aa r X i v : . [ m a t h . AG ] J u l L ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY
GUILLAUME VALETTE
Abstract.
Let X be a subanalytic compact pseudomanifold. We show a de Rhamtheorem for L ∞ forms on the nonsingular part of X . We prove that their cohomology isisomorphic to the intersection cohomology of X in the maximal perversity. Introduction
During the three last decades, many authors studied L p differential forms on singularvarieties. The history started with J. Cheeger who computed the cohomology of L formson pseudomanifolds with metrically conical singularities [Ch1]. He proved in [Ch2] thatthe L cohomology is actually isomorphic (for pseudomanifolds with metrically conicalsingularities) to intersection homology in the middle perversity (see also [CGM]).Intersection homology was introduced independently by M. Goresky and R. MacPhersonin [GM1] in order to study the topology of singular sets. Its main feature is to satisfyPoincar´e duality for a large class of singularities, sufficiently general to enclose all thecomplex projective analytic varieties (see [GM1, GM2]).Cheeger’s de Rham theorem thus provided a means to investigate the topology of sin-gular sets via differential geometry. It also enabled to carry out a Hodge theory on pseu-domanifolds with metrically conical singularities, which was developed by Cheeger himselfin a series of works [Ch1, Ch2, Ch3, Ch4]. L p cohomology, p = 2, turned out to be related to intersection homology as well. Letus mention some of the many related works which then appeared. In [Y], Y. Youssincomputes the L p cohomology groups of spaces with conical horns. He shows that the L p cohomology groups are isomorphic to intersection cohomology groups in the so-called L p perversity 1 < p < ∞ . He also describes quite explicitly the case of f -horns. The so-called f -horns are cones endowed with a metric decreasing at a rate proportional to a function f of the distance to the origin. L. Saper studies in [S2] the L cohomology for sets withisolated singularities with a distinguished K¨ahler metric.In [HP], the authors focus on normal algebraic complex surfaces (not necessarily met-rically conical). They also show that the L cohomology is dual to intersection homology(see also [S1]).In [BGM], the authors show, on a simplicial complex, an explicit isomorphism betweenthe L p shadow forms and intersection homology. The shadow forms are smooth formsconstructed by the authors in a combinatorial way, like Whitney forms [Wh]. Mathematics Subject Classification.
Key words and phrases. differential forms; de Rham cohomology; subanalytic sets; singular sets; inter-section homology.
It is striking that, all the above mentioned de Rham theorems include an assumption onthe metric type of the singularities or are devoted to low dimensional singular sets whosemetric type is easier to handle. In this paper, we focus on L ∞ forms, i.e., forms having abounded size. We prove a de Rham theorem for any compact subanalytic pseudomanifold,establishing an isomorphism between L ∞ cohomology and intersection homology in themaximal perversity (Theorem 1.2.2). We also prove that the isomorphism is providedby integration on subanalytic singular chains. The class of subanalytic pseudomanifoldscovers a large class of subsets such as all the complex analytic projective varieties. Fur-thermore, the theory presented in this paper could go over singular subanalytic subsetswhich are not pseudomanifolds and we could adapt the statement to arbitrary subanalyticsubsets.This theorem, which applies to any compact subanalytic pseudomanifold, is proved bylooking in details at the metric structure of subanalytic sets (see section 2). The sharpdescription of the metric type of singularities obtained [V1, V2] will make it possible towork without any extra assumption on the metric type of the singularities.As a consequence, we immediately see the L ∞ groups are finitely generated. Thepurpose is also, as in the case of L cohomology, to find a category of forms for which wecan carry out a Hodge theory for any compact subanalytic singular variety. Performinganalysis or differential geometry on singular spaces is much more challenging that onsmooth manifolds because the metric geometry of singular sets is much harder to handle. Acknowledgment.
The author is very happy to thank Pierre Milman for his encourage-ments and valuable discussions on this topic.1.
Definitions and the main result
This paper deals with subanalytic sets. We recall their definition and outline their basicproperties in an Appendix at the end of the paper (sections 5 and 6).1.1. L ∞ -cohomology groups. Before stating the main result, we need to define the L ∞ cohomology groups. Definition 1.1.1.
Let Y be a C ∞ submanifold of R n . As Y is embedded in R n , it inheritsa natural structure of Riemannian manifold. We say that a j -differential form ω on Y is L ∞ if there exists a constant C such that for any x ∈ Y : | ω ( x ) | ≤ C, where | ω ( x ) | denotes the norm of ω ( x ) (as a linear mapping). We will write d for theexterior differential operator.We denote by Ω j ∞ ( Y ) the real vector space constituted by all the differential C ∞ j -forms ω such that ω and dω are both L ∞ .Given ω ∈ Ω j ∞ ( Y ), we set | ω | ∞ := sup x ∈ Y | ω ( x ) | .The cohomology groups of this cochain complex are called the L ∞ cohomology groupsof Y and will be denoted by H •∞ ( Y ). ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY Intersection homology in the maximal perversity and the main theorem.
We recall below the definition of intersection homology (see [GM1]). Intersection homologyas defined in the latter article depends on a ”perversity”. The definition below correspondsto the case of the maximal perversity t = (0 , , . . . , l −
2) (the letter t stands for ”top”perversity).As we will be interested in the only case of the maximal perversity, we specify thisparticular case in the definition and shall not introduce the technical notion of perversity.But this accounts for the notation I t C j ( X ) (which is the notation of [GM1]) used below.Given a subanalytic set X , we denote by X reg the set of points of X at which X islocally a C ∞ manifold (without boundary, of any dimension) and we will write X sing forthe complement of X reg in X . Subanalytic singular simplices are subanalytic continuous maps c : T j → X , T j being the oriented j -simplex spanned by 0 , e , . . . , e j where e , . . . , e j is the canonicalbasis of R j . Given a globally subanalytic set X ⊂ R n we denote by C • ( X ) the resultingchain complex (with coefficients in R ). We will write | c | for the support of a chain c andby ∂c the boundary of c . Definition 1.2.1. An l -dimensional pseudomanifold is a globally subanalytic locallyclosed set X ⊂ R n such that X reg is a manifold of dimension l and dim X sing ≤ l − Y ⊂ X is called ( t ; i ) -allowable if dim Y ∩ X sing < i − I t C i ( X ) as the subgroup of C i ( X ) consisting of those chains ξ such that | ξ | is( t, i )-allowable and | ∂ξ | is ( t, i − j th intersection cohomology group of maximal perversity , denoted I t H j ( X ),is the j th cohomology group of the cochain complex I t C • ( X ) = Hom ( I t C • ( X ); R ).In this paper, we prove: Theorem 1.2.2.
Let X be a compact subanalytic pseudomanifold. For any j : H j ∞ ( X reg ) ≃ I t H j ( X ) . This theorem is proved in section 4. This requires to investigate in details the metrictype of subanalytic singular sets. This is accomplished in section 2.We then briefly recall the notion of normalization of pseudomanifolds in section 3.We will also show that the isomorphism is given by integration on simplices (section4 . X (whereas forms are only defined on X reg ) butintegration is well defined and gives rise to a cochain map if the simplices are subanalytic(see section 4 . Notations and conventions.
We denote by B n ( x ; ε ) the ball of radius ε centeredat x ∈ R n while S n − ( x ; ε ) will stand for the corresponding sphere. The symbol | . | willdenote the Euclidean norm while d ( ., . ) will stand for the Euclidean distance.We denote by k (0 + ) the field of Puiseux series P i ≥ m a i T ip , p ∈ N , a i ∈ R , i, m ∈ Z , with P i ∈ N a i t i convergent for t in a neighborhood of zero (see Appendix II). We can order thisfield by setting f ≤ g in k (0 + ) if f ( t ) ≤ g ( t ) for t positive real number in a neighborhood GUILLAUME VALETTE of zero. We write T for the indeterminate. The motivation for considering this field isclarified in Appendix II.Let R stand for either R or k (0 + ). By Lipschitz function , we will mean a function f : A → R , A ⊂ R n , satisfying for some integer N : | f ( x ) − f ( x ′ ) | ≤ N | x − x ′ | , for all x and x ′ in A . It is important to notice that we require the constant to be an integerfor k (0 + ) is not archimedean (see again Appendix II). A map h : A → R m is Lipschitz ifso are all its components; a homeomorphism h is bi-Lipschitz if h and h − are Lipschitz.Given two functions f, g : A → R , we write f ∼ g (and say that f is equivalent to g )if there exists a positive integer C such that fC ≤ g ≤ Cf .Given a function ξ : A → R , we denote by Γ ξ its graph and by ξ | B its restriction to asubset B of A . Given two functions ζ and ξ on a set A ⊂ R n with ξ ≤ ζ , we define the closed interval as the set:[ ξ ; ζ ] := { ( x ; y ) ∈ A × R : ξ ( x ) ≤ y ≤ ζ ( x ) } . The open and semi-open intervals are then defined analogously.Given A ⊂ R n , we respectively write cl ( A ) and Int ( A ) for the closure and interior of A (with respect to the Euclidean topology). We also define the (topological) boundary of A by δA := cl ( A ) \ Int ( A ). Convention.
All the sets and mappings considered in this paper will be assumed tobe globally subanalytic (if not otherwise specified), except the differential forms.For the convenience of the reader, all the necessary definitions and basic facts of suban-alytic geometry may be found in two Appendixes at the end of the paper, where referencesof proofs are also provided. 2.
Lipschitz retractions
This section provides some results about the metric geometry of globally subanalyticsets. These results will be very important to compute the L ∞ cohomology groups later on.Given a germ of subanalytic set X at x , we shall construct a Lipschitz strong deformationretraction r t , t ∈ [0 , r ≡ x , r = Id , of this germ onto x (Theorem 2.3.1). It is themain result of this section.By way of motivation for all the results of this section, let us briefly outline the strategyof the proof of Theorem 2.3.1. Let X be the germ of a singular (subanalytic) set. Replacing X by ˆ X (see (2.7) for ˆ X ), we may estimate the distance to the origin by the first coordinate.We proceed by induction on n , if X ⊂ R n . The result is therefore true for π n ( X ) if π n : R n → R n − is the canonical projection. By Corollary 2.1.4 and Lemma 2.1.5, up toa bi-Lipschitz homeomorphism preserving the first coordinate, we know that ˆ X may beincluded in the graphs of finitely many Lipschitz functions. We thus can lift the retractionobtained by induction, making use of the estimates of Lemma 2.2.3 so as to establish itsLipschitz character.The techniques of this section, especially Theorem 2.3.1, can have other applications(see [SV]). We start by recalling some results of [V1, V2]. ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY Given n > M we set: C n ( M ) := { ( x ; x ′ ) ∈ R × R n − : 0 ≤ | x ′ | ≤ M x } . For n = 1, we just set C ( M ) = R .2.1. Regular vectors.
In the definition below, R stands for either k (0 + ) or R . Definition 2.1.1.
Let X be a subset of R n . An element λ of S n − is said to be regularfor X if there is a positive real number α such that: d ( λ ; T x X reg ) ≥ α, for any x in X reg .Recall that the order relation in k (0 + ) was defined by comparing the series on a right-hand-side neighborhood of zero. Therefore, in the above definition, the inequality meansin the case R = k (0 + ) that for x ∈ X reg the limit at zero of the Puiseux series d ( λ ; T x X reg )cannot be smaller than α >
0. It is important to notice that α is required to be a positive real number and not a Puiseux series: it implies that the Puiseux series d ( λ ; T x X reg ) maynot tend to zero at zero.Regular vectors do not always exist, as it is shown by the simple example of a circle.Nevertheless, we can get a regular vector without affecting the metric type of a subanalyticset: Theorem 2.1.2. [V1]
Let X be a subset of k (0 + ) n of empty interior. Then there exists abi-Lipschitz homeomorphism h : k (0 + ) n → k (0 + ) n such that e n is regular for h ( X ) . For instance, if X is the circle (in k (0 + ) ) defined by x + y = 1 then the providedbi-Lipschitz homeomorphism may send X onto a triangle (in k (0 + ) ). We see (intuitivelyat least) that it is not possible to require h ( X ) to be a smooth manifold even if so is X . Such a mapping h is the generic fiber (see Appendix II, section 6) of a family ofhomeomorphisms sending the cylinder (0 , ε ) × C , where ε is a positive real number and C denotes the unit circle in R , onto the product of a triangle with the interval (0 , ε ). Thesituation gets more difficult when X is singular since it may have many different limits oftangent spaces at a singular point. Definition 2.1.3.
A map h : R n → R n is x -preserving if it preserves the first coordinatein the canonical basis of R n .It is shown in [V2] that, if the considered subset lies in C n ( M ), then the homeomorphismof Theorem 2.1.2 may be chosen x -preserving. In [V2], the result was for semialgebraicsets. Below, we prove it in the subanalytic framework.In the proof below, we consider subsets of R n as families of subsets of R n − parameter-ized by the first coordinate. Given t ∈ R , we write X t for the set of points of X havingtheir first coordinate equal to t . Corollary 2.1.4.
Let X be the germ at of a subset of C n ( M ) of empty interior, M > .There exists a germ of x -preserving bi-Lipschitz homeomorphism (onto its image) h : C n ( M ) → C n ( M ) such that e n is regular for h ( X ) . GUILLAUME VALETTE
Proof.
Apply Theorem 2.1.2 to the generic fiber: X + := { x : ( T ; x ) ∈ X k (0 + ) } , where X k (0 + ) denotes the extension of the set X to k (0 + ) (see Appendix II). This providesa bi-Lipschitz homeomorphism H : k (0 + ) n − → k (0 + ) n − which immediately gives rise(via the so-called transfer principle, see again Appendix II) to a x -preserving bi-Lipschitzhomeomorphism h : (0; ε ) × R n − → (0; ε ) × R n − , ( t, x ) ( t, h t ( x )), with h t bi-Lipschitz(with the same constant as H ) for every t < ε and such that there is a real number α > d ( e n , T x h ( X t )) ≥ α, for any x ∈ h ( X t ) reg and t positive small enough. Up to a translation, we may assumethat h t (0) ≡ h maps C n ( M ) into C n ( M ′ ), for some M ′ . Up to a x -preservinglinear mapping, we may assume M = M ′ .We now check that e n is regular for the germ of Y := h ( X ). Suppose not. It meansthat the element (0 , e n ) belongs to the closure of the set: { ( x, u ) : x ∈ Y reg and u ∈ T x Y reg } . As a matter of fact, by curve selection Lemma (see Appendix I), there exists an analyticarc γ : [0; ε ] → Y reg with γ (0) = 0 and e n ∈ τ := lim t → T γ ( t ) Y reg . On the other hand, by(2.1), we have e n / ∈ lim t → T γ ( t ) Y γ ( t ) . This implies that τ ∩ < e > ⊥ = lim t → ( T γ ( t ) Y reg ∩ < e > ⊥ ) , and consequently τ may not be transverse to < e > ⊥ (since otherwise the intersectionwith the limit would be the limit of the intersection), which means that τ ⊆ < e > ⊥ .This implies that the limit vector lim t → γ ( t ) | γ ( t ) | = lim t → γ ′ ( t ) | γ ′ ( t ) | ∈ τ is orthogonal to e .Therefore, lim t → γ ( t ) | γ ( t ) | = 0 , in contradiction with γ ( t ) ∈ C n ( M ).Let us now show that h is also Lipschitz with respect to the parameter x . Supposethat the germ of h fails to be Lipschitz. In this case, the element (0 , ,
0) belongs to theclosure of the set germ: { ( p, q, z ) : p ∈ C n ( M ) , q ∈ C n ( M ) , p = q, z = | p − q || h ( p ) − h ( q ) | } . Then, by curve selection Lemma (see Appendix I), we can find two analytic arcs in C n ( M ), say p ( t ) and q ( t ), tending to zero and along which:(2.2) | p ( t ) − q ( t ) | ≪ | h ( p ( t )) − h ( q ( t )) | . Recall that h preserves the fibers of π , the projection onto the first coordinate. Wemay assume that p ( t ) (and thus h ( p ( t )) too) is parameterized by its x -coordinate, i. e.,we may assume π ( p ( t )) = t , t > f ( t ) := π ( p ( t )) being a real analytic function, itinduces a homeomorphism in a right-hand-side neighborhood of the origin whose inverse f − is a Puiseux series). As p ( t ) and h ( p ( t )) are Puiseux arcs in C n ( M ) we have:(2.3) | h ( p ( t )) − h ( p ( t ′ )) | ∼ | t − t ′ | ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY and(2.4) | p ( t ) − p ( t ′ ) | ∼ | t − t ′ | ≤ | p ( t ) − q ( t ) | , where t ′ denotes the first coordinate of q ( t ).Therefore, by (2.2) and (2.3) and (2.4) we have for some constant C ∈ R : | h ( p ( t )) − h ( q ( t )) | ∼ | h ( p ( t ′ )) − h ( q ( t )) | ∼ | p ( t ′ ) − q ( t ) | ≤ C | p ( t ) − q ( t ) | , a contradiction. Arguing in the same way on h − , we could show that h is bi-Lipschitz. (cid:3) There is a close interplay between Lipschitz functions and regular vectors.
Lemma 2.1.5.
Assume that e n is regular for a set X ⊂ R n . Then X is contained in theunion of the respective graphs of some Lipschitz functions ξ i : R n − → R , i = 1 , . . . , k .Proof. Take a cell decomposition compatible with X . Since e n is regular for X , the set X is the union of some cells which are graphs (not bands, see Definition 5.2.1) of someanalytic functions η i : D i → R , i = 1 , . . . , k , where D i ⊂ R n − . Observe that, because e n is regular for their graph, the η i ’s have bounded derivatives.By Theorem 1 . D i into analytic manifolds,say D i, , . . . , D i,m i , and a constant M such that such that any given two points x and y inthe same D i,j may be joint by an arc whose length does not exceed M | x − y | . This impliesthat any given smooth function f : D i,j → R , j ≤ m i , which has bounded derivatives isLipschitz. In particular, η i induces a Lipschitz function on every D i,j , say η i,j .Now, the lemma follows from the fact that we can extend each η i,j : D i,j → R to aLipschitz function ξ i,j : R n − → R by setting: ξ i,j ( x ) := inf { η i,j ( y ) + L i,j | x − y | : y ∈ D i,j } , where L i,j denotes the Lipschitz constant of η i,j . (cid:3) Some preliminaries.
Before constructing the desired retraction, we need to put theset in a nice position. For this purpose, we will need yet another result whose proof maybe found in [V1] as well (Proposition 2.2.1 below). It is a consequence of the preparationtheorem [P], [LR], [vDS].Basically, this proposition says that distance functions (i.e. functions of type x d ( x, W ), W ⊂ R n ) may be used as a ”basis of valuations”, in the sense that every (globallysubanalytic) nonnegative function may be compared (up to constants) to a product ofpowers of distance functions (after a partition).We recall that, except the differential forms, all the sets and functions of this paper areassumed to be globally subanalytic. Proposition 2.2.1.
Let X ⊂ R n and let ξ : X → R be a nonnegative function. Thereexists a finite partition of X such that over each element of this partition the function ξ is ∼ to a product of powers of distances to subsets of X . The powers involved in the above proposition are always rational numbers.
Remark 2.2.2.
We now would like to formulate two observations that will be useful inthe proof of the next lemma.
GUILLAUME VALETTE (1) If X is the union of the graphs of finitely many Lipschitz functions ξ , . . . , ξ k over R n then,using the operators min and max, we may find an ordered family of Lipschitz functions θ ≤ · · · ≤ θ k such that X is the union of the graphs of these functions.(2) Given a family of Lipschitz functions f , . . . , f k defined over R n − , we can find someLipschitz functions ξ ≤ · · · ≤ ξ m on R n − and a cell decomposition D of R n − such thatover each [ ξ i | D ; ξ i +1 | D ], where D ∈ D , the family of functions | y − f ( x ) | , . . . , | y − f k ( x ) | , f ( x ) , . . . , f k ( x ) , (for ( x ; y ) ∈ [ ξ i | D ; ξ i +1 | D ]) is totally ordered (for relation ≤ ). Indeed, it suffices to choosea cell decomposition D of R n − compatible with the sets f i = f j and to apply (1) to thefunctions f i , ( f i − f j ), ( f i + f j ), and f i + f j , i ≤ k, j ≤ k .The lemma below somehow combines Corollary 2.1.4 and Proposition 2.2.1 in a singlestatement. We denote by π n : R n → R n − the orthogonal projection onto R n − . Lemma 2.2.3.
Given some germs X , . . . , X s ⊂ C n ( M ) at , there exist a germ of x -preserving bi-Lipschitz homeomorphism (onto its image) h : C n ( M ) → C n ( M ) and a celldecomposition E of R n such that for some representatives of the germs:(1) E is compatible with h ( X ) , . . . , h ( X s ) and h ( C n ( M )) .(2) e n is regular for any cell of E which is a graph (not a band, see Definition 5.2.1).(3) Given finitely many nonnegative functions ξ , . . . , ξ m on C n ( M ) , we may assumethat on each cell E ⊂ h ( C n ( M )) of E , each function ξ i ◦ h − is ∼ to a function ofthe form: (2.5) | y − θ ( x ) | r a ( x ) (for ( x ; y ) ∈ R n − × R ) where a, θ : π n ( E ) → R are functions with θ Lipschitz, r ∈ Q .Proof. It will be convenient to complete the family X , . . . , X s by setting X s +1 := C n ( M ).Apply Proposition 2.2.1 to the functions ξ j , j = 1 , . . . , m . This provides a partition E , . . . , E b of C n ( M ) together with some subsets of C n ( M ), say W , . . . , W c , such that oneach E i , i ≤ b , each function ξ j , j ≤ m , is equivalent to a product of powers of functionsof type q d ( q ; W k ), k ≤ c .Possibly refining the partition E i , we may assume that the W k ’s are unions of someelements of this partition (thanks to existence of cell decompositions, see Appendix I).Hence, on every E i , if d ( x, W k ) is not identically zero, then it is nowhere zero and d ( x, W k )is equivalent to d ( x, δW k ). Therefore, we may assume that the W k ’s have empty interior,possibly replacing them with their boundaries (if a function ξ j is identically zero on E i then (3) is trivial on E i ).Apply now Corollary 2.1.4 to the union of the δX i ’s, the δE i ’s, and the W k ’s. Thisprovides a germ of x -preserving bi-Lipschitz homeomorphism h : C n ( M ) → C n ( M ) whichmaps the latter subsets into the union of the graphs of some Lipschitz functions θ , . . . , θ d .By Remark 2.2.2 (2) applied to the family of functions constituted by the θ i ’s togetherwith all the ( n − x d ( x ; π n ( W k ∩ Γ θ ν )), ν ≤ d, k ≤ c , we know thatthere exist a finite number of functions η ≤ · · · ≤ η p and a cell decomposition D of R n − ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY such that for every D ∈ D , over each [ η i, | D ; η i +1 , | D ], i < p , the family constituted by allthe n -variable functions | y − θ ν ( x ) | , ν ≤ d, together with the functions x d ( x ; π n ( W k ∩ Γ θ ν )) , ν ≤ d, k ≤ c is totally ordered (for order relation ≤ , considering the latter functions as n -variablefunctions). By (1) of Remark 2.2.2, we can find a totally ordered finite family σ ≤ · · · ≤ σ µ such that ∪ µi =1 Γ σ i contains both the graphs of the θ i ’s and the graphs of the η i ’s.Consider a cell decomposition D ′ of R n compatible with the cells of D , the sets definedby all the equations σ j = σ i , i ≤ µ , j ≤ µ , as well as all the sets h ( X j ) ∩ Γ σ i , j ≤ s + 1, i ≤ µ . The graphs of the respective restrictions of the functions σ , . . . , σ µ , to the sets π n ( E ), E ∈ D ′ , define a cell decomposition E of R n .For a proof of (1), take a cell E ∈ E , E ⊂ C n ( M ). If E is a graph (not a band) then(1) for E follows from the fact that D ′ is compatible with the h ( X j ) ∩ Γ σ i ’s. Assume thusthat E is a band, say ( σ i | D , σ i +1 | D ) where i < µ , D ⊂ R n − . As δh ( X j ) ⊂ ∪ µk =1 Γ σ k , forall j , the set E ∩ h ( X j ) is open and closed in E . Hence, if E ∩ h ( X j ) is nonempty it isequal to E ( E is connected). This yields (1).Observe that e n is regular for any cell of E which is a graph, since the σ i ’s are Lipschitzfunctions. This already proves that (2) holds.To prove (3), fix a cell E ⊂ h ( C n ( M )) of E which is a band, say ( σ k | D , σ k +1 | D ) where k < µ , D ⊂ R n − ((3) is trivial if E is a graph). We first check that E is included in h ( E i ), for some i . As δh ( E i ) ⊂ ∪ µk =1 Γ σ k , for each i , the set E ∩ h ( E i ) is open and closedin E . Hence, if E ∩ h ( E i ) is nonempty it is equal to E . As h ( E ) , . . . , h ( E b ) constitute apartition of h ( C n ( M )), this shows that E ⊂ h ( E i ), for some i .Consequently, as h is bi-Lipschitz, each ξ j ◦ h − is equivalent to a product of powers offunctions of type q d ( q ; h ( W i )), i ≤ c . It is thus enough to show (2.5) for these latterfunctions.As the θ ν ’s are Lipschitz functions, we have for any ν ∈ { , . . . , d } :(2.6) d ( q ; h ( W i ) ∩ Γ θ ν ) ∼ | y − θ ν ( x ) | + d ( x ; π n ( h ( W i ) ∩ Γ θ ν ))where q = ( x ; y ) in R n − × R .By construction E ⊂ [ η k, | A , η k +1 , | A ], for some k < p and A ⊂ R n − . As a matter of fact,for every i , the terms of the right-hand-side are comparable with each other (for partialorder relation ≤ ) over the cell E . Therefore, the left-hand-side is ∼ to one of them on E .Note that, as each h ( W i ) is included in the union of the graphs of the θ ν ’s, we have: d ( q ; h ( W i )) = min ≤ ν ≤ d d ( q ; h ( W i ) ∩ Γ θ ν ) . The latter family of functions is totally ordered over E . Hence, by (2 . d ( q ; h ( W i )) is equivalent over E either to one of the functions x d ( x ; π n ( h ( W i ) ∩ Γ θ ν )) , or to some function ( x ; y )
7→ | y − θ ν ( x ) | , ν ∈ { , . . . , d } . Thus, (3) holds. (cid:3) Lipschitz retractions of subanalytic germs.
We are now ready to construct thedesired strong deformation retraction. Given X ⊂ R n we define:(2.7) ˆ X := { ( y ; x ) ∈ R × X : | x | = y } . Observe that ˆ X is a subset of C n +1 (1). In the theorem below we write d x r t for the derivative of r t which exists for x genericalthough r t is not smooth since, like all the mappings in this paper, r is implicitly assumedto be subanalytic and thus smooth on a (subanalytic) dense subset.By Lipschitz deformation retraction onto x , we mean a Lipschitz family of maps r t with r ( x ) ≡ x and r ( x ) ≡ x . Theorem 2.3.1.
Let X ⊂ R n be locally closed and let x ∈ X . Then, for any ε > smallenough there exists a Lipschitz deformation retraction r : X ∩ B n ( x ; ε ) × [0; 1] → X ∩ B n ( x ; ε ) , ( x, t ) r t ( x ) , onto x , preserving X reg for t > .Furthermore, the derivative d x r t tends to as t → for any x generic in X reg .Proof. We will assume for simplicity that x = 0. We will actually prove by induction on n the following statements.( A n ) Let X , . . . , X s be finitely many subsets of C n ( M ) and let ξ , . . . , ξ m be boundedfunctions on C n ( M ), with M >
0. There exists ε > U ε := { x ∈C n ( M ) : 0 ≤ x < ε } , there is a Lipschitz strong deformation retraction of U ε r : U ε × [0; 1] → U ε , ( x, t ) r t ( x ) , onto 0 such that for any j ≤ s :(1) r t preserves X j ∩ U ε for t ∈ (0; 1](2) d x r t goes to zero as t tends to 0 for any x generic in X j ∩ U ε (3) There is a constant C such that for any i and any 0 < t ≤ x ∈ U ε :(2.8) ξ i ( r t ( x )) ≤ Cξ i ( x ) . Before proving these statements, let us make it clear that this implies the desired result.If X ⊂ R n then ˆ X (see (2.7) for ˆ X ) is a subset of C n +1 (1) to which we can apply ( A n +1 ).Then, as ˆ X is bi-Lipschitz equivalent to X , the result immediately ensues. Thanks to (1),we may assume that the retraction preserves X reg .As the theorem obviously holds in the case where n = 1 (with r t ( x ) = tx ), we fix some n >
1. We also fix some subsets X , . . . , X s of C n ( M ), for M >
0, and some boundedfunctions ξ , . . . , ξ m : C n ( M ) → R .Before defining the desired map, we need some preliminaries: we first construct a familyof bounded ( n − σ , . . . , σ p to which we will apply (3) of ( A n − ).Apply Lemma 2.2.3 to the family constituted by the germs of the X i ’s and the zeroloci of the ξ i ’s. We get a x -preserving bi-Lipschitz map h : C n ( M ) → C n ( M ) and a celldecomposition E such that (1) and (2) of the latter lemma hold. Moreover, thanks to (3)of the latter Lemma, we may also assume that the ξ i ’s are like in (2.5) on every cell. Aswe may work up to a x -preserving bi-Lipschitz map we will identify h with the identitymap. ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY By Lemma 2.1.5, the union of the cells of E for which e n is regular may be includedin the union of the graphs of finitely many Lipschitz functions η , . . . , η v . Moreover, byRemark 2.2.2 (1), we can assume η ≤ · · · ≤ η v .In order to define the desired functions σ , . . . , σ p , let us fix a cell A of E , and set A ′ := π n ( A ), π n : R n → R n − denoting the projection onto the ( n −
1) first coordinates.Choose then j < v and set D := ( η j | A ′ ; η j +1 | A ′ ). By construction, D is included in a cellof E .Since the ξ k ’s are like in (2.5) on D , for every k = 1 , . . . , m , there exist some ( n − A ′ , say θ k and a k , such that for ( x ; y ) ∈ D ⊂ R n − × R :(2.9) ξ k ( x ; y ) ∼ | y − θ k ( x ) | α k a k ( x ) , where α k is a rational number (possibly negative).As E is compatible with the zero loci of the ξ k ’s, we have on A ′ : if ξ k is not identicallyzero on D then either θ k ≤ η j or θ k ≥ η j +1 . Fix k with ξ k = 0 on D . We will assume forsimplicity that θ k ≤ η j .It means that on D :(2.10) ξ k ( x ; y ) ∼ min(( y − η j ( x )) α k a k ( x ); ( η j ( x ) − θ k ( x )) α k a k ( x )) , if α k is negative, and(2.11) ξ k ( x ; y ) ∼ max(( y − η j ( x )) α k a k ( x ); ( η j ( x ) − θ k ( x )) α k a k ( x )) , in the case where α k is nonnegative.We are now ready to define the desired family σ , . . . , σ p of ( n − ξ k = 0 on D :(2.12) κ k ( x ) := | η j ( x ) − θ k ( x ) | α k a k ( x ) . Since ξ k is bounded, by (2.5), this defines a bounded function. Complete the family κ byadding the functions min( f ; 1) where f describes all the ( η j +1 − η j ) a k ’s.Doing this for all the cells A ∈ E and integers j < v , and collecting all the respectivefamilies κ obtained in this way, we eventually get a family of bounded functions σ , . . . , σ p .We now turn to the construction of the desired retraction. Consider a cylindrical celldecomposition D compatible with the cells of E and the graphs of the η j ’s. Apply theinduction hypothesis to the family of sets π n ( D ) ∩ C n − ( M ), D ∈ D . This provides adeformation retraction r : V ε × [0; 1] → V ε , ε >
0, where V ε := { x ∈ C n − ( M ) : 0 ≤ x < ε } .We are going to lift r to a retraction of [ η | V ε , η v | V ε ]. Thanks to the induction hypothesis,we may assume that the functions σ , . . . , σ p , as well as the ( η j +1 − η j )’s and the functions x ξ i ( x ; η j ( x )) satisfy (2.8).Now, we may lift r as follows. On ( η j ; η j +1 ), j = 1 , . . . , v −
1, we set ν ( q ) := y − η j ( x ) η j +1 ( x ) − η j ( x ) , if q = ( x ; y ) ∈ ( η j ; η j +1 ) ⊂ R n − × R , and then e r t ( q ) := ( r t ( x ); ν ( q )( η j +1 ( r t ( x )) − η j ( r t ( x ))) + η j ( r t ( x ))) . This mapping is then easily extended continuously on each Γ η j by setting if q = ( x ; η j ( x )): e r t ( q ) := ( r t ( x ); η j ( r t ( x ))) . For any j , the mapping ˜ r maps linearly the segment [ η j ( x ); η j +1 ( x )] onto the segment[ η j ( r t ( x )); η j +1 ( r t ( x ))]. Thanks to the induction hypothesis, the inequality (2.8) is fulfilledby the function ( η j +1 − η j ). Therefore, as r is Lipschitz, we see that e r is Lipschitz as well.As e r preserves the cells of E , it preserves the X j ’s, the zero loci of the ξ k ’s, and U ε .We have to check that the ξ k ’s fulfill (2.8) along the trajectories of e r . We check it on agiven cell E of D . If E ⊂ Γ η j for some j , this follows from the induction hypothesis sincewe have assumed that the functions x ξ k ( x ; η j ( x )) satisfy (2.8) on E .Otherwise, there exists j such that E sits in ( η j ; η j +1 ). Fix an integer 1 ≤ k ≤ m . Onthe cell E , the function ξ k may be estimated as in (2.9). By the induction hypothesis weknow that κ k (see (2.12), if ξ k = 0 on E then (2.8) is trivial for ξ k ) satisfies (2.8).If a function ξ is bounded and if min( ξ ; 1) satisfies (2 .
8) then ξ satisfies this inequalityas well. We will therefore check (2.8) for min( ξ k ; 1).Observe also that if two given functions ξ and ζ both satisfy (2.8) then min( ξ ; ζ ) andmax( ξ ; ζ ) both satisfy this inequality as well. Hence, by (2.10) and (2.11), it is enough toshow that the functions min(( y − η j ( x )) α k a k ( x ); 1) and the functions | θ k − η j | α k a k satisfy(2.8). The latter functions are nothing but the κ k ’s for which we already have seen thatthis inequality is true. Let us focus on the former functions.For simplicity we set F ( x ; y ) := ( y − η j ( x )) α k a k ( x )and G ( x ) := ( η j +1 − η j )( x ) α k a k ( x ) . We have to show the desired inequality for min( F ; 1). We have:(2.13) F ( x ; y ) = ν ( q ) α k · G ( x ) . Remark that the function ν ( e r t ( q )) is constant with respect to t . This implies that:(2.14) F ( e r t ( q )) = ν ( q ) α k · G ( r t ( x )) . We assume first that α k is negative. Thanks to the induction hypothesis (min( G ; 1) isone of the σ i ’s) we know that for some constant C we have for all x in π n ( E ):min( G ( r t ( x )); 1) ≤ C min( G ( x ); 1) . This implies (multiplying by ν α k and applying (2.13) and (2.14)) that for q ∈ E :min( F ( e r t ( q )); ν α k ( q ); 1) ≤ C min( F ( q ); ν α k ( q ); 1) . But, as α k is negative, min( F ; ν α k ; 1) = min( F ; 1), which yields the desired inequality formin( F ; 1), as required.We now assume that α k is nonnegative. Thanks to (2.13) and (2.14), it actually sufficesto show the desired inequality for G . But, as ξ k is bounded, by (2.11) so is G , and theresult follows from the induction hypothesis since min( G ; 1) is one of the σ i ’s (as G isbounded and min( G,
1) satisfies (2.8) then G satisfies this inequality as well). This yields(2.8) along the trajectories of e r .We now check that d q e r t tends to zero when t goes to zero. It follows from the inductionhypothesis that d x r t goes to zero as t goes to zero, for any x generic. As the η i ’s have ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY bounded derivatives, this already proves for almost every x :(2.15) lim t → d x [( η i − η i +1 ) ◦ r t ] = 0 . On the other hand, a straightforward computation shows that for q = ( x ; y ): | d e r t ( q ) ν | ≤ C | η i ( x ) − η i +1 ( x ) | , which, together with (2.15) and (2.8) for ( η i +1 − η i ), implies that d q e r t tends to zero as t goes to zero. (cid:3) Remark 2.3.2.
The mapping r t could be proved to be bi-Lipschitz for every t >
0. TheLipschitz constant of r − t may of course tend to infinity as t goes to zero. We neverthelesscould have a control on the way distances are contracted by r t , similarly as in [V1, V2].We could show that for a suitable basis of unit 1-forms θ , . . . , θ n and some functions ϕ , . . . , ϕ n on R n such that almost everywhere on X reg ∩ B n ( x ; ε ) × [0 , r ∗ t ρ ( x ) ≈ n X i =1 ϕ i ( x ; t ) θ i ( x ) , where ρ is the metric of R n and r ∗ t ρ its pull-back. Similarly as in [V1], the functions ϕ i ,which are the contractions of the metric that r operates, could be expressed as powers,products, and sums of distance functions in X (i. e. x d ( x ; W ) with W ⊂ X ∩ B n (0; ε ))and the function ( x ; t ) t . These powers may be negative which makes it difficult toget decreasing functions and accounts for the difficulty we have in the proof of the abovetheorem. 3. Normal pseudomanifolds.
In this section, we shall also deal with topological pseudomanifolds. Given X ⊂ R n ,denote by X reg the set of points of X near which X is a C -manifold (of any dimension).We say that a locally closed set X is an l -dimensional topological pseudomanifold ifdim X \ X reg < l − X reg is an l -dimensional manifold. Definition 3.0.3. An l -dimensional topological pseudomanifold X is called normal if forany x in X , dim H l ( X ; X \ { x } ) = 1.We shall recall some basic facts about normal pseudomanifolds. These may be found in[GM1] (section 4) and make normalizations very useful to investigate intersection homologyin the maximal perversity. Observe that if X ⊂ R n is a normal topological pseudomanifoldwhich is connected then H l ( X ) = R , since if there were two generators, say σ and τ ,dim | σ | ∩ | τ | < l , we would have H l ( X ; X \ x ) = R at any point of the intersection of thesupports.The main interest of normal spaces lies in the following Lemma. Denote by L ( x ; X reg )the set S n − ( x ; ε ) ∩ X reg . It is well known that the topology of L ( x ; X reg ) is independentof ε > Lemma 3.0.4. [GM1]
A topological pseudomanifold X ⊂ R n is normal if and only if L ( x ; X reg ) is connected at any point of X \ X reg . See for instance [GM1] section 4 for a proof. The very significant advantage of normalpseudomanifolds lies in the following proposition.
Proposition 3.0.5. [GM1]
Let X be a normal topological pseudomanifold. The mapping α : I t H j ( X ) → H j ( X ) , induced by the inclusion between the chain complexes, is anisomorphism for all j . Normalizations of pseudomanifolds.
We shall need some basic facts about nor-malizations.
Definition 3.1.1. A normalization of the topological pseudomanifold X is a normaltopological pseudomanifold e X together with a finite-to-one continuous mapping π : e X → X such that, for any p in X , π ∗ : ⊕ q ∈ π − ( p ) H l ( e X ; e X \ q ) → H l ( X ; X \ p )is an isomorphism.We are going to see that normalizations are useful to compute intersection homology inthe maximal perversity. Proposition 3.1.2.
Every pseudomanifold X admits a normalization π : e X → X . Themapping π then induces a homeomorphism above the regular locus of X .Proof. We follow the construction of [GM1]. Consider a triangulation of X (since X isglobally subanalytic, it admits a C triangulation, see Appendix I), i.e., a homeomorphism T : K → X , with K finite union of open simplices.Let L be the disjoint union of all the closures in K of the l -dimensional open simplicesof K (where l is the dimension of X ). Identify the closure in K of two ( l −
1) open facesof two elements of L if these two faces coincide in K . This provides a simplicial complex˜ X . Denote then by π : ˜ X → X the map induced by T .Observe that by construction the mapping π is a homeomorphism on the complementin X of the ( l − π induces a homeomorphism above X reg and that L ( x, ˜ X reg ) is connected at singularpoints. (cid:3) Remark 3.1.3.
It is possible to see that the normalization of a pseudomanifold is unique,up to a homeomorphism.It is not difficult to see from their construction that normalizations must identify ( t ; i )-allowable chains of e X with ( t ; i )-allowable chains of X , which implies that they yield anisomorphism between the intersection homology groups (see [GM1]): Proposition 3.1.4. [GM1]
Let π : ˜ X → X be a normalization of X . Then, for any j theinduced map π ∗ : I t H j ( ˜ X ) → I t H j ( X ) is an isomorphism. Computation of the L ∞ cohomology groups This section proves the main result of this paper, Theorem 1.2.2. ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY Weakly differentiable forms.
For technical reasons, we will need to work withnon smooth forms, which are weakly differentiable, i.e., differentiable as distributions.Therefore, the first step is to prove that the bounded weakly differentiable forms give riseto the same cohomology theory. We will follow an argument similar to the one used byYoussin in [Y].Let M be a smooth manifold. We denote by Λ j ( M ) the set of C j -forms on M withcompact support. Definition 4.1.1.
Let U be an open subset of R n . A continuous differential j -form α on U is called weakly differentiable if there exists a continuous ( j + 1)-form ω such thatfor any form ϕ ∈ Λ l − j − ( U ): Z U α ∧ dϕ = ( − j +1 Z U ω ∧ ϕ. The form ω is then called the weak exterior differential of α and we write ω = dα . Acontinuous differential j -form α on M is called weakly differentiable if it gives rise toweakly differentiable forms via the coordinate systems of M .We denote by Ω j ∞ ( M ) the set of weakly differentiable forms which are bounded andwhich have a bounded weak exterior differential. They constitute a cochain complexwhose coboundary operator is d . We denote by H •∞ ( M ) the resulting cohomology groups.It is well known that if ω is smooth then it is weakly differentiable and dω = dω .Therefore Ω j ∞ ( M ) ⊂ Ω j ∞ ( M ). Moreover, every L ∞ weakly differentiable form may beapproximated (for the L ∞ norm) by smooth bounded forms (with approximation of thedifferential if it is L ∞ ). Consequently, any weakly differentiable 0-form ω satisfying dω = 0is constant.We shall see that smooth and weakly differentiable forms give rise to isomorphic coho-mology theories. The lemma below addresses the case of compact manifolds with bound-ary.Given a smooth manifold with boundary K , we write H jdR ( K ) for the de Rham coho-mology of K , i.e., the cohomology of the C ∞ differential forms on K . Lemma 4.1.2.
Let K be a compact manifold with boundary. The mapping H jdR ( K ) → H j ∞ ( K \ ∂K ) induced by the inclusion between the respective cochain complexes is anisomorphism.Proof. As the smooth forms on K satisfy Poincar´e Lemma (see for instance [BT]), theygive rise to a fine torsionless resolution of the constant sheaf. By the uniqueness of the mapbetween sheaf cohomology theories with coefficient in sheaves of R -modules, it is enough toshow Poincar´e Lemma for weakly differentiable forms, i.e., it is enough to show that everypoint of K has a contractible neighborhood U in K such that for any ω ∈ Ω j ∞ ( U \ ∂K ), j >
0, there is α ∈ Ω j − ∞ ( U \ ∂K ), such that dα = ω .Poincar´e Lemma for Ω j ∞ ( K \ ∂K ) may be either derived by following the same argumentas for the smooth forms on compact manifolds with boundary or directly deduced from theproof of Theorem 4.2.1 which actually applies to any weakly differentiable bounded j -form ω (this theorem indeed states a more difficult result since it deals with every subanalyticset, possibly singular). (cid:3) For noncompact manifolds, we can now prove the following:
Proposition 4.1.3.
For any C ∞ manifold M (without boundary), the inclusion Ω •∞ ( M ) ֒ → Ω •∞ ( M ) induces isomorphisms on the cohomology groups.Proof. It is enough to show that, for any form α ∈ Ω j ∞ ( M ) with dα ∈ Ω j +1 ∞ ( M ) (i. e. α is weakly differentiable and dα is smooth ), there exists θ ∈ Ω j − ∞ ( M ) such that ( α + dθ ) is C ∞ (if j = 0 then θ ≡ K i ⊂ M , i ∈ N , suchthat for each i ≥ K i is included in the interior of K i +1 and ∪ K i = M .Fix a form α ∈ Ω j ∞ ( M ) with dα ∈ Ω j +1 ∞ ( M ). We are going to construct a sequence( θ i ) i ∈ N in Ω j − ∞ ( M ) such that for every i ∈ N , we have supp θ i ⊂ Int ( K i ) \ K i − as well as | θ i | ∞ + | dθ i | ∞ ≤ α i := α + P ik =0 dθ k is smooth in a neighborhoodof K i − .Before defining inductively the θ i ’s, observe that θ := P ∞ i =0 θ i is the desired form (thissum is locally finite).We now define the θ i ’s by induction on i . Let us assume that θ , . . . , θ i − have beenconstructed, i ≥ K − := K − := ∅ ). We will also argue by induction on j .For j = −
1, both cochain complexes vanish and the result is clear.Observe that by Lemma 4.1.2, there exists a smooth j -form β on K i such that dβ = dα i − . It means that ( α i − − β ) is d -closed, and thus again by Lemma 4.1.2 there is asmooth j -form β ′ on K i such that(4.16) α i − − β = β ′ + dγ, with γ ∈ Ω ( j − ∞ ( K i ).Thanks to the induction on i , we know that there exists an open neighborhood V of K i − on which α i − is smooth. This implies that dγ is smooth on V . Therefore, applyingthe induction hypothesis to γ (which is a ( j − dσ to γ to get a form smooth on a neighborhood of K i − . Multiplying σ by a smooth functionwhich has compact support included in V and which is 1 on a neighborhood W ⊂ V of K i − , we get a form σ ′ on M such that ( dσ ′ + γ ) is smooth on W . It means that we canassume that γ is smooth on an open neighborhood W of K i − . We will assume this factwithout changing notations.By means of convolution products with a bump function, for any ε >
0, we may con-struct a smooth form γ ε such that | γ ε − γ | ∞ ≤ ε and | dγ ε − dγ | ∞ ≤ ε on K i .Consider a smooth function φ which is 1 on a neighborhood of ( M \ W ) ∩ K i − andwith support in ( K i \ ∂K i ) \ K i − . Then, set:(4.17) θ i ( x ) := φ ( x )( γ ε − γ )( x ) . If ε is chosen small enough | θ i | ∞ + | dθ i | ∞ ≤
1. On a neighborhood of ( M \ W ) ∩ K i − ,because φ ≡
1, we have by (4.16) and (4.17): α i − + dθ i = β + β ′ + dγ ε which is clearly ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY smooth. The form ( α i − + dθ i ) is smooth on W as well, since α i − and θ i are bothsmooth. (cid:3) Proof of the De Rham Theorem for L ∞ cohomology. We are now ready toprove Poincar´e Lemma for L ∞ cohomology. Theorem 4.2.1. (Poincar´e Lemma for L ∞ cohomology) Let X ⊂ R n be locally closed andlet x ∈ X . There exists ε > such that for any closed form ω ∈ Ω j ∞ ( B n ( x , ε ) ∩ X reg ) , j ≥ , we can find α ∈ Ω j − ∞ ( B n ( x , ε ) ∩ X reg ) , such that ω = dα .Proof. Let r : X ∩ B n ( x , ε ) × [0; 1] → X ∩ B n ( x , ε ) be the map obtained by applyingTheorem 2.3.1 to X . For simplicity, as our problem is local, we will identify X with thesubset X ∩ B n ( x ; ε ). Let ω ∈ Ω j ∞ ( X reg ), with j ≥ dω = 0. By Proposition 4.1.3,it is enough to find α ∈ Ω j − ∞ ( X reg ) satisfying dα = ω .The problem is that r may fail to be weakly smooth. To overcome this difficulty, weshall work with an approximation of r . We need to be particular since we wish to preservethe property that the derivative of r t goes to zero (pointwise and generically) as t goes tozero.Consider a sequence of compact subsets, ( K i ) i ∈ N , such that ∪ K i = X reg × (0; 1) and K i ⊂ Int ( K i +1 ), for any i . Let Y be the set of points of X reg × (0; 1) at which r fails tobe smooth. Define then a sequence of compact subsets for i ≥ L i := { q ∈ K i : d ( q ; Y ) ≥ /i } . Define also X ′ := cl ( Y ) ∩ ( X reg × { } )and observe that, since Y is of positive codimension in X reg × (0 , X ′ is of positivecodimension in X reg (we will consider it as a subset of X reg ).As r is continuous, we may choose, for a given ε i >
0, a C ∞ approximation r i (notnecessarily subanalytic) of r on K i satisfying for any x in this set: | r t ( x ) − r i,t ( x ) | ≤ ε i . Furthermore, as r is smooth on L i , we may require that on this set(4.18) | d x r i,t − d x r t | ≤ ε i . Moreover, as the first derivative of r is bounded, the first derivative of r i may be assumedto be bounded as well.Let ( ϕ i ) i ∈ N be a partition of unity subordinated to the covering ( Int ( K i +2 ) \ K i ) i ∈ N of X reg × (0; 1). Set r ′ := P ϕ i r i . If the sequence ε i is decreasing fast enough, then astraightforward computation shows that the first derivative of r ′ is bounded above.Furthermore, given any positive continuous function ε : X reg × (0; 1) → R , we can haveif the sequence ε i is decreasing fast enough:(4.19) | r ′ t ( x ) − r t ( x ) | ≤ ε ( x ; t ) . Finally, we shall check that d x r ′ t tends to zero as t goes to zero for x / ∈ X ′ . Fix x in X reg \ X ′ . There exists a > { x } × [0; a ] does not meet Y . It means that forany i large enough(4.20) K i ∩ ( { x } × [0; a ]) = L i ∩ ( { x } × [0; a ]) . For t small enough, if ϕ i ( x ; t ) = 0 then ( x ; t ) ∈ K i +2 and thus by (4.20) belongs to L i +2 .By (4.18), this implies that if ε i tends to zero fast enough, lim t → | d x r ′ t | = 0 for every x ∈ X reg \ X ′ . We also see for the same reason that d x r ′ t tends to the identity as t goes to1 (for almost every x ).Let π : W → X reg be a retraction where W is a tubular neighborhood of X reg . Taking W small enough, we may assume that π has bounded first partial derivatives. By (4.19), r ′ t ( x ) belongs to W if the function ε is decreasing fast enough. Hence, composing r ′ with π if necessary we may assume that r ′ preserves X reg . We will assume this without changingnotations.Define two L ∞ forms ω and ω on X reg × (0; 1] by: r ′∗ ω := ω + dt ∧ ω . Now, we may set: α ( x ) := Z ω ( x ; t ) dt. As ω is L ∞ and r ′ has bounded first partial derivatives, the form α is clearly bounded. ByLebesgue’s dominated convergence theorem, it is continuous. We claim that it is weaklydifferentiable and that dα = ω .Let us fix a C -form ϕ ∈ Λ m − j ( X reg ) with compact support. We have, by definition of α :(4.21) Z X reg α ∧ dϕ = Z X reg Z ω ∧ dϕ = lim t → Z X reg × [ t ;1] r ′∗ ω ∧ dϕ. As r ′∗ ω is closed, by Stokes’ formula we have: Z X reg × [ t ;1] r ′∗ ω ∧ dϕ = ( − j Z x ∈ X reg ω ( x ) ∧ ϕ ( x ) − ( − j Z x ∈ X reg ω ( x ; t ) ∧ ϕ ( x ) , since lim t → r ′∗ ω ( x ; t ) = ω ( x ) for any x ∈ X reg . Recall that d x r ′ t tends to zero as t goes to0 for almost every x . This implies that ω ( x ; t ) goes to zero as t goes to 0. Hence, passingto the limit we get: Z X reg α ∧ dϕ = ( − j Z X reg ω ∧ ϕ, as required. (cid:3) Proof of Theorem 1.2.2.
Let π : ˜ X → X be a normalization of X (see Proposition 3.1.2).Let us define a presheaf on ˜ X by˜Ω j ∞ ( U ) := Ω j ∞ ( π ( U ) ∩ X reg ) , for every open set U of ˜ X ( π is a homeomorphism above X reg ). For every j , this presheafimmediately gives rise to a sheaf that we will denote by F j ∞ . We will write F j ∞ ,x for thestalk of F j ∞ at x ∈ ˜ X , i.e., the vector space obtained after identifying two sections whichcoincide near x . As ˜ X is compact, any global section of F j ∞ is bounded, so that, since π induces a homeomorphism above X reg :(4.22) F •∞ ( ˜ X ) ≃ Ω •∞ ( X reg ) , as cochain complexes. ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY We denote by R X the constant sheaf on X . Let x ∈ ˜ X and set U ε := B n ( x , ε ) ∩ ˜ X . As π is a normalization, π ( U ε ) ∩ X reg is connected, which means that H ∞ ( X reg ∩ π ( U ε )) = R ,for ε > j >
0, the germ at π ( x ) of a smooth bounded closed j -form ω on π ( U ε ) ∩ X reg is the exterior differential of the germ of a form α ∈ F j − ∞ ,x .Therefore, the sequence:(4.23) 0 −→ R X d −→ F ∞ d −→ F ∞ d −→ . . . is a fine torsionless resolution of the constant sheaf. By classical arguments of sheaftheory (see for instance [W]), the latter exact sequence of sheaves implies via (4.22) that H j ∞ ( X reg ) is isomorphic to the singular cohomology of ˜ X . But then, by Propositions 3.0.5and 3.1.4, we get:(4.24) H j ∞ ( X reg ) ≃ H j ( ˜ X ) ≃ I t H j ( ˜ X ) ≃ I t H j ( X ) . (cid:3) Integration on subanalytic singular simplices.
We are going to prove that theisomorphism is provided by integrating the forms on the allowable chains. We first checkthat integration gives rise to a well defined cochain map. This may be done because werestrict ourselves to the t -allowable subanalytic singular cochains. Let X be a compactpseudomanifold.Let L ⊂ X reg be an oriented manifold of dimension j with cl ( L ) ( t ; j )-allowable i. e.: dim cl ( L ) ∩ X sing ≤ ( j − . Set ∂L := cl ( L ) \ L . Then, for any given ω in Ω j − ∞ ( X reg ), R L dω and R ( ∂L ) reg ω arewell defined since ω is continuous almost everywhere on ( ∂L ) reg and bounded. We startby recalling a version of Stokes’ formula proved by Lojasiewicz in [L] who generalized aformula of Paw lucki [Pa]. Lemma 4.3.1. [L]
Take L and ω as in the above paragraph. Then: (4.25) Z ( ∂L ) reg ω = Z L dω. Lojasiewicz’s formula is actually devoted to bounded subanalytic forms, but the requiredproperty is indeed that they are bounded and extend continuously almost everywhere onthe closure of the manifold L , which obviously holds true when the form is L ∞ and cl ( L )is ( t ; j )-allowable.Next we turn to see that integration is well defined for any ( t ; j ) allowable subanalyticsingular simplex . Let σ : ∆ j → X be an oriented ( t ; j )-allowable (subanalytic) simplex.Denote by σ reg the set of points in σ − ( X reg ) near which σ induces a smooth mapping.Observe that the complement of σ reg in ∆ j has Lebesgue measure zero. Hence, it makessense to set for ω ∈ Ω j ∞ ( X reg ):(4.26) Z σ ω := Z ∆ j σ ∗ ω = Z σ reg σ ∗ ω. Stokes’ formula continues to hold for subanalytic singular ( t ; j )-allowable simplices: Lemma 4.3.2.
Let σ ∈ I t C j ( X ) and ω ∈ Ω j − ∞ ( X reg ) . Then the integral defined in (4.26)is finite and: (4.27) Z σ dω = Z ∂σ ω. Proof.
Let Γ := { ( x ; y ) ∈ X × ∆ j : x = σ ( y ) } , and consider a cell decomposition of R n + j compatible with Γ. Refining it, we can assumethat the boundary of a cell is a union of cells. For simplicity, we will identify ∆ j with Γand assume that σ is the canonical projection (restricted to Γ).Let C ⊂ Γ be a cell of this cell decomposition and let i := dim C . Observe that either σ | C is a diffeomorphism or dim σ ( C ) < i . In the former case, if we endow σ ( C ) with theorientation induced by ∆ j via σ , we have by definition:(4.28) Z C σ ∗ α = Z σ ( C ) α. for any α ∈ Ω i ∞ ( X reg ) (we shall need both the cases α = ω and α = dω ). If dim σ ( C ) < i ,then both vanish and this remains true. Note that, as a cell decomposition is a finitepartition, the latter formula already shows that the integral defined in (4.26) is finite.By Lemma 4.3.1, if C is a cell of dimension j :(4.29) Z C σ ∗ dω ( . ) = Z σ ( C ) dω ( . ) = Z ∂ ( σ ( C )) ω = Z σ ( ∂C ) ω ( . ) = Z ∂C σ ∗ ω, the third equality being true because σ is identified with a linear mapping (a projection)on the cell C . As ∆ j is a union of cells, the latter equality still holds if we replace C with∆ j . We conclude that for relevant orientations: Z σ dω = Z ∆ j σ ∗ dω ( . ) = Z ∂ ∆ j σ ∗ ω = Z ∂σ ω. (cid:3) In conclusion, we get that the isomorphism of Theorem 1.2.2 is given by integrating thedifferential forms on the simplices:
Theorem 4.3.3.
Let X be a compact pseudomanifold. The cochain maps ψ jX : Ω j ∞ ( X reg ) → I t C j ( X ) ,ω [ σ Z σ ω ] , induce isomorphisms between the cohomology groups. To prove it, observe that the cochain map ψ X induces a sheaf homomorphism (recallthat the L ∞ forms give rise to sheaf on the normalization of X , see the proof of Theorem1.2.2). By the uniqueness of the map between sheaf cohomology theories with coefficientin sheaves of R -modules, this map must coincide with the isomorphism (4.24). ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY Remarks 4.3.4.
Theorem 1.2.2 still holds if X is a pseudomanifold with boundary [GM1](indeed, our Poincar´e Lemma for L ∞ cohomology does not assume that X is a pseudoman-ifold). The relative version is then true as well, by the five lemma. Again, the isomorphismis provided by integration of forms on allowable chains.It is worthy of notice that the arguments of the proof of Theorem 1.2.2 also apply inthe noncompact case, establishing an isomorphism between the cohomology of the locally bounded forms (locally in X , not in X reg ) and intersection homology in the maximalperversity.The results of this paper remain true if we replace the subanalytic category with a poly-nomially bounded o-minimal structure [vDM]. We need the structure to be polynomiallybounded for we made use of the preparation theorem for proving Proposition 2.2.1. It isunclear (but not impossible) whether the results of this paper, especially Theorem 2.3.1,are valid on a non polynomially bounded o-minimal structure, especially for the log − exp sets, on which a generalized preparation theorem holds [LR].5. Appendix I: globally subanalytic sets
Basic definitions.
Let N be an analytic manifold. Recall that a subset E ⊂ N iscalled semianalytic if it is locally defined by finitely many real analytic equations andinequalities. More precisely, for each p ∈ N , there is a neighborhood U of p in N , and realanalytic functions f i , g ij on U , where i = 1 , . . . , r, j = 1 , . . . , s , such that E ∩ U = r [ i =1 s \ j =1 { x ∈ U : g ij ( x ) > f i ( x ) = 0 } . A subset E ⊂ N is subanalytic if it can be locally represented as the projection of asemianalytic set. More precisely, for every p ∈ N , there exist a neighborhood U of p in N ,an analytic manifold P , and a relatively compact semianalytic set Z ⊂ N × P such that E ∩ U = π ( Z ), where π : N × P → N is the natural projection. In particular, semianalyticsets are subanalytic.A subset of R n is globally subanalytic if it coincides with a subanalytic subset of P n after identifying R n with and open subset of P n via:( y , . . . , y n ) → (1 : y : · · · : y n ) : R n → P n . We will denote by S n the set of globally subanalytic subsets of R n .Clearly, a bounded subset is subanalytic if and only if it is globally subanalytic. We saythat a function is globally subanalytic if its graph is globally subanalytic.5.2. Basic properties of globally subanalytic sets.
Any real algebraic set is glob-ally subanalytic. Furthermore, globally subanalytic sets have the following very usefulproperties (see [vDM]):(1) S n is stable under unions, intersections and complement.(2) If A ∈ S m and B ∈ S n then A × B ∈ S m + n .(3) If π : R n +1 → R n is the projection on the first n coordinates and A ∈ S n +1 , then π ( A ) ∈ S n .(4) The elements of S are precisely the finite unions of points and intervals. When a family of sets has these properties, we say that it constitutes an o-minimalstructure . These properties are indeed all the basic properties we need to do most ofgeometric constructions (such as triangulations, stratifications, retracts, ...). Property (3)is the motivation for introducing subanalytic sets: semianalytic sets are not stable underprojection and thus do not fulfill (3).Property (4) is a finiteness assumption which makes it possible to derive all the finitenessproperties of globally subanalytic sets. The first one and the most important is existenceof cell decompositions (see definition below). Most of the results we give below are notreally proper to globally subanalytic sets and are shared by all the sets definable in ano-minimal structure. We will therefore often refer to [Co] for proofs.
Definition 5.2.1. A cell decomposition of R n is a finite partition of R n into globallysubanalytic sets ( C i ) i ∈ I , called cells , satisfying certain properties explained below. n = 1 : A cell decomposition of R is given by a finite subdivision a < · · · < a l of R . The cells of R are the singletons { a i } , 0 < i ≤ l , and the intervals ( a i , a i +1 ), 0 ≤ i ≤ l ,where a = −∞ and a l +1 = + ∞ . n > cell decomposition of R n is the data of a cell decomposition of R n − and,for each cell D of R n − , some globally subanalytic functions analytic on D (which is ananalytic manifold): ζ D, < ... < ζ D,l ( D ) : D → R . The cells of R n are the graphs { ( x, ζ D,i ( x )) : x ∈ D } , < i ≤ l ( D ) , and the bands ( ζ D,i , ζ
D,i +1 ) := { ( x, y ) : x ∈ D and ζ D,i ( x ) < y < ζ D,i +1 ( x ) } , for 0 ≤ i ≤ l ( D ), where ζ D, = −∞ and ζ D,l ( D )+1 = + ∞ .A cell decomposition is said to be compatible with finitely many sets A , . . . , A k if the A i ’s are unions of cells.Given some globally subanalytic sets A , . . . , A k , it is always possible to find a celldecomposition compatible with this family of sets. Detailed proofs may be found in [Co].This already describes very precisely the geometry of globally subanalytic sets. Below,we list some related extra basic properties, useful for us. ⋄ (curve selection Lemma) Let A ∈ S n and b ∈ cl ( A ). There is an analytic map γ :( − , → R n such that γ (0) = b and γ ((0 , ⊂ A . ⋄ ( subanalycity of the connected components ) Subanalytic sets have only finitely many con-nected components. They are subanalytic. ⋄ ( uniform bound ) Let f : A → B be a globally subanalytic map, with finite fibers. Thereis k ∈ N such that card f − ( b ) ≤ k for any b . ⋄ ( subanalytic choice ) Any globally subanalytic map f : A → B (not necessarily continuous)admits a globally subanalytic section, i.e., a globally subanalytic map s : B → A suchthat f ( s ( b )) = b . ⋄ ( subanalycity of the regular locus ) Let X ∈ S n . Then X reg is a finite union of analyticmanifolds; it is globally subanalytic and dense in X . ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY For a proof of curve selection Lemma or subanalycity of the regular locus we refer thereader to [BM]. A proof of all the other statements may be found in [Co].The set X reg is the union of finitely many analytic manifolds. The dimension of X ,denoted dim X , is the maximal dimension of these manifolds.6. Appendix II: Some basic model theoretic principles
Formulae.
We shall need some basic facts of model theory. We first define what wecall L -formulae. Basically, it is a sequence constituted by quantifier and some symbols,like for instance ∀ x, ∃ y, x ≤ yz . More precisely, L -formulae are defined inductively asfollows:(1) If f is a globally subanalytic function then f ( x ) > f ( x ) = 0 are L -formulae.(2) If Φ( x , . . . , x n ) and Ψ( x , . . . , x n ) are L -formulae, then ”Φ and Ψ”, ”Φ or Ψ”, and”not Φ”, are L -formulae as well.(3) If Φ( y, x ) is an L -formula, then ∃ x, Φ( y, x ) and ∀ x, Φ( y, x ) are L -formulae.The parameters x = ( x , . . . , x n ) in Φ( x ) denote the free variables (those which are notquantified). The symbol L stands for language: L -formulae are sentences ”in the languageof subanalytic geometry”. Roughly speaking, L -formulae are all the mathematical sen-tences that one can write using quantifiers, globally subanalytic functions, equalities andinequalities.As an example, consider the formula Φ( x ): ∀ ε > , ∃ α > , ∀ y, | x − y | ≤ α ⇒ | f ( x ) − f ( y ) | < ε. However, the formula ∃ n, n ∈ N and y = nx is not an L -formula ( N may not be describedby an L -formula). Observe that this formula does not define a subanalytic set of R .Indeed, the following fundamental result relates L -formulae to subanalytic sets: Proposition 6.1.1. If Φ( x ) is an L -formula, then the set { x ∈ R n : Φ( x ) } is globallysubanalytic. To briefly account for this proposition, let us point out that the sentence ∃ y, f ( x, y ) = 0defines the projection of the set defined by f ( x, y ) = 0. Thus, for this sentence the resultfollows from Property (3) of section 5.2. The proposition is then showed by inductionon the number of quantifiers (the case of the universal quantifier may be reduced to theexistential by considering the negation of the sentence, for more details see [Co], Theorem1 . L -formula, then this function is globally subanalytic. It enables to establish that afunction is subanalytic without much work.For instance, if ξ : R n × [0 , → [0 ,
1] is a globally subanalytic function then the functiondefined by ζ ( x ) := inf t ∈ [0 , ξ ( x, t ) is globally subanalytic. It is then easy to check that if A denotes a subanalytic set then the function x d ( x, A ), which assigns to x the Euclideandistance from x to A , is globally subanalytic. Field extensions.
Consider all the one variable globally subanalytic functions whichare defined in a right-hand-side neighborhood of the origin and identify any two of themwhich coincide in a small right-hand-side neighborhood of the origin. It follows fromPuiseux Lemma [Pa] that this set is indeed the field of real Puiseux series P i ≥ m a i T ik , m ∈ Z , a i ∈ R , with P i ∈ N a i t i convergent for t in a neighborhood of zero. We shalldenote this field k (0 + ). We may imbed R ֒ → k (0 + ), sending every real number onto thecorresponding constant series.We can order this field by setting f ≤ g in k (0 + ) if f ( t ) ≤ g ( t ) for t in a right-hand-sideneighborhood of the origin. Observe that the indeterminate T is smaller than any positivereal number in k (0 + ). Consequently, this field is not Archimedean.We may also define the Euclidean norm on k (0 + ) n , by | x | := P ni =1 x i ∈ k (0 + ). Thisgives rise to a topology. A good reference for all the results of this section is [Co]. Extension of sets and functions.
As the composite of globally subanalytic mappings isglobally subanalytic, any globally subanalytic function ξ : R n → R may be extended to afunction ξ k (0 + ) : k (0 + ) n → k (0 + ), ξ k (0 + ) ( x ( T )) := ξ ( x ( T )), x ( T ) ∈ k (0 + ). Any L -formulamay then be also ”extended” to k (0 + ). For instance the formula Φ, ∃ x ∈ R n , f ( x ) ≥ f is an analytic function, has the following extension Φ k (0 + ) to k (0 + ): ∃ x ∈ k (0 + ) n , f k (0 + ) ( x ) ≥ . In other words, we can extend a formula by extending the functions this formula involves.It is not very difficult to check that if two formulae define the same set in R n then theirrespective extensions define the same set in k (0 + ) n (see [Co]). Hence, we may define the extension of the set A := { x ∈ R n : Φ( x ) } by setting A k (0 + ) := { x ∈ k (0 + ) n : Φ k (0 + ) ( x ) } . In other words, the extension of a set is obtained by regarding the associated equationsand inequalities in the field of real analytic Puiseux series. This set is merely the set ofgerms of Puiseux arcs lying on A . For instance, the extension of the sphere S n − (generallystill denoted S n − ) is the set: { x ∈ k (0 + ) n : n X i =1 x i = 1 } . Generic fibers.
Let now A ⊂ R × R n be a globally subanalytic set. Regarding the firstvariable as a parameter, we will consider this set as a family. We define the generic fiber of this family of sets as (recall that T ∈ k (0 + ) stands for the indeterminate): A + := { x ∈ k (0 + ) n : ( T, x ) ∈ A k (0 + ) } . It is nothing but the set of germs of Puiseux arcs x ( t ) such that ( t, x ( t )) ∈ A for every t positive small enough. Observe that we can also define the generic fiber of a family ofglobally subanalytic functions f : A → R , which is the function f + : A + → k (0 + )which assigns to x ∈ A k (0 + ) the value f k (0 + ) ( T, x ).We then can define the subanalytic sets (resp. mappings) of k (0 + ) n as the collection ofall the generic fibers of subanalytic families of sets (resp. mappings). This constitutes afamily of Boolean algebras which enjoys the same properties as ( S n ) n ∈ N . Indeed, as they ∞ COHOMOLOGY IS INTERSECTION COHOMOLOGY satisfy (1 −
4) of section 5.2, they then satisfy all the other properties which come downfrom these properties.As a matter of fact, Proposition 6.1.1 is still true if we replace R with k (0 + ). Forexample, the function x d ( x, A ) is well defined and globally subanalytic if so is A ⊂ k (0 + ) n .We can also define the generic fiber of a formula . If Φ( t, x ) is a formula, with x and t free variables ( t considered as a parameter) we define the generic fiber of Φ( t, x )as the formula obtained by replacing t with the indeterminate T , i.e. we set Φ + ( x ) :=Φ k (0 + ) ( T, x ). This of course reduces the number of free variables.
Transfer principle.
The study of the generic fiber of a family A ⊂ R × R n can provideus information on what happens on the fiber A t for generic parameter t . More precisely,we have the following very important fact. Let Φ( t ) be an L -formula. The formula Φ + holds true in k (0 + ) if and only if Φ( t ) holds for every positive real number t small enough.This is a consequence of a more general theorem sometimes referred as Lo´s’s Theorem .To make it more concrete, let us illustrate it by some examples. One may easily derivefrom this fact that if the generic fiber of the family of functions f : R × R n → R is boundedby 1 then there is ε > x ∈ R n f ( t, x ) is not greater that 1 for any t ∈ (0 , ε ).This is due to the fact that ∀ x, f ( t, x ) ≤ L -formula.Less easy, but still true, is the fact that if f + is continuous then there is ε > f : (0 , ε ) × R n → R is continuous (the non obvious part is the continuity withrespect to the parameter t ∈ (0 , ε ) which is not guaranteed by the above transfer principle,see [Co] section 5 . k (0 + ) n and the geometry of families of globally subanalyticsets of R × R n , for generic parameters.To give another illustration, let us focus on the argument of the proof of Corollary2.1.4. Applying Theorem 2.1.2 provides a bi-Lipschitz globally subanalytic map H : k (0 + ) n − → k (0 + ) n − . This homeomorphism is the generic fiber of a family of mappings h : (0 , ε ) × R n − → (0 , ε ) × R n − . 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