Intersection differential forms

Intersection differential forms

GUIDOPOLLINI(*)

ABSTRACT- We present a survey about the complex IpV_X ofintersection differential forms, a sub complex of the de Rham complex of the regular locusXreg of a controlled pseudomanifold X endowed with a perversity function p, which computes the real intersection cohomology with respect to the fixed perversity.

Introduction.

The intersection homology theory developed by M. Goresky and R.

MacPherson in [GM1] has proven to be an useful tool in studying singular spaces; the original geometric definition goes roughly as follow. Consider a PL-pseudomanifoldX with a fixed stratification and its complex C X;R of PL-chains with real coefficients; ifSwere a stratum ofX andjani-chain in general position with S then their intersection would have dimension i‡cod S dimX. Thus one can assign to every codimension of the strata an integerp(cod S) that is the extra amount of this intersection dimension which is allowed. If the perversity functionpis fixed then the chainjis called p-perverse if for every stratumS,dimjjj \Si‡cod S dimX‡p(cod S);

the perversity measures how much a chain is free to intersect the singular part of the pseudomanifold. This leads to the definition of intersection homology as thehomology of thecomplex IpC X;R

of thep-perverse chains inX.

An alternative technical approach to intersection cohomology using sheaf theory is described in [GM2] where an axiomatic framework for IC is also proved (Deligne's axioms).

In this survey we describe another method originally due to Goresky and MacPherson to compute such cohomology using special differential

(*) Indirizzo dell'A.: Dipartimento di Matematica «G. Castelnuovo», UniversitaÁ di Roma «La Sapienza», Piazzale Aldo Moro, 5 - 00185 Roma. E-mail:

pollini@mat.uniroma1.it

web: http://www-studenti.dm.unipi.it/ pollini/

forms which is more geometrical than the latter; in a smooth manifold a differential form can thought as ashape evaluatorof chains (theflat cochain interpretation): every infinitesimal element of a smooth chain is evaluated using multilinear form which returns somehow its volume and then ev- erything is integrated in some sense. In a singular space there is not a geometric definition of tangent bundle, so infinitesimal elements and dif- ferential forms are meaningless; however, if the space is good enough there is an additional structure(a control data) consisting in a family of tubular neighborhoods and retractions for the strata with high compatibility de- gree. This family can be used to define on the regular part of X thedi- rection toward the singularities; using this extra informations the smooth part can remember the original singularities. Now, fix a stratumSwith tube T_S, retractionp_Sand associated perversityp: a smooth differentiali-formv inX_reghas perversitypwith respect toSif for everyx2X \T_Sevaluates as zero every «infinitesimal i-element» atx which contains morethat p

«infinitesimal directions» along the fibers of p_S which aresmooth sub- manifolds. Thus a form is said to have low perversity if it is unable to detect an infinitesimal element when is too directed toward the singular locus.

Fixing a perversitypand repeating this reasoning for every stratum lead to the complex of sheaves I_pV_X ofdifferential intersection forms.

The axiomatic framework is used to show that this complex has the same local properties of the ICcomplex and thus it computes intersection coho- mology. Moreover, some well-known properties of classic forms as softness, Poincare lemma, Mayer-Vietoris sequence and a partial KuÈnneth formula areproved.

Thedefinition of I_pV_X here presented is not new: it has been introduced in [BRY] where it is attributed to Goresky and MacPherson (unpublished);

such complex is reconsidered in [BHS] where the aim of the authors is to establish a pairing between a complex of singular intersection chains de- scribed by King [KIN] and the complex I_pV_X: they show that intersection differential forms with some extra properties can be integrated on parti- cular intersection homology chains, using some local computations typical in intersection homology to complete their program.

There also exists variations on this complex, for example the complex of S-regular forms of intersection introduced in the papers [BF] and also [SAR]; the definition is similar to the definition presented in this survey, but now forms are given on all strata (and not only the open one) with some gluing conditions on theneighborhood of thestrata. Using theaxiomatic theory of Deligne for intersection cohomology the authors also prove that their complex is quasi-isomorphic to the intersection cohomology sheaf.

Thetechnical bulk of this articleis a mixtureof such local computations and axiomatic approach.

Notations.

± IfM is a smooth manifold wedenotewithtanM,J[M] andV^j[M] respectively the tangent bundle, the family of smooth vector fields and the differentialj-forms.

± IfA is a sheaf ofR-vector spaces over a topological spaceX and SX we denote the section ofA overSwith G(S;A) orA[S] and the stalk ofA over a pointx2X asAx.

± IfA is a complex of sheaves thenH ^j(A) denote thej-th coho- mology sheaf; if M is a complex of R-vector spaces then H^j(M) isj-th cohomology vector space.

1. Controlled pseudomanifolds.

We briefly collect here the main definitions and theorems about stra- tifications and control theory; a detailed and modern description of such topics can be found in [PFL] (chapters 1-3) which is the main reference.

Other sources include the older [THO], [MA1], [MA2], [GWPL] and [VER].

Roughly speaking, acontrolled pseudomanifold(also known as aThom- Mather stratified space) is a nicetopological space[1.1-A] with a good de- composition into smooth pieces [1.1-B] where a way is given to describe approaching to singular locus [1.1-C]; although thefollowing formal defi- nition seems to be very restrictive and unpleasant, almost every good sin- gular space admits the structure of controlled pseudomanifold (see theorem [1.3] and [1.4]):

DEFINITION 1.1. (Controlled pseudomanifold) A triple X:ˆ(X;S; f(TS;pS;r_S)g_S2S) is called a controlled pseudomanifold iff the following properties hold:

A: X is a locally compact Hausdorff space with countable topology;

B: S is a locally finite partition ofX into subspaces (the strata ofX) that are smooth manifolds in the induced topology and satisfy the following conditions:

B1: if R;S2S and R\S6ˆ ;then RS (and we write RS);

B2: if the max dimension of the strata is n2N then X has no

stratum of dimension n 1and the union of all n-strata is open- dense inX (ifXkis the k-skeleton ofX, i.e. the union of strata of dimension k, then X ˆXn, X_{n 1} X_{n 2}ˆ; and X X_{n 2}is open-dense inX);

C: for every S2S the triple(T_S;p_S;r_S)(a tube for S inX) satisfies the following relations:

C1: pS:TS!S is a continuous retraction of an open neighborhood T_S of S inX such that for every stratum RS the restriction p_Sj_TS\R:T_S\R!S is smooth;

C2: r_S:T_S!R⁰is a continuous map such thatr ¹(0)ˆS and for each stratum RS the restriction r_Sj_TS\R:T_S\R!S is smooth;

C3: for each pair R>S of strata, for all x2T_S\T_R\p_R¹(T_S)we have:

p_Sp_R(x)ˆp_S(x), r_Sp_R(x)ˆr_S(x);

C4: for each pair R>S of strata, the restriction

(p_S;r_S)j:T_S\R!SR^>0

is submersive (in particular,p_Sj:T_S\R!S is submersive);

The setXreg:ˆXn X_{n 2}is called the regular part andX_sing:ˆX_{n 2} the singular part.

A topological spacewith data satisfying theconditions of definition [1.1], except perhaps for property B2, is called acontrolled space; such spaces are well behaved, however to achieve topological invariance of intersection cohomology(see [BOR] pp. 86-96, [GM1] and [GM2]) the pseudomanifold structureis fundamental (i.e. ifX has dimensionnthen no stratum of di- mensionn 1 is allowed and the regular part must be open-dense).

Standard point-set topology shows that the topological space underlying a controlled space is metrizable and so paracompact (and finite-dimen- sionalby definition); by the following lemma one can assume a good se- paration of tube-neighborhoods which will be used tacitly elsewhere ([PFL]

Prop. 3.6.7(1)):

LEMMA1.2. By shrinking the tubes we can assume that:

± if S;R2S and T_S\R6ˆ ;then RS;

± if R;S2S and T_S\T_R6ˆ ;then R and S are comparable (i.e. R<S, RˆS or R>S).

Everyopen set of a controlled pseudomanifold canonically inherits a

structure of controlled pseudomanifold; also everysmooth manifoldwith or without boundary has a natural controlled pseudomanifold structure (the latter requires a collar as tube for the boundary), but it is considerably more difficult to prove the following existence theorem (see [PFL] Th. 3.6.9):

THEOREM 1.3. Every locally compact Whitney-stratified subset of a given smooth manifold admits a structure of controlled space.

Consider asX a 2-torus with a circlecollapsed and with thecentral hole filled with a disk as in figure 2; the stratification isS :ˆ

S₀;S₁;S^I₂;S^II₂ , where S^I₂ andS^II₂ are diffeomorphic to an open 2-disk and S₁ to an open segment. It is easy to see that such space satisfies Whitney condition and thus by theorem [1.3] it admits a control structure; however condition [1.1- B2] is not satisfied since the singular setX_sing :ˆS₀[S₁ has real codi- mension 1 and henceX is not a pseudomanifold. It can be shown that the intersection cohomology of a pseudomanifold is a topological invariant (see [BOR] Cor. 4.18, 4.19) thus not depending on the chosen stratification and this is generally false for a generic stratified space; however this simple example is useful to visualize what is going on.

Fig. 1. ± A stratification.

Fig. 2. ± Control data.

The following theorem allows to construct control data for a variety of spaces ([BCR] pp. 184-214)) using theorem [1.3]:

THEOREM1.4. The following spaces can be Whitney-stratified:

± complex analytic varieties;

± real analytic varieties.

In particular, complex analytic varieties and real analytic varieties with dense singular locus of real dimension 2 admit a structure of controlled pseudomanifold.

The following constructions are needed to describe the local structure of a controlled pseudomanifold:

EXAMPLE 1.5 (Cylinder). Given a controlled pseudomanifold (X;S;f(TS;pS;r_S)g_S2S) and a smooth manifold M without boundary (e.g R^l) onecan form a control structureover thecylinder X M as follow: thestrata set is

SM _S2S and control data for a stratumSM is given by (T_SM;p_Sid_M;r_S0); it is easily seen that this datum verifies the conditions [1.1-A, B, C].

EXAMPLE1.6 (Cone). Let (L;S;f(TS;pS;r_S)g_S2S) be acompactcon- trolled pseudomanifold and consider the open coneconL :ˆ^L[0;‡1[_Lf0g ; the strata set is given by

S]0;‡1[ _S2S (lateral strata) plus thevertex, the unique0-stratum. Control data forlateral stratais built using thecylinder structureas in example[1.5] and for thevertex weproceed as follow, taking thetriple(^L_L^[0;e[_f0g;pvert;r_vert) whe re e>0, the mappvertcollapses the neighborhood into the vertex and r_vert is thedistancefrom vertex. As usual, the needed verifications are trivial, but it is important to remark that:

the only0-stratum in the cone conL is its vertex and the associated retractionpvert, being trivial, has just one fiber, namely the whole set

L[0;‡e[

Lf0g :

DEFINITION1.7 (Controlled isomorphism). A controlled isomorphism between two controlled pseudomanifolds (X;S;f(TS;pS;r_S)g_S2S) and (Y;T ;f(TR;pR;r_R)g_R2T)is an homeomorphismf:X !Y between the underlying topological spaces, mapping diffeomorphically strata of X into strata ofY and satisfying the following compatibility relations:

if S2S is mapped by f onto R2T then there exists an open neighborhood UTSfor S inX such that8x2U:

± fp_S(x)ˆp_Rf(x),

± r_S(x)ˆr_Rf(x).

The following is the main result due to Thom and Mather of stratifica- tion theory (for a proof see [PFL] Th. 3.9.2 and Cor. 3.9.3); it roughly says that a controlled space is locally a cylinder over a cone:

THEOREM1.8 (Local structure). IfX is a controlled pseudomanifold then it is topologically locally trivial with cones as typical fibers, i.e.: for every stratum S ofX and for every x2S there exist an open neighborhood U of x in X, a controlled compact pseudomanifold L of dimension dimX dim S 1and a controlled isomorphism:

U !^ R^dimSconL

mapping x to(0;vertex) (and then U\S toR^dimSfvertexg).

Thus a controlled pseudomanifold is in particular a topological pseudo- manifold in thesenseof articles [BOR] pp. 1-2 and [GM2] and it can be shown that it also carries a structure of PL-pseudomanifold as required in [GM1].

REMARK1.9. To construct the intersection differential form and work with intersection cohomology one really needs the good decomposition of X in strata, the tubular neighborhoods and retractions and the local to- pological triviality. Themapsfr_Sg_S2S ([1.1-C2]) will not be explicitly used in what follows; however, they are fundamental to prove theorem [1.8] and to ensure that the linkL is a controlled pseudomanifold too.

2. Deligne's axioms.

Here we briefly recall the formalism used by Deligne to give a sheaf theoretic definition of the intersection cohomology of Goresky &

MacPherson; the main reference for derived categories and homological algebra are [KS] and [IVE].

In this section X ˆ(X;S;f(TS;pS;r_S)g_S2S) will denote a fixedcon- trolled pseudomanifoldwithdimX ˆn; denoting thek-skeleton ofX by

Xkweget a filtration byclosedsets:

XnXn 1ˆXn 2Xn 3. . . .

Xk . . . .X1X0X 1:ˆ ;

such that for eachk2f0;. . .;ngthesetS_k:ˆX_k X_{k 1}if not empty is a smoothk-manifold (S_kis again called thek-stratum, although in general the condition in definition [1.1-B1] is not satisfied). The closure of each X_k derives from local finiteness of strata [1.1-B] and the fact that ifR>Sthen dim R>dim S, as follows from the submersiveness condition in [1.1-C4].

Setting duallyUk:ˆX Xn kfor eachk2f2;. . .;n‡1gweobtain a sequence ofopen sets:

X_reg ˆU₂U₃. . . .U_k U_k‡1. . . .U_n‡1ˆX:

By construction,U_k‡1ˆU_k[ S_{n k} and is important to notethatS_{n k} is closedinU_k‡1; in the sequel we will always denote byi_ktheopeninclusion ik :Uk !Uk‡1.

Thus thespaceX is constructed beginning with the n-stratum U2ˆSnˆXreg, the regular part; then we attach the (n 2)-stratumSn 2

obtainingU₃, heuristically adding toX_reg a smoothn 2-stratum of sin- gularities; such process continues until we reachU_{n 1}and after gluingS₀(a discrete set of point) we complete the construction ofX. Starting from this, the main idea is to check and control object defined over the wholeX (as chains, cochains or sheaves) and see how they change when a new stratum of singularities is added during the described construction. The control is imposed using an integer valued function (see [BOR] pp. 8-9):

DEFINITION 2.1 (Perversity). A perversity associated to X is a function p:f0;. . .;dimXg !Z⁰ such that p₍₀₎ˆp₍₁₎ˆp₍₂₎ˆ0 and p_(k)p_(k‡1)p_(k)‡1;the domain of p is the set of all the possible codi- mensions of strata ofX.

In the case of complexes of sheaves the control process is achieved with thefollowing: for eachk2f2;. . .;ngletD^B(U_k‡1) be the derived category of cohomologically bounded R-sheaf complexes over U_k‡1; thenatural transformationid_D^B_(Uk‡1) !Riki_kinduced by the adjunction gives for each sheaf complexAoverX themap:

att_k:Aj_Uk‡1 !Ri_ki_k Aj_Uk‡1

ˆRi_k Aj_Uk :

which is usually called theattaching mapsince it describes the cohomolo-

gical sheaf theoretic passage fromUktoUk‡1after the attaching of stratum Sn k. Its role in intersection cohomology theory is described in the fol- lowing (see [BOR] pp. 61-62 and [GM2]):

DEFINITION2.2 (Deligne's axioms). LetX be a controlled pseudoma- nifold of dimension n, and supposepis a fixed perversity relative to the filtration of X;a complex of sheaves A belonging to D^B(X) satisfies Deligne's axioms (relative to the constant sheafR_Xregand to perversityp) if:

± the complex A is zero in negative degree and the restriction Aj_Xreg is isomorphic inD^B(X_reg)to the constant sheafR_Xreg (seen as a complex concentrated in degree zero);

± for every k2f2;. . .;ng, x2Sn kand j>p_(k)we haveH ^j…A†_xˆ0;

± for every k2f2;. . .;ngand for each jp_(k)the attaching-map attk

induces a sheaf isomorphism:

H ^j attk

:H ^j Aj_Uk‡1

! H ^j RikAj_Uk :

REMARK2.3. The last axiom of Deligne is geometrically more trans- parent if we substitute toAa softresolution, which we still callA for simplicity of notation; then the last axiom can be reformulated as follows (notethat is fundamental thatSn kis closed inUk‡1):

8k2f2;. . .;ng;8jp_(k) and8x2Sn kthe restriction maps induces

an isomorphism (where V varies into a cofinal family of neighbor- hoods of x in U_k‡1):

lim_!

V3x

H^j…G(V;A)† !^ lim_!

V3x

H^j(G(V S_{n k};A))

Deligne's Axioms are patterned over the local properties of theinter- section cochaincomplex of sheaves I_pC_X ([BOR] p. 34) and their role in this context is given by the following fundamental theorem by Goresky and MacPherson ([GM2]):

THEOREM2.4. LetX be a controlled pseudomanifold and pa fixed perversity;ifAis a complex inD^B(X)satisfying Deligne's axioms then its hypercohomology is naturally isomorphic to intersection cohomology.

Again, ifAis a complex of soft sheaves, the previous theorem implies that it computes intersection cohomology via its global section coho- mology, i.e . H(G(X;A))I_pH(X;R).

Since all axioms are «local» to prove that a complex calculates inter- section cohomology is enough to do some local computations; however, absolutely no hint is given about the isomorphism between the cohomology modules.

3. The complex of intersection differential forms.

Every submersionp:M !N between smooth manifold induces the following well-known filtration:

V_k

0tanM V_k

1tanM . . . .V_k

dimM dimN tanM ˆV_k

tanM

whereV_k

ptanM:ˆL

jˆ0...pV_j

(kerdp)V_{k j}

(ptanN); if themapp is represented locally as a projection (x₁;. . .;x_n;t₁;. . .;t_{m n})7 !(x₁;. . .x_n) then for every pointa2M wehave(V_k

ptanM)_aˆspan_jIjpfdx_J^dt_I ; Using thecontraction operator between a vector field and a differ- ential form one obtains a coordinate-free description of the subbundle V_k

ptanM as follow:

V_k

ptanM ˆn v2V_k

tanMfor ally0;y1;. . .;yp2sect(kerdp) y0y1. . .ypvˆ0

o

Note that the tangent bundle of each submanifold-fiberp ¹(y) is exactly therestriction (kerdp)j_p 1(y), thus to discriminatesmooth forms inM one looks in local trivialization of thesubmersionpand counts how much terms dt directed along the fibers are present in wedge product. Thus if one considers the fibers ofpasparticular directions (ifR>Sarestrata of a pseudomanifold then by condition [1.1-C4] the map pj:T_S\R!S is submersive and its fibers can be thought as a description of how the stratum RapproachesS) one is lead to the following definition:

DEFINITION3.1 (Perversity condition). Let p:M !N be a submer- sion between two smooth manifolds;a smooth j-formv2V^j[M]overM has perversity p2f0;. . .;dimM dimNg with respect topif for every (p‡1)-uple of smooth vector fieldsy₀,y₁;. . .;yp2J[M]tangent to fibers of p(i.e. smooth sections of kerdp) we have:

y₀y₁. . .y_pvˆ0:

For example consider the projectionp(x;t)ˆxas in figure 3; every 0- perverse form with respect topis of theformf(x;t)dx. Heuristically, ifc is an e mbe dde d smooth 1-chain the n a 0-pe rve rse form can only de te ct

horizontal infinitesimal elements of c (i.e. elements not parallel to the fibers ofp):

Now let (X;S;f(T_S;p_S;r_S)g_S2S) be ann-dimensionalcontrolled pseu- domanifold as in definition [1.1] and pa fixed perversity for X; by hy- pothesis [1.1-B2], the setX Xn 2ˆXregis dense inX and this implies by [1.1-B1] that therelationSXregis true for everyS2S. Then, again by definition [1.1-C4], the map p_Sj:T_S\X_reg !S is a smoothsubmersion and this heuristically justifies the following:

DEFINITION 3.2 (Perverse and Intersection differential forms; [BRY]

def. 1.2.5, [BHS] chap. B). Let (X;S;f(T_S;p_S;r_S)g_S2S)be a controlled pseudomanifold of dimension n, p a fixed perversity for X and let j2 f0;. . .;ng;a smooth differential j-formv2V^j‰X_regŠis calledp-perverse forX iff 8S2S and8x2S there exists an open neighborhood U for x in TS such that v has perversity p_(codS) with respect to the submersion p_Sj:U\Xreg !S (see definition[3.1]). TheR-vector space of the intersec- tion j-forms overX relative to perversitypis defined as:

I_pV^j[X]:ˆn

v2V^j‰X_regŠbothvanddvare p-perverse forXo : Fig. 3. ± Perversity condition.

Fig. 4. ± Working in theregular part.

Figure4 shows what is happening in thesingular torus of figures 1 and 2; we work only in theregular part using theadditional data of thecontrol structure. The restrictions of the retractions p_S0j_TS

0\Xreg and p_S1j_TS

1\Xreg

originates thedirectionstowardS₀andS₁that aresmooth submanifold by submersiveness (sinceS₀is a point such direction is a whole neighborhood, and forS1they are represented by immersed lines, the «normal directions»

toward S1). Thus the geometric discrimination of definition [3.2] can be applied as in figure 3.

Since definition [3.2] is purely local inX and each openU ofX cano- nically inherits a controlled pseudomanifold structure we are lead to the following:

DEFINITION 3.3 (Intersection differential forms complex). The com- plex of sheaves of intersection forms of perversity p on the controlled pseudomanifoldX is the subcomplex ofaV_Xreg (a:X_reg,!X) defined by the assignment U7!I_pV[U]for every U open inM.

REMARK3.4. We are working with special differential forms defined over the regular part of the pseudomanifold and not with stratified forms (i.e. a smooth form on every stratum); see in particular [PFL] pp. 68-69 and the paper [BF] for a complete analysis of the latter approach.

REMARK3.5 (Invariance). The complex IpV_X is invariant under con- trolled isomorphisms [1.7] of X dueto thefact that such maps aredif- feomorphism over the regular partX_regand preserves tube retraction and fibers; if a smooth manifoldM is controlled trivially then I_pV[M] clearly coincides with the de Rham complexV[M]. Theorem [7.1] and theorem [2.4] implies that for every pseudomanifoldX thecohomology of IpV[X] is naturally isomorphic to the intersection cohomology and hence it is a topological invariant ofX (see [BOR]), i.e. it depends only on the under- lying topological structureand not on stratification or control data. This is highly not trivial even for a manifold since it can be unnaturally stratified in a very complex way.

4. The softness of the complexI_pV_X.

The complex of smooth differential forms V_M over a smooth mani- fold is themain exampleof a complex of soft sheaves; this is usually proved using the fact that every sheaf of modules over asoft sheaf of

rings with unitsis soft, thatV_M is aC¹_M-moduleand thatC¹_M is a soft sheaf; the latter requires the existence of smooth partitions of unity and in order to mimic thesameprocedurein singular spaces weneed ad hoc partitions.

Over a controlled pseudomanifold X;S;f(T_S;p_S;r_S)g_S2S

onecan work withreal controlled function(see [PFL]):

C^cnt_X :ˆ

f:X !R fis continuous,fj_Sis smooth for eachS2S and there exists an open neighborhoodUforS:8x2T_S\U;fp_S(x)ˆf(x)

Thus a controlled function must be constant along the retraction-fibers of every germ of tubular neighborhood; dealing with germs is fundamental to prove theorem [4.1].

To show that IpV_X is acomplex ofC^cnt_X -modulesIt is an open setUin X, an intersectionj-formv2I_pV^j[U] and a controlled functionf2C^cnt_U ;v can be seen as a particular smooth form in the differential manifold U\X_reg and thus multiplied by the smooth functionfj_U\Xreg: it is easy to show that fv:ˆfj_U\Xregv is indeed p-perverse for U (it respects all p- perversity condition even without any hypothesis about f). The exterior derivative d(fv):ˆd(fj_U\Xregv)ˆd(fj_U\Xreg)^v‡fj_U\Xregdv is p-per- verse due to the fact that the latter is true by definition for dvand that by controlledness d(fj_U\Xreg)ˆd(fj_U\Xreg)dp_Sfor every stratumSinU; thus yd(fj_U\Xreg)^v is automatically zero for every y vector field over U\Xregand tangent topS-fibers. This implies thatfv2IpV^j[U] and, since the restriction maps of such sheaves are indeed restrictions of forms and function, I_pV_X is aC^cnt_X -module; to continue, we recall the following result dueto Verona ([VER] pp.8-9):

THEOREM4.1. LetX be a controlled pseudomanifold;then every open cover of X has a subordinated partition of unit composed of controlled functions.

Let K bea compact ofX andf2C^cnt_X [K]; by metrizabilityofX the sectionfoverKcan be extended to a controlled functionef2C^cnt_X [U] over an open set UK. So if fc_U;c_{X K}g is a controlled partition of unit sub- ordinated to the open coverfU;X Kgthecontrolled functionefc_U is a well-defined element ofC^cnt_X [X] extendingf. By this wecan concludethat:

COROLLARY4.2. The sheafC^cnt_X of controlled functions is soft.

COROLLARY4.3 (Softness). IfX is a controlled pseudomanifold andp is a perversity then the complexIpV_X is a complex of soft sheaves.

The softness of the complex of intersection forms allows to make local computation required in Deligne's axioms without explicit reference to total derived functors (in particular without using injective or flabby-type re- solutions).

5. Intersection homotopy operator.

In order to accomplish easily computations over cylinders and cones one needs to develop a Poincare operator for singular case; let (X;S;f(T_S;p_S;r_S)g_S2S) bea controlled pseudomanifold of dimension n and an open segment Iˆ]a;b[ where a generic point t₀2I is fixed; we denote the t₀-section and projection respectively with s:X !XIand p:X I!X.

A perversitypassociated to the canonical filtrationfX_jg_jˆ0...ninduces a perversity forX Iwith respect tofX_jIg_jˆ0...n:

XnI (Xn 1IˆXn 2I) Xn 3I . . . . Xn kI . . . . X0I

Xn (Xn 1ˆXn 2) Xn 3 . . . . Xn k . . . . X0

p(0) p(1)ˆp(2) p(3) . . . . p(k) . . . . p(n)

Thelatter is dueto thefact thatX I, decomposed canonically as in [1.5], has no 0-dimensional stratum, so it is no use to assign p_(n‡1) for codimension n‡1 (just recall that codimensions of Xn k inX and of Xn kI in X I are the same); such argument can be inverted and I can be easily replaced with any boundaryless smooth manifold (e.g R^l).

Themorphismssepinducetwopullbacksover the complex of differ- ential forms defined over the regular part:

V_Xreg ^p! V_XregI ^s! V_Xreg;

next goal is the extension of these morphisms to the complex of intersection forms:

LEMMA5.1. The pullback operations for smooth forms over the regular part ofX andX I are compatible with every perversitypand induce

two pullbacks on the intersection forms complex:

p:I_pV_X !I_pV_XI; s:IpV_XI !IpV_X:

PROOF. By construction, eachk-codimension stratumSofX with tube p_S:T_S!Scorresponds to thek-codimension stratumSIofX Iwith tubep_Sid_I:T_SI!SI. So, after choosing a point (x;t)2SIand an open neighborhoodUJT_SIfor (x;t) the only vector fields over the regular part, parallel to fibers ofpSidI, arethefollowing:

(y;0):(UJ)\(XregI)!tan(U\Xreg)tan I

where the field y:U\X_reg !tan(U\X_reg) is parallel to fibers of p_Sj_U\Xreg; in other words,every vector in a smooth point of the composed tube must have a component tangent to the fiber of original tube and the other must be zero.

Supposenow that v2IpV^j[X]; v is p-perverse and if (y0;0);

(y₁;0);. . .;(y_p(codSI);0) are vector fields parallel to fibers of the tubeT_SI of a stratum SI3(x;t) evaluating inner products and the latter ob- servation gives:

(pv)_[(x;t)]((y₀(x);0);. . .;(y_p(codSI)(x);0); )ˆv_[x](y₀(x);. . .;y_p(codSI)(x); )ˆ0:

In thesameway, ifv2I_pV^j[X I] andy₀;y₁;. . .;y_p(codS)are vector fields parallel to fibers of the tubeTSfor a stratumS3xthen:

(sv)_[x](y₀(x);. . .;y_p(codS)(x); )ˆv_[(x;t0)]((y₀(x);0);. . .;(y_p(codS)(x);0); )ˆ0 Since by hypothesis even dv is p-perverse one obtains dsˆsd and dpˆpd; the same reasoning can be done for the exterior derivatives and consequentlysandpare well defined over intersection forms. p It is possibleto definean integration operatorover vertical fibers of X I, following mutatis mutandis thestandard construction over smooth manifolds; we begin to work with smooth forms defined overXregI, the regular part of the pseudomanifoldX I(recall thatIis anopen segment in R and t0 a chosen base point). So if v2V^j‰XregIŠ one defines u^jv2V^{j 1}‰X_regIŠas:

(u^jv)_[x;t](y1;. . .;yj 1):ˆ

[to;t]

v[x;t] @

@t;y1;. . .;yj 1

dt

for every point (x;t)2XregI and every (j 1)-uple of tangent vectors y1;. . .;yj 12tanxXregtantI (as usual u⁰ is by definition zero over 0- form) where_@t^@ is the standard tangent vector ofIR. This is thehomotopy operatorof Poincare and the following result is also a classic one (see [BT]

pp. 33-35), adapted in this case to the smooth manifoldX_regI:

THEOREM5.2. The following homotopic formula holds for every dif- ferential j-formvoverXregI, the regular part ofXI:

du^jv u^j‡1dvˆ( 1)^j (sp)v v :

In what follows, we describe how the latter operator naturally extends to intersection forms;

THEOREM 5.3 (Intersection homotopy operator). The Poincare homo- topy operatoruis compatible withp, i.e. the following relation continues to hold overI_pV^j[XI]for each intersection j-formv:

du^jv u^j‡1dvˆ( 1)^j (sp)v v

;

in particular,urealizes an algebraic homotopy between the endomorph- isms(sp)andid_IpV[XI]inI_pV[XI]:

PROOF. Since every intersection form is a differential form over the regular part (recall that IpV_XIaV_XregI), by [5.2] it suffices to show that u^j respects perversity; we proceed with a decreasing induction starting fromn‡1 (thedimension ofXI):

It must beshown that u^j I_pV^j[X I]

I_pV^{j 1}[XI]; supposethen v2IpV^j[X I] and consequently thatvisp-perverse. It is easily seen that

u^jvisp-perverse (just recall how inner product works; in a simpler way note that if v was f(x;t;l)dxj1^. . .^dxja^dti1^. . .^dtib^dl in R^a‡bR then u^a‡b‡1vwould be „

f(x;t;L)dL

dx_j1^. . .^dx_ja^dt_i1^. . .^dt_ib: integration cannot increase perversity).

Induction is needed to show that du^jvisp-perverse and one can proceed as follow:

± ifjˆn‡1 then the smooth homotopy relation gives du(n‡1)vˆ

ˆ( 1)^n‡1 (sp)v v

; the inductive start point is settled noting that (sp) respects the perversity ofvsinced(sp)_[x;t](v;l)ˆ(v;0) andvisp-perverse;

± ifj<n‡1 then dvisp-perverse and by inductive hypothesisu^j‡1 respects perversity; smooth homotopy formula allows to conclude.

This ends the proof of the theorem.

6. Local computations.

In this section we perform some basic computations on cones and cy- linders built from a given pseudomanifold (which can be found also in the paper [BHS], Chapitre C: Calculs locaux, page 230); if X is a controlled pseudomanifold of dimensionnby local structure theorem [1:8] every point xbelonging to ak-dimensional stratumSofX admits an open neighbor- hoodUinX isomorphic (as controlled space, [1.7]) toR^{n k}conL (where L is a compact pseudomanifold of dimension k 1) such that thepair (U\S;x) is mapped to (R^{n k};vertex). Since as noted in remark [3.5] the complex of intersection forms is invariant under controlled isomorphism this will suffice.

The following theorem is well-known in the smooth case ([BT] p. 35) and its proof in the singular case proceeds along the same line:

THEOREM6.1 (Cylinders).LetX be a controlled pseudomanifold andp an associated perversity;then the pullback s:IpV[X I]!IpV[X] induces an isomorphism for each cohomologicR-vector space:

H^j(IpV[XI]) !^ H^j(IpV[X]) 8j2 f0;. . .;dimXg:

PROOF. As in thesmooth casepsˆidX and consequently spˆ

ˆidIpV[X]; moreover, by theorem [5.3],u realizes an algebraic homotopy betweenps and id_IpV[XI]. Thussis a quasi-isomorphism between the

two given complexes. p

Roughly speaking, theorem [6.1] allows to represent a cohomology class of thecylinderX Ias a class of oneof its horizontal sliceXftg; it will be useful in theorem [6.4] tomoveclasses between different slices.

To deal with the coneconL is essential to recall that, as described in [1.6], its control structure makes heavy use of the control over L ]0;‡1[ and that:

the vertex is the only stratum of conL with max codimension (i.e.

dimL ‡1) and the corresponding retraction has only one fiber, namely T_vert:ˆL [0;e[.

Note that every perversity p over a cone conL (stratified and con- trolled via its base) naturally induces a perversity p over its cylinder L ]0;‡1[ and its base L (just discard the perversity for the vertex p_(dimL‡1)ˆp(cod vertex)); thecontrary is obviously false.

The crucial ingredient is the following immediately proved lemma:

LEMMA 6.2. Let M be a smooth manifold, v2V^j‰MŠ a differential form and fix an integer lj;then if the inner product ofvwith any l-uple of vector fields overM is always zero we can conclude thatvvanished. In other words:

(8y₁;. . .;y_l2J[M] y₁. . .y_lvˆ0)ˆ) vˆ0:

REMARK6.3. It is worthy to remark that if we insert via inner product l>jfields into aj-form we automatically obtain zero by definition, so the latter relation doesn't give any hint about the nullity of the original form.

As in the paper [BOR] the key step to apply the axiomatization of Deligne is to understand the cohomological relation between a cone and its baseas follow:

THEOREM6.4 (Cones). LetL be a compact controlled pseudomanifold of dimension k;endow the cone conL with the canonical control structure from its baseL as in‰1:6Šand fix an associated perversitypfor conL. Then the morphism sinduces the following isomorphisms:

H^j(I_pV[conL]) H^j(I_pV[L]) if jp_(k‡1);

0 if j>p_(k‡1):

PROOF. Using the previous remarks and lemma [6.2] we obtain the following relation between intersection forms over the cone conL and forms over the associated cylinderL _regR^>0:

I_pV^j[conL ]ˆ v2V^j‰L _regR^>0Š 8<

:

v2IpV^j[L R^>0] and:

ifj>p_(k‡1)then9e>0:vj_LregŠ0;e‰ˆ0 ifjˆp_(k‡1)then9e>0:dvj_LregŠ0;e‰ˆ0

9>

=

>; To provethis, webegin to notethatvand dvarein particularp-perverse

forL R^>0(i.e.v2IpV^j[L R^>0]) sincethecontrol structureof thecone

is an extension of the one of its cylinder (examples [1.5] and [1.6]); this is not enough, since there are the perversity conditions over the tube for the vertex of the coneT_vert:ˆL [0;e[ withe0.

This means that for every (p_(k‡1)‡1)-uple of vector fieldsy₀;y₁;. . .;y_p(k‡1) defined in Tvert\L_reg]0;e[ and tangent to fibers of pvert thefollowing relations must be satisfied:

y₀y₁. . .y_p(k‡1)vˆ0;

y₀y₁. . .y_p(k‡1)dvˆ0:

Next, recall that sincethefiber ofp_vertis by construction theregular part of thewholeretracting neighborhood (seetheremark in [1.6] and figure 5), the latter conditions must be verified foreveryvector field; now lemma [6.2] allows to conclude: ifvj_LregŠ0;e‰is a smoothj-form that becomes zero after inserting p_(k‡1)‡1 vector fields over L _regŠ0;e‰ and jp_(k‡1)‡1 thenvj_LregŠ0;e‰ˆ0 and consequently dvj_LregŠ0;e‰ˆ0. On thecontrary, one has just to imposea condition over dvonly ifj‡1ˆp_(k‡1)‡1, requiring that dvj_LregŠ0;e‰ˆ0; this is enough taking into account remark [6.3].

Concluding, one is able to deal with forms over a cone just working over its cylinder plus some nullity condition nearby the vertex since the vertex is theonlyk‡1-codimensional stratum and its associated retraction fiber is

Fig. 5. ± The control near the vertex.

thewholeneighborhood Tvert\L _regR^>0: every vector field approaches thesingularity.

We now use this characterization to prove the theorem, analyzing se- parately the two cases:

± thej>p_(k‡1)case±

Let v2IpV^j[conL ] a cocycle; by thelatter, thefollowing relations holds:

v2I_pV^j[L R^>0]

vˆ0 over L_regŠ0;e‰ (for somee>0)

Sovis just a closed intersection form over the cylinder, zero nearby the vertex; we are going to show thatv is indeed a coboundary moving the cohomology class via theorem [6.1] nearby the vertex, where some nullity condition holds. For this purpose, choose a pointa₀2]0;e[ and denote with s,panduthea0-section, projection and homotopyoperator related toa0

andL R^>0as in section [5]. The homotopy relation du^jvˆ( 1)^(j‡1)vis

truein I_pV^j[L R^>0] dueto thefact thatvis a cocycleandpsvˆ0 since vcan be pushed in a slice where it is zero; thus to obtain a primitive forvin I_pV[conL] it is enough to show thatu^jv2I_pV^{j 1}[conL]. By construction u^jv «lives» over the cylinder, but it is actually an intersection form be- longing to theconedueto thefact that j>p_(k‡1) and so vˆ0 over L_regŠ0;e‰. Finally u^jv is a primitive of our cocycle that becomes a co- boundary, trivializing theassociatecohomology:

H^j(IpV[conL])0 8j>p_(k‡1):

± thejp_(k‡1)case±

Again, using the initial characterization, we get:

IpV^j[conL ]ˆIpV^j[L R^>0] if j<p_(k‡1)

I_pV^j[conL ]\kerneldˆI_pV^j[L R^>0]\kerneld if jˆp_(k‡1) (

The former relation is trivial and to achieve the latter it is enough to observe that if jˆp_(k‡1) the compatibility between v and vertex tube retraction requires the vanishing of its exterior derivative nearby; this is trivially satisfied if dvˆ0. Since only closed forms are needed to compute coho- mology up to leveljp_(k‡1) the vertex gives no additional constraint and thecylindric computation [6.1] allows to conclude:

H^j(I_pV[conL ])ˆH^j(I_pV[L R^>0]) ^!H^j(I_pV[L]) 8jp_(k‡1):

p

Note that cohomology done with intersection forms is not homotopy invariant; for example, consider the contractible spacecon(S¹S¹) and fix a perversitypsuch thatp(3)ˆ1 (this is enough sincesuch spacehas only a singular point with codimension 3, the vertex). Theorem [6.4] im- plies that:

H¹ IpV[con(S¹S¹)]

H¹ IpV[S¹S¹]

H¹_DR S¹S¹

R¹R¹

7. Verification of Deligne's axioms.

Wearenow ready to show that I_pV_X satisfies Deligne's Axioms ([2.2]) and then by theorem [2.4] it has the same local cohomological properties of IpC_X; this will follow the results of the previous sections about thelocal structureof a pseudomanifold ([1.8]),softnessof IpV_X ([4.3]) andcylindric- conic computations([6.1], [6.4]).

THEOREM 7.1. LetX be a controlled pseudomanifold and p be an associated perversity;then the complex of sheaves IpV_X of intersection forms satisfies Deligne's axioms relative to perversitypand constant sheaf R_Xreg.

PROOF. Le t nbethedimension ofX; boundedness and triviality in negative degree are obvious by definition; the first axiom follows easily from thefact thatX_regis a smoothn-manifold and so that an intersection form is just a differential form on Xreg; Poincare Lemma allows to con- clude.

To simplify the remaining verifications, we setS_j:ˆS

fS2Sjdim Sˆjg and we use the notation introduced in section [2]; it is necessary to show that:

8k;8x2S_{n k} H ^j…I_pV_X†_xˆ0 if j>p(k) H ^j…IpV_Uk‡1†_x !H ^j…Ri_kIpV_Uk†_x if jp(k):

8<

:

Taking into account theremark [2.3] and thesoftnessresult of [4.3] for the complex IpV_X thediagramH ^j…IpV_Uk‡1†_x !H ^j…RikIpV_Uk†_xcan bere- placed by the following equivalent one:

lim_!

x2VX

H^j…I_pV[V]† ! lim_!

x2VX

H^j…I_pV[V S_{n k}]†

where the direct limit can be done over the directed set oftrivializing open neighborhoodsfor X over xand the arrow is simply the restric- tion map; this simple observation allows to avoid any explicit reference to total derived functors and consequently to apply theorems [6.1] and [6.4] with ease. Let V bea trivializing neighborhood of x and let V ! R^{n k}conL beacontrolled isomorphism(by thelocal structure theorem [1.8]), whereL denotes the controlled compactlinkforS_{n k}in x(S_{n k} has dimension k 1); wethereforecan form thefollowing com- mutativediagram:

(we need here to know that any perversity works withcodimension, and this implies that a p relative to R^{n k}conL induces a perversity for conL). Thecommutativity of thediagram is implied by thefact that ver- tical arrows are restrictions and the remaining ones are induced by inclu- sions and everyhorizontal arrowis a quasi-isomorphism of complexes ofR- vector spaces due to the computation over cylinders; the following rea- soning allows to conclude:

± ifj>p(k) then the first row gives the result after a direct limit;

± if jp(k) then the dashed arrow is a quasi-isomorphism of R- modules complexes till orderp(k) (by theorem [6.4]) and we deduce that the double arrowis a quasi-isomorphism up to levelp(k).

Notethat theisomorphism [2.4] is obtained in an abstract way and ab- solutely no hint is given on its concrete definition; however in the paper [BHS] pp. 223-224 a procedure to perform integration of perverse forms over perverse chains (with some additional conditions) is discussed in detail.

8. Mayer-Vietoris sequence and examples.

Here we collect some lemmas that help in computation of intersection cohomology via differential forms.

DEFINITION 8.1 (de Rham Intersection Cohomology). Let X be a controlled pseudomanifold andpa perversity forX;the j-th module of de Rham Intersection Cohomology ofX relative topis:

I_pH^j_DR(X):ˆH^j…X;I_pV_X† ˆH^jRG(X;I_pV_X)ˆH^j(I_pV[X]):

To identify the 0-th cohomology module note that an intersection 0-form f2I_pV⁰[X] is a smooth map defined overX_regsuch that its differential df is ap-perverse 1-form; iff is a cocyclethen df ˆ0 and thus I_pH⁰_DR X

is thevector spaceof thelocally constant mapsdefined inX_reg:

LEMMA 8.2. LetX be a controlled pseudomanifold andpa fixed per- versity;then the 0-th module of the de Rham intersection cohomology computes the number of connected components of the regular part ofX, i.e.:

I_pH⁰_DR X

Y

j2J

R withJ:ˆ fconnected components of X_regg

THEOREM8.3 (Mayer-Vetoris sequence). LetX be a controlled pseu- domanifold,pa perversity and U;V open subsets ofX such thatX ˆU[V;

then the following diagram of complex ofR-vector spaces is exact:

0 !I_pV[X] !^a I_pV[U]I_pV[V] !^b I_pV[U\V] !0 wherea:ˆhres_X;U

res_X;V

iandb:ˆ[resU;U\V resV;U\V]are induced by the re- striction morphisms.

PROOF ([BT] pp. 22-23). The exactness at first two nodes is easily proved as in the smooth case; to check the surjectivity ofbone can proceed as follow: letv2IpV^j[U\V] and using theorem [4.1] choose acontrolled partition of unityfc_U;c_Vgsubordinated to the open coverfU;VgofX. As shown in section [4], by controlledness, the forms (c_V)j_U\Vvand (c_U)j_U\Vv belong to I_pV^j[U\V] and as in the smooth case they can be extended by zero to elementsv_Uandv_Vof I_pV^j[U] and I_pV^j[V] respectively (note that to obtain a form inUonemust multiply forc_Vand viceversa). It is clear that bmaps (vU; vV)2IpV^j[U]IpV^j[V] tov.

For a non trivial example of computations consider the topological space S¹P

T³ where T³ˆS¹S¹S¹ is the3-torus and P T³:ˆ

:ˆ_{<f 1gT}^{[ 1;1]T}3;f1gT^{3 3}>is thesuspension ofT³ (see also [BOR] pp. 35-39 for a

similar computation using intersection homology); such space is a con-

trolled pseudomanifold since S¹ and T³ aresmooth manifolds trivially controlled and a suspension can be controlled doubling the cone con- struction. Thesingular part is thedisjoint union of twoS¹and it is enough to fix ap2f0;1;2gto assign a perversity relative to the 1-strata; by the- orems [6.1], [6.4] and remark [3.5] the following relations hold:

IpH_DR ] 1;1[T³

IpH_DR T³

ˆH_DR T³

 R;R³;R³;R;0;0;. . . I₀H_DR conT³

 R;0;0;0;0;. . . I1H_DR conT³

 R;R³;0;0;0;. . . I2H_DR conT³

 R;R³;R³;0;0;. . .

Now we can use Mayer Vietoris to compute for example I2H_DR P T³ as follow: consider P

T³ as con^‡T³[con T³ where con^‡T³:ˆ_<f1gT^{] 1;1]T}3>³, con T³:ˆ_<f^{[ 1;1[T}_1gT3³> and con^‡T³\con T³ˆ] 1;1[T³; by Mayer- Vietoris theorem the following diagram is exact:

A simple dimensional count is not enough to identify the cohomology of PT³, however in this case every double arrow)is clearly anepimorph- ism: in fact themap I2H^k_DR con^‡T³

!I2H^k_DR ] 1;1[T³

is an iso- morphism ifk2f0;1;2gas shown in the proof of theorem [6.4] and hence by construction of the Mayer-Vietoris sequence from [8.3] the maps ) are epimorphism. Thus by exactness the dimension of every unknown module can be computed and a similar reasoning can be made for the other per- versity giving the following results:

I0H_DR P T³

 R;0;R³;R³;R;0;0;. . . I1H_DR P

T³

 R;R³;0;R³;R;0;0;. . . I2H_DR P

T³

 R;R³;R³;0;R;0;0;. . .