aa r X i v : . [ m a t h . AG ] J a n VANISHING HOMOLOGY
GUILLAUME VALETTE
Abstract.
In this paper we introduce a new homology theory devoted to the studyof families such as semi-algebraic or subanalytic families and in general to any familydefinable in an o-minimal structure (such as Denjoy-Carleman definable or ln − exp definable sets). The idea is to study the cycles which are vanishing when we approach aspecial fiber. This also enables us to derive local metric invariants for germs of definablesets. We prove that the homology groups are finitely generated. Introduction
The description of the topology of a set nearby a singularity is a primary focus ofattention of algebraic geometers. We can regard a semi-algebraic singular subset of R n as a metric subspace. Then the behavior of the metric structure of a collapsing familyreflects implicit information on the geometry of the singularity of the underlying set whichis much more accurate than the one provided by the study of the topology.In [V1], the author proved a bi-Lipschitz version of Hardt’s theorem [H]. This theorempointed out that semi-algebraic bi-Lipschitz equivalence is a good notion of equisingularityto classify semi-algebraic subsets from the metric point of view. For this purpose, it isalso very helpful to find invariants such as homological invariants.In this paper we introduce a homology theory for families of subsets which providesinformation about the behavior of the metric structure of the fibers when we approacha given fiber. This enables us to construct local metric invariants for singularities. Weprove that these homology groups are finitely generated when the family is definable inan o-minimal structure. This allows, for instance, to define an Euler characteristic whichis a metric invariant for germs of algebraic or analytic sets.In [GM], M. Goresky and R. MacPherson introduced intersection homology and showedthat their theory satisfies Poincar´e duality for pseudo-manifolds which cover a quite largeclass of singular sets and turned out to be of great interest. They also managed to com-pute the intersection homology groups from a triangulation which yields that they arefinitely generated. In [BB1] L. Birbrair and J.-P. Brasselet define their admissible chainsto construct the metric homology groups. Both theories select some chains by puttingconditions on the support of the chains. Our approach is similar in the sense that ourhomology groups will depend on a velocity which estimates the rate of vanishing of thesupport of the chains.Our method relies on the result of [V1], where the author showed existence of a trian-gulation enclosing the metric type of a definable singular set. To compute the vanishinghomology groups we will not use the triangulation constructed in [V1] but Proposition Mathematics Subject Classification. . . R n , the case of an arbitrary real closed field will be required. Our approachwill be patterned on the one of the classical homology groups as much as possible. Somestatements (Theorem 3.2.2) are close to those given by Goresky and MacPherson forintersection homology but of course the techniques are radically different since the settingis not the same.The admissible chains depend on a velocity which is a convex subgroup v of our realclosed field R . For instance, if R is the field of real algebraic Puiseux series endowed withthe order making the indeterminate T smaller than any positive real number, v may bethe subgroup(0.1) { x : ∃ N ∈ N , | x | ≤ N T } . The v -admissible chains are the chains having a “ v -thin” support. Roughly speaking, if v isas above, v -thin subsets of R n are the generic fibers of families of sets whose fibers collapseonto a lower dimensional subset with at least the velocity N t (if t is the parameter of thefamily, N ∈ N ). For instance, let us consider the cycle given by Birbrair and Goldshtein’sexample. Namely, the subset of X ⊂ R defined by: x + x = T p ,x + x = T q . (0.2)This set is the generic fiber of a family of tori, such that the support of the generatorsof H ( X ) collapse onto a point at rate T p and T q respectively. Therefore, if for instance p = 0 and q = 2 then the 0-fiber is a circle and this family of torus is v -thin (with v likein (0.1)).Taking all the v -admissible chains of a definable set X , we get a chain complex whichimmediately gives rise to the v -vanishing homology groups H vj ( X ). We will show thatthese groups are finitely generated (Corollary 3.2.3).If X is the set defined by (0.2) with v like in (0.1), the v -vanishing homology groupsdepend on of p and q . For instance, we will prove (see Example 5.2.2) that if p = 0 and q = 2: H v ( X ) = Q (if Q is our coefficient group), and H v ( X ) = Q .We may summarize it by saying that we get all the T -thin cycles of X . The group H vj ( X ) is not always a subgroup of H j ( X ). In general we may also have cycles that do notappear in the classical homology groups, i. e. which are in the kernel of the natural map ANISHING HOMOLOGY H vj ( X ) → H j ( X ). The following picture illustrates an example for which such a situationoccurs: a Figure 1.
The cycle a is collapsing onto a point faster than the set itself is collapsing. We see thatwe have an admissible one dimensional chain a which bounds a two dimensional chainwhich may fail to be admissible (depending on the velocity v ). Therefore H v ( X ) = 0(while H ( X ) = 0).This homology theory is not a homotopy invariant. It is preserved by Lipschitz homo-topies but these are very hard to construct. For instance, given a function f : R n → R itis well known that there exists a topological deformation retract of f − (0; ε ) onto f − (0).It is easy to see that it is not possible to find such a retract which would be Lipschitzif f ( x ; y ) = y − x . The method used in this paper provides homotopies that are notLipschitz but which preserve admissible chains. It seems that one could define varioushomology theories for which this method could be adapted. The theory developed belowseemed to the author the simplest one and the most natural to start.We compute the vanishing homology groups in terms of some basic sets obtained byconstructing some nice cells decompositions (Theorem 3.2.2). For this we construct ahomotopy which carries a given singular chain to a chain of these basic sets (Proposition3.2.1). The homotopy has to preserve thin subsets. We are not able to construct such ahomotopy for any admissible chain. Chains for which we can construct such a homotopyare called strongly admissible and are chains for which the distances in the support areknown in a very explicit way. Therefore, the first step is to show that any class in H vj ( X )has a strongly admissible representant (Lemma 3.1.3). This is achieved by constructingsome rectilinearizations of v -thin sets (Proposition 2.2.4). These are maps which transformour set into a union of hyperplanes crossing normally while controlling the distances inthe transformation.A non trivial convex subgroup v may be regarded as an interval in R which has noendpoint. This fact will somewhat complicate our task. To overcome this difficulty, weintroduce an extra point u “at the end of v ” which will fill the gap. This point living inan extension k v of R , we will carry out most of the constructions rather in k v than in R .The precise definition of k v and the basic related notions are provided in the first sectionbelow. An advantage of using model theory is that we are able to carry out the theory forall the possible velocities (see example 1.1.2) in the same time. GUILLAUME VALETTE
We may use these homology groups to derive invariants for semi-algebraic singularities.Given a germ A of semi-algebraic subset of R n at the origin, the link of A is the subset L r := A ∩ B (0; r )for r small enough. It is known that the homology of the latter set is a topological invariantof A . The cycles of L r are collapsing to a single point with a certain “rate”. This rate isrelated to the metric type of the singularity.It is proved in [V2] that the metric type of the generic fiber of the family L r , namely L + , is a metric invariant of A . Therefore the vanishing homology groups H vj ( L + ) aresemi-algebraic bi-Lipschitz invariants of A (see section 4.). Content of the paper.
In section 1, we provide all the basic definitions about thevanishing homology. We prove in the next section some cell decomposition theorems andrectilinearization theorems necessary to compute the vanishing homology groups. In sec-tion 3, we compute the v -vanishing homology groups in terms of this cell decomposition.The main result is Theorem 3.2.2 which yields that the homology groups are finitely gener-ated. In section 4 we give an application: we find local metric invariants for singularities.The last section computes the vanishing homology groups on some examples.The reader is referred to [C] or [vD] for basic facts about o-minimal structures. Notations and conventions.
Throughout this paper we work with a fixed o-minimalstructure expanding a real closed field R . Let L R be the first order language of orderedfields together with an n -ary function symbol for each function of the structure. The worddefinable means L R -definable. The language L R ( u ) is the language L R extended by anextra symbol u .The letter G will stand for an abelian group (our coefficient group). Singular sim-plices will be definable continuous maps c : T j → X , T j being the j -simplex spanned by0 , e , . . . , e j where e , . . . , e j is the canonical basis of R j . Sometimes, we will work in anextension k v of R and simplices will actually be maps c : T j ( k v ) → k nv where T j ( k v ) isthe extension of T j to k v . Given a definable set X ⊂ R n we denote by C ( X ) the chaincomplex of definable chains with coefficients in a given group G . We will write | c | for thesupport of a chain c .By Lipschitz function we will mean a function f satisfying | f ( x ) − f ( x ′ ) | ≤ N | x − x ′ | for some integer N . It is important to notice that we require the constant to be aninteger for R is not assumed to be archimedean. A map h : A → R n is Lipschitz if all itscomponents are, and a homeomorphism h is bi-Lipschitz if h and h − are Lipschitz.We denote by π n : R n → R n − the canonical projection and by cl ( X ) the closure of adefinable set X . 1. Definition of the vanishing homology.
The velocity v . We shall use some very basic facts of model theory. We refer thereader to [M] for basic definitions.
ANISHING HOMOLOGY The vanishing homology depends on a velocity v which estimates the rate of vanishingof the cycles. This is a convex subgroup v of ( R ; +) (convex in the sense that it is a convexsubset of R ).We then define a 1-type by saying that a sentence ψ ( u ) ∈ L R ( u ) is in this type iff theset { x ∈ R : ψ ( x ) } contains an interval [ a ; b ] with a ∈ v and b / ∈ v . This type is complete due to the o-minimality of the theory.We will denote by k v an L R -elementary extension of R realizing this type.Roughly speaking we can say that the velocity is characterized by a cut in R , at whichthe gap is “bigger” than the distance to the origin. This is to ensure that the sum of twoadmissible chains will be admissible (see section 1.3). Notations.
Throughout this paper, a velocity v is fixed and u is the point realizing thecorresponding type in k v .We define a convex subgroup w of ( k v ; +) extending the group v in a natural way: w := { x ∈ k v : ∃ y ∈ v, | x | ≤ y } . Remark 1.1.1.
Given z ∈ R we may define a velocity N z by setting: N z := { x ∈ R : ∃ N ∈ N , | x | ≤ N z } . Example 1.1.2.
Let k (0 + ) be the field of real algebraic Puiseux series endowed with theorder that makes the indeterminate T positive and smaller than any real number (see[BCR] example 1.1.2). Then, as in the above remark, the element T k gives rise to asubgroup N T k which is constituted by all the series z having a valuation greater or equalto k . One could also consider the velocity v defined by the set of x satisfying | x | ≤ N T k for any N in N . In the field of ln − exp definable germs of one variable functions (in aright-hand side neighborhood) one may consider the set of all the L p integrable germs ofseries. Extension of functions.
On the other hand, as k v is an elementary extension of R ,it is well known that we may define X v , the extension of X to k v , by regarding theformula defining X in k nv . Every mapping σ : X → Y may also be extended to a mapping σ v : X v → Y v .1.2. v -thin sets. We give the definition of the v -thin sets which is required to introducethe vanishing homology. Definitions 1.2.1.
Let j ≤ n be integers. A j -dimensional definable subset X of R n iscalled v -thin if there exists z ∈ v such that, for any linear projection π : R n → R j , noball (in R j ) of radius z entirely lies in π ( X ).For simplicity we say that X is ( j ; v ) -thin if either X is v -thin or dim X < j . A setwhich is not v -thin will be called v -thick .Note that in the above definition it is actually enough to require that the property holdsfor a sufficiently generic projection π : R n → R j . As we said in the introduction, roughlyspeaking, N T -thin sets of k (0 + ) n are the generic fibers of one parameter families whosefibers “collapse onto a lower dimensional subset at rate at least t ” (if t is the parameter GUILLAUME VALETTE of the family). Also, by convention R = { } so that a 0-dimensional subset is never v -thin. This is natural in the sense that a family of points never collapses onto a lowerdimensional subset. Basic properties of ( j ; v ) -thin sets. (1) If a definable subset A ⊂ X is ( j ; v )-thin andif h : X → Y is a definable Lipschitz map then h ( A ) is ( j ; v )-thin.(2) Given j , ∪ pi =1 X i is ( j ; v )-thin iff X i is ( j ; v )-thin for any i = 1 , . . . , p .1.3. Definition of the vanishing homology.
Given a definable set X let C vj ( X ) be the G -submodule of C j ( X ) generated by all the singular chains c such that | c | is ( j ; v )-thinand | ∂c | is ( j ; v )-thin as well. We endow this complex with the usual boundary operatorand denote by Z vj ( X ) the cycles of C vj ( X ).A chain σ ∈ C vj ( X ) is said v -admissible . We denote by H vj ( X ) the resulting homologygroups which we call the v -vanishing homology groups .If v is N z , for some z ∈ R (see Remark 1.1.1), then we will simply write C zj ( X ) and H zj ( X ) (rather than C N zj and H N zj ). Remark 1.3.1. If X is v -thin and if j = dim X then every j -chain is v -admissible.Moreover every ( j + 1)-dimensional chain is admissible by definition. Hence the map H vj ( X ) → H j ( X ) induced by the inclusion of the chain complexes is an isomorphism.Note also that the map H vj − ( X ) → H j − ( X ) is a monomorphism.Every Lipschitz map sends a ( j ; v )-thin set onto a ( j ; v )-thin set. Thus, every Lipschitzmap f : X → Y , where X and Y are two definable subsets, induces a sequence of mappings f j,v : H vj ( X ) → H vj ( Y ). In consequence, the vanishing homology groups are preserved bydefinable bi-Lipschitz homeomorphisms.As we said in the introduction this homology gives rise to a metric invariant for families(preserved by families of bi-Lipschitz homeomorphisms) by considering the generic fiberas described in the following example. Example 1.3.2.
With the notation of example 1.1.2, given an algebraic family X ⊂ R n × R defined by f = · · · = f p = 0, we set X + := { x ∈ k (0 + ) n : f ( x ; T ) = · · · = f p ( x ; T ) = 0 } . Hence, H vj ( X + ) is a metric invariant of the family.1.4. The complex C vj ( X ; F ) . Given a finite family F , of closed subsets of X , we write C j ( X ; F ) for the j -chains of ⊕ F ∈F C j ( F ). Similarly we set: C vj ( X ; F ) := ⊕ F ∈F C vj ( F )and denote by H vj ( X ; F ) the corresponding homology groups. By Remark 1.3.1, if τ is achain of Z vj ( | σ | ) whose class is σ in H j ( | σ | ) then τ = σ in H vj ( | σ | ) as well. Therefore, as H j ( | σ | ; F ) = H j ( | σ | ) we get:(1.3) H vj ( X ; F ) ≃ H vj ( X ) . ANISHING HOMOLOGY Strongly admissible chains.
It is difficult to construct homotopies between v -admissible chains. To overcome this difficulty we introduce strongly v -admissible chains. Definition 1.5.1.
We denote by T qj the set of all ( x ; λ ) ∈ T j × R such that x + λe q belongsto T j . A simplex σ : T j → R is strongly v -admissible if there exists q such that for any( x ; λ ) ∈ T qj :(1.4) ( σ ( x ) − σ ( x + λe q )) ∈ v. A chain is strongly admissible if it is a combination of strongly admissible simplices.We denote by b C vj ( X ) the chain complex generated by the strongly admissible chains σ for which ∂σ is strongly admissible, and by b Z j ( X ) the strongly admissible cycles. Theresulting homology is denoted by b H vj ( X ). If F is a family of closed subsets of X , we alsodefine b C vj ( X ; F ), b Z vj ( X ; F ), and b H vj ( X ; F ) in an analogous way (see section 1 . Remark 1.5.2.
Let σ : T j → R n be a strongly admissible simplex with j ≤ n . Then bydefinition, there exists z ∈ v such that for any x ∈ T j : d ( σ ( x ); σ ( ∂T j )) ≤ z. As σ ( ∂T j ) is of dimension strictly inferior to j we see that the image of this set under aprojection onto R j contains no open ball in R j . In other words, if σ and ∂σ are stronglyadmissible chains then σ is admissible. In consequence, a strongly admissible cycle isadmissible. 2. Rectilinearizations of v -thin sets. Regular directions.
We recall a result proved in [V1] which will be very useful tocompute our vanishing homology. We start by the definition of a regular direction. Wedenote by X reg the set of points x ∈ X at which X is a C manifold. Definition 2.1.1.
Let X be a definable set of R n . An element λ of S n − is said regularfor X if there exists a positive α ∈ Q : d ( λ ; T x X reg ) ≥ α, for any x ∈ X reg .Not every definable set has a regular line. However, we have: Proposition 2.1.2. [V1]
Let A be a definable subset of R n of empty interior. Then thereexists a definable bi-Lipschitz homeomorphism h : R n → R n such that e n is regular for h ( A ) . Remark 2.1.3.
When e n is regular for a set X , we may find finitely many Lipschitzdefinable functions, say ξ i : R n − → R , i = 1 , . . . , s, satisfying(2.5) ξ ≤ · · · ≤ ξ s , and such that the set X is included in the union of their respective graphs. GUILLAUME VALETTE
Cell decompositions.
In order to fix notations we recall the definition of the cells,which, as usual, are introduced inductively. All the definitions of this section deal withsubsets of R n , but since R stands for an arbitrary real closed field, we will use them forsubsets of k nv as well. Definitions 2.2.1.
For n = 0 a cell of R n is { } . A cell E of R n is either the graph of adefinable function ξ : E ′ → R , where E ′ is a cell of R n − or a band of type:(2.6) { x = ( x ′ ; x n ) ∈ E ′ × R : ξ ( x ′ ) < x n < ξ ( x ′ ) } , where ξ , ξ : E ′ → R are two definable functions satisfying ξ < ξ or ±∞ . The cell E is Lipschitz if E ′ is Lipschitz and if ξ and ξ (or ξ ) are Lipschitz functions (and { } isLipschitz). A closed cell is the closure of a cell (which is obtained by replacing < by ≤ in the definition).Given z ∈ R , the Lipschitz cell E is z -admissible if(1) E ′ is z -admissible(2) If E is a band defined by two functions ξ and ξ , then either ( ξ − ξ )( x ) ≤ z forany x ∈ E ′ , or ( ξ − ξ )( x ) ≥ z for any x ∈ E ′ .Set also that the cell { } is z -admissible.A cell E of dimension j is canonically homeomorphic to (0; 1) j . The barycentricsubdivision of E is the partition defined by the image by this homeomorphism of thebarycentric subdivision of (0; 1) j .We shall need the following very easy lemma. Lemma 2.2.2.
Let E ′ be a w -thick Lipschitz cell of k n − v and let ξ s : E ′ → k v , s = 1 , ,be two Lipschitz functions such that ξ < ξ and ( ξ − ξ )( x ) / ∈ w , for any x ∈ E ′ . Thenthe band: E := { ( x ; y ) ∈ E ′ × k v : ξ ( x ) < y < ξ ( x ) } is w -thick.Proof. We may assume that E ′ is open in k n − v since we may find a bi-Lipschitz home-omorphism which carries E ′ onto an open cell. Then E is also an open and the cell E ′ contains a ball of radius z / ∈ w , say B ( x ; z ). Let t := ξ ( x ) − ξ ( x ); we have by assump-tion t / ∈ w . Taking t small enough we may assume B ( x ; t ) entirely lies in E ′ . Let N bethe Lipschitz constant of ( ξ − ξ ) and note that: ξ ( x ) − ξ ( x ) ≥ t , for x ∈ B ( x ; t N ). This implies that E contains a ball of radius t N . But, as t / ∈ w wehave t N / ∈ w . (cid:3) Definition 2.2.3.
The subset { } is an L -cell decomposition of R . For n >
0, an L -cell decomposition of R n is a cell decomposition of R n satisfying:(i) The cells of R n − constitute an L -cell decomposition of R n − (ii) There exist finitely many Lipschitz functions ξ , . . . , ξ s : R n − → R satisfying (2.5)such that the union of all the cells which are graphs of a function on a subset of R n − , is the union of the graphs of the ξ i ’s. ANISHING HOMOLOGY An L -cell decomposition is said compatible with finitely many given definable subsets X , . . . , X m if these subsets are union of cells. It is said z -admissible if every cell is z -admissible. Taking the barycentric subdivision of every cell, we get a barycentricsubdivision of an L -cell decomposition.We are going to show that, we may find a u -admissible L -cell decomposition which iscompatible with some given L R ( u )-definable subsets of k nv . This will be helpful to provethat the homology groups are finitely generated, since we will show that only the N u -thincells are relevant to compute the homology groups. The following proposition deals withsubsets of k v since we will apply it to k v but of course the proof goes over an arbitrarymodel of the theory. Proposition 2.2.4.
Let X , . . . , X m be L R ( u ) -definable subsets of k nv . There exists a L R ( u ) definable bi-Lipschitz homeomorphism h : k nv → k nv such that we can find a u -admissible L -cell decomposition of k nv compatible with h ( X ) , . . . , h ( X m ) .Proof. For n = 0 there is nothing to prove. Assume n > ∪ mj =1 ∂X j (where ∂ denotes the topological boundary). Then (see Remark 2.1.3) thereexist finitely many definable Lipschitz functions ξ i , i = 1 , . . . , s satisfying (2.5). Considera cell decomposition of k nv compatible with X , . . . , X m , all the graphs of the ξ i ’s, as wellas all the sets { x ∈ k n − v : ξ i +1 ( x ) − ξ i ( x ) = u } . Now apply the induction hypothesis to all the cells of this decomposition which lie in k n − v to get a cell decomposition E of k n − v . Then set ξ := −∞ , ξ s +1 := ∞ , and consider thecell decomposition of k nv constituted by the graphs of the restrictions of the functions ξ i ’sto an element of E on the one hand, and all the subsets of type: { ( x ; x n ) ∈ E × k v : ξ i ( x ) < x n < ξ i +1 ( x ) } , where E ∈ E , on the other hand. The required properties hold. (cid:3) Rectilinearization of v -thin sets. We introduce the notion of rectilinearization.This is a mapping which transforms a set into a union of coordinate hyperplanes and whichinduces an isomorphism in homology (the usual one). Admissible rectilinearizations willbe very helpful to construct strongly admissible chains (see section 1 . v -admissible rectilinearization compatible with a givenfamily of v -thin sets. Definitions 2.3.1. A hyperplane complex is a subset W of R n , which is a union offinitely many coordinate hyperplanes of type x j = s where, for each hyperplane, s is aninteger. There is a canonical cell decomposition of R n compatible with W . We refer tothe cells (resp. closure of the cells) as the cells of W (resp. closed cells of W ).Let X , . . . , X m be definable subsets. A rectilinearization of X , . . . , X m is a mapping h : R n → R n , such that the h − ( X i )’s are union of cells of W and such that for any i = 1 , . . . , m the mapping h i : h − ( X i ) → X i induces an isomorphism in homology (theusual one). If X , . . . , X m are v -thin, a rectilinearization of X , . . . , X m is v -admissible if for eachcell σ of W included in h − ( X i ) there exists an integer q with e q tangent to σ for which(2.7) ( h ( x ) − h ( x + λe q )) ∈ v for any x ∈ σ and λ ∈ R such that x + λe q ∈ σ . Remark 2.3.2.
After a barycentric subdivision of h − ( X i ), we get a simplicial complex K i and a map h i : K i → X i which induces an isomorphism in homology. Note that, thanksto (2.7) each simplicial chain gives rise (identifying each j -simplex to T j in a linear way) toa strongly admissible chain (see Definition 1.5.1). Moreover, as h induces an isomorphismin homology, this identification defines an isomorphism in homology H j ( K i ) → H j ( X i ). Proposition 2.3.3.
Let X , . . . , X m be closed definable v -thin subsets of R n . Then thereexists a v -admissible rectilinearization of X , . . . , X m .Proof. We start by proving the following statements ( H n ) by induction on n .( H n ) . Let E be a u -admissible L -cell decomposition of k nv and let Y , . . . , Y r denote the w -thin closed cells. Then there exists a N u -admissible rectilinearization h : k nv → k nv of Y , . . . , Y r such that, for every E in E , h − ( cl ( E )) is a union of closed cells of W and thereexists a strong deformation retract r E : h − ( cl ( E )) → C E , where C E is a closed cell of W .Note that it follows from the existence of this deformation retract that h induces anisomorphism in homology above any union of closed cells of E . Actually, the existence of r Y i implies H j ( h − ( Y i )) ≃ H j ( C Y i ) ≃ H j ( Y i ) , and the map h | h − ( Y i ) : h − ( Y i ) → Y i induces an isomorphism in homology. Therefore,thanks to the Mayer-Vietoris property and to the 5-Lemma, we see that for any subset X constituted by the union of finitely many closed cells the map h | h − ( X ) : h − ( X ) → X induces an isomorphism in homology.Note that nothing is to be proved for n = 0 and assume ( H n − ). Apply the inductionhypothesis to the family constituted by the closure of the cells of E in k n − v which are w -thin to get a rectilinearization h : k n − v → k n − v and a hyperplane complex W .Note that by definition, the cells of E on which the restriction of π n is one-to-one areincluded in the union of finitely many graphs of definable Lipschitz functions ξ , . . . , ξ s : k n − v → k v satisfying (2.5).We obtain a hyperplane complex f W by taking the inverse image of W by π n , and byadding the hyperplanes defined by x n = i , i = 1 , . . . , s .Define now the desired mapping e h as follows: e h ( x ; i + t ) = ( h ( x ); (1 − t ) ξ i ( h ( x )) + tξ i +1 ( h ( x )))for 1 ≤ i < s integer, x ∈ k nv and t ∈ [0; 1). Define also: e h ( x ; 1 − t ) = ( h ( x ); ξ ( h ( x )) − t )and e h ( x ; s + t ) = ( h ( x ); ξ s ( h ( x )) + t ) ANISHING HOMOLOGY for t ∈ [0; ∞ ). This defines a mapping e h : k nv → k nv . We are going to check that(2.8) | e h ( x ) − e h ( x + λe n ) | ≤ u when x and ( x + λe n ) belong to the same cell.Let σ be a cell of f W which is mapped into ∪ ri =1 Y i . If π n ( σ ) is w -thin (2.8) follows fromthe induction hypothesis. Otherwise e h ( σ ) must lie in the band delimited by the graphsof the restrictions of ξ i and ξ i +1 for some i ∈ { , . . . , s − } as described in (2.6). If e h ( π n ( σ )) fails to be w -thin then, thanks to Lemma 2.2.2 (recall that e h ( σ ) is w -thin) andthe u -admissibility of the cell decomposition, we necessarily have: | ξ i ( x ) − ξ i +1 ( x ) | ≤ u, for any x ∈ π n ( σ ). This, together with definition of e h , implies that e h satisfies (2.8) andyields that e h is N u -admissible. It remains to find the retraction r E for each cell E .Fix E ∈ E and observe that it follows from the definition of e h and the inductionhypothesis that e h − ( cl ( E )) is a union of cells of f W . If E is the graph of a function ξ : E ′ → k v (where E ′ := π n ( E )), then the result directly follows from the inductionhypothesis. Otherwise, since E is an L -cell decomposition, the cell E lies in the banddelimited by the graphs of two consecutive functions, say ξ i and ξ +1 . LetΓ i := { ( x ; x n ) ∈ k n − v × k v : i ≤ x n ≤ i + 1 } . We first define first a retract: r ′ E : e h − ( cl ( E )) × [0; 12 ] k v → Γ i ∩ e h − ( cl ( E )) , by setting for x n ≥ i + 1: r ′ E ( x ; x n ; t ) := ( x ; 2 tx n + (1 − t )( i + 1)) , and for x n ≤ i : r ′ E ( x ; x n ; t ) := ( x ; 2 tx n + (1 − t ) i ) , and of course r ′ E ( x ; x n ; t ) := ( x ; x n ) when i ≤ x n ≤ i + 1.Note that it follows from the definition of e h that if ( x ; x n ) belongs to e h − ( cl ( E )) thenfor any i + 1 ≤ x ′ n ≤ x n and any x n ≤ x ′ n ≤ i : e h ( x ; x ′ n ) = e h ( x ; x n ) . This implies that r ′ E preserves e h − ( cl ( E )).On the other hand, thanks to the induction hypothesis, there exists a retract r E ′ : h − ( cl ( E ′ )) × [0; 1] k v → C E ′ . Let us extend this r E ′ into a retract: r ′′ E ′ : π − n ( h − ( cl ( E ′ ))) × [ 12 ; 1] k v → π − n ( C E ′ )by r ′ E ( x ; x n ; t ) := ( r E ′ ( x ; 2 t − x n ) . Clearly, there exists a unique cell C E of f W which is included in Γ i and which projectson C E ′ . Now, these retracts give rise to a retract e r E : e h − ( cl ( E )) × [0; 1] k v → C E defined by e r E ( x ; t ) := r ′ E ( x ; t ) if t ≤ and e r E ( x ; t ) := r ′′ E ( r ′ E ( x ; 12 ); t )if t ≥ . This yields ( H n ).We return to the proof of the proposition. Apply Proposition 2.2.4 to X ,v , . . . , X m,v .This provides a bi-Lipschitz homeomorphism g : k nv → k nv such that we can find a u -admissible L -cell decomposition of k nv compatible with g ( X ,v ) , . . . , g ( X m,v ). Note that,as the g ( X i,v )’s are w -thin, each of them is the union of some w -thin cells. Then by ( H n ),there exists a N u -admissible rectilinearization of these cells h : k nv → k nv .Composing with g , the mapping h gives rise to a N u -admissible rectilinearization f of X ,v , . . . , X m,v . As the X i,v are extensions, there exist two families of rectilinearizations f z and h z for z ∈ [ a ; b ] with a < u < b and a, b ∈ R . Let us check that these rectilinearizationsare v -admissible for z ∈ v large enough.Note that each X i is the union of the images by h z of finitely many cells of W . Fur-thermore, as (2.8) is a first order formula we get that h z satisfies on any given cell in theinverse image of the X i ’s: | h z ( x ) − h z ( x + λe n ) | ≤ z, when x and ( x + λe n ) belong to this given cell.This implies that f satisfies (since g is bi-Lipschitz): | f z ( x ) − f z ( x + λe n ) | ≤ N z, for some N ∈ N and any z ∈ v large enough on any cell mapped into one of the X i ’s.Thus, (2.7) holds and f z is w -admissible. (cid:3) Remark 2.3.4.
Actually, working a little more, we could have proved that the constructedrectilinearization induces an isomorphism in homology above any subset A of R n . Namely,in the above proof, given a subset A of R n , the induced mapping e h : e h − ( A ) → A inducesan isomorphism in homology.Observe also that the constructed mapping is a homeomorphism above a dense definablesubset. If we take an algebraic hypersurface, the situation is fairly similar to the one whichoccurs with resolution of singularities in the sense that the inverse image of the set abovewhich the map is not one-to-one (the “exceptional divisor”) is constituted by finitely manycoordinate hyperplanes normal to the hyperplanes lying above our given set. We couldalso have a more precise description of how the mapping h modifies the distances (like in[V1]). More precisely, it is possible to see that on each cell, we have | h ( x ) − h ( x ′ ) | ∼ n X i =1 ϕ i ( x ) | x i − x ′ i | where ϕ i is a sum of product of powers of distances to cells of W . If we compare this resultwith Theorem 5 . . h is not a homeomorphism (contrarilyas in [V1]), but since it induces an isomorphism between the homology groups, it will beenough for the purpose of the present paper. ANISHING HOMOLOGY The vanishing homology groups are finitely generated
Some preliminary lemmas.
Every mapping σ : T j → X may be extended to amapping σ v : T j ( k v ) → X v (see subsection 1 . extj ( X v ) be the submodule of C wj ( X v )generated by the simplices which are extensions of an element of C vj ( X ). Clearly, for each j the mapping: ext : C vj ( X ) → ∆ extj ( X v ) , which assigns to every chain σ the chain σ v , induces an isomorphism in homology.The following Lemma says that the vanishing homology groups for the velocities N u and w coincide with the homology groups of ∆ extj when the considered set is L R -definable. Lemma 3.1.1.
Let X be a definable subset of R n . Then the maps induced by the inclusions H j (∆ ext ( X v )) → H uj ( X v ) and H j (∆ ext ( X v )) → H wj ( X v ) are isomorphisms for any j .Proof. We do the proof for u . To get the proof for w , just replace u by w . We firstcheck that this map is onto. Let σ = P i ∈ I g i c i ∈ Z uj ( X v ). By definition of u there existfinitely many L R -definable mappings, say τ i : T j ( k v ) × [ a ; u ] k v → X v , with a ∈ v suchthat c i ( x ) = τ i ( x ; u ) for any x ∈ T j ( k v ). Define θ i ( x ) := τ i ( x ; a ) and θ := P i ∈ I g i θ i ∈ C extj ( X v ). Observe that τ i gives rise to a N u -admissible ( j +1)-chain (after a subdivision of T j ( k v ) × [ a ; u ]). Moreover, as the property of admissibility may be expressed by a formulawith parameters in R and with u , we know that the obtained chain is N u -admissible if a is chosen large enough. Set τ := P i ∈ I g i τ i ∈ C uj +1 ( X v ) and note that since τ i ( x ; u ) = c i ( x )and τ i ( x ; a ) = θ i ( x ) we clearly have ∂τ = σ − θ . As θ belongs to C extj ( X v ), this impliesthat the inclusion C extj ( X v ) → C uj ( X v ) induces a surjection in homology.We now check that this map is injective by applying a similar argument. Let α ∈ C extj ( X v ) with α = ∂σ where σ belongs to C uj +1 ( X v ). The chain σ induces chains τ ∈ C uj +2 ( X v ) and θ ∈ C extj +1 ( X v ) such that ∂τ = σ − θ in the same way as in the previousparagraph. But this implies ∂θ = α which means that α ∈ ∂C extj +1 ( X v ), as required. (cid:3) Given a definable family Y of R n × R and t ∈ R , we denote by Y t the fiber at t : { x ∈ R n : ( x ; t ) ∈ Y } . We also define the restriction of the family to [ a ; b ] as follows: Y [ a ; b ] := { ( x ; t ) ∈ Y : a ≤ t ≤ b } . Lemma 3.1.2.
Let Y be a L R ( u ) -definable family of k nv × k v such that Y u is a N u -thinsubset of k nv and let j = dim Y u . Then there exists z in v such that for any t ∈ v greaterthan z the map induced by inclusion: H wk ( Y t ) → H uk ( Y [ z ; u ] ) , is an isomorphism for k = j and is one-to-one for k = j − .Proof. As Y is L R ( u )-definable and N u -thin there exists z in v such that for any t in v greater than z , Y t is w -thin. Thanks to Remark 1.3.1, this implies that the naturalmapping H wj ( Y t ) → H j ( Y t ) is an isomorphism. Furthermore, since the family Y is topologically trivial if the interval [ z ; u ] is chosensmall, the inclusion H j ( Y t ) → H j ( Y [ z ; u ] ) induces an isomorphism in homology as well.We have the following commutative diagram for t ∈ v greater than z :143 2 H wj ( Y t ) H j ( Y t ) H uj ( Y [ z ; u ] ) H j ( Y [ z ; u ] ) ✲✲❄ ❄ By the above, the arrows 1 and 2 are isomorphisms. Moreover as Y u is N u -thin thefamily Y [ z ; u ] is N u -thin. Thus, the arrow 4 is an monomorphism (see the last sentenceof Remark 1.3.1). This implies that the arrow 3 is an isomorphism and establishes thetheorem in the case k = j .Now, in the case where k = j − H j − . The arrows1 and 2 (of the obtained diagram) are still one-to-one (again thanks to Remark 1.3.1 andthe topological triviality of Y [ z ; u ] ), so that the arrow 3 is clearly one-to-one. (cid:3) The following lemma is a consequence of existence of v -admissible rectilinearizations. Lemma 3.1.3.
Given X ⊂ k nv L R ( u ) -definable and F finite family of closed L R ( u ) -definable subsets of X , the map b H wj ( X ; F ) → H wj ( X ) , induced by the inclusion, is onto.Proof. Let σ ∈ C wj ( X ). If the support of σ is of dimension < j then the class of σ is 0 in H wj ( X ). Thus, we may assume that dim | σ | = j .Let h : k nv → k nv be a w -admissible rectilinearization of | σ | and of all the elements of F . There exists a simplicial chain τ (see Remark 2.3.2), which is strongly w -admissiblesince h is w -admissible, such that σ = τ in H j ( | σ | ) = H wj ( | σ | ) (see Remark 1.3.1). Butthis means that the class of τ is that of σ also in H wj ( X ). This yields that the inclusioninduces an onto map in homology. (cid:3) It is unclear for the author whether the inclusion of the above lemma is one-to-one.Actually, it is even unclear whether b H vj ( X ) is finitely generated.3.2. The main result.
It is very hard to construct homotopies which are Lipschitz map-pings. To compute the homology, we actually just need to find a homotopy that carries achain σ to the cells of a given cell decomposition, and which preserves the v -admissibilityof the chain σ . We prove something even weaker: given a strongly w -admissible chain, wemay construct a homotopy which carries the chain σ to a strongly N u -admissible chain ofthe cells of dimension j . This is enough since we have seen that we had isomorphisms be-tween the theories defined by w and N u . This technical step is performed in the followingproposition. Proposition 3.2.1.
Let X be a closed L R ( u ) -definable subset of k nv and let E be a u -admissible L -cell decomposition compatible with X . Let F be the family constituted by the ANISHING HOMOLOGY closed cells of E and let Y j be the union of the closures of the ( N u ; j ) -thin elements of thebarycentric subdivision of E . Then, there exists a map ϕ : b C wj ( X ; F ) → b C uj ( Y j ) such that: ( i ) ϕ∂ − ∂ϕ = 0( ii ) For any σ ∈ b Z wj ( X ; F ) we have: ϕ σ = σ, in H uj ( X ) , ( iii ) If Y is the union of some elements of F , then for any σ ∈ b Z wj ( X ; F ) with | σ | ⊂ Y we have: ϕ σ = σ in H uj ( Y ) .Proof. We are going to prove the following statements:
Claim.
Given σ ∈ C j ( X ; F ), there exists a definable homotopy h σ : T j ( k v ) × [0; 1] k v → X, such that:(1) For each x the path t h σ ( x ; t ) stays in the same closed cell,(2) For each t the map x h σ ( x ; t ) is a strongly N u -admissible simplex if σ is astrongly w -admissible simples,(3) If σ is strongly w -admissible, the support of the simplex ϕ σ : T j ( k v ) → X definedby ϕ σ ( x ) = h σ ( x ; 1) entirely lies in Y j (4) We have ∂h ∗ ( σ ) − h ∗ ( ∂σ ) = ϕ σ − σ for any σ ∈ C j ( X ; F ) where (as usual) h ∗ : C j ( X ; F ) → C j +1 ( X ; F ) is the mappinginduced by h on the chain complexes.Note that ϕ is defined by (3). Observe that (4) implies ( i ), together with (2) implies( ii ), and together with (1) yields ( iii ).We prove that it is possible to construct such a homotopy by induction on n (thedimension of the ambient space). Let E ′ be the cell decomposition of k n − v constituted byall the cells of E lying in k n − v . Let σ in C j ( X ; F ) and write σ := ( e σ ; σ n ) ∈ k n − v × k v .Apply the induction hypothesis to e σ and E ′ to get a homotopy h e σ : T j ( k v ) × [0; 1] k v → k n − v .By definition, the union of the cells of E on which π n is one-to-one is given by thegraphs of finitely many Lipschitz functions ξ ≤ · · · ≤ ξ s . Note that we may retract thecells above (resp. below) the graph of ξ s (resp. ξ ) onto the graph of ξ s (resp. ξ ) so thatwe may assume that X entirely lies between these two graphs.By compatibility with F we know that the support of σ entirely lies in one single cell E ∈ E which is either the graph of a Lipschitz function ξ or a band which is delimitedby the graph of the restriction to E ′ := π n ( E ) of two consecutive functions ξ i and ξ i +1 ,with ξ i < ξ i +1 on E ′ . In the latter case, we may define a function ν σ : T j ( k v ) → [0; 1] k v by setting for x ∈ T j ( k v ) ν σ ( x ) := σ n ( x ) − ξ i ( e σ ( x )) ξ i +1 ( e σ ( x )) − ξ i ( e σ ( x )) . To deal with both cases simultaneously it is convenient to set ν σ ( x ) ≡ ξ i = ξ i +1 = ξ , if the cell is described by the graph of a single function ξ . To define h σ we first define a function s σ : T j ( k v ) → [0; 1] k v . We set: s σ ( e i ) = 0 if σ n ( e i ) − ξ i ( e σ ( e i )) ∈ w and ξ i +1 ( e σ ( e i )) − σ n ( e i ) = 0and s σ ( e i ) = 1 otherwise.Then we extend s σ over T j ( k v ) linearly.Now we can set for ( x ; t ) ∈ T j ( k v ) × [0; ] k v : θ ( x ; t ) = 2 ts σ ( x ) + (2 t − ν σ ( x ) . Set for simplicity: ξ ′ = ξ i +1 − ξ i and, for x = ( e x ; x n ) ∈ k n − v × k v and t ∈ [0; 1] k v , let: h σ ( x ; t ) := ( e σ ( x ); ξ i ( e σ ( x )) + θ ( x ; t ) ξ ′ ( e σ ( x ))) if t ≤ h σ ( x ; t ) := ( h e σ ( e x ; 2 t −
1) ; ξ i ( h e σ ( e x ; 2 t − s σ ( x ) ξ ′ ( h e σ ( e x ; 2 t − t ≥ . Note that as s σ (resp. ν σ ) satisfies: s ∂σ = ∂s σ (resp. ν ∂σ = ∂ν σ ), we see that the map induced by h σ is a chain homotopy. Moreover,it is clear from the definition of h σ that the path t h σ ( x ; t ) remains in the same closedcells. Therefore (1) and (4) hold.To check (2), fix a strongly admissible simplex σ . We have to check that there exists q ∈ { , . . . , n } such that:(3.9) ( h σ ( x + λe q ; t ) − h σ ( x ; t )) ∈ N u for any ( x ; λ ) ∈ T qj ( k v ) and any t in [0; 1] k v .If σ is the graph of one single function ξ then the result is immediate for t ≤ andfollows from the induction hypothesis for t ≥ .By definition of strongly admissible simplices there exists a vector of the canonical basis,say e q , such that:(3.10) ( σ ( x ) − σ ( x + λe q )) ∈ w, for any ( x ; λ ) ∈ T qj ( k v ). This implies that(3.11) ( σ (0) − σ ( e q )) ∈ w. We distinguish two cases:
First case: s σ (0) = s σ ( e q ). This implies that for any ( x ; λ ) ∈ T qj ( k v ) we have s σ ( x ) = s σ ( x + λe q ) , and therefore(3.12) | θ ( x ) − θ ( x + λe q ) | ≤ | ν σ ( x ) − ν σ ( x + λe q ) | . Note that if ξ ′ ( e σ ( x )) ∈ w then ξ ′ ( e σ ( x + λe q )) ∈ w , which means that in this case (3.9)follows immediately from (3.10) for t ≤ . Otherwise ξ ′ ( e σ ( x )) / ∈ w and then by (3.10):(3.13) 12 ξ ′ ( e σ ( x )) ≤ ξ ′ ( e σ ( x + λe q )) ≤ ξ ′ ( e σ ( x )) . ANISHING HOMOLOGY Recall that the functions ξ i and ξ i +1 are both Lipschitz functions. Hence, if σ is stronglyadmissible, for t ≤ a straightforward computation shows that thanks to (3.12) and (3.13)we have for any ( x ; λ ) ∈ T qj ( k v ):(3.14) ( h σ ( x + λe q ; t ) − h σ ( x ; t )) ∈ w. For t ≥ , (3.9) still holds thanks to the induction hypothesis and the Lipschitzness of ξ i and ξ i +1 . Second case: s σ (0) = s σ ( e q ). In this case we observe that if s σ (0) is 0 then( σ n (0) − ξ i ( e σ (0))) ∈ w which amounts to d ( σ (0); Γ ξ i ) ∈ w, (where Γ ξ i denotes the graph of ξ i ). By (3.11), this implies that d ( σ ( e q ); Γ ξ i ) belongs to w and so ( σ n ( e q ) − ξ i ( e σ ( e q )) ∈ w. As s σ ( e q ) is necessarily equal to 1 we see that σ n ( e q ) − ξ i +1 ( e σ ( e q )) = 0so that ξ ′ ( e σ ( e q )) ∈ w. But, as the cell E is u -admissible this implies that for any x ∈ E ′ : ξ ′ ( x ) ≤ u. This, together with the induction hypothesis, implies that h σ satisfies (3.9). This com-pletes the proof of (2).It remains to prove (3). First observe that all the e j ’s are sent by ϕ σ onto vertices of E . Note also that ϕ σ ( x ) = ( ϕ e σ ( x ); ξ i ( ϕ e σ ( x )) + s σ ( x ) ξ ′ ( ϕ e σ ( x )))and so, by the definition of the cells, the support of ϕ σ lies in cells of dimension at most j of F . Moreover we just checked that (3.9) holds in any case. This implies that ϕ σ isstrongly admissible and therefore its support must lie in Y j . This completes the proof ofthe claim. (cid:3) We are now able to express the v -vanishing homology groups in terms of the (usual)homology groups of some v -thin subsets constituted by the v -thin cells of the barycentricsubdivision of some L -cell decompositions. Theorem 3.2.2.
For any X ⊂ R n closed definable, there exist some definable subsets of X : X ⊂ · · · ⊂ X d +1 = X d such that: H vj ( X ) ≃ Im ( H j ( X j ) → H j ( X j +1 )) (where the arrow is induced by inclusion and Im stands for image). Proof.
We start by defining inductively the subsets X j ’s. Set X = ∅ and assume that X , . . . , X j − have already been defined. According to Proposition 2.2.4, up to a bi-Lipschitz homeomorphism, we can assume that we have a u -admissible L -cell decompo-sition compatible with X v and X j − ,v . Let E j be the barycentric subdivision of this celldecomposition and define Θ j as the union of all the ( j ; N u )-thin cells. There exists a L R -definable family Y j such that Y j,u = Θ j . Now, thanks to Lemma 3.1.2, there exists z in v , such that for any t in v greater than z : H wj ( Y j,t ) ≃ H uj ( Y j,u ) . Now define X j as the subset of R n defined by a L R -formula defining Y j,t for some t ≥ z in v . If t is chosen large enough, X j is v -thin. As bi-Lipschitz homeomorphismsinduce isomorphisms between the vanishing homology groups, we identify subsets withtheir image so that, for instance, we consider below the X j,v ’s and Y j,u as subsets of X v .Consider the following diagram: Im { H j ( X j ) → H j ( X j +1 ) } a ← Im { H vj ( X j ) → H vj ( X j +1 ) } b → H vj ( X ) , where again a and b are induced by the inclusions of the corresponding chain complexes.We shall show that a and b are both isomorphisms.a is an isomorphism: We have the following commutative diagram: H vj ( X j ) H vj ( X j +1 ) H j ( X j ) H j ( X j +1 ) ✲✲❄ ❄ where all the maps are induced by inclusion. By Remark 1.3.1, the first vertical arrow isan isomorphism and the second is one-to-one. This proves that a is an isomorphism. b is onto: Note that it is enough to prove that the inclusion X j → X induces an onto mapbetween the v -vanishing homology groups.We have the following commutative diagram: diag. 1. extextH vj ( X j ) H j (∆ ext ( X j,v )) H j +1 (∆ ext ( X v )) H vj ( X ) H wj ( X j,v ) H wj ( X v ) ✲✲ ✲✲❄ ❄ where the mapping ext , provided by extension of chains, is an isomorphism (see section3 . H wj ( X j,v ) → H wj ( X v ) (the last verticalarrow) is onto. ANISHING HOMOLOGY For t ≥ z in v , let α and β be the maps defined by inclusion: H wj ( Y j,t ) α → H j ( Y j, [ z ; u ] ) β ← H uj ( Y j,u ) . By Lemma 3.1.2, α and β are isomorphisms so that γ := β − α provides the followingcommutative diagram: γH wj ( Y j,t ) H wj ( X v ) H uj ( Y j,u ) H uj ( X v ) ✲✲❄ ❄ By Lemma 3.1.1 the second vertical arrow is onto. Thus, it is enough to show that H uj ( Y j,u ) → H uj ( X v ) is onto. By construction, Y j,u is the union of all the ( j ; N u )-thinclosed cells of the barycentric subdivision of E j . Note that it is enough to consider a chain σ ∈ b Z wj ( X v ; F ) where F is the family constituted by all the closure of the cells of E j (sincethe inclusion b H wj ( X v ; F ) → H uj ( X v ) is onto, thanks to Lemmas 3.1.1 and 3.1.3). By ( ii )of Proposition 3.2.1, there exists ϕ σ ∈ C uj ( Y j,u ) such that σ = ϕ σ in H uj ( X v ), as required. b is one-to-one: Note that as diag. 1 . holds for X j +1 as well (and the horizontal arrowsare isomorphisms as well), it is enough to show that the map induced by inclusion b ′ : Im ( H wj ( X j,v ) → H wj ( X j +1 ,v )) → H wj ( X v )is one-to-one. Recall that by definition X j +1 ,v is Y j +1 ,t , for some t and consider thefollowing commutative diagram: ν t ν u H wj ( X j,v ) H uj ( Y j +1 ,u ) H wj ( Y j +1 ,t ) H uj ( Y j +1 , [ z ; u ] ) ✲✲❄ ❄ where again ν u and ν t are induced by the respective inclusions. By Lemma 3.1.2 thesemaps are one-to-one.This implies that we have the following commutative diagram: b ′ b ′′ µIm ( H wj ( X j,v ) → H wj ( Y j +1 ,t )) Im ( H wj ( X j,v ) → H uj ( Y j +1 ,u )) H wj ( X v ) H uj ( X v ) ✲✲❄ ❄ where all the horizontal arrows are induced by the corresponding inclusions and µ isinduced by the restriction of ν − u ν t . Since µ is one-to-one, it is enough to show that b ′′ isone-to-one. To check that b ′′ is one-to-one, take σ in Z wj ( X j,v ) which bounds a chain of C uj +1 ( X v ).As the inclusion H wj ( X v ) → H uj ( X v ) is an isomorphism, there exists τ in C wj +1 ( X v ) suchthat σ = ∂τ . Consider a w -admissible rectilinearization of | τ | , | σ | and F where F is thefamily constituted by the closure the cells of the barycentric subdivision of E j +1 . Thechain σ is equal in H j ( X j,v ) ≃ H wj ( X j,v ) (see Remark 2.3.2) to a simplicial chain σ ′ whichis strongly w -admissible and compatible with F . The class of the chain σ is zero in H j ( | τ | )and therefore σ ′ bounds a simplicial chain τ ′ which is also strongly w -admissible (againby Remark 2.3.2).By construction, X j,v is a union of cells of E j +1 and the union of all the closure of thecells of dimension ( j + 1) of the barycentric subdivision of E j +1 which are ( j + 1; N u )-thinis precisely Y j +1 ,u . Therefore we may apply Proposition 3.2.1 to X v . This provides a map ϕ : b C wj ( X v ; F ) → b C uj ( Y j +1 ,u ; F ) such that ∂ϕ τ ′ = ϕ ∂τ ′ = ϕ σ ′ . As by ( iii ) of this proposition σ ′ = ϕ σ ′ in H uj ( X j,v ), this implies that the class of σ iszero in H uj ( Y j +1 ,u ) and yields that b ′′ is one-to-one. (cid:3) Corollary 3.2.3.
For any closed definable subset X , the vanishing homology groups H vj ( X ) are finitely generated. Note that the above corollary enables us to define an Euler characteristic which is adefinable metric invariant by setting: χ v ( X ) := ∞ X i =1 ( − i dim H vi ( X ) . This invariant for definable subsets of R n gives rise to a metric for definable families orfor germs of definable sets (see example 1.3.2 and section 4 below). Remark 3.2.4.
The hypothesis closed is assumed for convenience. We could shrink anopen tubular neighborhood of radius z ∈ v of the points lying in the closure but not in X so that we would have a deformation retract of our set onto the complement of thisneighborhood which is very close to the identity, and hence which preserves thin subsets,identifying the vanishing homology groups of our given set with those of a closed subset.4. Local invariants for singularities.
In [V2], we introduced the link for a semi-algebraic metric space. Let us recall itsdefinition. We recall that we denote by k (0 + ) the field of algebraic Puiseux series endowedwith the order that makes the indeterminate T positive and smaller than any real number.We denote by d the Euclidian distance. Given the germ at 0 of a semi-algebraic set X let: L X := { x ∈ X k (0 + ) : d ( x ; 0) = T } where T ∈ k (0 + ) is the indeterminate and X k (0 + ) the extension of X to k (0 + ).In [V2] we proved that the set L X is a metric invariant which characterizes the metrictype of the singularity in the sense that: ANISHING HOMOLOGY Theorem 4.0.5. [V2] L X is semi-algebraically bi-Lipschitz homeomorphic to L Y iff Y issemi-algebraically bi-Lipschitz homeomorphic to X . This theorem admits the following immediate corollary.
Corollary 4.0.6.
For any convex subgroup v ⊂ k (0 + ) , the groups H vj ( L X ) are semi-algebraic bi-Lipschitz invariants of X . Note that by Corollary 3.2.3 these groups are finitely generated and that χ v ( L X ) is asemi-algebraic bi-Lipschitz invariant of the germ X . Remark 4.0.7.
We assumed in this section that X is a semi-algebraic set because thiswas the setting of [V2]. Nevertheless, the main ingredient of the proof of Theorem 4.0.5 isTheorem 5 . . L X may fail to be a metric invariant of the singularity when the set is definable in anon-polynomially bounded o-minimal structure as it is shown by the following example. Example 4.0.8.
Let X := { ( x ; y ) ∈ R : | y | = e − x } and Y = { ( x ; y ) ∈ R : | y | = e − x } .Note that X and Y are both definable in the ln − exp structure (see [vDS], [LR], [W]).Furthermore X and Y are definably bi-Lipschitz homeomorphic. However the links of X and Y are constituted by two points of k + (where k + is the corresponding residue field)whose respective distances are clearly not equivalent.Note that a revolution of these subsets about the x -axis provides two subsets whoselinks have different vanishing homology groups (for a suitable velocity).5. Some examples.
We give two examples of computations of the homology groups. It is convenient todevelop ad hoc techniques to compute the homology groups such as the excision property.5.1.
The excision property.
It follows from the definition that we may have c + c ′ in C vj although neither c nor c ′ belong to this set. This is embarrassing since it makes itimpossible the splitting of a chain of X into a chain of X \ A plus a chain of A , whichis crucial for the excision property. To overcome this difficulty we are going to considermore chains. This will not affect the resulting homology groups.We defined the vanishing homology groups by requiring for a chain σ that | σ | and | ∂σ | to be both ( j ; v )-thin. We may work with another chain complex.Let A and X be closed definable subsets of R n with A ⊂ X and denote by F the pair { X \ Int ( A ); A } where Int ( A ) is the interior of A . Let ∆ vj ( X ) the subset of C vj ( X ; F )constituted by the j -chains having a ( j ; v )-thin support. Of course, such a family ofmodules is not preserved by the boundary operator but, if we want to have a chain complex,we may add the boundaries by setting:∆ ′ vj ( X ) := ∆ vj ( X ) + ∂ ∆ vj +1 ( X ) . This provides a chain complex with obviously H j (∆ ′ v ( X )) = H vj ( X ). The inconvenient is that we are going to work with non admissible chains but theadvantage is that we have now more freedom to work since we have more chains. Forinstance if ( c + c ) ∈ ∆ ′ vj ( X ) then c and c both belong to ∆ ′ vj ( X ).To state the excision property we need to introduce the homology groups of a pair. Forthis purpose, we first set:∆ vj ( X/A ) := { c ∈ ∆ vj ( X ) : ( ∂c − ∂ A c ) ∈ ∆ vj − ( X ) } , where ∂ A takes the boundary and projects it onto C j ( A ).Define also ∆ v,Xj ( A ) := ∆ vj ( A ) + ∂ A ∆ vj +1 ( X/A ) . First observe that by definition if c ∈ ∆ vj +1 ( X/A ) then ∂ A c ∈ ∆ vj ( X ) + ∂ ∆ vj +1 ( X ) . Therefore, by definition of ∆ v,Xj we get∆ v,Xj ( A ) ⊂ ∆ vj ( X ) + ∂ ∆ vj +1 ( X ) = ∆ ′ vj ( X ) . Thus, we may set ∆ vj ( X ; A ) := ∆ ′ vj ( X )∆ v,Xj ( A ) , and H vj ( X ; A ) := H j (∆ v ( X ; A )) . Remark 5.1.1. If X is a v -thin set of dimension j then H vj ( X ; A ) = H j ( X ; A ) (seeRemark 1.3.1).Let i : ∆ v,Xj ( A ) → ∆ ′ vj ( X ) be the inclusion. Clearly, we have the following exactsequences:(5.15) 0 → ∆ v,Xj ( A ) i → ∆ ′ vj ( X ) q → ∆ ′ vj ( X ; A ) → , (where q is the quotient map) and therefore we get the following long exact sequence: ... → H vj ( δ X A ) → H vj ( X ) → H vj ( X ; A ) → H vj − ( δ X A ) → ... Remark 5.1.2.
We could have defined the homology groups of a pair by H vj ( X ; A ) := H vj ( C v ( X ; A )) where C v ( X ; A ) := C vj ( X ) C vj ( A ) , and of course the latter exact sequence wouldhold for H vj ( A ) (instead of H vj ( δ X A )). However the excision property would not hold.As we said, if ( c + c ′ ) belongs to ∆ vj ( X ) then c and c ′ both belong to C vj ( X ). Therefore,the excision property holds for H vj ( X ; A ). Let ( X ; A ) and W be definable such that W lies in the interior of A . Then for any j :(5.16) H vj ( X ; A ) = H vj ( X \ W ; A \ W ) . ANISHING HOMOLOGY Two examples.
We are going to compute the vanishing homology groups on twoexamples which are semi-algebraic families. Let us take Q as our coefficient group. Example 5.2.1.
We first compute the homology groups on the example sketched on fig1. We consider two spheres from which we shrink a little disk which collapses into a pointand which intersects along the boundaries of these disks.Let X ( ε ) := { ( x ; y ; z ) ∈ k (0 + ) : ( x − ε (1 − T )) + y + z = 1 , εx ≥ } for ε = ±
1. Then let X := X (1) ∪ X ( −
1) and A = X (1) ∩ X ( − T . The computation could actually be carried outfor any velocity. Since the set A is N T -thin we have: H T ( δ X A ) = H T ( A ) = H ( A ) = Q , and H T ( δ X A ) = 0.Note that, thanks to the excision property, we have: H v ( X ; A ) ≃ H v ( X (1); A ) ⊕ H v ( X ( − A ) . If we add the disk D = { ( x ; y ; z ) ∈ k (0 + ) : ( x − ε (1 − T )) + y + z = 1 , εx ≤ } to X (1), we get the sphere S . Thus, by the excision property, H T ( X (1); A ) ≃ H T ( S ; D ) = 0 , and so H T ( X ; A ) = 0. Examining the exact sequence of the pair ( X ; A ) we see that: H T ( X ) ≃ H T ( δ X A ) ≃ Q . Observe also that we have: H T ( X ) ≃ H T ( X ) ≃ Example 5.2.2.
Let X be the set defined by (0.2) assume that p < q . Let us computefor instance the vanishing homology groups for the velocity T q . We could use here theexcision property and follow the classical methods for computing the homology groups ofthe torus but it is actually simpler to derive it from the classical homology groups of X since it is N T q -thin. This implies that the inclusion H T q ( X ) → H ( X ) is an isomorphismand that the inclusion H T q ( X ) → H ( X ) is one-to-one. Therefore H T q ( X ) ≃ Q and dim H T q ( X ) ≤
2. Actually, one generator of H ( X ) has a representant with T q -thinsupport and every 1-chain representing a different class has a support whose length isclearly bigger than T p . This proves that dim H T q ( X ) = 1. Acknowledgements.
The author is very grateful to Pierre Milman for valuable discus-sions on related questions and would like to thank the Department of Mathematics of theUniversity of Toronto where this work has been carried out.
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