View of Cech homology and the Novikov Conjectures for K- and L-Theory

CïECH HOMOLOGY AND THE NOVIKOV CONJECTURES FOR K- AND L-THEORY

GUNNAR CARLSSON AND ERIK KJR PEDERSEN

1. Introduction.

In [5], we studied the assembly map in algebraic K- and L-theory, and showed that the assembly map splits for a class of groupsÿ with ¢nite Bÿ for whichEÿadmits a metrizable, contractible, equivariant compacti¢cation such that theÿ-action is ``small at in¢nity''. This means that every compact subset of Eÿ when translated out near a point in the boundary becomes small i.e. for everyy2@Eÿ and for every neighborhoodUofyinEÿ, there is a neighborhoodV of y so that K\V6ˆ ;impliesKU. The method used in [5] was to use continuously controlledK- andL-theory.

Given a spectrum S one may de¢ne homology with coe¤cients in the spectrumSby the formula

h…X;x₀;S† ˆ…X^S†

for any ¢nite pointed CW-complexX. If X does not come exhibited with a basepoint we add a disjoint basepoint and get what is usually called un- reduced homology by the formula h…X;S† ˆh…X_‡;‡;S† ˆ…X_‡^S†.

When X is not a CW-complex this does not give a good de¢nition of the homology ofX. The main theme of this paper is a Cïech construction which gives a homotopy theoretically de¢ned extension of such a functor to all compact Hausdor¡ spaces. Boris Goldfarb [10] has pointed out to us, that this construction is close to the constructions used by Edwards and Hastings [8, ½8.2], and can be seen as one possible solution to a problem posed by Edwards and Hastings [8, p. 251]. We refer the reader to [10] for further discussion of the history of this subject. We also construct natural transfor- mations from various continuously controlled theories such asK-theory and L-theory to Cïech theory. The theory satis¢es the Steenrod axioms [13].

Hence using [14] the natural tranformation will be an isomorphism on the smaller category of compact metrizable spaces.

Research of the ¢rst author was supported in part by NSF DMS 8907771.A01.

Research of the second author was supported in part by NSF DMS 9104026.

Received December 15, 1995.

As an application of this we show how this may be used to split assembly maps for various groups. We still need compacti¢cations ofEÿ but we relax both the condition that the given compacti¢cation must necessarily be me- trizable and the condition that the action be small at in¢nity.

The Cïech homotopy type of a spaceZis the homotopy limit of the nerve of the ¢nite coverings of Z. We say Z is Cïech contractible if the Cïech homotopy type is contractible (in particular if Z is contractible). If we are given a familyfof open subsets ofZ, the Cïech homotopy type with respect tof, is the homotopy limit of the nerve of ¢nite coverings of Z, where we only use open sets fromfin the ¢nite coverings.

We prove the following theorems

Theorem A. Assume ÿ is a group with a finite Bÿ and that Eÿ has an equivariant compact Hausdorff compactification which is C¯ech contractible and such that the action is small at infinity then

a)If R is any ring then the assembly map

Bÿ_‡^K^ÿ1…R† !K^ÿ1…Rÿ†

is equivalent to an inclusion of a direct summand of spectra.

b)If R is a ring with involution such that K_ÿi…R† ˆ0 for i su¤ciently large then the assembly map

Bÿ_‡^L^ÿ1…R† !L^ÿ1…Rÿ† is equivalent to an inclusion of a direct summand of spectra.

We also have results about splitting assembly maps when the action is not small at in¢nity

Theorem B. Assume ÿ is a group with a finite Bÿ and that Eÿ has an equivariant compact Hausdorff compactification which is Cïech contractible and such that there exists a family of coveringsf of@Eÿ by sets which are boundedly saturated (see Definition 8.16 ), which is invariant under the group action and such that the Cïech homotopy type defined by the family f is homotopy equivalent to theCïech homotopy type of@X. Then

a)If R is any ring then the assembly map

Bÿ‡^K^ÿ1…R† !K^ÿ1…Rÿ†

is equivalent to an inclusion of a direct summand of spectra.

b)If R is a ring with involution such that Kÿi…R† ˆ0 for i su¤ciently large then the assembly map

Bÿ_‡^L^ÿ1…R† !L^ÿ1…Rÿ† is equivalent to an inclusion of a direct summand of spectra.

Boris Goldfarb in his Cornell thesis [10] has veri¢ed these conditions for various groups. Speci¢cally he treats groups ÿ such that ÿ is the (tor- sion-free) fundamental group of a complete non-compact ¢nite-volume Riemannian manifold with pinched negative sectional curvatures:

ÿb²K…M† ÿa²<0. TheL-theory assembly map was known to be split for this class of groups, but not the algebraicK-theory assembly map.

Throughout this paper we shall use the language of algebraic K-theory, the modi¢cations needed to deal withL-theory are immediate using [5]. The results of this paper do work to split assembly maps inA-theory and topo- logicalK-theory as well, by using the excision results in [7] , [12] and [6]. For the readers convenience we state

TheoremC. Assumeÿ is a group as in Theorem A or B. Then

a)IfX is a space withKÿi…1…X†† ˆ0forisu¤ciently large, then the assem- bly map

Bÿ‡^Aÿ1…X† !Aÿ1…BÿX† is equivalent to an inclusion of a direct summand of spectra.

b)IfCis aC-algebra then the assembly map

Bÿ_‡^K^top…C† !K^top…C_rÿ†

is equivalent to an inclusion of a direct summand of spectra.

We would like to thank the referee for numerous useful suggestions.

2. Preliminaries.

Throughout this paper,sswill denote the category of based simplicial sets andk will denote the full subcategory of Kan complexes. We will assume familiarity with the standard properties of homotopy inverse limits, also called homotopy limits, as presented in [3]. We shall conventionally only consider homotopy limits of Kan complexes, so if we ever encounter a homotopy limit in the category of simplicial sets, it is to be understood that we ``Kan-ify'' before taking the homotopy limit. Homotopy limits and coli- mits are extended to the category of spectra, by doing the constructions in each degree. IfCandDare categories,:D!kis a functor, andf:C!D is a functor, the induced map limD!limCf will be referred to aspull- backorrestriction. Recall also that ifN:! is a natural transformation of k-valued functors on D, then N induces a map limD!limD . As mentioned above, we will only consider homotopy inverse limits of functors with values in k. This means that the cosimplicial spaces used in de¢ning

the homotopy inverse limits are always ¢brant, and all homotopy inverse limits will themselves be Kan complexes. A map of simplicial sets will be called aweak equivalenceif its geometric realization is a homotopy equiva- lence. A map between Kan complexes is a weak equivalence if and only if it is a homotopy equivalence, i.e. there is a two-sided homotopy inverse map.

We will refer to a weak equivalence between Kan complexes as an equiva- lence.

Similarly, if :C !ss is any functor, we may construct the homotopy colimit of over C, hocolimC. In this case we have pushforward maps corresponding to functors f:C!D, and natural transformations again in- duce maps of homotopy colimits. If ; :C !ss are functors, and N:! is a natural transformation, we say N is a weak equivalence if N…c†is a weak equivalence for each objectcofC. Weak equivalences ofss- valued functors induce weak equivalences on homotopy colimits, and weak equivalences of k-valued functors induce weak equivalences of homotopy limits. We also recall from [4] that ahomotopy natural transformationfrom a functor to is a sequence of functors _i and _i, for iˆ0;1;. . .;k, to- gether with a family of natural transformations _i ! _i and a family of natural equivalences _i! _iÿ1 for i>0, where ₀ ˆ and _k ˆ . A homotopy natural transformation induces a homotopy class of maps on homotopy colimits and homotopy limits.

If we have a diagram

Y ^f X!^g Z

in ss, its homotopy pushout will be the double mapping cylinder Y`

XI`

Z=, where …x;0† f…x† and …x;1† g…x†. Note that this is the homotopy colimit of the diagram. We say a commutative diagram of simplicial sets

X ! Y

# #

Z ! W

is homotopy co-Cartesian if the natural map Y[XZ!W is a weak equivalence. Similarly, given a diagram

Y!^s W ^t Z

in k we de¢ne the homotopy pullback of the diagram as the subspace of YW^IZ of points…y; ;z† so that…0† ˆy and…1† ˆZ. W^I here de- notes the function complex of maps from the simplicial unit interval toW. This pullback is denoted byYW Z. We say a commutative diagram

X ! Y

# #

Z ! W

is homotopy Cartesian if the natural mapX!Y_WZis an equivalence.

Traditionally a homology theory was de¢ned [1] to be a functor h…X†

from CW-complexes to graded abelian groups satisfying the Eilenberg Steenrod axioms except for the dimension axiom. Using Brown's represent- ability theorem [1] a homology theory has a representing spectrum S. This means that a homology theory can be written as …X_‡^S† for a suitable spectrum S or …X^S†for the corresponding reduced theory h…X;x₀;S†

on pointed spaces. The functorX !X^Sfrom ¢nite CW-spaces to spectra is homotopy invariant and sends co¢brations of spaces to co¢brations of spectra and the one-point space to a contractible spectrum. The functor X!¹¹…X^S†from spaces to spaces sendingX to the 0-th space of the in¢nite loop spectrum corresponding to X^S is homotopy invariant and sends co¢brations to ¢brations. On the other hand if f is a homotopy in- variant functor sending co¢brations to ¢brations and a point to a con- tractible space, it follows that…f…X††is a homology theory in the classical sense. It is easy to see thatf…X††is homotopy equivalent tof…X†, sof…X†

is the 0-th space of an in¢nite loop spectrum. This spectrum is the re- presenting spectrum of the homology theory …f…X†† as is shown in [19].

We may thus think of a homology theory in various equivalent ways. Since we shall need the results of [3] on homotopy limits and colimits it is practical to work simplicially rather than with spaces. We shall use the following de-

¢nition:

Definition 2.1. A functorT:ss!kis said to be a homology theory if a) The induced map T…X0† !T…X ‰0;1Š†is an equivalence for all sim-

plicial setsX. b)T…†is contractible.

c) For any homotopy co-Cartesian diagram X ! Y

# #

Z ! W the induced diagram

TX ! TY

# #

TZ ! TW is homotopy Cartesian.

It follows that if we have an inclusioni:X,!Y, we obtain a sequence TX !TY!T…Y[XCX†

which is a ¢bration ``up to homotopy'' in the sense that the natural map from TX to the homotopy ¢ber of the map TY!T…Y[_XCX† is an equivalence.

Remark 2.2. Given a (simplicial) spectrum swe get a homology theory in this sense, by definingT…X†to be the zero'th space of¹S¹…X^sfol- lowed by a functor turning a simplicial set into a weakly equivalent Kan simplicial set.

We will also need standard information concerning the construction of spectra from category theoretic data. The following theorem covers what we will need. For more detail on the terminology in the statements, see [18].

Theorem2.3. There is a functor from the category of symmetric monoidal categories and lax symmetric monoidal functors to the category of spectra sa- tisfying the following conditions.

1.If f:C!D is a lax symmetric monoidal functor and N.…f†, the induced map on the nerve, is a weak equivalence of simplicial sets, thenSpt…f†is a weak equivalence of spectra.

2.For any symmetric monoidal category C, letSpt₀…C†denote the zeroth space of the spectrumSpt…C†. There is a natural map N.C!Spt₀…C†, which in- duces an isomorphism

…0N.C†^ÿ1H…N.C†H…Spt₀…C††

3.Let f:C!D be a unital symmetric monoidal functor between unital sym- metric monoidal categories C and D, and suppose₀…C†contains a co¢nal submonoid M so that₀…f†…M†is also a co¢nal submonoid of₀…D†. Sup- pose further that for every object x2D lying in an equivalence class belong- ing to 0…f†…M†, x#f ( or f #x) has a weakly contractible nerve. Then i……f††is an isomorphism for i>0.

4.If:AB!C is a symmetric monoidal pairing, then there is an induced pairing of spectra

Spt…†: Spt…A† ^Spt…B† !Spt…C†

so that the composite

N.AN.B!Spt₀…A† ^Spt₀…B† ! …Spt…A† ^Spt…B††₀!Spt₀…C†

is equal to the composite

N.AN.Bÿÿ!^N. N.C!Spt₀…C†

Proof. The first 2 points are [18, Lemma 2.3] and [18, Condition 2.2], and 3. and 4. are proved in [4, Theorem I.6].

We also recall the notationh…X;x₀;sfor the homology of the based space …X;x0† ``with coe¤cients in the spectrum s'', or the smash product of the spaceX with the spectrums.

We will be using homotopy inverse limits over certain categories of cov- erings of topological spaces in our de¢nition of Cïech homology. These cate- gories have certain properties which make them convenient to work with, and we discuss these now. Recall from [9] that a categoryCis said to beleft

¢lteringif (i.) for every pair of objectsc;c⁰inC, there exists an objectc⁰⁰to- gether with maps c ÿc⁰⁰!c⁰, and (ii.) for every pair of mapsf;g:c⁰!c, there exists a morphism h:c⁰⁰!c⁰ so that f h and gh are equal. If the category C happens to be a partially ordered set, i.e. there is at most one morphism between any pair of objects, then this reduces to the requirement that for any pair of objectsc;c⁰ ofC, there is ac⁰⁰so that there are morph- isms c⁰⁰!c and c⁰⁰!c⁰. We will adopt the convention that a partially or- dered set in the usual sense is made into a category by declaring that there is a morphism fromx toy if and only ifxy. If this category is left ¢ltering we shall say the partially ordered set is left directed. Note that it follows as in [16] that the nerve of any left ¢ltering category is weakly contractible, and hence the nerve of any left directed partially ordered set is weakly con- tractible.

Proposition 2.4. Let C and D be two left directed partially ordered sets, and suppose we have an order preserving map f:C!D. Further, suppose that is a functor from D to the category of Kan simplicial sets, and that for every element z2D there is an element x2C so that f…x† z. Then the pullback map

holim

D

!holim

C

f

is a weak equivalence.

Proof. From [3], it will suffice to show that the category f #zhas con- tractible nerve for eachz2D. But it is clear that each categoryf #zis itself a left directed partially ordered set, and the hypothesis of the proposition shows that it is non-empty. Therefore, its nerve is contractible.

We also have the following standard fact(see [3]).

Proposition 2.5. Let C be a category, and suppose that we have functors F;G;H and K from C to the category of based Kan complexes, and that we have a commutative diagram of natural transformations as follows.

F ! G

# #

H ! K

Suppose further that for each c2C, the diagram ishomotopy Cartesian, i.e.

that the natural map from F…c† to the homotopy pullback of the diagram H…c† !K…c† ÿG…c†is an equivalence. Then the diagram

holimF ! holimG

C C

# #

holimH ! holimK

C C

is also homotopy Cartesian.

We will also need some conditions which assure that a natural transfor- mation between functors from a left directed partially ordered set to the ca- tegory of Kan complexes induces a weak equivalence on homotopy inverse limits.

Proposition2.6. Let C denote a left partially ordered set, and suppose that we are given a natural transformation :F !G of functors from C to k.

Suppose further that for every c2C, there is a c⁰c so that …c⁰† is an equivalence. Then the mapholim_CF !holim_CG induced by is an equiva- lence.

Proof. This is a straightforward consequence of [3].

We must also understand the behavior of restriction maps on inverse lim- its. Preparing for this we state

Lemma 2.7. Let f :E!C be an order preserving map of partially ordered sets, and T :E!

Lemma k a functor. For everyy2C letEy be the full subcategory of E consisting ofy⁰so thatyf…y⁰†then

holimT'holim holimTjEy

E C E y

Proof. Let Z denote the partially ordered set consisting of pairs …c;e†

with cf…e†. Then the iterated homotopy limit may be identified with the homotopy limit over Z of the functor T⁰:Z!k sending …c;e† to T…e†.

Moreover there is a forgetful functor i:Z!E sending …c;e† to e, and T⁰ˆTi. It is readily checked thatisatisfied the hypothesis of [3, Theorem XI.9.2]. Henceiinduces an equivalence on homotopy inverse limits.

Lemma 2.8. Let CD be an inclusion of left directed partially ordered sets. Let F be a functor from D to the category of Kan complexes. Suppose that for every x2D, there exists x⁰2D and a y⁰2C, with x⁰x and x⁰y⁰ and with F…x⁰† !F…y⁰† a weak equivalence. Then the restriction map holimDF !holimCF is a weak equivalence.

Proof. Let E be the partially ordered set whose objects are pairs …x;y†, withx2D,y2C, andxy. We have functorsr:E!Dandi:C!E, with r…x;y† ˆx and i…y† ˆ …y;y†. Note that ri is equal to the inclusion C,!D.

Let E^ E be the full subcategory on all …x;y† so that F…xy†is a weak equivalence. Of course,i…C† E. The hypothesis shows that the restriction^ ofrtoE^ satisfies the hypotheses of 4, so the natural map

holimF!holimFr

D E^

is an equivalence. It now su¤ces to show that the restriction map holimFr!holimFri

^

E C

is an equivalence. LetT denote the functor fromE^ to Kan complexes given by T…x;y†ˆF…y†. There is an evident natural equivalence of functors on E^ from Fr to T, given by F…xy† so when restricted to i…C† it gives the identity equivalence. Hence

holimFr'holimT

^

E ^E

Note thatTCˆFri. Consequently, it su¤ces to show that the restriction map holim_E^T!holim_CTi is a weak equivalence. To prove this, given any y2C, we let E^y denote the full subcategory on those…x;y⁰†for which yy⁰. We use Lemma 2.7 to express holimE^T as an iterated homotopy limit

holimT'holim holimTjE^_y

^

E y2C ^_{E y}

it is easy to see that the diagram

holimT !holim holimTjE^_y

^

E y2C ^_{E y}

# #

holimTiˆholim T…y†

C y2C

is commutative. The right hand vertical map is given by restriction along the inclusion f…y;y†g E^_y. In view of 2.6, it thus su¤ces to check that holim_E^yTjE^_y!T…y† is an equivalence. Now, by 2.4, it is clear that if E_y E^y is the subset on objects of the form…x;y†, then the restriction func- tor holimE^yT!holimE_yT is an equivalence. Consequently, it su¤ces to check that the map holim_EyT !T…y†is an equivalence. ButT isconstanton E_y with valueT…y†, so the homotopy limit overE_y can be identi¢ed with the function complex F…N.E_y;T…y††, and the restriction map is simply restric- tion along the inclusion of nerves of the one object category…y;y†intoN.E_y. ButN.E_yis weakly contractible since it has a ¢nal object…y;y†.

3. Lemmas on Coverings.

By a covering of a topological space X, we mean a parameterized family uˆ fUg_2Aof open subsets ofX, whereAis a set, so thatXˆS

2AU. A map of coverings from fUg_2A to fVg_2B is a set map f:A!B so that UV_f…† for all2A.

By a simplicial complexwe mean, as usual, a vertex setV and a family of ¢nite subsetsp, so that ifU2p andU⁰UthenU⁰2p. Simplicial maps are de¢ned in the usual way. Two simplicial maps f;g:1!2 are said to bes-homotopic if for every U 21, f…U† [g…U† 22. The join of two simplicial complexes and t, t, has V`

VT as its vertex set, and a subset ofV_tis inp_tif and only if it is the union of an element of V with an element ofV_t.

For any setX, let f…X†denote the partially ordered set of nonempty ¢- nite subsets ofXand inclusions. As usual we may view af…X†as a category , which by our conventions is the opposite category of the category of non- empty subsets and inclusions. The functorX !N.f…X† gives a covariant functor fromSetstoss. Given a simplicial complex, let the realization of , R., be the nerve of the full subcategory of f…V† with objects the subsets belonging to p. This is the simplicial version of the barycentric subdivision of the usual realization. The functorR. preserves pushouts and carriess-homotopic maps to simplicially homotopic maps.

For any coveringuˆ fUg_2Aof a topological space, let _u be the sim- plicial complex whose vertex set isA, and wheref1;. . .; kgis a simplex of uif and only ifU1\. . .\Uk6ˆ ;. By the nerve of the covering,N.u, we will meanR.u.

Lemma 3.1. If uandv are any coverings of X, and f;g:u!vare any maps of coverings, then N.f and N.g are simplicially homotopic.

Proof. This result follows directly from the above discussion ofs-homo-

topies, since if f and g are both maps of coverings from u to v, and f₁;. . .; _kg 2_u, then

ff…₁†;. . .;f…_k†;g…₁†;. . .;g…_k†g 2_v since

U₁\. . .\U_k U_f…1†\. . .\U_f…k†\U_g…1†\. . .\U_g…k† Thus,f andgares-homotopic.

Corollary 3.2. Let f:u!v be any map of coverings. Suppose further that there is a map of coverings fromvtou. Then N.f is a weak equivalence.

We will also de¢ne certain other simplicial sets associated tou. Suppose BA.

Definition 3.3. ByN.^Buwe mean the realization of the simplicial com- plex _u^B, whose simplices are the subsets f₁;. . .; _kg B so that U1\. . .\Uk 6ˆ ;. We define e.^Bu to be the realization of the simplicial complex t^B_u, whose vertex set is B, and so that any finite subset f1;. . .; kg Bis a simplex.

Of course,e.^Buis weakly contractible. All these simplicial sets are viewed as subsimplicial sets ofe.uˆe.^Au.

Lemma 3.4. Let uˆ fUg_2A be a covering of a space X, and let B⁰BA be subsets of A. Suppose further that for each2B, there is a ⁰2B⁰ so that UU0. Then the evident inclusion N.^B0u!N.^Bu is an equivalence.

Proof. Letf:B!B⁰ be any function so thatUU_f…†, andfjB⁰ˆid_B0. f induces a map of nervesN.f, and it is clear that the composite

N.^B0u!N.^Bu!N.^B0u is equal to the identity. On the other hand, the composite

N.^Bu!N.^B0u!N.^Bu

is simplicially homotopic to the identity in view of the fact that U₁\. . .\U_k6ˆ ; )U₁\. . .\U_k\U_f…1†\. . .\U_f…k†6ˆ ; This gives the result.

Consider also the following situation. Letuˆ fUg_2Abe a covering of a spaceX, and letBAbe a subset. LetW_u…B† X be the setS

2BU. Let ^B…u† ˆ fWg_2A be the covering of X given by WˆU if =2B, and

WˆWu…B†if2B. LetP…B† ˆPu…B† Abe the set of all2Aso that U\W…B† 6ˆ ;. We may view^B_u† as a subcomplex oft_u, and as such it is contained in the subcomplex_u[P…B†

u t^P…B†_u . This is true since it is clear from the de¢nitions that a simplex f1;. . .; kg of ^B_u† either contains a vertex inB, in which casei2P…B†for alli, or it does not, in which case it is in_u. Also it is clear that

u[_u^Bt^B_u^B_u†

Lemma 3.5. Let u;A, and B be as above. Suppose that for any f1;. . .; kg Pu…B†,

U₁\. . .\U_k 6ˆ ; )U₁\. . .\U_k\W_u…B† 6ˆ ; Then the inclusion

^B_u†,!_u[P…B†

u t^P…B†_u induces an equivalence on nerves.

Proof. We first note that ^B_u† is the union of _u with ^P…B†B…u†; the overlap is_u^P…B†. Consequently, we have a map of pushout diagrams

u ÿ_u^P…B†!^P…B†B…u†

# # #

u ÿ_u^P…B†!t^P…B†_u

of simplicial complexes. Since the realization and nerve constructions pre- serve pushouts, it will su¤ce to show that ^P…B†B…u†!t^P…B†_u induces a weak equivalence on nerves. Since N.t^P…B†_u is contractible, it will su¤ce to show thatN.^PB^u…u†^B†is contractible. WriteP_u…B† ˆB`

Q…B†. A typical simplex in ^PB^u…u†^B† is of the form f1;. . .; s;q1;. . .;qtg, with i2B and qi2Q…B†.

Clearly f1;. . .; s;q1;. . .;qtg is a simplex of ^PB^u…u†^B† if and only if fq1;. . .;qtg is a simplex in_u^Q…B†. Consequently^PB^u…u†^B† is the join of^Q…B†B…u†

with ^BB…u†. Since ^BB…u†ˆt^B_u, whose nerve is contractible, the result fol- lows.

We wish to consider subspaces also. A relative covering of a subspace YX is a parameterized familyuˆ fUg_2A of open subsets U of X so that Y S

2AU. _u and N.u are de¢ned precisely as before, i.e. the simplices of _u are ¢nite subsets f₁;. . .; _kg so that U₁\. . .\U_k 6ˆ ;.

For any relative coveringuofY X, we have the coveringuofY, where uˆ fU\Yg_2A. There is an evident map u!u given by

f1;. . .; kg ! f1;. . .; kg, and hence a map of simplicial sets N.u!N.u.

Lemma 3.6. Suppose X is a compact Hausdorff space and Y is a closed subspace. Let uˆ fUg_2A be a finite open covering of Y. Suppose further that for each, we are given a closed set WUand thatfWg_2Ais also a covering of Y. Then there is a relative covering of Y in X,fVg_2A, so that WY\V\U, and so that for every subsetf1;. . .; kg A,

V1\. . .\Vkˆ ; ,W1\. . .\Wk ˆ ;

Proof. This is a straightforward generalization of the fact that in a com- pact Hausdorff space, any two disjoint closed subspaces are contained in disjoint open sets. We leave the proof to the reader.

4. Rigid Coverings and C¯ech Homology.

In our construction of Cïech homology, it will be important that the category of coverings used as parameter category for certain homotopy inverse limits is a left directed partially ordered set. We will use an analogue to the ``rigid coverings'' used by Friedlander [9] in his construction of the ``etale topolo- gical type '' associated to a scheme.

Definition 4.1. LetX be a topological space. A rigid covering of X is a function from the underlying set of X to the collection of open subsets of X satisfying the following three conditions.

a)x2…x†

b) For any open setUX,^ÿ1U U

c) Only ¢nitely many distinct open sets occur among the sets…x†. That is, the image of is a ¢nite collection of the collection of open subsets ofX.

Let RC…X†denote the set of all rigid coverings ofX. If₁; ₂2RC…X†, we say ₁ re¢nes ₂ and write ₁₂ if and only if ₁…x† ₂…x† for all X, RC…X† now becomes a partially ordered set, and hence can be viewed as a category. We also de¢ne a relative version, where Y X is a subspace. A relative rigid covering ofY inXis a functionfrom the underlying set ofY into the open subsets ofX, which satis¢es properties a)^ c) above. We simi- larly obtain a category RC…X;Y†.

Thus, a rigid covering is an open covering ofX, with parameter setX, so that only ¢nitely many distinct subsets occur. It turns out thatX!RC…X†

de¢nes a contravariant functor from the category of pointed topological spaces to the category of pointed small categories.

Proposition 4.2. For any map of topological spaces f:X !Y, and 2RC…Y†, we define a function f^! from the underlying set of X into the collection of open subsets of X by the formula f^!…x† ˆf^ÿ1…f…x††. Then f^!is a rigid covering of X, and the formulae X!RC…X†and f !f^!make RC…ÿ†

into a contravariant functor.

Proof. All conditions defining RC…X†are clear except b). To check b), let U be an open subset occurring in the image of , i.e. Uˆ…y†. Then f^!ÿ1…f^ÿ1U†is equal to the union of the inverse images underf of all the sets ^ÿ1V, as V ranges over all the sets in the image of for which f^ÿ1V ˆf^ÿ1U. Note that this is clearly a finite union, and the closure of each setf^ÿ1V is contained inf^ÿ1U. Since for a finite union,

V₁[. . .[V_kˆV₁[. . .[V_k condition b) follows.

We will also need a kind of product of rigid coverings.

Proposition 3. Let₁ and₂ denote rigid coverings of a space X. We de- fine a new function_1X2 from the set X into the collection of open subsets of X by the formula 1X2…x† ˆ1…x† \2…x†. Then 1X2 is a rigid covering of X. Furthermore, it refines both1 and2.

Proof. As in the preceding proposition, all is clear except the fact that 1X2 satisfies condition b) in the definition of rigid coverings. To check this condition, we construct first the function 12 fromXX into the collection of open subsets ofXX by the formula

…12†…x1;x2† ˆ1…x1† 2…x2†

₁₂is evidently a rigid covering ofXX. Now, if we let:X !XX denote the diagonal map, then

1X2ˆ^!12

and the result follows directly from 2.

It thus follows that RC…X†is a left directed partially ordered set. Given an arbitrary covering uˆ fUg_2A, the reader may wonder if there is a rigid covering of X, so that for allx2X,…x† ˆU_x† for some …x†. Indeed, this question is important for us for technical reasons. The following lemma will be useful.

Lemma4. Given any coveringuˆ fUg_2Aof a compact Hausdorff space, there is a rigid coveringof X so that for each x2X, there is an…x† 2A so that…x† ˆU_x†.

Proof. Since X is compact, we may assume that A is finite. Since X is compact Hausdorff, there is a coveringfVg_2A, withVUfor all. Let :X !A be such that x2V_x† for all x. Then define by …x† ˆU_x†. This gives the required rigid covering.

Lemma 4.5. For every2RC…X†and closed subset Y, there is an open set U, with Y U, and a refinement 2RC…X†, , so that for every x2U,…x† …y†for some y2Y.

Proof. Consider jY2RC…X;Y†. We first construct an open setV and ~2RC…X† so that jY~ ˆjY, and so that for all x2V, …x† ~ …y†~ for some y. Let UˆS

y2Y…y†. This is an open set containing Y. Since X is compact Hausdorff, there is an open setV, withY V V U. Consider the open covering consisting of all the sets…y†together withXÿV. Again since X is compact Hausdorff, we may select an open covering fZyg_y2Y[ fWg so that Zy …y† and WXÿV. For every x2XÿY, choose…x†~ to be either(i.)…y†, whereyis such thatx2Zy, or(ii.) XÿV if x2W. (Note that both possibilities can occur simultaneously, so the construction of involves choices.) If x2Y, set …x† ˆ~ …x†. With this choice of V and , we clearly have that if~ x2V, …x† ~ …y† for some y2Y. Now setˆ~X. With the same choice ofV, this clearly has the required properties.

Lemma 4.6. Let X be a compact Hausdorff space, and Y X a closed subspace. Let2RC…X;Y†. Then there is a rigid covering^2RC…X†so that jY^ ˆ.

Proof. Let Z denote the open set S

y2Y…y† X. Of course, Y Z.

SinceX is compact Hausdorff and Y is closed, we may choose an open set V, so thatV XÿZ, and so that V\Yˆ ;. LetfUg_2Abe an indexing of the sets which occur in the image of , together with V, so A is finite.

SinceX is compact Hausdorff andY is closed, there is a family of open sets fWg_2A with WU and W^ÿ1…U† for all 2A, and so that fWg_2A is a covering of X. For each y2Y, set …y† ˆ^ …y†. For each x2XÿY, find an2Aso thatx2U, and set…x† ˆ^ U. This construc- tion gives the required rigid covering.

Lemma 4.7. Let X be a compact Hausdorff space, and let Y X be a closed subspace. Let be any rigid covering. Then there is a rigid covering , so that

…x₁† \. . .\…x_k† ˆ ; )…x₁† \. . .\…x_k† ˆ ;

for allfx1;. . .;xkg. Further,can be chosen so that

…x1† \. . .\…xk† \Yˆ ; )…x1† \. . .\…xk† \Y ˆ ; Proof. LetfUg_2A be a finite listing of all the subsets of the form…x†

for some x2X. Let VU be ^ÿ1…U† U. Since X is compact Haus- dorff, we can choose open setsW, so thatVWWU, and define a new rigid covering^1†by^1†…x† ˆWif and only if…x† ˆU. Note that

…x₁† \. . .\…x_k† \Yˆ ; )^1†…x₁† \. . .\^1†…x_k† \Y ˆ ; Note also thatwˆ fWg_2Ais now a finite covering, and the identity map onAproduces an inclusion of finite simplicial complexesw,!u. Repeat- ing this process gives a descending chain of finite simplicial complexes on the same vertex setA, which must eventually stabilize. This means that we have open setsW⁰ andW, with

VW⁰ W⁰ WWU so that

W⁰₁\. . .\W⁰_k ˆ ; )W1\. . .\Wk ˆ ;

which implies that W⁰₁\. . .\W⁰_kˆ ;. Define by …x† ˆW⁰ if and only if…x† ˆU.

Definition 4.8. Let …X;x₀† be a topological space, and let _X be the functor from RC…X† to ss given by !N.. For any homology theory T:ss!k from based simplicial sets to based Kan complexes, we define the Cïech homology ofX with ``coefficients inT'' ,h…X;x₀;T†, to be

holim

RC…X†TX

This de¢nes h on objects. If f:X!Y is a map of topological spaces, we de¢neh…f;T†to be the composite

holimTX!holimTXRC…f†!holimTY

RC…X† RC…Y† RC…Y†

where the left arrow is pullback of homotopy inverse limits along the functor RC…f† and where the right hand arrow is induced by the evident natural transformation_XRC…f† !_Y.

Notice that the basepointx₀2X determines a basepoint inh…X;x₀;T†, so h…ÿ;T†can be viewed as a functor from the category of based spaces tok.

We will occasionally suppress the basepoint when no confusion will result.

5. Excision.

Throughout this section, let X denote a compact Hausdor¡ space and let YX denote a closed subspace. Let T denote a functor from the category of based simplicial sets to the category of based Kan complexes which is a homology theory in the sense of De¢nition 2.1. Let X be any compact Hausdor¡ space, and letYXbe a closed subspace. In this section, we will prove ``strong excision '' for the functor h…ÿ;T†, i.e. that the sequence of maps

h…Y;y₀;T† !h…X;x₀;T† !h…X=Y;‰y₀Š;T†

is a ¢bration up to homotopy in the sense that the evident map from h…Y;y0;T†to the homotopy ¢ber of the maph…X;x0;T† !h…X=Y;‰y0Š;T†is an equivalence of Kan complexes.

We ¢rst observe that for any based pair of spaces…X;Y†, we have a com- mutative diagram

h…Y;y0;T† ! h…;;T†

# #

h…X;x₀;T† ! h…X=Y;;T†

Theorem 5.1. Let X be a compact Hausdorff space, and Y X a closed subspace. Then the above diagram is homotopy Cartesian.

Proof. The strategy will be to find a weakly equivalent diagram which is induced by a diagram of functors over RC…X†, and to apply 2.5 suitably. Let i:Y,!X and j:,!X=Y denote the inclusions and let p:X !X=Y and q:Y!denote the projections onto the quotient space. The above diagram may be written as follows.

holim TY!holim T

RC…Y† RC…†

# #

holim TX!holim T_X=Y

RC…X† RC…X=Y†

It follows from Lemma 4.6 that the conditions of Proposition 2.4 are sa- tis¢ed, so this diagram is weakly equivalent to the new diagram

holim TYRC…i†!holim TRC…j†

…A† RC…X† RC…XY†

# #

holim TX!holim T_X=Y

RC…X† RC…X=Y†

where the vertical arrows are induced by natural transformations _Y RC…i† !_X andRC…j† !_X=Y. The horizontal arrows are pull- back maps along RC…p†composed with maps induced by the natural trans- formation from Y RC…i† RC…p† ˆY RC…q† RC…j† !RC…j†

and _XRC…p† !_X=Y. Exhibiting only the natural transformations we get the diagram of functors and natural transformations

_Y RC…pi† !RC…j†

# #

_XRC…p† !_X=Y

Denoting the functor RC…Z† !sssending toR.t byC_Zfor any space Z, and the constant functor with value the one point simplicial set bye, this diagram of natural transformations factors as described in the following diagram.

Y RC…pi† ÿÿÿÿÿÿÿÿÿ!CY RC…pi† ÿÿÿÿÿÿÿÿÿ!RC…j†

# # #

XRC…p† ÿÿ!XRC…p† [_YRC…pi†CY RC…pi† ÿÿ!_X=Y and consequently we have a map of diagrams from the diagram…B†below to diagram…A†.

holim _Y RC…i† ÿÿÿÿÿÿÿ! holim C_YRC…i†

…B† RC…X† RC…X†

# #

holim _X!ÿÿÿÿ! holim _X[_YRC…i†C_YRC…i†

RC…X† RC…X=Y†

The maps on the left hand part of the diagram are identity maps, and the map in the upper right hand corner is an equivalence since both entries are contractible. Further, the last diagram is homotopy Cartesian in view of Proposition 2.5. Consequently, it su¤ces to show that the map

holim T …X[_YRC…i†CYRC…i†† !holim T_X=Y

RC…X† RC…X=Y†

is an equivalence. The proof of this result is in two stages, since this map is actually a composite of two maps, one

holim T…X…† [_YRC…i†…†CYRC…i†…†† !

2RC…X†

holim T…XRC…p†…† [_{YRC…pi†…†}CYRC…pi†…††

2RC…X=Y†

and the other

holim T…XRC…p†…† [_{YRC…pi†…†}CYRC…pi†…†† !holim _X=Y:

2RC…X=Y† RC…X=Y†

The ¢rst is restriction along the inclusion RC…p†, and the second is induced by a natural transformation.

We analyze the restriction map ¢rst. For any 2RC…X†, let ⁰ be the rigid covering of Y given by ⁰…y† ˆ…y† \Y. Thus, YRC…i†…† ˆ⁰. We have a natural inclusion of simplicial complexes⁰,!^Y. We will show

¢rst that for every2RC…X†, there is a so that the natural inclusion N.⁰!N.is an equivalence. To construct, letfUg_2Abe a listing, with

¢nite index set, of the distinct subsets occurring in the image of⁰jY, and let WU be the closed sets…⁰†^ÿ1…U†. According to 3.6, there exist open setsVinX so thatWV\YU, and so that

V₁\. . .\V_k 6ˆ ; )W₁\. . .\W_k 6ˆ ;

Let~2RC…X;Y†be de¢ned by …y† ˆ~ V if and only if ⁰…y† ˆU. Then by 4.6, it is possible to ¢nd a rigid covering^ofX, withjY^ ˆ.~ will now be taken to be_X.^ _XjY^ ˆ, and from the construction of~ , it is~ clear that the map ⁰ !^Y is an isomorphism of simplicial complexes.

This is the required result. From the de¢nition of homotopy pushouts, it follows that the natural map

holim T…R.[R.0R.tY† !holim T…R.[_R.^YR.t^Y†

2RC…X† 2RC…X†

is an equivalence. Further, it is clear that if2RC…X=Y†,⁰ ˆ^Y, and it will therefore su¤ce to show that the restriction map

holim T…R.[_R.^Y

†R.t^Y† !holim T…R.[_R.^Y

R.t^Y†

2RC…X† 2RC…X=Y†

is an equivalence. We will show this via a series of equivalences. We have a commutative diagram

holim T…R.[_R.^Y

R.t^Y† ÿÿÿ!holim T…R.[^W…Y†

R.t^{W …Y†}†

2RC…X† # 2RC…X† # holim T…R.[_R.^Y

R.t^Y† !holim T…R.[W…Y†

R.t^{W …Y†}†

2RC…X=Y† 2RC…X=Y†

where the horizontal arrows are induced by natural inclusions of simplicial

complexes, and where the vertical arrows are restriction maps. We will show that the horizontal arrows are equivalences. It will then follow that if the right hand vertical arrow is an equivalence, then so is the left hand arrow, and we will then proceed to show that the right hand vertical arrow is an equivalence.

To show that the upper horizontal arrow is an equivalence, it will su¤ce, in view of the fact that T is a homology theory and that R.t^Y and R.t^W…Y† are contractible, to show that

holim T…R.^Y† !holim T…R.^W…Y††

2RC…X† 2RC…X†

is an equivalence. In view of 4.5, for any2RC…X†, there is a , so that for any x2W…Y†, there is a y2Y so that …x† …y†. From 3.4, it follows that the inclusion ^Y !^W…Y† induces a weak equivalence on nerves, and the result now follows from 2.6. To see that the lower horizontal arrow is an equivalence, it similarly su¤ces to show that holim_2RC…X=Y†T…R.^Y† !holim_2RC…X=Y†T…R.^W…Y†† is an equivalence.

This follows as above with the additional observation that we may taketo lie in RC…X=Y†.

In order to prove that the restriction map holim T…R.[W…Y†

t^{W …Y†}† !holim T…R.[W…Y†

t^{W …Y†}†

2RC…X† 2RC…X=Y†

is an equivalence, consider the following commutative diagram.

v2

holim T…^W…Y†_‰Š†

holim T…[W…Y†

t^{W …Y†}† !holim T…[PW…Y†

t^PW…Y† †

holim T…^W…Y†_‰Š†

holim T…[W…Y†

t^{W …Y†}† !^hl holim T…[PW…Y†

t^{PW…Y †}† The triangles at either end of the ``prism'' are induced by natural inclusions of simplicial complexes, as described in section 5. The vertical arrows are all

f^u

f^l

v₁ v₁

g^u

gⁱ

2RC…X†

2RC…X†

2RC…X=Y†

2RC…X†

2RC…X=Y† 2RC…X=Y†

restriction maps. We wish to show that v1 is an equivalence; for this it will su¤ce to show that f^u;f^l, and v₂ are equivalences. We ¢rst deal with f^u. From the diagram, it will clearly su¤ce to show that h^u and g^u are equiv- alences. To show that h^u is an equivalence, we observe that in view of the fact that T is a homology theory and the contractibility of t^W…Y† and t^PW…Y† it will su¤ce to show that the inclusion

holim T…R.^W…Y†† !holim T…R.^PW…Y††

2RC…X† 2RC…X†

is an equivalence. SinceT is a homology theory, it will su¤ce to show that holim T…R.^PW…Y†=R.^W…Y††

2RC…X†

is contractible. For this, it will su¤ce by 2.6 to show that for every 2RC…X†, there is a 2RC…X†, with, so that the map

R.^PW…Y† =R.^W…Y† !R.^PW…Y†=R.^W…Y†

is simplicially homotopic to a constant map. We ¢rst choose⁰, so that for everyx2Xwith⁰…x† \Y6ˆ ;, there is ay2Y with⁰…x† …y†. Now, let U be an open set, with Y US

fxj⁰…x†\Y6ˆ;g⁰…x†. Also, choose V open, withY V V U. These choices are possible sinceX is compact Hausdor¡. Consider the open coveringfU;WÿV;XÿUg. Letbe a rigid covering ofX so that for eachx2X,…x†is one of these sets, the existence of which is guaranteed by 4.4, and consider ˆ⁰X. We have ⁰, and furthermore it is clear from the de¢nitions that if we let Wˆ fxj…x† \Y6ˆ ;gandW^ ˆ fxj…x† \W6ˆ ;g, thenW^ W. From this it follows that we may de¢ne a function fromP…W…Y; †† to Y so that…x† ……y††. We may also insist that…y† ˆyfor ally2W…Y; †, since re¢nes . Since maps of coverings induce maps of the associated simplicial complexes, we obtain a map of simplicial complexes ^PW…Y† !^W…Y†, so that the composite

^PW…Y† !^W…Y†!^PW…Y†

induces a map on realizations which is simplicially homotopic, rel R.^W…Y;†to the mapR.^PW…Y† !R.^PW…Y† induced by the identity map onX. This clearly gives the result, so the map h^u is an equivalence. That

h^l:holim T…R.^W…Y†† !holimT …R.^PW…Y††

2RC…X=Y† 2RC…X=Y†

is an equivalence follows from the same argument, again by observing that may be taken to lie in RC…X=Y†.

We must now deal with g^u. According to 2.6 and 3.5, it will su¤ce to construct, for every2RC…X†, a⁰, so that for any fx₁;. . .;x_kg X, with⁰…x_i† \W…Y; ⁰† 6ˆ ;for alli,

⁰…x1† \. . .\⁰…xk† 6ˆ ; )⁰…x1† \. . .\⁰…xk† \W…Y; ⁰† 6ˆ ; To construct⁰, ¢rst construct^so that

…x^ ₁† \. . .\…x^ _k† 6ˆ ; )…x^ ₁† \. . . ^…x_k† 6ˆ ;

For eachfx1;. . .;xkgsuch that …x^ i† \W…;^ Y† 6ˆ ;for alli, and such that …x^ ₁† \. . . ^…x_k† \W…;^ Y† 6ˆ ;, we have the closed set …x^ ₁† \. . .\…x^ _k†.

Since

…x^ ₁† \. . .\…x^ _k† 6ˆ ; )…x^ ₁† \. . .\…x^ _k† ˆ ;

we see that…x^ 1† \. . .\…x^ k†is disjoint fromW…;^ Y†. LetS be the family of all subsets fx1;. . .;xkg, such that …x^ i† \W…;^ Y† 6ˆ ; for all i, and so that…x^ 1† \. . .\…x^ k† ˆ ;. Then

Zˆ [

fx1;...;xkg2S

…x^ 1† \. . .\…x^ k†

is a closed set disjoint fromW…;^ Y†. Let V ˆXÿZ, so V is an open set containingW…;^ Y†. Choose an open setZ1 so that

W…;^ Y† Z1Z1V

and letUˆXÿZ₁. ThenfU;Vg is an open covering ofX, and we letbe a rigid covering so that …x† ˆU or V for all X. Let ⁰ˆ^_X. ⁰ now clearly has the required properties, sog^u is an equivalence. As usual,g^l fol- lows by the identical argument, with the observation that ^ and may be taken to be in RC…X=Y†. The conclusion is thatf^u andf^l are equivalences.

Thus, for our purposes, it will su¤ce to show thatv₂ is an equivalence. But this follows directly from 2.8, since it is easily checked that ^W…Y† is an object of RC…X=Y† for all . The point is that for any x so that …^W…Y††…x† \Y 6ˆ ;,Y …^W…Y††…x†, so if we let^rbe the rigid covering ofX=Y given by^r…‰xŠ† ˆp……^W…Y††…x††, then^W…Y†ˆRC…p†…^r†.

Finally, then, we must show that the natural transformation XRC…p† [_YRC…pi†CYRC…pi† !_X=Y

is a weak equivalence of functors. To see this, we note that the inclusion XRC…p† !XRC…p† [_YRC…pi†CYRC…pi†

is a weak equivalence of functors, sinceYRC…pi†…†is evidently con-

tractible for each 2RC…X=Y†. We claim XRC…p† !_X=Y is a weak equivalence of functors. But this is clear from 3.2, since_XRC…p†…†is the covering obtained from_X=Y…† by repeating the set …‰y₀Š†once for every y2Y, and the natural map is the map of coverings sending each of these copies to…‰y0Š†.

6. Homotopy Invariance of C¯ech Homology.

We wish to demonstrate that the inclusion i:X0,!X ‰0;1Š induces an equivalence h…X;T† !h…X ‰0;1Š;T†for any homology theoryT. Lemma 4.6 shows that the conditions of Proposition 2.4 are satis¢ed for the functor RC…i†:RC…XI† !RC…X0†, hence

holim T_X0 !holim T_X0RC…i†

RC…X0† RC…XI†

is a homotopy equivalence, so showing h…X;T† !h…X ‰0;1Š;T† is a homotopy equivalence is equivalent to the assertion that the natural trans- formationX0RC…i† !_X‰0;1Š induces a homotopy equivalence

holim TX0RC…i† !holim T_X‰0;1Š:

RC…XI† RC…XI†

Notice that since T is homotopy invariant T applied to a weak homotopy equivalence will be a homotopy equivalence.

We will ¢rst establish some preliminaries.

Proposition6.1. Letuˆ fUg_2Aandvˆ fVg_2Bbe open coverings of spaces X and Y, respectively, and let uv be the open covering fUVg_;†2AB of XY. Then N.…uv† is naturally equivalent to N.uN.v.

Also, for anyn and >0, with <₂n‡1¹ , letvn;denote the open covering of ‰0;1Š given by f…₂^knÿ;^k‡1₂n ‡†gkˆ0;1;...;2ⁿÿ1. Of course, …ÿ;₂¹n‡† and …1ÿ₂¹nÿ;1‡† are to be interpreted as ‰0;₂¹n‡† and …1ÿ₂¹nÿ;1Š, re- spectively. Note thatN.v_n; is weakly contractible.

Proposition6.2. Letbe any rigid covering of X ‰0;1Šwhose underlying covering is of the form uv_n;, for some open covering u of X. Then X0RC…i†…† !_X‰0;1Š…†is a weak equivalence.

Proof. Clear from the preceding proposition and the contractibility of N.vn;