Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata

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1 Toology and its Alications 134 (2003) Stratified fibrations and the intersection homology of the regular neighborhoods of bottom strata Greg Friedman Yale University, Deartment of Mathematics, 10 Hillhouse Ave, P.O. Box , New Haven, CT 06520, USA Received 6 August 2002; received in revised form 5 March 2003 Abstract In this aer, we develo Leray Serre-tye sectral sequences to comute the intersection homology of the regular neighborhood and deleted regular neighborhood of the bottom stratum of a stratified PL-seudomanifold. The E 2 terms of the sectral sequences are given by the homology of the bottom stratum with a local coefficient system whose stalks consist of the intersection homology modules of the link of this stratum (or the cone on this link). In the course of this rogram, we establish the roerties of stratified fibrations over unfiltered base saces and of their maing cylinders. We also rove a folk theorem concerning the stratum-reserving homotoy invariance of intersection homology Elsevier B.V. All rights reserved. MSC: rimary 55N33, 55R20; secondary 57N80, 55T10 Keywords: Intersection homology; Sectral sequence; Regular neighborhood; Stratified sace; Stratified seudomanifold; Stratified fibration; Homotoy link (holink); Stratum-reserving homotoy equivalence 1. Introduction The urose of this aer is to rove the existence of a Leray Serre-tye sectral sequence for the intersection homology (with local coefficients) of the regular neighborhood (and deleted regular neighborhood) of the bottom stratum of a stratified PLseudomanifold. Although it is well known that the regular neighborhood of a sace X is homotoy equivalent to X, this homotoy equivalence will not necessarily reserve the stratification of a filtered sace and so it is not ossible to determine the intersection homology of the neighborhood from that of X alone as would be ossible for ordinary homology. address: friedman@math.yale.edu (G. Friedman) /$ see front matter 2003 Elsevier B.V. All rights reserved. doi: /s (03)

2 70 G. Friedman / Toology and its Alications 134 (2003) This reflects, of course, that intersection homology is a finer invariant than homology and designed to suit stratified henomena. On the other hand, the neighborhood does retain stratified versions of certain other essential roerties of regular neighborhoods. For instance, it is homotoy equivalent to the maing cylinder of a fibration in a manner that reserves the essential roerties of the stratification. The articular fibration involved can be given in terms of the homotoy links of Quinn [17] and rovides a secial case of the stratified fibrations of Hughes [10], whose roerties we must first develo. In fact, we will show that any such regular neighborhood is stratum-reserving homotoy equivalent to the maing cylinder of such a stratified fibration and then emloy the stratum-reserving homotoy invariance of intersection homology. This last seems to be a well-known fact, but its roof aears to be unavailable in the literature and is thus resented below in full generality, including local coefficient systems. Since some of the saces involved in carrying out our rogram do not aear to be even toological seudomanifolds (in the sense of [7]), it is not clear that there exists a roof of our main theorem (at least along these lines) that utilizes the sheaf-theoretic machinery of intersection homology, so instead we use singular intersection homology as develoed in [12]; note that on a stratified seudomanifold, the two theories are isomorhic if we assume sheaf homology with comact suorts or work on a comact sace. Once we have established the roer stratified analogues of fibration theory and homotoy invariance of intersection homology, many of the subsequent details directly mirror the standard roof of the existence of sectral sequences for the homology of ordinary fibrations, though there are still several minor modification necessary in order to handle the nuances of intersection homology and local coefficients. Our main theorem then is the following; see below for further exlanations of the notations: Theorem 1.1 (Theorem 6.1). Let X be a finite-dimensional stratified seudomanifold with locally finite triangulation and filtration =X 1 X 0 X n = X such that X i = for i<k.letn = N(X k ) be an oen regular neighborhood of X k, and let L be the link of the stratum X k (if X k is not connected, then we can treat each comonent searately and each comonent will have its own link). Then, for any fixed erversity and local coefficient system G defined on X X n 2, there are homological-tye sectral sequences E,q r and Er,q that abut (u to isomorhism) to IHi (N X k; G) and IH i (N; G) with resective E 2 terms E,q 2 = H ( Xk ; IHq (L; G L) ), E,q 2 = H ( Xk ; IHq (cl; G cl) ) (cl = the oen cone on L),whereIHq (L; G L) and IHq (cl; G cl) are local coefficient systems with resective stalks IHq (L; G L) and IHq (cl; G cl) (see below). Furthermore, the ma i : IH i (N X k; G) IH i (N; G) induced by inclusion induces a ma of sectral sequences which on the the E 2 terms is determined by the coefficient homomorhism IHq (L; G L) IHq (cl; G cl) given by the ma on the stalk intersection homology modules induced by the inclusion L cl.

3 G. Friedman / Toology and its Alications 134 (2003) The stes to the roof of this theorem roceed as follows: In Section 2, we recall the definition of intersection homology on a filtered sace and show that intersection homology is a stratum-reserving-homotoyinvariant. In Section 3, we introduce a secial case of the stratified fibrations of Hughes [10] and examine some of its roerties. In articular, we show that these satisfy various stratum-reserving analogues of the standard roerties of fibrations. We also show, in Section 4, how a stratified fibration can determine a bundle of coefficients on its base sace with coefficient stalks given by the intersection homology of the stratified fiber. In Section 5, we resent some notations and facts from the works of Quinn [17,16] and Hughes [10] on manifold weakly stratified saces and homotoy links. Finally, in Section 6, we ut this all together to rove the theorem. We have also rovided Aendix A which contains a corrected version of the roof of the theorem (see [2,17]) that certain deformation retract neighborhoods are homotoy equivalent to the maing cylinders of certain ath saces. The sectral sequences develoed here can be viewed as fundamental tools for the comutation of intersection homology modules, not just for regular neighborhoods but for entire saces. Couled with the intersection homology analogue of the Mayer Vietoris theorem, these sectral sequences may be utilized to carry out comutations by working u through a stratified seudomanifold one stratum at a time. We refer the reader to [4] for an examle of such techniques as alied to study the roerties of intersection Alexander olynomials of PL-knots which are not locally-flat. For references to intersection homology theory, the reader is advised to consult [6,7,12, 3]. 2. Stratum-reserving homotoy invariance of intersection homology In this section, we review some of the relevant definitions of intersection homology on a filtered sace (note that we do not yet assume the saces to be stratified in the sense of seudomanifolds with normally locally trivial strata (e.g., [7])). Then we fill in some of the details, which are left to the reader in [16], on the invariance of intersection homology under stratum-reserving homotoy equivalence. We also show how this works for intersection homology with local coefficients. We first recall some standard definitions. As defined in [6], a (traditional) erversity is a sequence of integers { 0, 1, 2,...} such that i i+1 i + 1 and such that 0 = 1 = 2 = 0 (much of the following would hold using more general erversities (see [12,1,5,4]), but for the other common case, in which 2 = 1 (suererversities), it is not clear that intersection homology with local coefficients can be well defined in a geometric manner, i.e., without sheaves; hence we omit discussion). A filtered sace is a sace, X, together with a collection of closed subsaces =X 1 X 0 X 1 X n 1 X n = X. If we want to emhasize both the sace and the filtration, we will refer to the filtered sace (X, {X i }). Note that X i = X i+1 is ossible. We will refer to n as the (stratified) dimension of X and X n k as the n k skeleton or the codimension k skeleton. ThesetsX i X i 1

4 72 G. Friedman / Toology and its Alications 134 (2003) are the strata of X. We call a sace either unfiltered or unstratified if we do not wish to consider any filtration on it. We can now define the singular intersection homology of X for a erversity, IH (X), in the usual manner: as the homology of the chain comlex IC (X), which is the submodule generated by the -allowable chains of the singular chain comlex, C (X). A singular i-simlex σ : i X is -allowable if σ 1 (X n k X n k 1 ) is contained in the i k + (k) skeleton of the (olyhedral) simlex i (with the usual filtration by olyhedral skeletons), and an i-chain is -allowable if it and its boundary (in the chain comlex sense) are linear combinations of -allowable i-simlices and i 1 simlices, resectively. Similarly, we can define the intersection homology with coefficients in a grou or with local coefficients (see below for more details). If X is a stratified PLseudomanifold with filtration determined by the stratification, then this definition of intersection homology is the standard one (with comact suorts, in the sheaf language) and is a toological invariant [12]. Note, however, that for a general filtered sace, X, the intersection homology may not be a toological invariant, e.g., it may deend on our choice of filtration. Since the intersection homology definitions deend only on the codimensions of the strata, we see that we can reindex a filtration by addition of a fixed integer, accomanied by the same change to the stratified dimension, without affecting the intersection homology of the sace. In what follows, we shall often defer from the norm by not reindexing in certain situations where it is standard to do so when working with stratified seudomanifolds. In articular, for simlicity we will usually give filtered subsaces, Y X, thesame dimension as X and filtration indexing Y i = Y X i, even though the common ractice for stratified seudomanifolds is to reindex to the simlicial dimensions of Y and Y i by subtracting the codimension of Y in X from each index. If X and Y are two filtered saces, we call a ma f : X Y stratum-reserving if the image of each comonent of a stratum of X lies in a stratum of Y (comare [17]). In general, it is not required that strata of X matostrataofy of the same (co)dimension, but unless otherwise noted we will assume that X and Y have the same stratified dimension and that f(x i X i 1 ) Y i Y i 1. We call f a stratum-reserving homotoy equivalence if there is a stratum-reserving ma g : Y X such that fg and gf are stratum-reserving homotoic to the identity (where the filtration of X I is given by the collection {X i I}). We will sometimes denote the stratum-reserving homotoy equivalence of X and Y by X s..h.e. Y and say that X and Y are stratum-reserving homotoy equivalent or s..h.e. A slightly more general notion which we will need below is the following [16]: Consider two filtered saces X and Y with a common subset U.Amaf : X Y of filtered saces is a stratum-reserving homotoy equivalence near U rovided: (1) f is stratum-reserving; (2) f U is the identity; (3) There exists a neighborhood V Y of U and a stratum-reserving ma g : V X such that fg and gf, where defined, are stratum-reserving homotoic rel U to the inclusions (i.e., fg is stratum-reserving homotoic rel U to the inclusion V Y and gf is stratum-reserving homotoic rel U to the inclusion of f 1 (V ) into X).

5 G. Friedman / Toology and its Alications 134 (2003) We now study in some detail the effect of a stratum-reserving homotoy equivalence on intersection homology. If X s..h.e. Y, then it follows that there must be a bijective corresondence between non-emty strata of X and non-emty strata of Y determined by the maing. We rove the following roosition which is imlied in [16]: Proosition 2.1. If f : X Y is a stratum-reserving homotoy equivalence with inverse g : Y X, X and Y have the same dimension n (as filtered saces), and f(x i X i 1 ) Y i Y i 1 and g(y i Y i 1 ) X i X i 1 for each i,thenf and g induce isomorhisms of intersection homology f : IH i (X) IHi (Y ) and g : IH i (Y ) IHi (X). Proof. Note that, since X and Y are the disjoint unions of their strata, the assumtions that f(x i X i 1 ) Y i Y i 1 and g(y i Y i 1 ) X i X i 1 for each i imly that f 1 (Y i Y i 1 ) = X i X i 1 for each non-emty stratum and similarly for g. We observe that f and g are well-defined mas on intersection homology: If σ : i X is a -allowable singular i-simlex of X, thenf σ = fσ: i Y is a singular i- simlex of Y. To see that it is allowable, we note that if Y n k Y n k 1 is a non-emty stratum of Y,then(f σ ) 1 (Y n k Y n k 1 ) = σ 1 f 1 (Y n k Y n k 1 ).Butf 1 (Y n k Y n k 1 ) = X n k X n k 1,so(f σ ) 1 (Y n k Y n k 1 ) = σ 1 (X n k X n k 1 ),which lies in the i k + (k) skeleton of i since σ is -allowable in X. Hence f induces a ma f : IC i (X) IC i (Y ) which clearly also commutes with boundary mas so that it takes cycles to cycles and boundaries to boundaries, inducing a well-defined ma on intersection homology. We next show that (fg) and (gf ) are isomorhisms, which will suffice to comlete the roof. By assumtion, there exists a stratum-reserving homotoy H : X I X such that, for x X, H(x,0) is the identity and H(x,1) = gf (x). Now consider a allowable i-cycle σ = j a jσ j in X, wherethea j are coefficients and the σ j are i- simlices. For the (olyhedral) i-simlex i,let[v 0,...,v i ] and [w 0,...,w i ] be the simlices i 0and i 1in i I.Then i I can be triangulated by the i + 1 simlices i+1,l =[v 0,...,v l,w l,...,w i ]. Now consider the singular i + 1 chain in X given by F = a j ( 1) l H (σ j id) σ jl, (1) j l where (σ j id) : i I X I is given by σ j on the first coordinate and the identity on the second and σ jl is the singular i + 1 simlex given by the inclusion i+1,l i I. Then, by a simle comutation (see [8, 2.1] for the details), F = (gf )σ σ. Thus (gf ) σ = σ in ordinary homology. To show that this holds in intersection homology, it remains only to check that F is a -allowable i + 1 chain. Since σ is -allowable by assumtion and gf σ is allowable as the image of σ under a stratum-reserving ma, F is -allowable, and it suffices to check that each H (σ j id) σ jl : i+1,l X is a -allowable singular simlex, i.e., that σ 1 jl (σ j id) 1 H 1 (X n k X n k+1 ) lies in the i + 1 k + (k) skeleton of i+1,l. We must consider the inverse image of each stratum X n k X n k 1. Since we filter X I by {X k I} and H is a stratum-reserving homotoy to the identity,

6 74 G. Friedman / Toology and its Alications 134 (2003) H 1 (X n k X n k 1 ) = (X I) n k (X I) n k 1 = (X n k X n k 1 ) I.Then (σ j id) 1 [(X n k X n k 1 ) I]=[σj 1 (X n k X n k 1 )] I.Sinceσ j is -allowable, this is contained in the roduct of I with the i k + (k) =: r skeleton of i (denoted i 1 r ). Finally, we show that σjl ( i r I) lies in the r + 1 skeleton of the (i + 1)-simlex i+1,l by construction. In fact, σjl 1 ( i r I) = ( i r I) [v 0,...,v l,w l,...,w i ].But i r I = [v α0,...,v αr ] I, where the union is taken over all sets of ordered r- tules of integers with 0 α 0 < <α r i. Furthermore, each [v α0,...,v αr ] I = [v α 0,...,v α,w α,...,w αr ]. That each [v α0,...,v αr ] I is a union of such simlices follows as for our original decomosition of i I, and each such simlex lies in the given triangulation of i I since each is clearly a face of one of the i+1,l. Therefore, any simlex in the intersection of i+1,l and i r I is sanned by a set of vertices which is a subset of both {v 0,...,v l,w l,...,w i } and one of the collections {v α0,...,v α,w α,...,w αr }. Thus, each such simlex must have dimension less than or equal to r + 1, and ( i r I) i+1,l must lie in the r + 1 skeleton of i+1,l. Hence H (σ j id) σ jl is a -allowable (i + 1)-simlex. This shows that (gf ) = g f is an isomorhism of intersection homology (in fact the identity). Similarly, (fg) = id,andsof and g must each be isomorhisms. This roof clearly extends to hold for any constant coefficient module. To show that this also works for local systems of coefficients, it is erhas simlest to work with an analogue of the following version of ordinary homology with local coefficients as resented in [8, 3.H] (note that Hatcher confines his definition to bundles of Abelian grous, but, at least for homology, there is no difficulty extending the definition to bundles of modules): Given a sace X with local coefficient system G with stalk module G and given action of the fundamental grou, we use this data to form a bundle of modules (or sheaf sace in the language of sheaves), which we also denote G. Then an element of the singular chain comlex C i (X; G) is a finite linear combination j n j σ j, where each σ j is a singular simlex σ j : i X and n j is a lift of σ j to G. Ifn j and m j are both lifts of the same singular simlex, then n j + m j is defined using the addition on the bundle, which is continuous. We also take n j + m j = 0 if it is the lift to the zero section. Similarly, we can define scalar multilication by elements r in the ground ring R of the bundle of (left) R-modules by defining n j rn j ointwise and noting the continuity of the scalar multilication oeration in a bundle (clearly similar definitions can be made for bundles of right R-modules). The coefficient of a boundary face of a simlex is given by ( j n j σ j ) = j,k ( 1)k n j σ j [v 0,...,ˆv k,...,v i ], where each n j on the right-hand side is the restricted lift n j [v 0,...,ˆv k,...,v i ]. With these definitions, the homology of the chain comlex C (X; G) is the usual homology of X with local coefficients G. Note that given a ma between saces f : X Y covered by a bundle ma of coefficient systems f : G H, thenf ( j n j σ j ) is defined as j ( f n j ) (f σ j ) and induces a ma on homology f : H (X; G) H (Y ; H). Before getting to intersection homology, let us first show the following: Proosition 2.2. Suose we are given homotoy equivalent (unfiltered) saces X and Y, homotoy inverses f : X Y and g : Y X, and a local coefficient system G over

7 G. Friedman / Toology and its Alications 134 (2003) X. Theng covered by the induced ma g : g G G induces a homology isomorhism g : H (Y ; g G) H (X; G). Proof. Consider the following diagram: g f g G g ḡ f g G f f g G g g Y X Y X where g, f,andḡ are the mas induced by the ullback constructions. We will first show that the ma gf covered by the ma g f induces an isomorhism (gf ) : H (X; f g G) H (X; G).Todoso,letH denote a homotoy H : X I X from the identity to gf.then we have an induced bundle H G over X I and a ma H : H G G such that, over X 0, this is simly the identity ma G G and, over X 1, it is the induced ma f g G G. To show the surjectivity of the homology ma (gf ) = g f induced by gf covered by g f,let[c] be an element of H i (X; G). Then, by our definitions, [C] is reresented by acyclec = n j σ j,wheretheσ j are singular simlices, i.e., mas of i into X, and the n j are lifts of the σ j to G. We can also then consider these simlices and their lifts as mas into X 0andH G X 0. Next, we can extend each simlex ma σ j to the ma σ j id : i I X I. Since bundles of coefficients are covering saces, there exist unique lifts, σ j id, of each of these mas extending the lift over X 0. Together these rovide a homotoy from C over X 0 to a new chain, say C = n j σ j,wherethese σ j are the same as those above identifying X and X 1, and the n j are given by the lifts σ j id i 1. Furthermore, C is a cycle: since C is a cycle, the sum of the lifts over each oint x 0 X 0of σ i lies in the zero section of the bundle, but, by the roerties of bundles of coefficients, the liftings of the aths x I X I collectively rovide a homomorhism from G x 0 G x 1. So the sum of the lifts over x 1 at the other end of the homotoy is also 0. Now, we can comose each of the σ j id covered by σ j id with H covered by H to obtain a homotoy into (X, G) from the mas reresenting C to the mas reresenting the image of C under gf covered by g f. By the rism construction emloyed above in the roof of Proosition 2.1 to break a homotoy into simlices (and the obvious extension to coefficient lifts), this gives a homology between [C] and the homology class of the image g f ( C). Hence (gf ) = g f is surjective. For injectivity, let C = n i σ i be a cycle reresenting an element of H i (X; f g G) whichmasto0inh i (X; G) under (gf ). As in the last aragrah, we can lift σ j id : i I X I to an extension of n i in H G (although this time we extend in the other direction). Again, this induces a chain C in (X 0, G), and comosing the homotoies and their lifts with H covered by H induces a homotoy and, by the rism construction and its lift, a homology from the image of C to C (recall that H and H are the identity on and over X 0, resectively). But since the image of C is0byassumtion,c must bound another chain with local coefficients, say D = m k τ k,wheretheτ k are singular i + 1 simlices in X and the m k are their lifts to G. Now, we roceed as in the last aragrah: we can consider D as a chain with coefficients in X 0 covered by G and lift the mas τ k id : i+1 I X I to mas into H G extending the lifts m k. Again this induces G (2)

8 76 G. Friedman / Toology and its Alications 134 (2003) achain D = m i τ i in (X 1,f g G), and, due to the unique ath lifting roerty of covering saces, it is readily verified that D = C. Thus, C reresents the 0 element of H i (X; f g G), and it follows that the homology ma (gf ) is injective. We have thus shown that the ma (gf ) = g f on homology induced by gf covered by g f is an isomorhism. By the same reasoning, the ma fgcovered by f ḡ (see diagram (2)) induces a homology isomorhism (f ḡ) = f ḡ : H (Y ; g f g G) H (Y ; g G). But together these imly that the homology ma induced by f covered by f is injective and surjective and hence an isomorhism H (X; f g G) H (Y ; g G). Finally, since f covered by f induces an isomorhism and gf covered by g f induces an isomorhism, g covered by g must induce an isomorhism. For intersection homology with local coefficients (and traditional erversities), we can aly the same ideas. As in the stratified seudomanifold case, we assume that the singular set has codimension at least two (i.e., X n 1 = X n 2 ) and that a local coefficient system G is given on X X n 2. We can also think of G as a covering sace over X X n 2. Now, given a -allowable i simlex, σ, it will no longer be ossible, in general, to lift the entire simlex, as σ may intersect X n 2. However, we can choose for coefficients lifts of σ (X X n 2 ), which consists of σ minus, at most, ieces of its i 2 skeleton. So, in articular, we can lift at least the interior of the simlex and the interior of any facet (i 1 face). The coefficient of the boundary σ can again be defined in terms of the signed (i.e., lus or minus) restrictions of the lift of the simlex to its boundary ieces, again with the limitation that some lower-dimensional ieces (and only lower-dimensional ieces) might be missing. In this manner, the definition of the local coefficient of a simlex by its lift is well-defined for -allowable intersection simlices, and it is further welldefined (modulo lower-dimensional skeleta) on all boundary ieces by taking restrictions multilied by ±1 since the allowability conditions ensure that the interior of each facet of an allowable simlex lies in the to stratum. Defining the chain comlex IC (X; G) as the -allowable chains formed by linear combinations of singular simlices with such lifts as coefficients yields a chain comlex whose homology is the intersection homology of X with coefficients in G. This aroach is clearly equivalent to the more common definition of intersection homology with local-coefficient which involves the lifts of given oints of the simlices; it has the disadvantage of the lift not being defined for all oints (though it is for enough!) and the advantage of not having to nitick about which oints and how to change oints under boundary mas. We should, however, note the following concerning the definitions of the last aragrah: When considering σ of a -allowable i-chain σ = n j σ j with local coefficients as defined above, a given boundary iece of a simlex σ j may not itself be allowable, just as is the case with constant coefficients. However, by restriction and multilication by ±1, it will have a coefficient lifting defined over its intersection with the to stratum of X, which includes at least its interior. But, again as in the constant coefficient case, all of the non- -allowable ieces must cancel in σ since each non-zero simlex of σ must be -allowable since σ was -allowable. In other words, the coefficients, where defined, over each oint of each of the non-allowable simlices in σ must add u to zero. Since the remaining boundary ieces are -allowable, their coefficient lifts are again defined, at the least, over their interiors and the interiors of their facets. Therefore, σ is well-

9 G. Friedman / Toology and its Alications 134 (2003) defined. It is then routine to check that σ = 0 just as for ordinary homology with or without coefficients, σ being equal to the boundary of what remains of σ after the ieces which cancel have been removed (i.e., set to zero). Alternatively, we could note that n i σ i is zero where the lift is defined and aly linearity (where it is not well-defined, it must be zero in σ anyway, by its non-allowability), or we could aeal directly to the equivalence of this definition of intersection chains with coefficients and the more standard definition involving lifts over well-chosen oints. Suose now that the filtered sace X is given a local coefficient system G, as above, and that f : X Y is a stratum-reserving homotoy equivalence with inverse g. Wealso again assume that X and Y have the same stratified dimension and that f and g take strata to strata of the same (co)dimension. Since f and g induce homotoy equivalences of X X n 2 and Y Y n 2 (again assuming X n 1 = X n 2 and Y n 1 = Y n 2 ), they also induce isomorhisms of the fundamental grous and determine local coefficients on Y via the ullback g. Furthermore, it is clear that f g G is bundle isomorhic to G. We want to show that g covered by g : g G G over the to stratum induces an isomorhism of intersection homology with local coefficients. But this now follows from a combination of the techniques from the roofs of Proositions 2.1 and 2.2. In articular, we simly aly the roof of Proosition 2.2, but all bundles and lifts are restricted to lie over the to stratum. Once again, we will not necessarily be able to lift entire simlices, but the homology theory works out as described in the last few aragrahs. Furthermore, since all of the homotoies in that roof can be taken to be stratum-reserving in this context, there is no trouble with extending lifts: any oint that is maed to a to stratum remains in the to stratum under all of the necessary homotoies and so its lift can be uniquely extended. Also, there is no difficulty showing that the homotoies induce homologies by breaking the homotoies of simlices u into triangulated risms. The allowability issues are taken care of just as in the roof of Proosition 2.1, and therefore each rism iece is allowable and its coefficient is determined over, at the least, its interior and the interiors of its facets as the restriction of the lift over the entire homotoy on the to stratum. Therefore, we have shown the following: Corollary 2.3. Suose that g : Y X is a stratum-reserving homotoy equivalence, G is a local coefficient system on X X n 2 (which is the to stratum), X and Y have the same dimension n (as filtered saces), and f(x i X i 1 ) Y i Y i 1 and g(y i Y i 1 ) X i X i 1 for each i. Then the intersection homology ma g : IH i (Y ; g G) IH i (X; G), induced by g covered over X X n 2 by the natural ma g G G, is an isomorhism. Although we have focused on stratum-reserving homotoy equivalences, which will be our rimary use of the above theory, these same arguments can be easily generalized to rove the following corollary. Corollary 2.4. If f 0,f 1 : X Y are stratum-reserving homotoic by a homotoy H, G is a local coefficient system over X X n 2, G is a local coefficient system over Y Y n 2, X and Y have the same stratified dimension, f 0 (X i X i 1 ) Y i Y i 1 (or, equivalently, f 1 (X i X i 1 ) Y i Y i 1 ) for each i, and f 0,f 1 are covered over X X n 2

10 78 G. Friedman / Toology and its Alications 134 (2003) by mas f 0, f 1 : G G which are homotoic over H (X Xn 2 ) I by a homotoy H,then f 0,f 1 : IH i (X; G) IHi (Y ; G ) are identical mas on intersection homology. Proof (Sketch). For simlicity, let us omit the direct mention of the fact that the covers only lie over the to stratum; it will be assumed. Once again, we can emloy a rism so that if σ : i X is a singular cycle with coefficient lift σ,thenh (σ id) covered by H ( σ id) rovides a homotoy between f 0 σ with lift f 0 σ and f 1 σ with lift f 1 σ. Hence, we can again emloy a rism as in the roof of Proosition 2.1 to show that f 0 and f 1 induce identical mas on homology with local coefficients. 3. Stratified fibrations In this section, we study a secial case of the notion of a stratified fibration as introduced by Hughes [10] (our definition will be less general than that of Hughes, which assumes that both of the saces in a stratified fibration are filtered, but it will suit our uroses). Definition 3.1 (Hughes, [10,. 355]). If Y is a filtered sace, Z and A unfiltered saces, amaf : Z A Y is stratum-reserving along A if, for each z Z, f(z A) lies in a single stratum of Y. Definition 3.2. Suose Y is a filtered sace and B is an unfiltered sace (equivalently a sace with one stratum). A ma : Y B is a stratified fibration if, given any sace Z and the commutative diagram 0 Z Z I f F Y B there exists a stratified solution which is stratum-reserving along I, i.e., a ma F : Z I Y such that F = F and, for each z Z, F(z,0) = f(z)and F(z I) lies in a single stratum of Y. We want to show that stratified fibrations satisfy stratified analogues of several of the standard roerties of fibrations, such as the existence of ullbacks and triviality over a contractible base. We obtain these results largely by reroducing the roofs for standard fibrations found in Whitehead [19], adding the relevant details where the stratifications come into lay. First, we resent the following two lemmas which are secial cases of [10, Lemmas 5.2 and 5.3] of Hughes and follow immediately from them: Lemma 3.3. Given two solutions, F and G, of a stratified lifting roblem (3), then there exists a homotoy H : Z I I Y from F to G rel Z {0} such that H(z,t,s)= F(z,t) and H is stratum-reserving along I I. (3)

11 G. Friedman / Toology and its Alications 134 (2003) Lemma 3.4 (Stratified relative lifting). Suose : Y B is a stratified fibration, B unstratified, (Z, A) an unstratified NDR-air, and we are given the commutative diagram: Z {0} A I Z I F f Y B. If there is a deformation retraction H t : Z I I Z I of Z I to Z {0} A I such that fh 1 : Z I Y is stratum-reserving along I, then there exists a stratified solution F : Z I Y which is stratum-reserving along I, satisfies F = F, and F Z {0} A I = f. (Note: Hughes assumes in his lemma that Z is a metric sace, but I do not believe this is ever used in the roof, at least suosing the definition of NDR as given in [19].) In fact, Hughes roof of Lemma 3.4 alies to the following slightly more general situation: Lemma 3.5. Suose : Y B is a stratified fibration, (Z I,A) is a DR-air, i : A Z I is the inclusion, and we are given the commutative diagram: i A Z I f F Y B. If there is a deformation retraction H t : Z I I Z of Z I to A such that fh 1 : Z I Y is stratum-reserving along I, then there exists a stratified solution F : Z I Y which is stratum-reserving along I, satisfies F = F, and F A = f. We now define the induced stratified fibration in analogy with induced fibrations. Definition 3.6. Suose : Y B is a stratified fibration and f : B B is a ma of saces. We denote by Y (or f Y if we want to emhasize the ma f )theset {(b,y) B Y f(b ) = (y)} and call it the induced stratified fibration or ullback stratified fibration. We filter Y as a subset of B Y, i.e., Y i ={(b,y) Y y Y i }. We obtain mas f : Y Y and : Y B induced by the rojections. Note that f is obviously stratum-reserving and takes the fiber over b B to the fiber over f(b ).If f is an inclusion, then so is f. Assuming this notation, we rove the following basic facts: Lemma 3.7 (Universality). If g : Z Y is a stratum-reserving ma and q : Z B is a ma such that g = fq, then there is a unique stratum reserving ma h : Z Y such that f h = g and h = q.

12 80 G. Friedman / Toology and its Alications 134 (2003) Proof. Z h g q Y f Y B f B Based on the hyotheses, q and g define a ma (q, g) : Z B Y which is stratumreserving with resect to the roduct filtration on B Y. Clearly, Im(q, y) Y, and thus (q, g) induces a stratum-reserving ma to Y. The commutativity relations are obvious. Uniqueness follows obviously from Y being a subset of Y B : the two rojections f and define the image oint uniquely in Y, but these are fixed by the hyotheses. Lemma 3.8. The induced ma : Y B is a stratified fibration. Proof. Z 0 g Y f Y G G Z I G B f B If g : Z Y and G : Z I B are the data for a stratified lifting roblem for the ma, then the mas fgand f g are the data for a stratified lifting roblem for, which has a stratified solution G by assumtion. Now, stratify Z I by (Z I) i = G 1 (Y i ).Note that if (z, t) (Z I) i for some z Z and t I,thenz I (Z I) i by the definition of a stratified lifting. With this stratification, G is clearly a stratum-reserving ma. But then G : Z I Y and G : Z I B satisfy fg= G, the hyothesis of the Universality Lemma. Hence there is a unique stratum-reserving ma G : Z I Y such that G = G and f G = G, and, since G Z {0}=f g and G Z 0 = g, G Z {0} Y is given by G(z {0}) = ( G(z 0), f G(z 0)) = ( g(z {0}), f g(z {0})) Y B Y. But since a oint in B Y is determined uniquely by its rojections, this is also g(z {0}), so G extends g. Finally, since G is stratum-reserving on the strata we have imosed on Z I, each z I, lying in a single stratum of Z I, must ma into a single stratum of Y under G. Hence, G is a stratified solution to the lifting roblem and is a stratified fibration. Corollary 3.9. The restriction of a stratified fibration is a stratified fibration. Proof. Pull back over the inclusion which induces the restriction.

13 G. Friedman / Toology and its Alications 134 (2003) Corollary Suose that : Y B is a stratified fibration, f : B B and g : B B are mas, : Y B is the stratified fibration induced by and f, and : Y B is the stratified fibration induced by and g. Then u to natural stratified isomorhism, is also the stratified fibration induced from by the ma fg. Proof. As for ordinary fibrations, φ : (fg) Y g f Y given by φ(b 2,y)= (b 2,(g(b 2 ), y)) rovides the isomorhism. It is clearly stratum-reserving. Definition Suose that i : Y i B, i = 1, 2, are stratified fibrations (note that these subscrits do not indicate strata). A ma H : Y 1 I Y 2 is a stratum-reserving fiber homotoy (s..-homotoy) if it is stratum-reserving and 2 H(y,t)= 1 (y) for all t I. f : Y 1 Y 2 is a stratum-reserving fiber homotoy equivalence if f is a stratumreserving ma such that 2 f(y 1 ) = 1 (y 1 ) for all y 1 Y 1 and there is a stratumreserving ma g : Y 2 Y 1 such that 1 g(y 2 ) = 2 (y 2 ) for all y 2 Y 2 and fg and gf are stratum-reserving fiber homotoic to the resective identity mas on Y 2 and Y 1. Lemma Suose that f 0,f 1 : B B are homotoic mas, : Y B is a stratified fibration, and let t : Y t B, t = 0, 1 be the induced ullback stratified fibrations. Then Y 0 and Y 1 are stratum-reserving fiber homotoy equivalent, i.e., there exist stratum-reserving fiber homotoy equivalences g : Y 0 Y 1 and h : Y 1 Y 0 and stratumreserving fiber homotoies between gh and hg and the resective identity mas. In this situation, we will say that 0 and 1 have the same stratified fiber homotoy tye or are stratum-reserving fiber homotoy equivalent (s..f.h.e.). Proof. The roof follows [19, I.7.25] closely. Let i t : B B I, t = 0, 1, be the inclusions of the ends, and let f : B I B be a homotoy from f 0 to f 1.Thenfi t = f t. Using Corollary 3.10, it then suffices to rove that for any stratified fibration : Y B I, if t : Y t B are the stratified fibrations induced by i t : B B I and Y t has the filtration induced as a subset of Y,then 0 and 1 have the same stratified fiber homotoy tye. So suose that this is the set-u. Let i t be the inclusion mas Y t Y over the inclusions i t. The mas id 0 : I Y 0 I B and the inclusions f 0 : Y 0 Y form the data for a stratified lifting roblem, and, by assumtion, there exists a stratified solution H 0 : I Y 0 Y such that H 0 (0,y 0 ) = i 0 (y 0) and H 0 (s, y 0 ) = (s, 0 (y 0 )). Sincei 0 is the inclusion, H 0 takes each I y 0 into the stratum of Y in which y 0 lies. Similarly, there is an H 1 such that H 1 (1,y 1 ) = i 1 (y 1), H 1 (s, y 1 ) = (s, 1 (y 1 )),andi y 1 mas into the stratum containing y 1.Defineφ: Y 0 Y 1 and ψ : Y 1 Y 0 by i 1 φ(y 0) = H 0 (1,y 0 ) and i 0 ψ(y 1) = H 1 (0,y 1 ). From the roerties of H t, φ and ψ are stratum-reserving, 1 φ = 0,and 0 ψ = 1. To see that ψφ is stratum-reserving fiber homotoic to the identity, define g :1 I Y 0 I I Y 0 Y and K : I I Y 0 I B by K(s,t,y 0 ) = (s, 0 (y 0 )) and H 0 (s, y 0 ), t = 0, ( g(s,t,x) = H 1 s,φ(y0 ) ), t = 1, i 1 φ(y 0), s = 1.

14 82 G. Friedman / Toology and its Alications 134 (2003) g is well defined and stratum-reserving on the induced strata. These mas form the data for a stratified lifting extension roblem. By Lemma 3.4, it has a solution if there is a deformation retraction R t : I I Y 0 I I I Y 0 to 1 I Y 0 I İ Y 0 such that gr 1 : I I Y 0 I Y is stratum-reserving along I. But we can take as R the ma r id Y0 where r is any standard deformation retraction of I I into 1 I I İ. Then each I I y 0 I clearly gets maed into a single stratum of I I Y 0 under R and hence under R 1,andg is stratum-reserving from its definition. Hence the conditions are met for there to be a stratified extension of g, namely G : I I Y 0 Y,such that G = K and G is stratum-reserving along the first I factor. But it is also stratumreserving along the second I factor since clearly G is stratum-reserving along it for s = 1. Then G 0 I Y 0 is a stratum-reserving fiber homotoy between ψφ and the identity. The other case, φψ, is handled similarly. Corollary If : Y B is a stratified fibration and B is contractible then is stratified fiber homotoically trivial, i.e., there is a stratum-reserving fiber homotoy equivalence between : Y B and : F B B for some stratified fiber F. Proof. Clearly the ullback of a ma from B to a oint in B gives the roduct stratified fibration. We will call a stratum-reserving fiber homotoy equivalence given by the ma φ : F B Y a stratified trivialization. We call it a strong stratified trivialization if, for some b 0 B with F = 1 (b 0 ),wehaveφ(x,b 0 ) = x for all x F. Corollary If (B, b 0 ) is a DR-air and : Y B is a stratified fibration, then there exists a strong stratified trivialization φ : 1 (b 0 ) B Y. Any two such strong stratified trivializations are stratum-reserving fiber-homotoic rel 1 (b 0 ) b 0. Proof. The roofs are analogous to those given for ordinary fibrations in [19, I ]; the necessary modifications to the stratified case are either obvious or emloy the same techniques used in the receding roofs. Finally, we will need some results on the maing cylinders of stratified fibrations. Thesearethesubjectofthenextfewlemmas. We will always use maing cylinders (including cones) with the teardro toology (see [9, 2] or [11, 3]). The teardro toology on the maing cylinder I of the ma : Y B is the toology generated by the sub-basis consisting of oen subsets of Y (0, 1] and sets of the form [ 1 (U) (0,ε)] U, whereu is an oen set in B. This is, in fact, a basis since if U and V are two oen sets of B, then within the intersection ( 1 (U) (0,ε 1 ) U) ( 1 (V ) (0,ε 2 ) V), we have the basis element 1 (U V) (0, min(ε 1,ε 2 )) (U V). The intersection of any basis elements of the first and second tyes or of two elements of the first tye is a basis element of the first tye. If : Y B is a stratified fibration, Y filtered by the sets {Y i }, then the maing cylinder I is naturally filtered by B, as the bottom stratum, and the maing cylinders of restricted to each Y i. There remains, however, the question of how to label the dimensions.

15 G. Friedman / Toology and its Alications 134 (2003) Even if we create a consistent lan for how to handle the dimension of each I Yi,itis not clear what the dimension of B should be. Fortunately, in the alications below, it will actually be the maing cylinder which comes equied which a natural filtration of this form, and the filtration of Y can be considered to be induced by the intersection of that filtration with Y t I, t (0, 1]. So, although this aroach aears somewhat backwards, we will assume in what follows that the filtrations arise in this manner. We will also have need to consider a local coefficient system on the to stratum I I,n 2 = (Y Y n 2 ) (0, 1]. By standard bundle theory, any such system, G, will be isomorhic to (G Y t) (0, 1], for any choice of t (0, 1]. Therefore, we will always assume that G has the form of such a roduct. Conversely, if we start with a coefficient system, G, ony Y n 2,thenG (0, 1] gives a coefficient system on I, unique u to isomorhism. We will denote this coefficient system by I G.IfI is a closed cone, cy, we will sometimes denote this induced coefficient system by cg (note that we reserve the symbol cx for the oen cone on the sace X). Lemma Suose f : X Y is a continuous ma with maing cylinder I f and that Z is another sace. Then I f Z = I f idz, the maing cylinder of f id Z : X Z Y Z. Proof. Both saces clearly have the same underlying sets, so we focus on the toologies. I f Z has a basis of the form V W,whereV is a basis element of I f and W is a basis element of Z. In articular, these all have the form either V 1 W or V 2 W, where V 1 is oen in X (0, 1] and V 2 has the form [f 1 (U) (0,ε)] U for an oen subset U Y. On the other hand, I f idz has basis elements of the form A = {an oen subset of (X Z) (0, 1]} or B =[(f id Z ) 1 (C)] (0,ε) C for some oen C Y Z. So we need to show that A and B are oen in I f Z and V 1 W and V 2 W are oen in I f idz. Since the subset toologies on X (0, 1] Z I f Z and X Z (0, 1] I f idz all include the standard basis elements for roducts, these toologies are finer than (in fact equal to) the obvious roduct toologies, and so the sets of tye A and V 1 W are oen in the desired saces. The set V 2 W =[f 1 (U) (0,ε) U] W can be rewritten as [(f id Z ) 1 (U W) (0,ε)] (U W). This is an element of tye B, so the inclusion of V 2 W into I f idz is oen. Finally, the sets of the form U W for U oen in Y and W oen in Z form a basis for Y Z, so we can restrict to looking at sets of tye B in which C has the form U W.But then B =[(f id Z ) 1 (U W) (0,ε)] U W =[(f 1 (U) W (0,ε)] U W = (f 1 (U) (0,ε) U) W, which is of the form V 2 W. Hence this inclusion is also oen. Lemma Suose f : Y 1 Y 2 is a stratum- and fiber-reserving ma of stratified fibrations, i : Y i B, i = 1, 2.Letf be the ma from I 1 to I 2 induced on the maing cylinders by the identity on B and f id (0,1] : Y 1 (0, 1] Y 2 (0, 1]. Thenf is a continuous stratum-reserving ma which, furthermore, takes each cone I (b), for b B, into the cone I (b).

16 84 G. Friedman / Toology and its Alications 134 (2003) Proof. The only inobvious art of the statement of the lemma is that f should be continuous at oints in B. Ifb B, then the basis elements of the toology of I 2 containing f(b)= b have the form V =[2 1 (U) (0,ε)] U,whereU is a neighborhood of b in B. Then from the definitions, f 1 (V ) =[f (U) (0,ε)] f 1 (U) = [1 1 (U) (0,ε)] U.Herewehaveusedthat f B is the identity on B and that 2 f = 1. So f 1 (V ) is an oen neighborhood of b that mas into V,andf is continuous. Corollary The ma f, as defined in the receding lemma, induces a ma on intersection homology f : IH (I 1 ) IH (I 2 ).Iff is covered by a bundle ma on the to stratum f : G H, thenf id (0,1] gives a ma I G I H covering f on the to stratum, and together these induce a ma f : IH (I 1 ; I G ) IH (I 2 ; I H ). Lemma Suose that f : Y 1 Y 2 and g : Y 2 Y 1 are stratum-reserving fiber homotoy inverses for the stratified fibrations i : Y i B, i = 1, 2. Thenf and ḡ as defined in the revious lemma are stratum- and cone-reserving homotoy inverses between the I i, meaning that the homotoies from f ḡ and ḡf to the identities are stratumreserving and take I i i 1 (b) I to I i i 1 (b). Proof. Let H : Y 1 I Y 1 be the stratum-reserving fiber homotoy from gf to the identity. By Lemma 3.15, I 1 I is equal to the maing cylinder of 1 id I : Y 1 I B I.Define H : I 1 I I 1 so that for each t I, H I 1 t is the ma determined as in the revious lemma by extending the ma H Y 1 t to the cylinder. Again, continuity is clear excet for oints (b, t) B I I 1 I. Consider H(b,t)= b. Again, the basis neighborhoods of b in B have the form V =[1 1 (U) (0,ε)] U. Since by Lemma 3.15, we can also consider I 1 I as the maing cylinder of 1 id I, (b, t) has a neighborhood of the form W =[1 1 (U I) (0,ε)] (U I).SinceH is a fiber homotoy, H takes U I to U by rojection, and since H is cone-reserving and takes each oint in the cylinder to a oint of the same height, it mas oints in 1 1 (U I) (0,ε) to 1 1 (U) (0,ε). Hence the neighborhood W mas into the neighborhood V,and H is continuous. Furthermore, over B 0, since the restriction of H is the identity ma of Y 1,the restriction of H is the identity ma of the maing cylinder I 1, while the restriction over B 1ofH is gf so that the restriction of H is gf, which is clearly ḡf from the definition of these mas. Therefore, H is a homotoy from gf to the identity. That it is cone-reserving follows immediately from the definitions and the fact that H was fiber reserving. Similarly, the fact that H was stratum-reserving and the definition of H shows that H is stratum-reserving. The roof that f ḡ is stratum- and cone-reserving homotoic to the identity on I 2 is the same. Corollary If a stratified fibration : Y B ossesses a (strong) stratified trivialization f : F B Y, then there is also a (strong) stratum- and cone-reserving homotoy equivalence cf B I.

17 G. Friedman / Toology and its Alications 134 (2003) Proof. By the receding lemma, there exists a stratum- and cone-reserving homotoy equivalence f from the maing cylinder of the rojection F B B to I.Using Lemma 3.15, this maing cylinder is exactly cf B. If the trivialization is strong so that f restricted to the fiber over some b 0 is the identity ma F b 0 F b0, then it is clear from the construction in the lemma that f cf b0 will be the identity ma cf b 0 cf b0. Furthermore, if the homotoies determining the equivalence of Y and F B are stationary over b 0, then so will be the homotoies which determine the equivalence of I and cf B as constructed in the roof of the lemma. Corollary If H : Y 1 I Y 2 is a stratum-reserving fiber homotoy from f to g, f,g : Y 1 Y 2 stratum- and fiber-reserving mas, then H : I 1 I Y 2, defined so that H (I 1 t) = H I 1 t for each t I, is a stratum-reserving homotoy H : I 1 I I 2 which mas c 1 1 (b) t to c 1 2 (b) for each b B and t I. Proof. The method of roof is the same as that emloyed in the roof of the lemma to construct the homotoy from ḡf to the identity. Corollary If f and g are stratum-reserving fiber homotoic mas of stratified fibrations, then f and ḡ induce the same ma on intersection homology f = g : IH (I 1 ) IH (I 2 ). If the homotoy, H,fromf to g is covered on the to stratum by a homotoy of local coefficients H : G I H from f to g, then H can be covered on the to stratum by a homotoy H id (0,1] : I G I I H from f id (0,1] to g id (0,1].Thenf covered by f id (0,1] and ḡ covered by g id (0,1] induce the same ma on intersection homology f = g : IH (I 1 ; I G ) IH (I 2 ; I H ). Proof. This follows from Corollaries 2.4 and Bundles of coefficients induced by a stratified fibration To construct bundles of modules from a stratified fibration, we will again need to generalize some of the standard arguments from the case of ordinary fibrations. We again modify some of the relevant results from [19]. Given a stratified fibration : Y B, with B unfiltered, and a ath u : I B from b 0 to b 1,letF t = 1 (b t ) and i t : F t Y (obviously these are stratum-reserving mas if we define the stratifications on the F t by their intersections with the strata of Y ). We call a stratum reserving ma h : F 1 F 0 u-s.. admissible if there exists a stratum-reserving homotoy H : I F 1 Y from h to i 1 such that H(t,y) = u(t) for all (t, y) I F 1. Just as for ordinary fibrations, the following are clear: (1) If u is the constant ath and h is the identity ma F 0 F 0,thenh is u-s.. admissible. (2) If v : [0, 1] B and u : [1, 2] B are aths I B with u(1) = v(1), h : F 1 F 0 is v-s.. admissible, and k : F 2 F 1 is u-s.. admissible, then h k : F 2 F 0 is v u- s.. admissible. The roof, as usual, consists of adjoining two homotoies: If H and K

18 86 G. Friedman / Toology and its Alications 134 (2003) are the homotoies which rovide the admissibility of h and k, then the admissibility of hk is given by the homotoy L : [0, 2] F 2 Y defined by { ( ) H t,k(x), t [0, 1], L(t, x) = K(t,x), t [1, 2]. Since H, K, andk are stratum-reserving, so is L. Given u π 1 (B; b 0,b 1 ), there always exists a u-s.. admissible ma: Let f : I 1 (b 1 ) B be given by f(t,y)= u(t),andletg: 1 1 (b 1 ) be given by g(1,y)= y. Then this defines a stratified homotoy lifting roblem, which has a solution G : I 1 (b 1 ) Y by assumtion. Then define g : 1 (b 1 ) 1 (b 0 ) by g (y) = G(0,y). This is a stratum-reserving ma. We next show that if u 0 and u 1 are homotoic rel I and h 0 and h 1 are resectively u 0 - and u 1 -s.. admissible, then h 0 and h 1 are stratum-reserving homotoic: Let F : I I B be a homotoy from u 0 to u 1 rel I, and, for s = 0, 1, let H s be the homotoy from h s to i 1 lying over u s. Define a ma G : I I F 1 B by G(s, t, y) = F(s,t) and a ma H : (İ I I 1) F 1 Y by { Hs (t, y), s = 0, 1, H(s,t,y)= y, t = 1. These rovide the data for a stratified lifting extension roblem. Since is a stratified fibration, there is a stratified solution, H, by Lemma 3.4: We can deformation retract I I F 1 to (İ I I 1) F 1 as the roduct of F 1 with any deformation retraction I I I I I 1, and clearly this retraction comosed with H is stratum-reserving. H I 0 F 1 then gives a stratum-reserving homotoy from h 0 to h 1. Note, incidentally, that together the above roerties imly that the fibers of Y are stratum-reserving homotoy equivalent. The ushot of this discussion is that given a homotoy class of aths rel endoints in B, we have defined a unique stratum-reserving homotoy class of mas from the fiber over the terminal oint to the fiber over the initial oint. Furthermore, this is functorial with resect to comosition of classes of aths. Hence, we have established a functor from the fundamental grouoid of B to the category of filtered saces and stratum-reserving homotoy classes of mas. This, in turn, induces a functor F to Abelian grous or modules by taking, for b B, F(b) = IH i ( 1 (b)) and as ma F(u) : IH ( 1 (b 1 )) IH ( 1 (b 2 )) the ma on intersection homology induced by the stratum-reserving homotoy class of mas determined by u (see Corollary 2.4). As defined in [19, Chater VI], such a functor determines a bundle of grous (modules) over B with fiber isomorhic to IH i ( 1 (b)) for any b in B. Now suose that Y = (Y, {Y n }), itself, is given a local coefficient system, G, over Y n Y n 2. It will again be convenient to think of G as a sace (in fact a covering sace of Y Y n 2 with rojection π : G Y ). In the following, G will always lie over the to stratum, but for convenience we suress this from the notation. Let b 0, b 1 continue to denote oints of B and F s = 1 (b s ), and consider the restrictions G F s. As above, given aathu in B from b 0 to b 1, we can construct a stratum-reserving ma g : F 1 F 0 as the end of a stratum-reserving homotoy G : F 1 I Y from the inclusion of F 1.The