the semialgebraic set S = { ( x, ǫ ) ∈ R d × n × R | s ij ( x ) ≥ ǫ ∀ ≤ i < j ≤ n, ǫ k x k ≤ } together with the semialgebraic map f : S ! R given by ( x, ǫ ) ǫ . For ǫ > we havethat W σ,d ( ǫ ) = f − ( ǫ ) and, by Semialgebraic Triviality, there exists ǫ ( d ) > such that for all ǫ < ǫ < ǫ ( d ) the inclusion W σ,d ( ǫ ) ֒ −! W σ,d ( ǫ ) is a homotopy equivalence and the directlimit (3.2.1) stabilizes. This proves the statement. (cid:3) The set W σ,d ( ǫ ) is now compact and for ǫ < ǫ ( d ) has the same homology of W σ,d . Fortechnical reasons, this is not yet the set we will work with. Instead we will work with itsdouble cover V σ,d ( ǫ ) ⊂ S N : V σ,d ( ǫ ) = { x ∈ S N | h ij ( x, z ) ≥ ǫz ∀ ≤ i < j ≤ n, k x k ≤ ǫ − z } ⊂ S N . This will not be an obstacle for computing the Betti numbers of W σ,d ( ǫ ) , because of nextlemma. Lemma 30. For every ǫ > the set V d,ǫ ( ǫ ) ⊂ S N consists of two disjoint copies of W σ,d ( ǫ ) .In particular for all k ≥ b k ( W σ,d ( ǫ )) = 12 b k ( V σ,d ( ǫ )) . Proof. Let { z = 0 } ≃ S N − be the equator in S N and observe that { z = 0 } ∩ V σ,d ( ǫ ) = ∅ . Thisimplies that V σ,d ( ǫ ) = ( V σ,d ( ǫ ) ∩ { z > } ) ⊔ ( V σ,d ( ǫ ) ∩ { z < } ) . The involution ( x, z ) ( − x − z ) on the sphere S N restricts to a homeomorphism between V σ,d ( ǫ ) ∩ { z > } and V σ,d ( ǫ ) ∩ { z < } . Each of these sets is homeomoprhic to its projectionto the projective space R P N , which is the set W σ,d ( ǫ ) . This concludes the proof. (cid:3) Systems of quadratic inequalities. The set V σ,d ( ǫ ) defined above is the set of solutionsof a system of quadratic inequalities, and we will now use the spectral sequence from Section2.4 for computing its Betti numbers with Z coefficients.In order to reduce to the framework of Section 2.4, let us introduce the homogeneousquadrics h ij,ǫ , h ,ǫ : R N +1 ! R k +1 defined for all ≤ i < j ≤ n by: h ij,ǫ ( x, z ) = σ ij k x i − x j k − σ ij z − ǫz and h ,ǫ ( x, z ) = k x k − ǫ − z . These quadrics can be put as the components of a quadratic map defined by h ǫ = ( h ,ǫ , h ,ǫ , h ,ǫ , . . . , h k,ǫ ) : R N +1 ! R k +1 , where we are using the identification of sets of indices { , , . . . , k } = { (1 , , (1 , , . . . , ( n − , n ) } . Inside the space R k +1 we can consider the closed convex cone K = { y ≤ , y ≥ , . . . , y k ≥ } , so that our original set can be written as V σ,d ( ǫ ) = h − ǫ ( K ) . In this case the set Ω ⊂ S k is the set Ω = { ( ω , . . . , ω k ) ∈ S k | ω ≥ , ω ≤ , . . . , ω k ≤ } . For every point ω = ( ω , . . . , ω k ) we can consider the quadratic form ωh ǫ defined by: ωh ǫ = ω h ,ǫ + · · · + ω k h k,ǫ . Using this notation, for every j ≥ we define the sets: Ω j ( ǫ ) = { ( ω , . . . , ω k ) ∈ Ω | ind + ( ωH ǫ ) ≥ j } . These are just the sets Ω j defined in Section 2.4, in the case of the quadratic map h ǫ .For every ≤ i < j ≤ n let us also denote by U ij ∈ Sym( n, R ) the symmetric matrixrepresenting the quadratic form u ij : R n ! R defined by: u ij ( t , . . . , t n ) = σ ij ( t i − t j ) . Then, if H ij ∈ Sym( dn, R ) is the matrix representing the quadratic form x σ ij k x i − x j k ,we have: H ij = U ij ⊗ d Lemma 31. The index function ind + : Ω ! N for our family of quadrics can be written as: ind + ( ωH ǫ ) = d · ind +1 ( ω ) + ind +0 ,ǫ ( ω ) , where ind +1 ( ω ) = ind + ω n + X i Observe that, for ω = ( ω , ω ij ) ∈ Ω , the matrix ωH ǫ is a block matrix: ωH ǫ = − ω ǫ − P i If a matrix Q ∈ Sym( n, R ) has eigenvalues λ ( Q ) ≥ · · · ≥ λ n ( Q ) (possibly with repetitions),the matrix Q ⊗ d has eigenvalues: λ i,j ( Q ⊗ d ) = λ i ( Q ) i = 1 , . . . , n, j = 1 , . . . , d. In particular ind + ( Q ⊗ d ) = d · ind + ( Q ) , and the result now follows. (cid:3) Corollary 32. For d ≥ n + 1 the set Ω nd ( ǫ ) is contractible and Ω nd +1 ( ǫ ) is empty.Proof. Let us first show that Ω nd +1 ( ǫ ) = ∅ . To this end consider the set: B ( ǫ ) = { ( ω, [ x ]) ∈ Ω × R P N | ωh ǫ ( x ) ≥ } . By [AL12, Lemma 24] the projection π = p | B ( ǫ ) on the second factor gives a homotopyequivalence between B ( ǫ ) and its image π ( B ( ǫ )) = R P N \ W σ,d ( ǫ ) . Since W σ,d ( ǫ ) is nonempty, we know that(3.4) π ( B ( ǫ )) = R P N . If now there was ω ∈ Ω such that ind + ( ω ) = N + 1 , then ωh ǫ > and { ω } × R P N ⊂ B ( ǫ ) .This would imply π ( B ( ǫ )) = R P N , which contradicts (4.1).Let us now prove that Ω nd ( ǫ ) is contractible. For d ≥ n + 1 , since Ω nd +1 ( ǫ ) = ∅ , then theset Ω nd ( ǫ ) can be described as: Ω nd ( ǫ ) = { ind + = nd } = { d · ind +1 + ind ,ǫ ≥ nd } = { ind +1 = n } ∩ { ind ,ǫ = 0 } . Observe that the point ω = (1 , , . . . , ∈ Ω belongs to both the sets { ind +1 = n } and { ind ,ǫ = 0 } , and their intersection is nonempty.Now, { ind +1 = n } and { ind ,ǫ = 0 } are obtained by intersecting a convex set in R k +1 with Ω ∩ S k , as they coincide with the set of the points where the linear families of symmetricmatrices ω n + P i There exists a cohomology spectral sequence ( E r ( ǫ ) , d r ( ǫ ) r ≥ ) converging to H ∗ ( S N \ V σ,d ( ǫ ); Z ) such that: (1) the second page of the spectral sequence is given, for j > , by E i,j ( ǫ ) = H i (Ω j +1 ( ǫ ) , Ω j +2 ( ǫ ); Z ) . For j = 0 , the elements of the second page of the spectral sequence fit into a long exactsequence: (3.6) · · · ! H i (Ω ( ǫ ); Z ) ! E i, ( ǫ ) ! H i (Ω ( ǫ ) , Ω ( ǫ ); Z ) ( · ) ⌣ν ( ǫ ) −! H i +1 (Ω ( ǫ ); Z ) ! · · · . (2) for j ≥ the second differential d i,j ( ǫ ) : H i (Ω j +1 ( ǫ ) , Ω j +2 ( ǫ )) ! H i +2 (Ω j ( ǫ ) , Ω j +1 ( ǫ )) is given by d i,j ( ǫ ) ξ = ∂ ( ξ ` ν j +1 ( ǫ )) + ∂ξ ` ν j ( ǫ ) . Remark . As explained in [AL12, Introduction], the second differential only depends on therestriction of ν j ( ǫ ) to the set Ω j ( ǫ ) \ Ω j +1 ( ǫ ) . Remark . In the previous spectral sequence the coefficient group for the various cohomologiesis the field Z . There is an analogous spectral sequence for coefficients in Z , but the descriptionof its differentials is less clear.3.2.3. The analysis of the spectral sequence and its asymptotic structure. We will start byproving the following proposition, which deals with the stabilization of entries of the secondpage of the spectral sequence of Theorem 33. Proposition 36. There exist semialgebraic topological spaces Ω = A ⊇ B ⊇ A ⊇ B ⊇ · · · ⊇ A n ⊇ B n = ∅ , vector spaces N , , . . . , N k, and ǫ > such that for all ǫ ≤ ǫ the second page of the spectralsequence of Theorem 33 has the following structure (3.7) E i,j ( ǫ ) ≃ H i ( B ℓ , A ℓ +1 ) if j = ℓdH i ( A ℓ , B ℓ ) if j = ℓd − N i, if j = 00 otherwise RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 23 Proof. Observe first that the second page of the spectral sequence is zero in the region { ( i, j ) | i ≥ k + 1 , j > } , because all the sets Ω j ( ǫ ) are semialgebraic and of dimension atmost k (since they are contained in Ω ⊂ S k ). The j = 0 row of the spectral sequence is alsozero for i ≥ k + 2 , since for the same reason all the groups in the exact sequence in (1) arezero.Observe now that Lemma 31 implies that the only possible values of the function ind + :Ω ! N are , , d, d + 1 , . . . , nd, nd + 1 and in particular: Ω = Ω ( ǫ ) ⊇ Ω ( ǫ ) ⊇ Ω ( ǫ ) = Ω ( ǫ ) = · · · =Ω d ( ǫ ) ⊇ Ω d +1 ( ǫ ) ⊇ Ω d +2 ( ǫ ) = Ω d +3 ( ǫ ) = · · ·· · · = Ω nd − ( ǫ ) =Ω nd ( ǫ ) ⊇ Ω nd +1 ( ǫ ) ⊇ ∅ . (3.8)In particular, for every ℓ = 0 , . . . , n , we deduce the vanishing of the homology of all the relativepairs: H ∗ (Ω dℓ +2 ( ǫ ) , Ω dℓ +3 ( ǫ )) = · · · = H ∗ (Ω ( ℓ +1) d − ( ǫ ) , Ω ( ℓ +1) d ( ǫ )) = 0 . This proves the “otherwise” part of the claim in (36).We now defined the sets A ℓ ( ǫ ) = { ind + ≥ dℓ } and B ℓ ( ǫ ) = { ind + ≥ dℓ + 1 } and observethat: A ℓ ( ǫ ) = { ind +1 ≥ ℓ } B ℓ ( ǫ ) = (cid:16) { ind +1 ≥ ℓ } ∩ { ind +0 ,ǫ = 1 } (cid:17) ∪ { ind +1 ≥ ℓ + 1 } , where the index functions ind +0 ,ǫ , ind +1 : Ω ! N are defined in Lemma 31. Since ind +1 does notdepend on d nor on ǫ and ind ,ǫ does not depend on d , by semialgebraic triviality it followsthat there exists ǫ > such that the homotopy of the sequence of inclusions Ω = A ( ǫ ) ⊇ B ( ǫ ) ⊇ A ( ǫ ) ⊇ B ( ǫ ) ⊇ · · · ⊇ A n ( ǫ ) ⊇ B n ( ǫ ) = ∅ stabilizes for ǫ ≤ ǫ . We define A ℓ = A ℓ ( ǫ ) and B ℓ = B ℓ ( ǫ ) . With this notation we have that the sequence ofinclusions (3.2.3) for ǫ ≤ ǫ becomes (up to natural homotopy equivalences): Ω = A ⊇ B ⊇ A = A = · · · = A ⊇ B ⊇ A = A = · · ·· · · = A n = A n ⊇ B n ⊇ ∅ . This proves the statement for the term E i,j ( ǫ ) of the spectral sequence with j = ℓd − , ℓd. In the case j = 0 , we observe that the dimension of E i, ( ǫ ) is determined by the exactsequence: ! ker ! H i − (Ω ( ǫ ) , Ω ( ǫ )) ! H i (Ω ( ǫ )) ! E i, ( ǫ ) !! H i (Ω ( ǫ ) , Ω ( ǫ )) ! H i +1 (Ω ( ǫ )) ! coker ! , where ker and coker refer to the map x x ⌣ ν ( ǫ ) . The homotopy of the first, the third andthe fourth element of the above sequence stabilizes for ǫ ≤ ǫ ; moreover (possibly choosing asmaller ǫ ) also the homotopy of the bundle P ( ǫ ) ! D ( ǫ ) from (3.2.2) stabilizes for ǫ ≤ ǫ and therefore the map x x ⌣ ν ( ǫ )) stabilizes as well, and consequently the ranks of ker and coker stabilize. This gives the stabilization of dim Z ( E i, ( ǫ )) to a finite number for ǫ ≤ ǫ .We set: N i, := Z dim Z ( E i, ( ǫ ))2 ∀ ǫ ≤ ǫ . This concludes the proof. (cid:3) Next we deal with the stabilization of the second differential. Proposition 37. The second differential of the spectral sequence (36) is zero.Proof. Observe that the only possible nonzero differential of the spectral sequence is (for d ≥ ) d ∗ ,ℓd ( ǫ ) : E ∗ ,ℓd ( ǫ ) ! E ∗ +2 ,ℓd − ( ǫ ) . Let us recall that we have defined ωH ǫ = ωq + ωq where ωq = ( ω dn + P i Hom( Z a , Z b ) ≃ Z a × b isa finite set, then up to subsequences d ( ǫ ) ∗ ,ℓd is eventually constant. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 25 n · d − · d − · d − d − H ( B , A ; Z ) H ( A , B ; Z ) d (cid:0) n (cid:1) + 1 d Figure 3. This is a schematic image of the E ( ǫ ) term of the spectral sequencewe are describing. The coloured parts correspond to the elements E i,j ( ǫ ) of thespectral sequence which are possibly non-zero.3.2.4. The asymptotic for the Betti numbers of the chamber. We are now in the position ofproving the main theorem of this section, namely Theorem 7. Proof of Theorem 7. The proof of this theorem is based on the analysis of the structure of thespectral sequence and its last page. First observe that by Proposition 29, for all k ≥ and forall ǫ < ǫ ( d ) we have: b k ( W σ,d ) = b k ( W σ,d ( ǫ )) . For the rest of the proof we will take ǫ ≤ min { ǫ ( d ) , ǫ } , where ǫ ≤ ǫ is given by Proposition37. Lemma 30 implies now that: b k ( W σ,d ) = 12 b k ( V σ,d ( ǫ )) . On the other hand, since the involved spaces are semialgebraic sets (hence triangulable), theBetti numbers of V σ,d ( ǫ ) are related to those of S N \ V σ,d ( ǫ ) through Alexander duality: ˜ b k ( V σ,d ( ǫ )) = ˜ b N − k − ( S N \ V σ,d ( ǫ )) . Finally, denoting by e i,j ∞ ( ǫ ) the dimension of E i,j ∞ ( ǫ ) , where E ∞ ( ǫ ) is the last page of thespectral sequence from Theorem 33, we have: ˜ b N − k − ( S N \ V σ,d ( ǫ )) = X i + j = N − k − e i,j ∞ ( ǫ ) . Collecting all this together, for ǫ ≤ min { ǫ ( d ) , ǫ } , we have: b k ( W σ,d ) = 12 X i + j = N − e i,j ∞ ( ǫ ) if k = 0 X i + j = N − k − e i,j ∞ ( ǫ ) if < k < N − − e , ∞ ( ǫ ) if k = N − Observe now that Proposition 36 implies that in the second page of the spectral sequenceonly the first (cid:0) n (cid:1) + 1 columns are nonzero (i.e. those with ≤ i ≤ (cid:0) n (cid:1) ); moreover in thesecond page only the rows with j = ℓd and j = ℓd − are potentially nonzero. Therefore, for d ≥ (cid:0) n (cid:1) + 2 all the higher differentials are zero and: E ∞ ( ǫ ) = E ( ǫ ) . On the other hand Proposition 37 implies that E ( ǫ ) = E ( ǫ ) = E ∞ ( ǫ ) , with the last equalityfor d ≥ (cid:0) n (cid:1) + 2 .Looking now at the top two rows of E ∞ , by Corollary 32 we know that: E i,dn ( ǫ ) = 0 ∀ i ≥ , E ,dn − ( ǫ ) ≃ Z and E i,dn − ( ǫ ) = 0 ∀ i ≥ . Thus, for d ≥ (cid:0) n (cid:1) + 2 , e i,dn ∞ ( ǫ ) = 0 ∀ i ≥ , e ,dn − ∞ ( ǫ ) = 1 and e i,dn − ∞ ( ǫ ) = 0 ∀ i ≥ . From this we immediately see that, for d ≥ (cid:0) n (cid:1) + 2 we have: b ( V σ,d ( ǫ )) = 1 and b k ( V σ,d ( ǫ )) = 0 ∀ ≤ k ≤ (cid:18) n (cid:19) . This already proves:(3.9) b ( W σ,d ) = 1 and b k ( W σ,d ) = 0 ∀ ≤ k ≤ (cid:18) n (cid:19) . Observe now that the fact that the rows with j = ℓd and j = ℓd − in E = E ∞ are the onlypossibly non-zero rows for ℓ = 0 , . . . , n , influences the Betti numbers b k ( W σ,d ) with(3.10) k = md, . . . , md − (cid:18) n (cid:19) − , where m = n − ℓ. We define now, for m = 1 , . . . , n − : Q G,m ( t ) = 12 · e , ( n − m ) d − ∞ t ( n ) +1 + ( n ) X i =1 (cid:18) e ( n ) − i, ( n − m ) d ∞ + e ( n ) − i +1 , ( n − m ) d − ∞ (cid:19) t i + e ( n ) , ( n − m ) d ∞ . The i –th coefficient of the polynomial Q G,m is b md − ( n ) + i − ( W G,d ) . In principle we would haveto consider also the case m = n , but Theorem 11 (which we prove below) guarantees thatthere is no homology in dimension greater than ( n − d − n + 1 . By (3.2.4), the conclusion ofthe theorem follows. (cid:3) RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 27 Remark . As we noticed in the introduction, since the polynomials Q G, , . . . , Q G,n − do notdepend on d , but only on the graph, and since these polynomials are the same for isomorphicgraphs, they define a graph invariant. Similarly the same is true for the Floer number β ( G ) ,which is just the sum of their coefficients. Of course the polynomials are finer invariants,however we do not have a clear interpretation of these quantities. Remark . From (3.2.4) it immediately follows that for d ≥ (cid:0) n (cid:1) + 2 each sign condition isconnected. In particular, if d ≥ (cid:0) n (cid:1) + 2 , two R d –geometric graphs on n vertices are isomorphicif and only if they are rigid isotopic. We will actually sharpen this in Corollary 45 below.4. Homotopy groups of the chambers We turn our attention now to proving Theorem 2, and we start by introducing some notation.For every ≤ r ≤ max { n, d } let us denote by ( R d × n ) r the set of matrices of rank r : ( R d × n ) r = n P ∈ R d × n (cid:12)(cid:12) rk ( P ) = r o ⊆ R d × n . When r = n we have that ( R d × n ) n deformation retracts the Stiefel manifold of orthonormal n –frames in R n (the retraction is given by the Gram-Schmidt procedure; in the case d = n thisis simply the deformation retraction of GL( n, R ) onto O ( n ) ). We recall that for k ≤ d − n − this Stiefel manifold is k –connected, see [Hat02, Example 4.53]:(4.1) π k (( R d × n ) n ) = 0 if k ≤ d − n − . We will need the next elementary lemma. Lemma 41. The complement of ( R d × n ) n can be written as a finite union of smooth subman-ifolds of codimension at least d − n + 1 .Proof. Recall that for every ≤ r ≤ max { n, d } the codimension of ( R d × n ) r in R d × n equals ( n − r )( d − r ) (see [Hir94, Chapter 3, Section 2, Exercise 4]) and, in particular if r ≤ n − : codim R d × n ( R d × n ) r ≥ d − n + 1 . Now, the complement of ( R d × n ) n in R d × n is a semialgebraic set that can be written as: R d × n \ ( R d × n ) n = n − a r =0 ( R d × n ) r and it is therefore a semialgebraic set of codimension at least d − n + 1 . (cid:3) We will now prove a sequence of results on the homotopy groups of the chambers. Theseresults will imply Theorem 2. Since W G,d might not be connected if d ≤ (cid:0) n (cid:1) +1 , part of theseresults are formulated using the set [ S k , W G,d ] of homotopy classes of continuous maps from S k to W G,d , instead of the homotopy group π k ( W G,d ) . As soon as W G,d becomes connectedand simply connected, we can endow [ S k , W G,d ] with a group structure. To stress this subtletywe will keep both notations. Proposition 42. If d ≥ k + n + 1 the inclusion i : W G,d ∩ ( R d × n ) n ֒ −! W G,d induces a bijection between [ S k , W G,d ∩ ( R d × n ) n ] and [ S k , W G,d ] . Proof. We need to prove that the map i ∗ : [ S k , W G,d ∩ ( R d × n ) n ] ! [ S k , W G,d ] induced by theinclusion is a bijection if k ≤ d − n − . We first prove the surjectivity of i ∗ . Let f : S k ! W G,d be a map representing an elementof [ S k , W G,d ] . Since W G,d is open, up to homotopies, we can assume that the map f issmooth. Moreover, by [Hir94, Chapter 3, Theorem 2.5], the map f is homotopic to a map f : S k ! W G,d which is transversal to all the strata of the complement of ( R d × n ) n . If now k < d − n + 1 , the transversality condition and Lemma 41 imply that the image of f does notintersect these strata; therefore f : S k ! W G,d ∩ ( R d × n ) n and i ∗ is surjective. (Notice thatfor the surjectivity we only need d ≥ k + n .)For the injectivity we argue similarly. Let f , f : S k ! W G,d ∩ ( R d × n ) n be two maps suchthat i ◦ f : S k ! W G,d is homotopic to i ◦ f : S k ! W G,d . This means that there exists a map F : S k × I ! W G,d such that F ( · , 0) = i ◦ f and F ( · , 1) = i ◦ f . Now, we can approximate F with a new map e F : S k × I which is smooth, homotopic to F and C arbitrarily close to it, andtransversal to all the strata of the complement of ( R d × n ) n . By Lemma 41, if k + 1 < d − n + 1 this implies that the image of e F does not intersect the complement of ( R d × n ) n . In particularwe have a homotopy between e F ( · , and e F ( · , all contained in W G,d ∩ ( R d × n ) n . On theother hand, since both F ( · , and F ( · , miss the complement of ( R d × n ) n , which is closed,by compactness of S k , any two maps C sufficiently close to these maps will be homotopicto them and will also miss this complement. In particular, if e F is sufficiently close to F , F ( · , is homotopic to e F ( · , , F ( · , is homotopic to e F ( · , and these homotopies miss thecomplement of ( R d × n ) n . In this way we have build a homotopy between f and f already in W G,d ∩ ( R d × n ) n , i.e. i ∗ is injective. (cid:3) Some useful maps. We introduce now some useful maps. First recall the “Gram-Schmidt” map σ : R d × n ! R d × n which orthonormalizes the columns of a matrix P ∈ R d × n and is defined by: σ ( P ) = P ( P T P ) − / . Since the columns of σ ( P ) are orthonormal, it follows that(4.2) σ ( P ) T σ ( P ) = n . Moreover σ ( P ) σ ( P ) T is the orthogonal projection on the span of the columns of P .Let now G be a geometric graph on n vertices and d ≥ n . Our first useful map is:(4.3) W G,n × ( R d × n ) n W G,d ( Q, P ) ( σ ( P ) Q ) α . This map essentially takes a n –tuple of points in R n and a linear space of dimension n in R d andputs these points isometrically on this linear space. We need to verify that the isomorphismclass of the labeled graph is not changed, i.e. that G ( σ ( P ) Q ) ≃ G ( Q ) ≃ G. This is truebecause all the relative distances of the points in σ ( P ) Q are the same as the distances of thepoints in Q . More precisely, denote by Q = ( q , . . . , q n ) and by σ ( P ) Q = ( p ′ , . . . , p ′ n ) , where RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 29 p ′ i = σ ( P ) q i . Then, using (4.0.1), we have: k p ′ i − p ′ j k = k σ ( P )( q i − q j ) k = ( q i − q j ) T σ ( P ) T σ ( P )( q i − q j )= ( q i − q j ) T ( q i − q j )= k q i − q j k . Since the relative distances between the points are the same, and d ≥ n it follows by Lemma28 that G ( σ ( P ) Q ) ≃ G ( Q ) .A second useful map is: W G,d ∩ ( R d × n ) n W G,n × ( R d × n ) n P ( σ ( P ) T P, P ) β . This map “decouples” the geometric graph into its part in the span of its vertices and thespan of the vertices. Also for this map we need to check that its first component has target in W G,n . Again this follows from the fact that the mutual distances of the corresponding pointsare preserved. Denoting by P = ( p , . . . , p n ) , we have: k σ ( P ) T p i − σ ( P ) T p j k = ( p i − p j ) T σ ( P ) σ ( P ) T ( p i − p j )= ( p i − p j ) T ( p i − p j )= k p i − p j k , where we have used the fact that σ ( P ) σ ( P ) T is the orthogonal projection onto the span of thecolumns of P . The claim follows again from Lemma 28.Observe that it follows immediately from the definition of the maps α and β that: α ◦ β = i : W G,d ∩ ( R d × n ) n ֒ −! W G,d . Stabilization. Proposition 43. If d ≥ k + n + 1 the map j ∗ : [ S k , W G,d ] −! [ S k , W G,d +1 ] induced by theinclusion is injective.Proof. Let g , g : S k ! W G,d be two continuous maps such that j ◦ g : S k ! W G,d +1 and j ◦ g : S k ! W G,d +1 are homotopic. Thanks to Proposition 42 we can assume g and g tobe elements of [ S k , W G,d ∩ ( R d × n ) n ] . We want to prove that g and g are homotopic. For amap f : S k ! W G,d , we consider the following commutative diagram of maps: ( W G,n ∩ ( R n ) n ) × ( R dn ) n ( W G,n ∩ ( R n ) n ) × ( R ( d +1) n ) n S k W G,d ∩ ( R dn ) n W G,d +1 ∩ ( R ( d +1) n ) nα u α fβ ◦ f =( f ,f ) jβ β Here the maps α and α are the restriction of the maps defined in (4.0.1) to the set of pairs ( Q, P ) with rk ( Q ) = n ; the values of these maps are in the set of matrices of rank n .Since α ◦ β = id , we can write the map f = as: f = α ◦ ( β ◦ f ) = α ◦ ( f , f ) , where ( f , f ) are the components of β ◦ f . We apply now the diagram to the map f = g and f = g , writing them as: g i = α ◦ ( β ◦ g i ) = α ◦ ( g i, , g i, ) , i = 0 , . We will prove that both components are homotopic g , ∼ g , and g , ∼ g , , which impliesthat g is homotopic to g .Since the map j ◦ g is homotopic to j ◦ g , then also the first component of β ◦ j ◦ g ishomotopic to the first component of β ◦ j ◦ g . But the first component of β ◦ j ◦ g equals g , ,the first component of β ◦ g , and similarly for the first component of β ◦ j ◦ g . Therefore g , ∼ g , . On the other hand the second components of β ◦ g , β ◦ g : S k ! ( R d × n ) n are homotopicsimply because π k (( R d × n ) n ) = 0 for k ≤ d − n − . This concludes the proof. (cid:3) Proposition 44. For every d ≥ n the inclusion W G,d ֒ ! W G,d + n is homotopic to a constantmap. Before we give the proof, let us observe that, since we do not know if W G,d is path-connected,then there are several constant maps up to homotopy, one for each component; this Propositiontells us that all the maps [ S k , W G,d ] are mapped to the same constant map in W G,d + n . Thisalso tells us that W G,d is contained in just one connected component of W G,d + n . Proof. Since for d ≥ n every graph is realizable as a geometric graph, pick R = ( r , . . . , r n ) ∈ W G,n by the previously-cited result of Maehara [Mae84].Consider the homotopy f t : W G,d ! R ( d + n ) × n , defined for t ∈ [0 , by f t ( P ) = (cid:18) √ − tP √ tR (cid:19) . With this choice: f = i : W G,d ֒ −! W G,d + n ⊂ R ( d + n ) × n and f ≡ (cid:18) R (cid:19) ∈ W G,d + n . We only need to prove that f t ( W G,d ) ⊆ W G,d + n for all t ∈ [0 , . To this end let us write f t ( P ) = ( p ( t ) , . . . , p n ( t )) = (cid:18) √ − tp · · · √ − tp n √ tr · · · √ tr n (cid:19) . Because of Lemma 28, in order to show that G ( f t ( P )) ≡ G it is enough to show that the signsof the family quadrics {k p i − p j k − R ( d + n ) × n ! R } evaluated on f t ( P ) are constants. Wehave k p i ( t ) − p j ( t ) k = (1 − t ) k p i − p j k + t k r i − r j k and therefore as sign (cid:0) k p i − p j k − (cid:1) = sign (cid:0) k r i − r j k − (cid:1) , it must be the case that sign (cid:0) k p i ( t ) − p j ( t ) k − (cid:1) = sign (cid:0) k p i − p j k − (cid:1) = sign (cid:0) k r i − r j k − (cid:1) . This concludes the proof. (cid:3) We are now ready to prove Theorem 2. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 31 Proof of Theorem 2. We first prove that for d ≥ n + 1 the set W G,d is path connected. ByProposition 44 the map i ∗ : [ S , W G,d ] ! [ S , W G,d + n ] is the map that sends everything to theclass of a constant map. On the other hand this map factors through the sequence of mapsinduced by the inclusions W G,d ֒ −! W G,d +1 [ S , W G,d ] ! [ S , W G,d +1 ] ! · · · ! [ S , W G,d + n ] . Each map in the previous sequence is an injection for d ≥ n + 1 by Proposition 43, thereforealso i ∗ : [ S , W G,d ] ! [ S , W G,d + n ] is an injection, and [ S , W G,d ] consists of only one element.Therefore W G,d is path connected.We prove now that, for d ≥ n +2 , W G,d is also simply connected. By Proposition 44 the map i ∗ : [ S , W G,d ] ! [ S , W G,d + n ] is the map that sends everything to the class of a constant map.On the other hand this map factors through the sequence of maps induced by the inclusions W G,d ֒ −! W G,d +1 [ S , W G,d ] ! [ S , W G,d +1 ] ! · · · ! [ S , W G,d + n − ] ! [ S , W G,d + n ] . Each map in the previous sequence is an injection for d ≥ n + 2 by Proposition 43, thereforealso i ∗ : [ S , W G,d ] ! [ S , W G,d + n ] is an injection, and [ S , W G,d ] consists of only one element.Recall now that [ S , W G,d ] consists of the set of conjugacy classes in π ( W G,d ) (we can omit thebase point because W G,d is path-connected): the fact that [ S , W G,d ] consists of one elementimplies that there is only one conjugacy class in π ( W G,d ) , which means that W G,d is simplyconnected.Let now k ≥ and d ≥ k + n + 1 . Since π ( W G,d ) = 0 for d ≥ n + 2 , it follows that [ S k , W G,d ] = π k ( W G,d ) /π ( W G,d ) = π k ( W G,d ) , by [Hat02, Proposition 4A.2]. By Proposition 44 the map i ∗ : π k ( W G,d ) ! π k ( W G,d + n ) is thezero map. On the other hand this map factors through the sequence of maps induced by theinclusions W G,d ֒ −! W G,d +1 π k ( W G,d ) ! π k ( W G,d +1 ) ! · · · ! π k ( W G,d + n − ) ! π k ( W G,d + n ) . Each map in the previous sequence is an injection for d ≥ k + n +1 by Proposition 43, thereforealso i ∗ : π k ( W G,d ) ! π k ( W G,d + n ) is an injection, and π k ( W G,d ) = 0 . (cid:3) Notice that thanks to this theorem we have the following corollary. Corollary 45. For d ≥ n + 1 , for each labeled graph G on [ n ] the isomorphism class W G,d isconnected. The infinite–dimensional case. The space R ∞× n is a pre-Hilbert space (since it isnot complete) with respect to the natural scalar product . The notion of geometric graphand discriminant also makes sense in this infinite dimensional space. More precisely, givenan element P = ( p , . . . , p n ) ∈ R ∞× n , we build the graph G ( P ) whose vertices and edges aredefined as in Definition 12. The discriminant ∆ ∞ ,n consists of points P = ( p , . . . , p n ) ∈ R ∞× n such that there exists a pair ≤ i < j ≤ n with k p i − p j k = 1 . The chambers are now definedas follows: for a given graph G on n vertices, we set W G, ∞ = (cid:8) P = ( p , . . . , p n ) ∈ R ∞× n \ ∆ ∞ ,n (cid:12)(cid:12) G ( P ) ≃ G (cid:9) . It is easy to see that W G, ∞ is the direct limit of the sequence of inclusions in (1.1). Inparticular, from Theorem 2 we deduce the following. Theorem 46. For every graph G , the set W G, ∞ = lim −! W G,d is contractible. The completion of R ∞× n is ( ℓ ( N )) n = (cid:26) p = ( x , x , . . . ) (cid:12)(cid:12)(cid:12)(cid:12) P ∞ k =1 x k < ∞ (cid:27) . Proof. We first observe that, by Lemma 28, for d ≥ n each W G,d is described by a list ofquadratic inequalities and therefore it is semialgebraic and it has the homotopy type of aCW–complex. Since W G, ∞ = lim −! W G,d , it follows by [Mil63, Corollary on pag. 253] that also W G, ∞ has the homotopy type of a CW–complex. By Whitehead’s Theorem ([Hat02, Theorem4.5]), in order to prove that W G, ∞ is contractible it is enough to prove that all its homotopygroups are zero.To this end, let f : S k ! W G, ∞ . We need to prove that f is homotopic to a constant map,which implies π k ( W G, ∞ ) = 0 . We will prove that f is homotopic to a map g : S k ! W G,d for some large enough d . Then we can use Theorem 2 to conclude that g is homotopic to aconstant map, and so is f .We first observe that, since S k is compact and W G, ∞ is open, there exists ǫ > such that(4.4) [ θ ∈ S k B ( f ( θ ) , ǫ ) ⊂ W G, ∞ . In fact, for every θ ∈ S k there exists ǫ ( θ ) > such that B ( f ( θ ) , ǫ ( θ )) ⊂ W G, ∞ (because W G, ∞ is open), and the existence of a uniform ǫ > follows by compactness of f ( S k ) . The inclusion (4.1) implies that if g : S k ! R ∞× n is a continuous map with(4.5) sup θ ∈ S k k f ( θ ) − g ( θ ) k ≤ ǫ, then f t = (1 − t ) f + tg is a homotopy between f and g with f t ( S k ) ⊂ W G, ∞ . Our scope is tobuild a map g : S k ! W G, ∞ satisfying (4.1) and such that g ( S k ) ⊂ W G,d with d ≥ k + n + 1 .Given a point P = ( p , . . . , p n ) ∈ R ∞× n let us denote by P ≤ d the point given by the same n –tuple of sequences as in P , but where we have set to zero all the terms of the sequences pastthe d –th one. In other words P ≤ d is the inclusion in R ∞× n of the projection of P to R d × n .The definition of R ∞× n implies that for every P there exists d ≥ such that P = P ≤ d . Thismeans that for every θ ∈ S k there exists d ( θ ) such that f ( θ ) = f ( θ ) ≤ d ( θ ) . We claim now that there exists d f ≥ such that for all θ ∈ S k we have(4.6) k f ( θ ) − f ( θ ) ≤ d f k ≤ ǫ. If this was false, then there would be a sequence { θ n } n ≥ ⊂ S k such that for all n ≥ thefollowing inequality would be satisfied: k f ( θ n ) − f ( θ n ) ≤ n k > ǫ. Up to subsequences, we may assume that θ n ! θ and f ( θ n ) ! f ( θ ) . Let d ( θ ) > be suchthat f ( θ ) = f ( θ ) ≤ d ( θ ) and pick θ n with n ≥ d ( θ ) and such that k f ( θ ) − f ( θ n ) k ≤ ǫ/ . Then ǫ < k f ( θ n ) − f ( θ n ) ≤ n k≤ (cid:0) k f ( θ n ) − f ( θ n ) ≤ n k + k f ( θ ) ≤ n − f ( θ n ) ≤ n k (cid:1) / = k f ( θ ) − f ( θ n ) k ≤ ǫ/ . (4.7)The last line follows from the fact that both f ( θ ) and f ( θ n ) can be written as f ( θ ) = ( f ( θ ) ≤ n )+( f ( θ ) − f ( θ ) ≤ n ) and f ( θ n ) = ( f ( θ n ) ≤ n ) + ( f ( θ n ) − f ( θ n ) ≤ n ) with the two summands belongingin both cases to the same orthogonal subspaces; this tells that(4.8) f ( θ ) − f ( θ n ) = (cid:2) f ( θ ) ≤ n − f ( θ n ) ≤ n (cid:3) + (cid:2) ( f ( θ ) − f ( θ ) ≤ n ) − ( f ( θ n ) − f ( θ n ) ≤ n )) (cid:3) , with the two summands in the square brackets belonging to orthogonal subspaces. Moreover,since f ( θ ) = f ( θ ) ≤ n by construction, the summand in the second square brackets of (4.1) RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 33 simply equals f ( θ n ) − f ( θ n ) ≤ n and this gives (4.1). Equation (4.1) gives a contradiction, andtherefore such a d f ≥ exists. Define now d = max { d f , k + n + 1 } and set: g ( θ ) = f ( θ ) ≤ d . The condition (4.1) gives now (4.1), which is what we wanted. (cid:3) Increasing the number of points Geometric graphs on the real line. We now want to study the number of possibleisotopy classes of geometric graphs on the real line when the number of points is large: this isprecisely the case d = 1 , n large. If we look at the discriminant ∆ ,n , this is an arrangementof hyperplanes, namely ∆ ,n = { ( x , . . . , x n ) ∈ R × n | ∃ i, j | x i − x j | = 1 } . Remark . There is a way to compute explicitly the number b ( R × n \ ∆ ,n ) using a gen-eralized version of the Mayer-Vietoris spectral sequence for semialgebraic sets. This gives for n = 3 , for n = 4 and for n = 5 . The computations becomes tricky for larger n ,however this numbers are the beginning of a known integer sequence, which is the sequence of labeled semiorders on [ n ] .The remark leads us to an obvious observation. An interval order for intervals { I i } ni =1 of unitlength (=semiorder for [ n ] ) is the partial order corresponding to their left-to-right precedencerelation, i.e. one interval I i being considered less than another I j iff I i is completely to the leftof I j . In the case d = 1 the number of components of the complement of ∆ ,n is exactly thenumber of possible semiorders for [ n ] . This because, once we defined the intervals [ p i − , p i ] for all i , each component of R × n \ ∆ ,n is uniquely determined by whether p i < p j − or p i ≮ p j − (see [S + 04, pag. 73]). Remark . The type of semiorders introduced are usually addressed as semiorders on n labeleditems. The number of distinct semiorders on n unlabeled items is given by the Catalan numbers { C n } n . Let us define f ( n ) := number of labeled semiorders of [ n ] . There is an explicit generatingfunction for this sequence (see [S + 04, pag.78, Corollary 5.12]). We have G ( x ) := X n ≥ f ( n ) x n n ! = C (1 − e − x ) where C is the generating function of the known sequence of Catalan numbers { C n } . Moreexplicitly we have: X n ≥ C n x n = C ( x ) = 1 − √ − x x . Theorem 49. The number of rigid isotopy classes of R –geometric graphs on n vertices equals: b (cid:0) R × n \ ∆ ,n (cid:1) = 1 n · r log · ne log ! n (cid:16) O ( n − ) (cid:17) . Proof. First of all let us notice that the Theorem 25 can be applied to G ( x ) since we have onlyone singularity in log and we can extend the function to a log D -domain, and actually tothe whole C \ [log , + ∞ ) . The function C ( x ) has a unique singularity at x = . It is easy tosee C ( x ) = 2 − √ − x + O (1 − x ) . By composition we get G ( x ) = 2 − p e − x − O (4 e − x − and from this G ( x ) = 2 − r log · r − x log + O − x log ! = F x log ! + O − x log ! . We can now apply Theorem 25. We get f ( n ) n ! = (cid:18) log (cid:19) − n · σ n + O (cid:18) log (cid:19) − n n ! where F ( x ) = P ∞ n =0 σ n z n . Using Remark 26 and Γ (cid:0) − (cid:1) = − √ π we get f ( n ) n ! = (cid:18) log (cid:19) − n · √ πn · r log 43 + O (cid:18) log (cid:19) − n n ! and by Stirling approximation f ( n ) = ne log ! n · n · r log 43 + O ne log ! n n − ! (cid:3) With these computations we know asymptotically the number of isotopy classes of geo-metric graphs on the real line. However, as we discussed before, different isotopy classes cancorrespond to the same isomorphism class. It is therefore natural to ask for the number ,n of isomorphism classes of R –geometric graphs, for n large. In [Han82], Hanlon computes theexponential generating function for this sequence (the author calls the corresponding graphs labeled unit interval graphs ).The exponential generating function for { ,n } n is Λ( x ) = exp (Γ( x )) − where Γ( x ) is the generating function for the sequence { b n } n of isomorphism classes of con-nected R –geometric graphs on n vertices. More explicitly, we have Γ( x ) = 14 (1 − z ) − r − z z where z = e x − . Reasoning as before, we prove the following theorem. Theorem 50. The number of isomorphism classes of R –geometric graphs on n vertices equals: ,n = e · n · r log · ne log ! n (cid:16) O ( n − ) (cid:17) . Proof. First of all let us notice that the Theorem 25 can be applied to Λ( x ) since we haveonly one singularity at log and we can extend the function to a log D -domain, actually to C \ [log , + ∞ ) . We start with x ) = exp (cid:18) · (3 − e x ) (cid:19) · exp (cid:18) − p e − x − (cid:19) Then, x ) = e + O − x log !! · (cid:18) − p e − x − O (4 e − x − (cid:19) RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 35 x ) = e − e r r − x log + O − x log ! . Finally, ,n n ! = e · (cid:18) log (cid:19) − n · √ πn · r log 43 + O (cid:18) log (cid:19) − n n ! ,n = e · ne log ! n · n · r log 43 + O ne log ! n n − ! . (cid:3) Even though in the general case we still do not have a clear understanding of the relationbetween b (cid:0) R × n \ ∆ ,n (cid:1) and ,n , in the case d = 1 we have the following corollary. Corollary 51. We have b ( R × n \ ∆ ,n ) = 8 e · ,n (cid:16) O ( n − ) (cid:17) where / √ e = 7 . ... . The number / √ e can be roughly interpreted as the average number of rigid isotopy classesrealizing a particular R -geometric graph isomorphism type.5.2. Asymptotic enumeration in higher dimensions. While the situation for isotopyclasses of geometric graphs on the real line is given by the number of semiorders on [ n ] , sucha closed form description apparently does not exist for larger values of d . Nonetheless weare able to obtain reasonable bounds on the asymptotics following methods of McDiarmidand Müller from [MM14] who study asymptotic enumeration of labeled disk graphs in R . Adisk graph in R is a graph given by an arrangement of open disks in R where the verticesare the disks and there is an edge between a pair of them if and only if the correspondingdisks intersect one another. In the case that all the disks have the same radius this is exactlythe setting of our geometric graphs in the case that d = 2 . McDiarmid and Müller showthat the number of labeled graphs on n vertices which are unit disk graphs in R , in ournotation ,n , is order exp(2 n log( n )+Θ( n )) , and adapting their method we prove the followingtheorem. In particular we prove that the right asymptotic rate of growth both for d,n andfor b (cid:0) R d × n \ ∆ d,n (cid:1) is exp( dn log( n ) + Θ( n )) . We will prove the following theorem, which is aconsequence of Theorem 54 and Theorem 56. Theorem 52. For d ≥ fixed and n ≥ d + 1 one has the following bounds: (cid:18) d + 1) e (cid:19) dn n dn ≤ d,n ≤ b (cid:16) R d × n \ ∆ d,n (cid:17) ≤ dn (cid:18) e d (cid:19) dn n dn General case: the upper bound. While McDiarmid and Müller are primarily inter-ested in enumerating labeled geometric graphs in the plane, in our notation ,n , their upperbound holds for general d , as they point out in [MM14]. The key lemma in their proof of theirupper bound is the following result of Warren [War68]. Theorem 53 ([War68]) . If P , ..., P m are polynomials of degree at most t in real variables z , ..., z k then the number of distinct sign patterns (sign( P ( z ) , ..., sign( P m ( z )) ∈ {− , } m thatoccur in R k \ ∪ mi =1 { z : P i ( z ) = 0 } is at most (cid:18) etmk (cid:19) k . Now given n, d we take the (cid:0) n (cid:1) polynomials in variables ( x , ..., x n ) ∈ R dn given by q i,j ( x ) = k x i − x j k − , defined in (3.1). Then each sign pattern of these (cid:0) n (cid:1) degree 2 polynomials in dn variable corresponds to a unique isomorphism class of labeled geometric graphs on n verticesin R d . Therefore we have the following bound: d,n ≤ (cid:18) ed (cid:19) nd n nd . However, a single sign pattern could be a disjoint union of several rigid isotopy classes, so weneed a different argument to bound b ( R d × n \ ∆ d,n ) . We prove the following theorem. Theorem 54 (Upper bound) . For fixed d and for n ≥ d + 1 , we have the bound: b ( R d × n \ ∆ d,n ) ≤ dn (cid:18) e d (cid:19) dn n dn Remark . Let us denote with ˆ∆ Gd,n the one point compactification of the set ∆ Gd,n definedin (2.2) where G is any graph on [ n ] . This is an algebraic set X of R nd +1 = ( x , . . . , x n , z ) defined by k + 1 equations which are k x i − x j k = (1 − z ) with ( i, j ) edge of G and the equation of the sphere k x k + · · · + k x n k + z = 1 . In fact, ifwe look at the explicit expression of the stereographic projection we get an homeomorphismbetween ∆ Gd,n and X \ ( { (0 , } ) and from this the claim. Proof. By Alexander duality (see Section 2.2) b ( R d × n \ ∆ d,n ) = b dn − ( ˆ∆ d,n ) + 1 , where ˆ∆ d,n is the one-point compactification of the discriminant. Therefore, it is sufficient to bound b dn − ( ˆ∆ d,n ) . Let us consider the Mayer–Vietoris spectral sequence for simplicial complexes(see [Bas03, Section 3.2] for a complete construction). Thanks to the previous remark ˆ∆ d,n isan algebraic set and we can use the mentioned spectral sequence with respect to the algebraiccovering { ˆ∆ Gd,n } G where G varies over nonempty labeled graphs on [ n ] . The E page of thespectral sequence has E i,j = M G a graph on [ n ] with exactly i + 1 edges H j ( ˆ∆ Gd,n ) , and b dn − ( ˆ∆ d,n ) ≤ dn − X i =0 dim Z ( E i, ( dn − − i )1 ) . Using Theorem 2 of [Mil64] and the fact that for any labeled graph G on [ n ] the topologicalspace ˆ∆ Gd,n is an algebraic set defined by equations of degree in R dn , we get that its total RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 37 Betti number is at most dn − . Using this, we have: b dn − ( ˆ∆ d,n ) ≤ dn X k =1 (cid:18)(cid:0) n (cid:1) k (cid:19) dn ≤ dn dn X k =1 (cid:18)(cid:0) n (cid:1) k (cid:19) ≤ dn dn (cid:18)(cid:0) n (cid:1) dn (cid:19) ≤ dn dn (cid:18) n e dn (cid:19) dn , where in the third inequality we used n ≥ d + 1 . (cid:3) General case: the lower bound. For the lower bound on the number of labeled diskgraphs, McDiarmid and Müller give a procedure for inductively generating many distinctlabeled disk graphs. Here we generalize this procedure to higher dimensions.For each k ≥ d + 1 we construct a family of non-isomorphic labeled geometric graphs on k vertices in R d , U k,d . If we let u k,d denote the number of graphs in U k,d , we show that for k ≥ d + 1 , u k +1 ,d ≥ (cid:18)(cid:22) kd + 1 (cid:23)(cid:19) d u k,d . This recursion implies the following which we prove in Section 5.4.1. Theorem 56 (Lower bound) . We have for n > d + 1 that (cid:18) n ( d + 1) e (cid:19) dn ≤ d,n For the base of the recursion, we start with the regular d -simplex in R d with edges of length1 and vertices given by P , P , ..., P d +1 . The -skeleton of the d -simplex is a geometric graphin R d this will be the singleton element of U d +1 ,d . Though this graph is degenerate it willstill contribute to d,n which is always a lower bound for b ( R d × n \ ∆ d,n ) by the discussionfollowing Lemma 15.To construct the families U k,d for d + 1 < k ≤ n we need the following technical lemmawhich generalizes Lemma 4.1 of [MM14]. Lemma 57. There exist constants ǫ > and C > such that for all < ǫ < ǫ and all p i ∈ B ( P i , ǫ ) for all i ∈ [ d ] there exists a unique point q ( p , . . . , p d ) ∈ B ( P d +1 , Cǫ ) with k q − p i k = 1 for all i ∈ { , ..., d } . In other words for ǫ small enough, this lemma tells us that there is a well-defined Lipschitzcontinuous function, with Lipschitz constant C , q on B ( P , ǫ ) × B ( P , ǫ ) ×· · ·× B ( P d , ǫ ) mapping ( x , ..., x d ) to the unique point of the intersection of sphere S ( x , ∩ S ( x , ∩ · · · ∩ S ( x d , closest to P d +1 . The d = 2 case is Lemma 4.1 of [MM14] and is essentially proved directly viaa closed form for q in terms of p , p ∈ B ( P , ǫ ) × B ( P , ǫ ) that is well-defined and Lipschitzcontinuous for ǫ small enough. For larger values of d writing down the closed form of q wouldbe much more complicated. Therefore, we instead describe algorihmically how one wouldcompute q given p , ..., p d sufficiently close respectively to P , P , ..., P d and show that q willultimately be a combination of Lipschitz continuous functions. Proof of Lemma 57. We show that for ǫ small enough, the intersection S ( p , ∩ · · · ∩ S ( p d , with p i ∈ B ( P i , ǫ ) for all i is two points q + ( p , ..., p d ) and q − ( p , ..., p d ) with q + ( p , ..., p d ) thecloser of the two to P d +1 , and that q + : B ( P , ǫ ) × · · · × B ( P d , ǫ ) ! R d is Lipschitz continuous.We will prove that q + is well-defined and Lipschitz continuous close to P , P , ..., P d bydescribing the algorithm one would use to compute q + and show that each step of the algorithmis given by composition or addition of Lipschitz continuous functions. Given a tuple of points ( p , p , ..., p d ) ∈ B ( P , ǫ ) × B ( P , ǫ ) × · · · × B ( P d , ǫ ) , with ǫ sufficiently small one could compute q + via the following recursive procedure.First find the ( d − -dimensional sphere given by the intersection of S ( p , and S ( p , .Now for any ≤ k ≤ ( d − a k -dimensional sphere in R d may be described completely by itscenter, its radius, and the affine subspace of dimension k + 1 in which it is contained. In otherwords a k -dimensional sphere in R d is described by a point in R d , a positive real number, andan element of the Grassmannian Gr ( k + 1 , d ) . Given p and p in R d with the distance from p to p smaller than 2 the intersection S ( p , ∩ S ( p , is a ( d − -dimensional sphere. Itfollows that taking ǫ small enough so that for any p , p ∈ B ( P , ǫ ) × B ( P , ǫ ) , || p − p || < we have a continuous function ( C, R, G ) : B ( P , ǫ ) × B ( P , ǫ ) ! R d × R + × Gr ( d − , d ) . Thismap sends ( p , p ) to the ( d − -dimensional sphere S ( p , ∩ S ( p , with center C ( p , p ) radius R ( p , p ) living in the affine hyperplane C ( p , p ) + G ( p , p ) . Now given ( p , ..., p d ) ∈ B ( P , ǫ ) × · · · × B ( P d , ǫ ) with ǫ small enough we have that theintersection of C ( p , p ) + G ( p , p ) with S ( p , ∪ S ( p , ∪ · · · ∪ S ( p d , ⊆ R d gives anarrangement of ( d − many ( d − -dimensional spheres in the affine hyperplane C ( p , p ) + G ( p , p ) . The center and radii of these spheres will be determined by how C ( p , p )+ G ( p , p ) intersects each S ( p i , . By induction we find the two points of intersection of these ( d − -dimensional spheres in the ( d − -dimensional Euclidean space given by the affine hyperplane C ( p , p ) + G ( p , p ) . Once these two points of intersection have been found we pick the onethat is closest to P d +1 to be q + ( p , ..., p d ) .It can be verified routinely that ( C, R, G ) as defined above is Lipschitz continuous ineach coordinate. Moreover the arrangement given by intersecting C ( p , p ) + G ( p , p ) with S ( p , ∪ S ( p , ∪· · ·∪ S ( p d , ⊆ R d when p i is sufficiently close to P i for all i , can be describedby a d − tuple of points ( c , r , ..., c d , r d ) where each c i belongs to C ( p , p )+ G ( p , p ) and r i ∈ (0 , . Here c i and r i are respectively the center and the radius of the ( d − -dimensionalsphere given by S ( p i , ∩ ( C ( p , p ) + G ( p , p )) for i ≥ with c and r respectively thecenter and radius of the ( d − -dimensional sphere given by the intersection S ( p , ∩ S ( p , i.e. c and r are C ( p , p ) and R ( p , p ) .By continuity C ( p , p ) can be made arbitarily close to C ( P , P ) , G ( p , p ) can be madearbitrarily close to G ( P , P ) , and R ( p , p ) can be made arbitrarily close to R ( P , P ) . Fromhere it may be verified that for ǫ small enough there is a Lipschitz continuous function from φ : B Gr ( d − ,d ) ( G ( P , P ) , ǫ ) × B R d ( C ( P , P ) , ǫ ) × B R ( R ( P , P ) , ǫ ) × B R d ( P , ǫ ) ×· · ·× B R d ( P d , ǫ ) ! ( R d − × R ) d − mapping an element of the domain to the arrangement of ( d − many ( d − -dimensional spheres in the affine hyperplane as described above. By induction we have thatthe S at the intersection of the arrangement is Lipschitz continuous on the image of φ , andfinally picking the closest of the two points to the fixed point P d +1 is Lipschitz continuous too.Note that the base case for the induction can simply be the d = 1 case; given two points in R picking the one closest to a fixed point is always Lipschitz continuous when the center of thetwo points lives in some small enough interval around a second fixed point. (cid:3) Let us take the sequence < ǫ < · · · < ǫ n defined by ǫ i = ǫ /C n − i , where ǫ and C as in theprevious Lemma and we are assuming C > . To construct elements of U k,d recursively from RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 39 U k − ,d we will also use as an inductive hypothesis that all the elements P = ( p , . . . , p k ) ∈ U k,d satisfy the following two properties:P k p i − P j k < ǫ i with i ≡ j mod d + 1 P S ( p i , ∩ · · · ∩ S ( p i d +1 , 1) = ∅ for all distinct { i , . . . , i d +1 } These two conditions hold for U d +1 ,d whose vertices are P , ..., P d +1 .Condition P above naturally partitions the vertices of G ∈ U k,d into d + 1 distinct classesgiven by the clustering of the vertices of G around the points P , ..., P d , P d +1 . The proofof the claimed recursive lower bound on u k,d will be that if G ∈ U k − ,d and, without loss ofgenerality, k ≡ d + 1) then picking a transversal σ = { p i , p i , · · · , p i d } of vertices of G where i l ≡ l mod ( d + 1) for every i , we give a procedure to choose a position for a newvertex p k to be added to G so that the neighborhood of p k is unique for each choice of σ andso that P and P are satisfied still satisfied after adding p k . As p k will depend on σ andeach choice of σ gives a distinct neighborhood for p k we have that there are at least as manycombinatorially distinct ways to extend G as there are choices for σ .If we denote by P ( G ) for G ∈ U k − ,d the set of all such transversals, that is the number of d -tuples ( i , . . . , i d ) with ≤ i j < k such that i j ≡ j mod d + 1 , we have |P| ≥ ⌊ ( k − / ( d + 1) ⌋ d . Now, let M π be defined as the intersection of the open balls, M π := B ( p i , ∩ · · · ∩ B ( p i d , . We have the following lemma, which in the 2-dimensional case is Claim 4.3 of [MM14]. Theproof, which we omit, is exactly analogous to the 2-dimensional case and relies on the factthat for each π = ( i , ..., i d ) ∈ P , S ( p i , ∩ · · · ∩ S ( p i d , ∩ B ( P d +1 , ǫ k ) is a single point dueto Lemma 57 and that single point is unique for each choice of π by condition P . Lemma 58. There exists nonempty open sets O π ⊂ M π , such that for all π = σ ∈ P we haveeither O π ∩ M σ = ∅ or O σ ∩ M π = ∅ . Now, for each π ∈ P let us pick an arbitrary q π = O π \ k − [ i =1 S ( p i , , and we have that the R d -geometric graph obtained by adding the vertex q π to G , which wewill denote ( G, q π ) , satisfies conditions P and P . Moreover for every choice of π we obtaina unique way to extend G by the following lemma. Lemma 59. If π = σ ∈ P ( G ) for G ∈ U k − ,d the geometric graphs ( G, q π ) and ( G, q σ ) arenot isomorphic. This holds because we have that q π and q σ will have different sets of neighbors, generalizingClaim 4.4 of [MM14]. Lemma 60. If π = σ ∈ P ( G ) for G ∈ U k − ,d , N ( q π ) = N ( q σ ) where N ( v ) denotes theneighbors of a point v in R d in the geometric graph ( G, v ) , that is the vertices of G at distanceless than 1 from v .Proof. For π = σ we have that σ ⊆ N ( q π ) if and only if q π ∈ M σ . Clearly σ ⊆ N ( q σ ) and π ⊆ N ( q π ) but by Lemma 58 it cannot be the case that both σ ⊆ N ( q π ) and π ⊆ N ( q σ ) . (cid:3) Now, for each G ∈ U k − ,d and π ∈ P ( G ) we construct a geometric graph ( P, q π ) whichsatisfies conditions P and P and which satisfies Lemma 59. Then, u k,d ≥ |P| · u k − ,d ≥ ⌊ ( k − / ( d + 1) ⌋ d · u k − ,d . Proof of Theorem 56. We have n,d ≥ u n,d ≥ n − Y i = d +1 (cid:22) id + 1 (cid:23)! d ≥ n − Y i = d +1 i − dd + 1 ! d ≥ (cid:18) ( n − d − d + 1) n − d − (cid:19) d . Using the estimate k ! ≥ (cid:0) ke (cid:1) k we get n,d ≥ (cid:18) n − d − d + 1) e (cid:19) d ( n − d − ≥ (cid:18) n ( d + 1) e (cid:19) dn (cid:18) nd + 1 (cid:19) − d ( d +1) , where for the last inequality we used (cid:0) − d +1 n (cid:1) d ( n − d − ≥ e − d ( d +1) , which derives from (1 + d +1 n − d − ) d ( n − d − ≤ e d ( d +1) . We then use (cid:16) nd +1 (cid:17) d ( d +1) ≤ (cid:0) e d (cid:1) n to obtain the lower bound.5.5. The top Betti numbers. The goal of this section will be to prove Theorem 11 regardingthe low degree Betti numbers of the one point compactification of the discriminant.We will prove this using the generalized Nerve Lemma of Björner. Theorem 61 (Special case of Theorem 6 of [Bjo03]) . Let X be a regular CW complex and ( X i ) i ∈ I a family of subcomplexes such that X = S i ∈ I X i . Suppose that every non-empty finiteintersection X i ∩ · · · ∩ X i t is ( k − t + 1) -connected then X is k -connected if and only if thenerve N ( X i ) is k -connected. Recall that given a CW-complex X and a covering by subcomplexes ( X i ) i ∈ I the nerve ofthe covering N ( X i ) is the simplicial complex on the vertex set I where σ = [ i , ..., i t ] is a faceof the nerve if and only if X i ∩ · · · ∩ X i t is nonempty.The covering that we will use for ˆ∆ d,n will be given by ( ˆ∆ i,jd,n ) ≤ i For any graph G with β ( G ) connected components ∆ Gd,n is a direct product of acompact set K := K ( G ) of dimension at most ( d − n − β ) and R d × β . Therefore ˆ∆ Gd,n is dβ − connected. Toward proving this result it will be helpful to observe that ∆ Gd,n is the space of graphhomomorphisms from G into the unit distance graph on R d . The unit distance graph on R d isthe graph whose vertices are the points in R d with an edge between two points if and only ifthe two points are at distance 1 from each other. Proof of Lemma 62. Let H be a component of G and let T be a spanning tree of H . Clearly ∆ Hd, | V ( H ) | ⊆ ∆ Td, | V ( H ) | as any homomorphism from H to the unit distance graph on R d inducesa homomorphism from T to the unit distance graph on R d . Moreover, we have that ∆ Td, | V ( H ) | ∼ R d × ( S d − ) | E ( T ) | . Indeed we may regard T as being a rooted tree and we can map the root of RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 41 T to any point of R d and from there every vertex may live anywhere on the sphere of radius1 centered at the image of its parent vertex. Now taking ∆ Hd, | V ( H ) | and modding out by the R d factor coming from the choice of image for the root in T we have a closed subset of thecompact set ( S d − ) | E ( T ) | = ( S d − ) | V ( H ) |− . Thus, without fixing the image of the root, wehave that ∆ Hd, | V ( H ) | is the direct product of R d and the compact space K ( H ) given by theclosed subset of ( S d − ) | V ( H ) |− . It is clear that we may describe any homomorphism from G to the unit distance graph on R d as a product of graph homomorphisms on the connectedcomponents. We have that ∆ Gd,n ∼ = K ( G ) × R dβ where K ( G ) is the direct product of K ( H ) over all connected components H . Thus K ( G ) iscontained in some n − β fold product of ( d − -dimensional sphere so it is at most ( d − n − β ) dimensional.We now turn our attention to the compactification of ∆ Gd,n . By the description of ∆ Gd,n ,we have that ∆ Gd,n is a dβ -ranked vector bundle over a compact CW complex (since we areworking with semialgebraic sets), so its compactification is the Thom space of this vectorbundle which is dβ − connected by Lemma 18.1 of [MS74]. (cid:3) Proof of Theorem 11. We apply Lemma 61 to prove that ˆ∆ d,n is n + d − connected. Weconsider the cover of ˆ∆ d,n by ( ˆ∆ i,jd,n ) ≤ i 2) = ( n − d − reduced homology class. The path componentsof R d × n \ ∆ d,n are the rigid isotopy classes of graph on n vertices in R d . We consider the rigidisotopy classes of the empty graph on n vertices. The rigid isotopy classes of the empty graphon n vertices gives all the configurations of n points in R d so that the distance between anypair of them is larger than 1. This is homotopy equivalent to Conf n ( R d ) , see Examples 3 and8, which has its top positive reduced Betti number in dimension ( n − d − . (cid:3) Examples Here we work out a few examples for computing the Betti numbers of R d × n \ ∆ d,n for smallvalues of d and n . We start with the case that d = 1 . In the case that d = 1 , computing thetopology of ˆ∆ G ,n across all graphs G on n vertices is sufficient to compute b ( R × n \ ∆ ,n ) .While we have shown the the number of rigid isotopy classes of R -geometric graphs on n verticesis given by the number of labeled semiorders on n elements, we work out a computation for n = 3 here primarily to show how spectral sequences and Alexander–Pontryagin duality canbe used to compute the exact number of rigid isotopy classes of R -geometric graphs. Example 63 ( n = 3 , d = 1 ) . By Alexander–Pontryagin duality, it suffices to compute theBetti numbers of ˆ∆ , . This is a union of three compactified quadrics given by the solutionsin R to | x i − x j | = 1 for ≤ i < j ≤ . Each of these is simply the disjoint union of twohyperplanes in R × . Thus ˆ∆ , is a 2-dimensional cell complex and we know by Theorem 11 that this complex is 1-connected so only the 2nd Betti number is interesting. This is notsurprising as the dual in S of ˆ∆ , is R \ ∆ , which is a disjoint union of finite intersectionsof halfspaces, so only its zeroth Betti number is interesting.Now by computing the E page of the Mayer–Vietoris spectral sequence given by the cov-ering of ˆ∆ , by (cid:16) ˆ∆ i,j , (cid:17) ≤ i RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 43 and Greenhill [DG00] shows that the problem of enumerating graph homomorphism into afixed bipartite graph H is H is a special type of bipartite graph, whichdoes not include the unit distance graph on R . Given these results, we should look for otherways to exactly count the number of isotopy classes given d and n than computing the fullMayer–Vietoris spectral sequence. To have at least some explicit examples for larger d wework out the Betti numbers for n = 3 , and any d ≥ , though the methods we use are ratherad hoc and doesn’t generalize to higher values of n . Example 64 ( n = 3 , d ≥ ) . Here we explicitly compute the topology of each rigid isotopyclass on 3 vertices. The space W G,d for G the complete graph on 3 vertices is contractible, infact it is easy to see that it is convex.For G on two edges we observe that W G,d has the topology of S d − . To see this we consider G as a path on 3 vertices, the first vertex on the path can go anywhere, the second vertex can goanywhere in the punctured ball of radius 1 around the first vertex; it must be punctured sinceonly vertices with identical closed neighborhoods can map to the same point. After mappingthe first two vertices the final vertex can be moved freely in some contractible subspace of R d determined by the position of the first two vertices.Now for G on 1 edge we observe have that W G,d has the topology of the configuration spaceof two points in R d , which is just S d − . Finally if G is the empty graph then W G,d is homotopyequivalent to the configuration space of 3 points in R d which is known to have b = 1 , b d − = 3 and b d − = 2 .Putting this all together, we have that the empty graph contributes 1 to b , to b d − and to b d − , the three graphs on one edge each contribute to b and to b d − , the three graphson two edges also each contribute to b and to b d − and the complete graph contributes to b . So we have x d − + 2 x d − as the Poincaré polynomial for R d × \ ∆ d, . Example 65 ( n = 4 , d ≥ ) . The case n = 4 is more interesting but again the computation israther ad hoc. We examine each graph G on 4 vertices and compute the Poincaré polynomialfor W G,d summarized in Table 1. These Poincaré polynomials are essentially computed byinspection; we don’t give the full details for the computations. For example the graph givenby a path on 3 vertices and an isolated vertex is has the homotopy type of S d − × S d − . Withone vertex of the path fixed, the next vertex is free to go anywhere in the punctured ballaround the first vertex. Next the final vertex of the path may be place freely inside somecontractible set. So the path contributes a factor of S d − Finally after the path is placed in R d the union of the balls around its vertices gives a contractible space in R d and the isolatedvertex may be place anywhere outside of this contractible space so this contributes the other S d − factor. From Table 1 we can determine β ( G ) for any graph on vertices, and determinethat the Poincare Polynomial of R d × \ ∆ d, is 64 + 7 x d − + 92 x d − + 7 x d − + 35 x d − + 6 x d − In particular in the case of the plane there are 71 rigid isotopy class of graphs on 4 vertices,while there are 64 labeled graphs on 4 vertices all of which can be realized as geometric graphsin the plane. 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Geometric combinatorics ,13:389–496, 2004.[Szu92] Andrzej Szulkin. Cohomology and Morse theory for strongly indefinite functionals. Math. Z. ,209(3):375–418, 1992.[War68] Hugh E. Warren. Lower bounds for approximation by nonlinear manifolds. Transactions of the Amer-ican Mathematical Society , 133(1):167–178, 1968. RAPH INVARIANTS FROM THE TOPOLOGY OF RIGID ISOTOPY CLASSES 45 SISSA (Trieste) E-mail address : [email protected] SISSA (Trieste) E-mail address : [email protected] A.N. Technische Universität Berlin, Chair of Discrete Mathematics/Geometry, Strasse des17. Juni 136, 10623 Berlin E-mail address : [email protected]@math.tu-berlin.de