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 EulerPoincar'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 DCW homology functor which allows for easy computation of the ordi
nary Dequivariant homology defined by E. Dror Farjoun in [1]. Our approach is
a generalization of the GCW (co)homology functor constructed by S.J. Willson *
*in
[13] for the case of G being compact Lie group.
Then we apply the DCW homology functor to obtain:
(i) Equivariant EulerPoincar'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 Dspace, 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 finvariant 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. Dspaces. 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 Dspaces. The arrows in T opD are natur*
*al
transformations of functors or equivariant maps.
2.2. Dhomotopy. An equivariant homotopy between two Dmaps f, g : Xe! Ye,
where Xe,Ye are Ddiagrams, is a Dmap H : Xex I ! Ye, where I denotes the
constant Dspace I(d) = [0, 1]. A homotopy equivalence f : Xe! Yeis a map with
a (two sided) Dhomotopy inverse.
2.3. Dorbits. We recall now the central concept of the Dhomotopy theory (in
troduced in [1],[3])  that of Dorbit. A Dorbit is a Dspace T : D ! T op, su*
*ch
that colimDT = {*}. A free Dorbit 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 Dorbit. A Dspace Xeis called free iff for any s 2 colim*
*CXe
the full orbit Ts lying over s is free.
2.4. DCW complexes. A Dcell is a Dspace of the form T x en, where T is a
Dorbit and en is the standard ncell. An attaching map of this Dcell to some
Dspace Xeis a map OE : T x @en ! Xe.
A (relative) DCW complex (Xe, Xe1) is a Dspace Xetogether with a filtrati*
*on
Xe1 Xe0 . . .Xen Xen+1 . . .Xe = colimnXen, such that Xen+1 is
obtained from Xenby attaching a set of ndimensional Dcells. Namely one has a
pushout diagram of Dspaces:
` n OE
i(Tix?@e )! Xen1?
?y ?y
` n
i(Tix e ) ! Xen
If Xe1= ? we call the DCW complex absolute.
Let Xebe a DCW complex. A Dsubspace Ye Xeis called the cellular subspace
if Yehas a DCW 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 Dorbits.
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 Dspaces of arbitrary orbit
type in [2]. We will be interested in the diagrams which are homotopy equivalent
to the finite DCW 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 Dorbits. 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 Dspace *
*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 Dspaces, then there exist an
Oopequivariant 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 Dspace Xethe Oop
space (Xe)O is Oopfree [1, 3.7].
Example 2.1. Consider a free Dspace 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 DCW 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 Oopspace and produce a Dspace 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 DCW 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. OopCW structure on the orbitopointpspace of a DCW 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 categoriesop: 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 ResC.
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 Dspaces, 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) DCW space of orbit type O,where Ois a
small category of orbits. Consider the Oopspace XeO to be the orbit point spac*
*e of
Xe.
Then XeO has OopCW structure which corresponds to the DCW structure of
Xein the following sense: let ptD X 0 X 1 . . .X n . . .X = colimnXn
is a DCW filtration of X , suech theat eeach X nisea pushouet: e
e e
` n1 OE
iTix?S ! Xen1?
?y ?yin
` n
iTix D ! Xen
then there exist a OopCW 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 n1 OEO O
iF ?x S ! Xen1?
?y ?yiO
n
` Ti n O O
iF x D ! Xen
is a pushout 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 pushouts and products.
3.2. DCW homology functor. The construction of the (co)homology functor
in [1, 4.16] depends on the specific DCW decomposition of XeO. We apply this
construction to the cellular structure of XeO, which was constructed in 3.3 and
obtain the required DCW homology functor.
3.3. Isotropy ring I. In [13] a universal coefficient system for the Gequivari*
*ant
homology have been developed(where G is a compact Lie group). Let us generalize
this approach to the coefficient systems for the classical Dhomology theory. S*
*up
pose O0 is a small, full subcategory of the orbit category O. Let Xebe a Dspace
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 Imodules which satisfy:
M
(3) 8M 2 ob(M), M = 1TM
T2ob(hO0)
(where {1TM}T2ob(hO0)are left Rmodules) and the category of R(hO0)mod of
functors from hO0 to the category of left Rmodules 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 leftRmodule.
T2ob(hO0)
Define the left Imodule 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 Imodule is an object of M, because
I ~= T2ob(hO0)1TI (as left Rmodules) 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 twosided 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 Dhomotopy 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
EulerPoincar'e formula.
6 BORIS CHORNY
4. Applications
Let Xebe a finite DCW complex of type O0 for some orbit category O0 with
obj(O0) a finite set.
4.1. Equivariant EulerPoincar'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 HattoriStallings 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 DCW complex. Suppose Xehas nq qdimensional
cells and t1 + . .+.ts = nq, ti is the number of qdimensional cells of the same
homotopy type Ti2 Iso(obj(hO0)). Then Cq(Xe) O0iI ~=iI(T1)t1 . . .iI(Tn)ts
as a left Zmodule.
Proof.Let ti= ri1+. .+.rik, where rijis the number of qdimensional 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
Zmodules and right Imodules 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 Imodules and the equivariant homology is endowed with the
right Imodule 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 Icomplex, 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 HattoryStallings 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 Jdiagram:
EQUIVARIANT CELLULAR HOMOLOGY 7
__________
Ze__________




_____?___
.. ...
Ze has two 0cells of type T2 = [#] and one 1cell 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 Jequivariant 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 EulerPoincar'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 Dorbits 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 DCW complex Xe will be called the triangulated Dspace 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 DCW 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 pullback:
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 DCW complex Xe0. Continuing in the same
way for the rest of the simplices of Y completes the construction of Xe0. Hen*
*ce
DCW complex Xe0has the same underlying topological diagram as Xe, therefore
they are Dhomeomorphic.
Definition 4.7. Let f : Xe! Xebe a map of the finite triangulated Dspace 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
Dmap. D (f) = (~1, . .,.~n) 2 U(D)  Lefschetz number of f. Then if there is
no finvariant 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 Imodule. 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 mth 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, 93134, Spr*
*ingerVerlag
1987.
[2]E. Dror Farjoun, Homotopy theories for diagrams of spaces, Proc. Amer. Math*
*. Soc. 101,
181189, 1987.
EQUIVARIANT CELLULAR HOMOLOGY 9
[3]E. Dror Farjoun, A. Zabrodsky, Homotopy equivalence between diagrams of spa*
*ces, J. Pure
Appl. Alg. 41, 169182, 1986.
[4]T. tom Dieck, Transformation groups, de Gruyter Studies in Mathematics 8, W*
*alter de
gruyter, 1987
[5]A. Dold, Lectures on Algebraic topology, SpringerVerlag, 1972
[6]W.G. Dwyer, D.M. Kan, Singular functors and realization functors, Proc. Kon*
*. Akad. van
Wetensch. A87 = Ind.Math. 46, 139146, 1984.
[7]A.D. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 227, 2*
*75284, 1983.
[8]A. Hattori, Rank element of a projective module, Nagoya Math. J. 25, 11312*
*0, 1965.
[9]G.M. Kelly, Basic concepts of enriched category theory, London Math. Soc., *
*Lecture Note
Series 64, Cambridge University Press, 1982.
[10]W. Lück, Transformation groups and algebraic Ktheory, Lecture Notes in Mat*
*h. 1408,
SpringerVerlag, 1989.
[11]J.R. Stallings, Centerless groupsan algebraic formulation of Gottlieb's th*
*eorem, Topology
4, 129134, 1965.
[12]Ch. Watts, A homology theory for small categories, Proceedings of the Confe*
*rence in Cate
gorical Algebra, LaJolla, SpringerVerlag, 1965.
[13]S.J. Willson, Equivariant homology theories on Gcomplexes, Trans. Amer. Ma*
*th. Soc. 212,
155171, 1975.
Einstein Institute of Mathematics, Edmond Safra Campus, Givat Ram, The Hebrew
University of Jerusalem, Jerusalem 91904, Israel
Email address: chorny@math.huji.ac.il