EQUIVARIANT CELLULAR HOMOLOGY AND ITS APPLICATIONS BORIS CHORNY Abstract.In this work we develop a cellular equivariant homology functor and apply it to prove an equivariant Euler-Poincar'e formula and an equi* *variant Lefschetz theorem. 1.Introduction Let D be an arbitrary small topologically enriched category. In this paper we develop a D-CW -homology functor which allows for easy computation of the ordi- nary D-equivariant homology defined by E. Dror Farjoun in [1]. Our approach is a generalization of the G-CW -(co)homology functor constructed by S.J. Willson * *in [13] for the case of G being compact Lie group. Then we apply the D-CW -homology functor to obtain: (i) Equivariant Euler-Poincar'e formula: 1X (1) ØD (Xe) = (-1)nfrkHS(HDn(X ; I)) n=0 e This formula establishes a connection between the equivariant homology and an equivariant Euler characteristic; frkHS(.) is a slight modification of Hattori-* *Stallings rank (originally defined in [8],[11]). (ii) Equivariant Lefschetz theorem: Let Xebe a triangulated D-space, f : Xe! * *Xe an equivariant map. If the equivariant Lefschetz number 1X (2) D (f) = (-1)nterHS(HDn(f; I)) n=0 is not equal to zero, then there are f-invariant orbits in Xe, moreover the orb* *it types of the invariant orbits may be recovered from D (f). Acknowledgments. I would like to thank E.Dror Farjoun, V.Halperin, A.Libman, Sh. Rosset for many stimulating conversations and helpful ideas. 2.Preliminaries 2.1. D-spaces. Let T opdenote the category of the compactly generated Hausdorff topological spaces. Fix an arbitrary small category D enriched over T op. We wo* *rk in the category T opD of functors from D to T op. The objects of this category ____________ Date: August 30, 2001. 1991 Mathematics Subject Classification. Primary 55N91; Secondary 55P91, 57S* *99. The author acknowledges the support of Sonderforschungsbereich (SFB) 478 of * *the University of Münster and Edmund Landau Center for Research in Mathematical Analysis. 1 2 BORIS CHORNY are called topological diagrams or just D-spaces. The arrows in T opD are natur* *al transformations of functors or equivariant maps. 2.2. D-homotopy. An equivariant homotopy between two D-maps f, g : Xe! Ye, where Xe,Ye are D-diagrams, is a D-map H : Xex I ! Ye, where I denotes the constant D-space I(d) = [0, 1]. A homotopy equivalence f : Xe! Yeis a map with a (two sided) D-homotopy inverse. 2.3. D-orbits. We recall now the central concept of the D-homotopy theory (in- troduced in [1],[3]) - that of D-orbit. A D-orbit is a D-space T : D ! T op, su* *ch that colimDT = {*}. A free D-orbit generated in d 2 ob(D) is T opD 3 F d= homD (d, .), i.e. F d(d0) = hom D(d, d0) and F d(d0 ! d00) is given by the comp* *osi- tion. Clearly F dis a D-orbit. A D-space Xeis called free iff for any s 2 colim* *CXe the full orbit Ts lying over s is free. 2.4. D-CW -complexes. A D-cell is a D-space of the form T x en, where T is a D-orbit and en is the standard n-cell. An attaching map of this D-cell to some D-space Xeis a map OE : T x @en ! Xe. A (relative) D-CW -complex (Xe, Xe-1) is a D-space Xetogether with a filtrati* *on Xe-1 Xe0 . . .Xen Xen+1 . . .Xe = colimnXen, such that Xen+1 is obtained from Xenby attaching a set of n-dimensional D-cells. Namely one has a push-out diagram of D-spaces: ` n OE i(Tix?@e )----! Xen-1? ?y ?y ` n i(Tix e ) ----! Xen If Xe-1= ? we call the D-CW -complex absolute. Let Xebe a D-CW -complex. A D-subspace Ye Xeis called the cellular subspace if Yehas a D-CW -structure such that each cell of Yeis also a cell of Xe. 2.5. The category of orbits. The category of orbits O is a full topological sub- category of T opD generated by all D-orbits. Usually O is not a small category. For example for D = J = (o-!o), then O ~= T op. A model category has been constructed for the D-spaces of arbitrary orbit type in [2]. We will be interested in the diagrams which are homotopy equivalent to the finite D-CW -complexes, i.e. only finite number of orbit types appear in* * such diagram. We collect those orbits into the a full subcategory O0 of O with a fin* *ite amount of objects. 2.6. Orbit point (.)O and realization | . |D functors. Suppose Ooispa small category of D-orbits. An orbit point functor (.)O : T opD ! T opO is a gener- alization to the diagram case of Bredon's fixed point functor. For any D-space * *Xe (usually of type O) (Xe)O is a Oop diagram such that (Xe)O (T ) = homD (T, Xe) * *for all T 2 ob(O) and the arrows of the diagram are induced by composition with the maps between orbits. If f : Xe! Yeis an equivariant map between two D-spaces, then there exist an Oop-equivariant map fO : XeO! YeO, which is obtained from f by composition: O XeO(T ) = homD (T, Xe) 3 g f7-!f O g 2 homD (T, Ye) = YeO(T ) EQUIVARIANT CELLULAR HOMOLOGY 3 The fundamental property of (.)O functor is that for any D-space Xethe Oop- space (Xe)O is Oop-free [1, 3.7]. Example 2.1. Consider a free D-space Xe. And let the orbit category O consists of all the free orbits. Then O is isomorphic to Dop as a category and (Xe)O ~=Xe (Yoneda's lemma). Another easy case occurs then Xeis a D-CW -space. We shall discuss it in the next section. There exist a left adjoint to (.)O . It is called realization functor |.|D , * *since it takes an Oop-space and produce a D-space with the prescribed orbit point data (up to local weak equivalence). Realization functor in the group case has been constru* *cted by A.D.Elmendorf in [7] and has been generalized to the arbitrary diagram case * *by W.Dwyer and D.Kan in [6]. Compare also [1]. 2.7. Equivariant Euler characteristic. Let Xebe a finite D-CW -complex, then Xeis of type O for some category of orbits with finite amount of objects. We define the Equivariant Euler characteristic to be the Universal Additive Invari* *ant [10, I.5] (U(D), ØD ). Or, equivalently, we say that ØD (Xe) 2 U(D) is equal to* * the alternating sum of theLorbit types over the dimensions of the cells of Xein the* * free abelian group U(D) = T2Iso(hO)Z generated by the homotopy types of orbits. 3. Equivariant cellular homology 3.1. Oop-CW structure on the orbitopointpspace of a D-CW -complex. Let C denote a subcategory of T opO which is obtained as the image of T opD under * *the functor (.)O . Recall that there is the inclusion of the categorieso-p: D ,! Oo* *p, where -(d) = F d, for each d 2 ob(D). Hence there is a functor Res: T opO ! T opD. By abuse of notation we denote by Res also Res|C. Lemma 3.1. The functor (.)O is fully faithful. Proof.The faithfulness is clear. We have to show only that for any map f0 : XeO! YeOthere exists a map f : Xe! Yes.t. f0 = fO . Take f = Res(f0), then if Xeand Yewere orbits the result follows from the bijective correspondence induced by t* *he Yoneda's lemma: hom D(Xe, Ye) = homOop(XeO, YeO). The general claim will follow from the comparison of f0 and fO orbitwise, i.e. by their action on each full o* *rbit. Fortunately the functor (.)O , being right adjoint, commutes with taking full o* *rbit (pullback). Lemma 3.2. The pair of functors Res(.) : C $ T opD : (.)O induce the equivalence of the categories C and T opD. Proof.We need to construct the natural isomorphisms of the functors idT opD~= Res((.)O ) and idC~=(Res(.))O . Let Xe2 T opD, then Res(XeO) ~=Xe because the generalized lemma of Yoneda [9] induces the objectwise homeomorphisms and the equivariance is preserved by the naturality of the Yoneda's isomorphism. But an equivariant map which is the objectwise homeomorphism is an isomorphism of D-spaces, hence the first isomor- phism of functors. Let XeO 2 C, then Res(XeO) ~= Xe by the first homeomorphism, then (Res(XeO))O ~=XeO. Hence the second isomorphism. 4 BORIS CHORNY Proposition 3.3. Let Xebe a (pointed) D-CW -space of orbit type O,where Ois a small category of orbits. Consider the Oop-space XeO to be the orbit point spac* *e of Xe. Then XeO has Oop-CW structure which corresponds to the D-CW structure of Xein the following sense: let ptD X 0 X 1 . . .X n . . .X = colimnXn is a D-CW filtration of X , suech theat eeach X nisea push-ouet: e e e ` n-1 OE iTix?S ----! Xen-1? ?y ?yin ` n iTix D ----! Xen then there exist a Oop-CW -filtration: ptOop X O0 X O1 . . .X On . . . X O= colimnXO , such that X O = (X n)O e, and e e e e en e n e ` Ti n-1 OEO O iF ?x S ----! Xen-1? ?y ?yiO n ` Ti n O O iF x D ----! Xen is a push-out square. Proof.We proceed by the induction on the skeleton of Xe. a a a XeO0(T ) = ( Ti)O (T ) = ((Ti)O (T )) = homD (T, Ti) = a a homOop(F T, F Ti) = F Ti(T ). Hence the base of the induction. Suppose we know the claim for Xen. Then it follows for Xen+1since (.)O is both left and right adjoint, so commutes both with push-outs and products. 3.2. D-CW -homology functor. The construction of the (co)homology functor in [1, 4.16] depends on the specific D-CW -decomposition of XeO. We apply this construction to the cellular structure of XeO, which was constructed in 3.3 and obtain the required D-CW -homology functor. 3.3. Isotropy ring I. In [13] a universal coefficient system for the G-equivari* *ant homology have been developed(where G is a compact Lie group). Let us generalize this approach to the coefficient systems for the classical D-homology theory. S* *up- pose O0 is a small, full subcategory of the orbit category O. Let Xebe a D-space of orbit type O0. Then a coefficient system for the ordinary (co)homology is a homotopy (co)functor M : O0! (R - mod). 0 Definition 3.4. Let R be a commutative ring. An isotropy ring I = IR,OD is generated by mor (hO0) as a free R - mod . Define the multiplication on the generators by æ fg = f Og,0,ifcodom(g)o=tdom(f)herwise and extend the definition to the rest of the elements of I by linearity. EQUIVARIANT CELLULAR HOMOLOGY 5 Proposition 3.5. The category M of the left I-modules which satisfy: M (3) 8M 2 ob(M), M = 1TM T2ob(hO0) (where {1TM}T2ob(hO0)are left R-modules) and the category of R(hO0)-mod of functors from hO0 to the category of left R-modules are equivalent. Proof.Let us define a pair of functors which induce the required equivalence: i : M ø R(hO0)-mod : ,. Let M 2 ob(M), T 2 ob(hO0), then define iM(T ) = 1TM. If mor(hO0) 3 f : T1 ! T2, then define iM(f)(1T1m) = f1T1m = (1T2f)1T1m 2 1T2M. Obviously the morphisms of the left I modules correspond to the natural transfo* *r- mations of the functors. Given a R(hO0)-module N, then M ,N = N(T ), as a leftR-module. T2ob(hO0) Define the left I-module structure on ,N by f(. .,.n, . .).= (. .,.fn, . .)., w* *here æ N(codom (f)) 3 fn = f(n),0ifn,2oN(domt(f))herwise Now it is clear that the defined functors provide the equivalence of the catego* *ries. RemarkL3.6. The ring I considered as a left I-module is an object of M, because I ~= T2ob(hO0)1TI (as left R-modules) by the construction. But it also carries an obvious structure of the right I module, so the iI(T ). RemarkP3.7. If ob(hO0) is finite then the ring I has a two-sided identity eleme* *nt 1 = T2ob(hO0)1T together with its decomposition into the sum of the orthogonal idempotents and the condition (3) is redundant. L Definition 3.8. The augmentation ' : I ! T2Iso(ob(hO0))R is defined for any X X X I 3 g = rff + shh T2ob(hO0)f2mor(T,T) h2mor(T1,T2),T16=T2 (only a finite number of rf, sh 2 R is non equal to zero) to be X M '(g) = (. .,. rf, . .).2 R f2mor(T,T) T2Iso(ob(hO0)) Remark 3.9. The idempotents in I which correspond to the D-homotopy equiv- alent orbits are identified under '. Apparently, ' is an epimorphism of rings. Consider the abelinization functor Ab : (Rings) ! Ab which corresponds to a ring itsLadditive group divided by the commutator subgroup. Then Ab(') : Ab(I) ! T2Iso(ob(hO0))R. The last map will be used to obtain a generalization of the Euler-Poincar'e formula. 6 BORIS CHORNY 4. Applications Let Xebe a finite D-CW -complex of type O0 for some orbit category O0 with obj(O0) a finite set. 4.1. Equivariant Euler-Poincar'e formula. We remind thatLthe equivariant Eu- ler characteristic lies in the abelian group U(D) ~= Iso(obj(hO0))Z, so in ord* *er to apply Hattori-Stallings machinery we need to choose a coefficient system for the equivariant homology such that the resulting chain complex and homology groups will be endowed with the module structure over some ring S which allows an epi- morphism " : Ab(S) -! U(D). Our choice of the coefficient0system for the equivariant homology will be the isotropy ring I = IZ,ODtaken over itself as a left module. Lemma 4.1. Let Xebe a finite D-CW -complex. Suppose Xehas nq q-dimensional cells and t1 + . .+.ts = nq, ti is the number of q-dimensional cells of the same homotopy type Ti2 Iso(obj(hO0)). Then Cq(Xe) O0iI ~=iI(T1)t1 . . .iI(Tn)ts as a left Z-module. Proof.Let ti= ri1+. .+.rik, where rijis the number of q-dimensional cells of ty* *pe Tij2 ob(O)Lof homotopy type Ti. By the construction of the equivariant homology Cq(Xe) = si=1( kj=1Z(hom O0(?, Tij)rij)). The dual Yoneda isomorphism [9, p.7* *4] implies: Ms Ms Cq(Xe) O0 iI ~= ( kj=1iI(Tij)rij) ~= kj=1(1TijI)rij, i=1 i=1 If Tij1is isomorphic to Tij2in hO0then there is an obvious isomorphism of the l* *eft Z-modules and right I-modules 1Tij1I ~=1Tij2I. Let us choose a representative Ti of each isomorphism class of objects in hO0, then Ms P k Ms Ms Cq(Xe) O0 iI ~= (1TiI)( j=1rij)~= (1TiI)ti~= (iI(Ti))ti i=1 i=1 i=1 Because of 3.6 the equivariant chain complex {Cq(Xe) O0 iI}dimXfq=0is a comp* *lex of projective right I-modules and the equivariant homology is endowed with the right I-module structure. Notation: ØHS(.) means Euler characteristic of a I differential complex with re* *spect to rkHS(.). Proposition 4.2. Let K* = C*(Xe) O0 iI be a right I-complex, then ØD (Xe) = Ab(')(ØHS(K*)) whenever left side is defined. Proof.It is easy to see that rkHS (1TI) = 1T 2 Ab(I). Lemma 4.1 together with 3.9 completes the proof. Now we combine 4.2 with the additivity properties of the Hattory-Stallings ra* *nk and obtain the following P 1 Theorem 4.3. ØD (Xe) = Ab(')( n=0(-1)nrkHSHDn(Xe; iI)), whenever the left side is defined. Example 4.4. Consider the J-diagram: EQUIVARIANT CELLULAR HOMOLOGY 7 __________ Ze__________ | | | | _____|?___ .. ... Ze has two 0-cells of type T2 = [#] and one 1-cell of type T3 = [# ], hence .. ... . . ØJ(Ze) = 2[#] -.[# ] . . The category O0 of orbits contains two objects: T2, T3. The cellular chain complex tensored with the coefficients I = IZ,{T2,T3}Jbecomes: . .!.0 ! 1T3I @1!(1T2I)2 and @1 = 0 from the orbit type considerations. U(J) = Z Z in that case. And HJ0(Ze, I) = (1T2I)2, HJ1(Ze, I) = 1T3I are ri* *ght I - modules. Hence, ØJ(Ze) = (2, 0) - (0, 1) = (2, -1). 0 Let us, for comparison,0calculate the J-equivariant homology of Ze with ZO coefficients: HJi(Ze, ZO ) = Hi(colimJZe, Z) (see [1, 5.2]). Then colimJZe= I = [0, 1] and æ 0 Z, i = 0 HJi(Ze, ZO ) = 0, otherwise We can see that ZO0 coefficients are inappropriate to the Euler-Poincar'e formu* *la. 4.2. Equivariant Lefschetz theorem. Using cellular equivariant homology func- tor we are able now to proof a version of the equivariant Lefschetz theorem. Some result of the Lefschetz type in the equivariant setting may be obtained already by applying the ordinary Lefschetz theorem: consider an equivariant map f : Xe ! Xe, where Xe is a diagram over small category D, then if the Lefschetz number (colimDXe) 6= 0 there are f- invariant D-orbits in Xe. However the advantage of using the equivariant homology and equivariant Lefschetz number D (Xe) 2 U(D) is that we obtain the specific information about orbit type of t* *he invariant orbit. First we give a technical Definition 4.5. A D-CW -complex Xe will be called the triangulated D-space if the natural CW -structure of colimXealso triangulates colimXe. The following lemma will be used in the proof of the equivariant Lefschetz th* *e- orem. Lemma 4.6. Let Xe be a triangulated diagram, then for any refinement Y of the triangulation of colimXe, there exists a D-CW -complex Xe0, such that Xe0is D- homeomorphic to Xe and colimXe0= Y (as the triangulated spaces). Xe0will be called the refinement of Xe. 8 BORIS CHORNY Proof.Consider a new simplex in the triangulation of Y . It lies in some old simplex of colimXe: 2 0. Then consider the pull-back: 0 1 X lim@ e# A = T x , ,! colimDXe where T is the orbit which lies over 0. We've obtained the cell of the new D-CW -complex Xe0. Continuing in the same way for the rest of the simplices of Y completes the construction of Xe0. Hen* *ce D-CW -complex Xe0has the same underlying topological diagram as Xe, therefore they are D-homeomorphic. Definition 4.7. Let f : Xe! Xebe a map of the finite triangulated D-space Xeof orbit type O0,0where O0 is an orbit category with the finite number n of object* *s. Let I = IZ,OD. Then the equivariant Lefschetz number of f: X1 U(D) 3 (~1, . .,.~n) = D (f) = Ab(')( (-1)ktrHS(Hk(f; I))) k=0 Theorem 4.8. Let Xe be a finite triangulated diagram over D. f : Xe! Xebe a D-map. D (f) = (~1, . .,.~n) 2 U(D) - Lefschetz number of f. Then if there is no f-invariant orbit of type Tm , ~m = 0. Proof.A simplex in colimXewill be called of type T if the overlaying orbit is of type T in Xe. Then the condition that there are no invariant orbits of type Tm * *is equivalent to the condition that there are no fixed points in the simplices of * *type Tm . Since Xeis a finite triangulated diagram, colimXeis a finite triangulated spa* *ce, hence it is a compact metric space. If there are no fixed points of type Tm , t* *hen there exists a refinement Y of the triangulation such that if is a simplex of* * type Tm in Y , \ (colimf)( ) = ?. Consider the refinement Xe0of Xe, which exists by lemma 4.6. Since Xe0~=Xe, HD*(Xe0; I) = HD*(Xe; I), (f0) = (f), where f0 : Xe0! Xe0is equal to f, ~0m= * *~m . Therefore, it is enough to show that ~0m= 0. Now, X1 X1 D (f0) = Ab(')( (-1)ktrHS(Hk(f0; I))) = Ab(')( (-1)ktrHS(Ck(f0; I))), k=0 k=0 where Ck(f0; I) is the map induced by f on the chains Ck(Xe; I) = Ck(Xe) O0 I = (1T1I)t1 . . .(1TnI)tnas I-module. Because of the property: \(colimf)( ) = ? for any simplex of type Tm , the induced map on Ck(Xe; I) will take the gen* *er- ator 1Tm corresponding toP outside the submodule 1Tm I, that it generates. Then the m-th entry of Ab(')( 1k=0(-1)ktrHS(Ck(f0; I))) will be zero. This is true * *for all k, hence ~m = 0. References [1]E. Dror Farjoun, Homotopy and homology of diagrams of spaces, Proceedings o* *f Conference on Algebraic Topology, Seattle 1985. Lecture Notes in Math.1286, 93-134, Spr* *inger-Verlag 1987. [2]E. Dror Farjoun, Homotopy theories for diagrams of spaces, Proc. Amer. Math* *. Soc. 101, 181-189, 1987. EQUIVARIANT CELLULAR HOMOLOGY 9 [3]E. Dror Farjoun, A. Zabrodsky, Homotopy equivalence between diagrams of spa* *ces, J. Pure Appl. Alg. 41, 169-182, 1986. 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Watts, A homology theory for small categories, Proceedings of the Confe* *rence in Cate- gorical Algebra, LaJolla, Springer-Verlag, 1965. [13]S.J. Willson, Equivariant homology theories on G-complexes, Trans. Amer. Ma* *th. Soc. 212, 155-171, 1975. Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel E-mail address: chorny@math.huji.ac.il