MEASURE HOMOLOGY

MEASURE HOMOLOGY

SREN KOLD HANSEN

Abstract.

LetXbe a topological space, Sink…X†the space of singulark-simplices with the compact-open topology, and letck…X†be the real vector space of all compactly supported signed Borel Mea- sures of bounded total variation on Sink…X†. There are linear operators@:ck…X† !ckÿ1…X†, so thatfc…X†; @gis a chain complex. The homologyH…X†is the measure homology ofXof Thurston and Gromov. The main results in this paper are thatH…ÿ†satis¢es the Eilenberg- Steenrod axioms for a wide class of topological spaces including all metric spaces, and is ordi- nary homology with real coe¤cients for CW-complexes.

1. Introduction.

Measure homology was introduced by Gromov and Thurston in [T] ½6 in connection with Gromov's theorem that the Gromov norm of a closed oriented hyperbolic n-manifold M equals the volume of M divided by the supremum of the volumes of the geodesic n-simplices in the hyperbolic n- space.

For a measurable space …X; }†, let v…X; }† be the vector space of all signed measures of bounded total varation. The total variation of a signed measureon…X; }†iskk ˆ^‡…X† ÿ^ÿ…X†whereˆ^‡ÿ^ÿis the Jor- dan decomposition ofinto its positive and negative variation. A measure on …X; }† has support in A2}, Supp…† A, if …A\B† ˆ…B† for all B2}. We writeb…X†for the Borel-algebra on the spaceX, and de¢ne a linear subspace ofv…X;b…X††by

m_c…X† ˆf2v…X;b…X†† jhas compact supportg:

A continuous map f :X !Y induces a linear map f :mc…X† ! mc…Y†, namely the image measure ofunderf.

Let Sin_k…X†be the set of continuous maps from the standardk-simplex^k to the spaceX with the compact-open topology, and set

c_k…X† ˆm_c…Sin_k…X††:

MATH. SCAND. 83 (1998), 205^219

Received July 2, 1996.

The ith face map ⁱ:^kÿ1 ! ^k induces a continuous map

@_i: Sin_k…X† !Sin_kÿ1…X†, @_i…† ˆ_i, and hence a linear map …@i†:ck…X† !ckÿ1…X†. The measure chain complex is the spaces ck…X†

together with the boundary operators@ˆP_k

iˆ0…ÿ1†ⁱ…@i†. The homology of c…X†is denoted H…X†and is the measure homology of X, cf. [T]. Actu- ally, in [T], the authors only de¢ned c…X† when X is a smooth manifold and used the setsSin¹_k…X†of singulark-simplices of classC¹ with theC¹to- pology instead of Sink…X†. We shall see that this makes no di¡erence. The main theorem of this paper is the following result, listed without proof in the case of smooth manifolds in [T] ½6 p. (6.7):

Theorem 1.1. The measure homology functor satis¢es the Eilenberg- Steenrod axioms on the category of metric spaces.

Remarks. 1) Actually we prove thatH…X†satis¢es the Eilenberg-Steen- rod axioms for all Hausdor¡ spaces X such that Sin_k…X† and Sin_k…A† are normal for all k0 and all AX. This is indeed satis¢ed if X is a metric space. Note that normality ofX does not imply normality of Sin_k…X†. Ac- tually A. H. Stone showed in [S] that if I ˆ ‰0;1Š and Y is the product of uncountably many copies ofI then Y^I is not normal, whereY^I is the space of maps ofI intoY with the compact-open topology.

2) If X is a smooth manifold theorem 1.1 and the proof we give for it is still valid if one uses the setsSin^r_k…X†of singulark-simplices of classC^rwith theC^r topology instead of Sink…X† ˆSin⁰_k…X†to de¢ne measure homology, 1r 1.

I would here like to thank H. J. Munkholm for drawing my attention to this problem and I. Madsen and J. Tornehave for guidance.

2. The measure homology functor.

In the preceding section we introduced the measure chain complexc…X†for an arbitrary topological space X. A map f :X!Y induces linear maps f :c_k…X† !c_k…Y† by f ˆ …f_#† where f_#: Sin_k…X† !Sin_k…Y† is as usual.

Instead of f we usually write f :c_k…X† !c_k…Y†. This makes c…ÿ† a co- variant functor and turnsH…ÿ†into a covariant functor in a standard way.

One can generalize the above to pairs of Hausdor¡ spaces…X;A†. We have b…A† ˆfZ\AjZ2b…X†g, so that b…A† ˆfZ2b…X† jZAgb…X† if A2b…X†. For an arbitrary setE,p…E† ˆ fAjAEgdenotes the power set of E. Taking direct images, f :E!F induces p…f†:p…E† !p…F† and makes p…ÿ† a covariant functor. For a homeomorphism f :X!Y, p…f†j_b…X†:b…X† !b…Y† is a bijection. We use below that Sink…X† is a Hausdor¡ space if and only ifX is.

Lemma2.1. For…X;A†a pair of Hausdor¡ spaces, the inclusion i:A!X induces a monomorphism i:c…A† !c…X†.

Proof. IfK is a compact subset ofSin_k…A†then Lˆi_#…K† is a compact subset ofSink…X†andi#j_K :K!Lis a homeomorphism. Let1,2 2ck…A†

with compact supports K1;K2, and i…1† ˆi…2†. If B2b…Sink…A†† then B\ …K2nK1† ˆK2\ …BnK1† 2b…K2†, so

₂…B\ …K₂nK₁†† ˆi…₂†…i_#…B\ …K₂nK₁††† ˆi…₁†…i_#…B\ …K₂nK₁†††

ˆ1…B\ …K2nK1†† ˆ0:

Thus K1\K2 is a support for 2, and, symmetrically for 1. If B2b…Sink…X††,

…i^ÿ1_#…B†† ˆ…i_#^ÿ1…B† \K1\K2† ˆ…i^ÿ1_#…B\L1\L2††

whereLˆi#…K†,ˆ1;2. But

b…K₁\K₂† ˆni^ÿ1_#…D† jD2b…L₁\L₂†o

ˆni^ÿ1_#…B\L1\L2† jB2b…Sin_k…X††o so₁ˆ₂ onb…K₁\K₂†hence on all ofb…Sin_k…A†.

We letc…X;A†be the cokernel ofi:c…A† !c…X†, so that we have an exact sequence

0ÿ!c…A† ÿ!ⁱ c…X† ÿ! c…X;A† ÿ!0

of chain complexes. The homology groups ofc…X;A†are the relative mea- sure homology groups of …X;A† and are denoted H…X;A†. A map f :…X;A† ! …Y;B† of pairs of Hausdor¡ spaces induces a commutative diagram

0 ÿ! c…A† ÿ!ⁱ c…X† ÿ! c…X;A† ÿ! 0

#j j

# j

# 0 ÿ! c…B† ÿ!

i c…Y† ÿ!

c…Y;B† ÿ! 0

of chain maps. Thus we get as usual a long exact homology sequence, nat- ural in…X;A†:

ÿ!^@ H_k…A† ÿ!ⁱ H_k…X† ÿ!^j H_k…X;A† ÿ!^@ H_kÿ1 …A† ÿ!ⁱ :

measure homology 207

3. Proof of theorem 1.1.

In this section we verify the homotopy axiom, excision and the dimension axiom, i.e.

i) If f0;f1:…X;A† ! …Y;B† are homotopic as maps of pairs of Hausdor¡

spaces, then H…f0† ˆH…f1†:H…X;A† !H…Y;B†.

ii) If (X,A) is a pair of metric spaces and UA hasU Int…A†,then the inclusion map i:…XÿU;AÿU† ! …X;A† induces an isomorphism on homology.

iii) If X is a one-point space, then H_k…X† ˆ0for k6ˆ0and H₀…X† ˆR.

If one works with complex measures instead of real measures the only dif- ference is thatH₀…X† ˆCfor a one-point space X. We start by showing the easy i) and iii).

Proof of iii). Since Sin_k…X† has only one element '_k, b…Sin_k…X†† ˆ f;;f'kgg and2ck…X†is completely determined by the value …f'kg†. If r2Rwe get an element^k_r 2ck…X†de¢ned by^k_r…;† ˆ0,^k_r…f'kg† ˆr. This shows thatck…X† R, and a simple calculation shows that

@…^k_r† ˆ 0 ;kodd

^kÿ1_r ;keven andk>0:

Since@ˆ0 for all2c0…X†by de¢nition the result follows.

Proofof i). We just do the absolute case, AˆBˆ ;. Lett:X!XI be given by t…x† ˆ …x;t†,I ˆ ‰0;1Š and let F :XI!Y be a homotopy betweenf₀ and f₁. Then F₀ˆf₀ and F₁ˆf₁ and it su¤ces to show that H…₀† ˆH…₁†:H…X† !H…XI†. To show this we construct a chain homotopy P:c…X† !c…XI†between 0 and1. Foriˆ0;1;. . .;kwe de¢ne mapsQi: Sink…X† !Sink‡1…XI†by

Q_i…†…t₀;. . .;t_k‡1† ˆ…t₀;. . .;t_iÿ1;t_i‡t_i‡1;t_i‡2;. . .;t_k‡1† 1ÿXⁱ

lˆ0

t_l

!

for2Sink…X†and…t0;. . .;tk‡1† 2^k‡1. TheQi are continuous and induce linear maps…Qi†:c_k…X† !c_k‡1…XI†. De¢ne P_k:c_k…X† !c_k‡1…XI†

by P_kˆP_k

iˆ0…ÿ1†ⁱ…Q_i†. A tedious calculation shows that the P_k form a natural chain homotopy between₀and₁. The general case now follows in a standard way, by using naturality ofP.

We now begin the proof of ii). Let U be an open subset of X and i:U!X the inclusion map. Then V ˆi#…Sin_k…U†† is open in Sin_k…X†

and i#: Sink…U† !V is a homeomorphism. It follows that p…i#†: b…Sink…U†† !b…V†is a bijection so that

b…Sin_k…U†† ˆni^ÿ1_#…B† jB2b…V†o

ˆni^ÿ1_#…B† jB2b…Sin_k…X††o : For a family uˆ fUj2Ig of (not necessarily open) subsets of X we consider the subchain complex ofc…X†of ``u-small'' measures

c^u_k…X† ˆX

2I

c^U_k…X†; c^U_k…X† ˆi…c_k…U††:

For the family uwe let Int…u† be the collection of interiors of elements of u. We then have

Theorem3.1. Let X be a metric space and letube a family of subsets of X such that Int…u† is a covering of X. Then the inclusion chain map I:c^u…X† !c…X†induces an isomorphism on homology.

The proof for this theorem is deferred to ½4. As in the case of singular homolog the excision axiom follows at once. Letuˆ fUj2IgandX be as in theorem 3.1 and let u\Aˆ fU\Aj2Ig. The commutative dia- gram of inclusion maps

U\A ÿ!^m U

#j

j j

#ⁱ A ÿ!_i X

shows thati…j…c_k…U\A††† i…c_k…U††soi…c^u\A …X††is a subcomplex of c^u…X†. Settingc…u;u\A† ˆc^u…X†=i…c^u\A …A††, we have a commutative diagram of chain maps

0 ÿ! c^u\A …A† ÿ!ⁱ c^u…X† ÿ! c…u;u\A† ÿ! 0

#j

I j

I# j

#

0 ÿ! c…A† ÿ!

i c…X† ÿ!

c…X;A† ÿ! 0:

By the preceding theorem the inclusions I induce isomorphisms on homo- logy, and the ¢ve-lemma yields that induces an isomorphism on homo- logy.

Proof of ii). Let uˆ fXÿU;Int…A†g. Then u\Aˆ fAÿU;Int…A†g and we have that Int…u†and Int…u\A†cover respectivelyXandA. Now let

Int…A† ÿ!^jX X AÿU ÿ!^m XÿU

#j

jA j

#^id j

iA# j

#^iX

A ÿ!

i X A ÿ!

i X

measure homology 209

be commutative diagrams of inclusion maps. Then we have c^u\A_k …A† ˆi_A…c_k…AÿU†† ‡j_A…c_k…Int…A†††

c^u_k…X† ˆi_X…c_k…XÿU†† ‡j_X…c_k…Int…A†††

and therefore i…c^u\A_k …A†† ˆi_Xm…c_k…AÿU†† ‡j_X…c_k…Int…A†††.This im- plies the isomorphisms

c…u;u\A† ˆc^u…X†=i…c^u\A …A†† i_X…c…XÿU††=i_X…m…c…AÿU†††

c…XÿU†=m…c…AÿU†† ˆc…XÿU;AÿU†

The ¢rst isomorphism follows by the fact that

‰i_Xm…c_k…AÿU†† ‡j_X…c_k…Int…A††Š \i_X…c_k…XÿU†† ˆi_Xm…c_k…AÿU††

Since :c…u;u\A† !c…X;A† induces an isomorphism on homology the result follows.

4. Proof of theorem 3.1.

The proof of theorem 3.1 uses the standard ideas from barycentric subdivi- sion in singular theory, which we begin by recalling, cf. [D]. The subdivision homomorphisms q:Sq…X† !Sq…X†, q2Z, are inductively de¢ned in the following way:

Let {_q 2S_q…^q† denotes the identity map of the standard q-simplex ^q with vertices the standard basis fe_igin R^q‡1 and letB_qˆP_q

iˆ0 1

q‡1e_i be the barycenter of^q. WriteB_qfor the cone construction (see [D] chap. III (4.7) p. 34), and set

₀ˆid _q…{_q† ˆB_qqÿ1…@{_q†; q>0:

…1†

One de¢nesq : Sinq…X† !Sq…X†byq…† ˆ#…q…{q††. Then :S…X† ! S…X†

…2†

is a natural chain map, [D] p. 41. For later use we need to explicate the natural chain homotopys:'id_S…X†, [D] p. 42. It is 0 forqˆ0 and is gi- ven by

s_q…{_q† ˆB_{q …q}…{_q† ÿ{_qÿs_qÿ1…@{_q†† 2S_q‡1…^q†

on{_q2S_q…^q†forq>0. For a general2Sin_q…X†,s_q…† ˆ_#…s_q…{_q††.

We now want to de¢ne a ``subdivision'' homomorphism :c…X† ! c…X† for the measure theoretical chain complex and a chain homotopy s:'id_c…X†. To this end we write out the construction in (1) in a form _qˆP

2Aqr_q where _q : Sin_q…X† !Sin_q…X† are continuous and induce

linear maps …_q†:c_q…X† !c_q…X†. Thus we can de¢ne our measure theo- retical ``subdivision'' homomorphism by_qˆP

2Aqr…_q†. This procedure will also be used to de¢ne the chain homotopy s:'id_c…X†. Forq1 the explicit formula is

_q…{_q† ˆX^q

kqˆ0

X^qÿ1

kqÿ1ˆ0

X¹

k1ˆ0

…ÿ1†^k1‡‡kqk_q^1...kq where

^k_q^1...kqˆBq …^kq …Bqÿ1 …^kqÿ1 B2 …^k2 …B1^k1†† ††† 2Sinq…^q†:

Thus we get mapsq: Sinq…X† !Sq…X†, qˆX^q

kqˆ0

X^qÿ1

kqÿ1ˆ0

X¹

k1ˆ0

…ÿ1†^k1‡‡kq_q^k1...kq …3†

where^kq^1...kq…† ˆ^kq^1...kq. Similarly when q1 we have s_q…{_q† ˆX^q

ˆ1

…ÿ1†^qÿX^q

kqˆ0

X^qÿ1

kqÿ1ˆ0

X¹

k1ˆ0

…ÿ1†^k1‡‡kqf_q;…k₁;. . .;k_q†

ÿX^qÿ1

ˆ1

…ÿ1†^qÿX^q

kqˆ0

X^qÿ1

kqÿ1ˆ0

X^‡1

k‡1ˆ0

…ÿ1†^k‡1‡‡kqg_q;…k_‡1;. . .;k_q† ÿg_q;q where

fq;…k1;. . .;kq† ˆBq …^kq …Bqÿ1 …^{kqÿ1 …k‡1} …B^k1k†† †††

gq;…k‡1;. . .;kq† ˆBq …^kq …Bqÿ1 …^{kqÿ1 …k‡1} …B{†† †††;

ˆ1;. . .;qÿ1, and

f_q;q…k₁;. . .;k_q† ˆB_q^k_q^1kq g_q;qˆB_q{_q

are elements ofSin_q‡1…^q†, sos_q: Sin_q…X† !S_q‡1…X†is given by sqˆX^q

ˆ1

…ÿ1†^qÿX^q

kqˆ0

X^qÿ1

kqÿ1ˆ0

X¹

k1ˆ0

…ÿ1†^k1‡‡kqf_q;^k1...kq …4†

ÿX^qÿ1

ˆ1

…ÿ1†^qÿX^q

kqˆ0

X^qÿ1

kqÿ1ˆ0

X^‡1

k‡1ˆ0

…ÿ1†^k‡1‡‡kqg^k_q;^‡1...kqÿhq;q

wherefq;^k1...kq;g^kq;^‡1...kq;hq;q: Sinq…X† !Sinq‡1…X†are de¢ned by

measure homology 211

f_q;^k1...kq…† ˆfq;…k1;. . .;kq† g^k_q;^‡1...kq…† ˆgq;…k‡1;. . .;kq† h_q;q…† ˆq_q;q:

In the following we shorten notation and write (3) as _qˆP

2Aqr_q where A_q is the set of q-tuples …a₁;. . .;a_q†, a_iˆ0;1;. . .;i, and r_k1;...;kq†ˆ …ÿ1†^k1‡‡kq. Forq2 we then have

@_qˆX^q

jˆ0

…ÿ1†^j@_j X

2Aq

r_q 0

@

1 AˆX^q

jˆ0

X

2Aq

…ÿ1†^jr@_jq

_qÿ1@ˆ X

2Aqÿ1

r_qÿ1 X^q

jˆ0

…ÿ1†^j@_j

!

ˆX^q

jˆ0

X

2Aqÿ1

…ÿ1†^jr_qÿ1 @_j: Now@qˆqÿ1@by (2) so in particular@q…{q† ˆqÿ1@…{q†, i.e.

X^q

jˆ0

X

2Aq

…ÿ1†^jr_q^jˆX^q

jˆ0

X

2Aqÿ1

…ÿ1†^jr^j_qÿ1: SinceSinqÿ1…^q†is a basis forSqÿ1…^q†we can write

X^q

jˆ0

X

2Aq

…ÿ1†^jr_q^jˆ X

2Mq

t_q X^q

jˆ0

X

2Aqÿ1

…ÿ1†^jr^j_qÿ1ˆ X

2Nq

s!_q;

where

_qj2Mq

n o

n_q^jj…;j† 2Aq f0;1;. . .;qgo

!_qj2N_q

n o

n^j_qÿ1j…j; † 2 f0;1;. . .;qg A_qÿ1o

;

and1 6ˆ2)_q¹6ˆ_q² and1 6ˆ2)!_q¹6ˆ!_q² andt6ˆ0 for all2Mq

and s6ˆ0 for all 2N_q. We observe that M_q and N_q contain the same number of elements and that for all2M_q there exists a2N_q such that sˆt and!_q ˆ_q. Now letT_q;_q : Sin_q…X† !Sin_qÿ1…X†be given by

T_q…† ˆ_q; _q…† ˆ!_q: Then we have that

T_qj2M_q

n o

n@_jqj…;j† 2A_q f0;1;. . .;qgo _qj2Nq

n o

_qÿ1@jj…j; † 2 f0;1;. . .;qg Aqÿ1

n o

and @_qˆP

2MqtT_q, _qÿ1@ˆP

2Nqs_q. These results are also true for qˆ1 with some small, obvious changes in the notation (put A₀ˆ f0g,r₀ˆ1,⁰₀ ˆid and⁰₀ˆid).

De¢ne_q :c_q…X† !c_q…X†by_qˆP

2Aqr…_q† forq1 and₀ˆid.

Lemma4.1. :c…X† !c…X†is a natural chain map.

Proof. The map is natural by de¢nition. Forq1 we have

@qˆX^q

jˆ0

…ÿ1†^j…@j† X

2Aq

r…_q† 0

@

1 AˆX^q

jˆ0

X

2Aq

…ÿ1†^jr…@j_q†

_qÿ1@ˆX^q

jˆ0

X

2Aqÿ1

…ÿ1†^jr…_qÿ1 @_j† ₀@ˆX¹

jˆ0

…ÿ1†^j…@_j†

! :

From the remarks before the lemma we conclude that

@_qˆ X

2Mq

t…T_q†; _qÿ1@ˆ X

2Nq

s…_q† which implies@qˆqÿ1@.

In the following we write (4) as sqˆP

2Bqrs_q where r2 fÿ1;1g and fs_qj2B_qg ˆM_f^q[M_g^q. Here

M_f^qˆ f_q;^k1...kqj2 f1;. . .;qg;…k1;. . .;kq† 2Aq

n o

M_g^qˆng^k_q;^‡1...kqj 2 f1;. . .;qÿ1g;…k_‡1;. . .;k_q† 2A^‡1_q o [h_q;q where A^p_q is the set of tuples …ap;. . .;aq†, aiˆ0;1;. . .;i, pˆ1;2;. . .;q. We then have that

@s_q ˆX^q‡1

jˆ0

X

2Bq

…ÿ1†^jr@_js_q; s_qÿ1@ˆX^q

jˆ0

X

2Bqÿ1

…ÿ1†^jrs_qÿ1@_j so@sq‡sqÿ1@ˆqÿid is equivalent to

X^q‡1

jˆ0

X

2Bq

…ÿ1†^jr@_js_q ‡X^q

jˆ0

X

2Bqÿ1

…ÿ1†^jrs_qÿ1@_jˆX

2Aq

r_qÿid:

…5†

measure homology 213

De¢ne s_q:c_q…X† !c_q‡1…X† by s_qˆP

2Bqr…s_q† for q1. For q0, s_qˆ0.

Lemma4.2. s:'id_c…X†is a natural chain homotopy.

Proof. Naturality follows from the de¢nition. We have _qÿidˆ P

2Aqr…_q†ÿid and

@sqˆX^q‡1

jˆ0

X

2Bq

…ÿ1†^jr…@js_q†; sqÿ1@ˆX^q

jˆ0

X

2Bqÿ1

…ÿ1†^jr…s_qÿ1@j†: Now evaluate (5) on{_q and use the same procedure as in the remarks before lemma 4.1 to deduce thatsde¢nes a chain homotopy @s_q‡s_qÿ1@ˆ_qÿid.

Let uˆ fUj2Ig be a family of subsets of X. We put W^kˆ …i†_#…Sin_k…U††wherei:U!X is the inclusion map. IfUis open in X then W^k is open in Sink…X† and …i†_#: Sink…U† !W^k is a home- omorphism.

Lemma 4.3. Let n2N. Then we have a natural chain homotopy c:ⁿ'id_c…X†. If2cq…X†and@2c^u_qÿ1…X†then cqÿ1…@† 2c^u_q…X†.

Proof. Let cqˆsq…id‡q‡. . .‡_q^nÿ1†:cq…X† !cq‡1…X†. Then c is a natural chain homotopy between ⁿ and id_c…X†. Now let q1 and 2cq…X† and assume that @2c^u_qÿ1…X†. Write @ˆP_n

jˆ1rjj,rj2R, j2c^U_qÿ1^j…X†and choose j2cqÿ1…Uj†such that j ˆij…j†. By naturality ofc

cqÿ1…@† ˆXⁿ

jˆ1

rjcqÿ1…ij…j†† ˆXⁿ

jˆ1

rjij…cqÿ1…j†† 2c^u_q…X†:

The main lemma is

Lemma 4.4. Let uˆ fUj2Ig be a family of subsets of X such that Int…u†is a covering of X. Then for all2cq…X†there exists a natural number n so that the n'th iterate…q†ⁿ…† 2c^u_q…X†.

Proof. We may assume that U is open since c^Int…u†_q c^u_q…X†. Now let q1. SinceqˆP

2Aqr…_q† we have …_q†^kˆ X

12Aq

X

k2Aq

r_1...k…q^k† …_q¹†

ˆ X

12Aq

X

k2Aq

r_1...k…q^k _q¹†

for every k2Nwhere r_1...kˆQ_k

jˆ1r_j. From standard singular theory (cf.

[D] (6.3) p. 41) we know that for given >0 …_q†^k…{_q† is a formal linear combination of simplices of diameter less thanifkis su¤ciently large, say kn0. Now

…_q†^k…{_q† ˆ X

12Aq

X

k2Aq

r_1...k…q^k _q¹†…{_q†

for every k2N so diam…_q^k _q¹…{q†…^q††< for all kn0 and all …₁;. . .; _k† 2 …A_q†^k. Here diam…C† denotes the diameter of C. For 2Sin_q…X†, wˆ f^ÿ1…U†j2Ig is an open covering of ^q. This being compact there exists an>0 such that forC^qof diameter less than, there exist an index with C2^ÿ1…U† ( is the Lebesgue number of the coveringw). Choosen such that for all kn and all…1;. . .; k† 2 …Aq†^k we have the implications

diam…_q^k _q¹…{_q†…^q††< ) 92I :_q^k _q¹…{_q†… ^q† ^ÿ1…U† ) 92I :_q^k _q¹…† 2W^q:

Now let kn and …₁;. . .; _k† 2 …A_q†^k and choose 2I such that _q^k _q¹…† 2W^q. Since W^q is an open subset of Sinq…X† and _q^k _q¹ : Sinq…X† !Sinq…X† is continuous, it follows that there is an open neighborhood U^1...k of in Sin_q…X† such that _q^k _q¹…U^1...k† W^q. Set U^kˆT

12Aq T

k2AqU^1...k. It is an open neigh- boorhood ofinSin_q…X†and for all…₁;. . .; _k† 2 …A_q†^kthere is an index with _q^k _q¹…U^k† W^q. Let 2cq…X† be a chain with Supp…† K where KSinq…X† is compact and set OˆUⁿ. Since fOg₂Sinq…X† is a covering ofK with open subsets of Sinq…X†we can ¢nd1;. . .; l2Sinq…X†

such that KO₁[. . .[O_l. Set n_jˆn_j for j2 f1;. . .;lg and set nˆmaxfn₁;. . .;n_lgand let 2K. Choose aj2 f1;. . .;lgsuch that 2O_j. Then for all …1;. . .; nj† 2 …Aq†^nj there is an index with q^nj _q¹…† 2W^q. Now let knj and …1;. . .; k† 2 …Aq†^k. Choose 2I such that q^nj _q¹…† 2W^q. Then _q^k _q¹…† 2W^q. (Let

!2W^q. Then!…^q†U. We therefore have

_q…!†…^q† ˆ!_q…^q† !…^q† U

for 2A_q so _q…!† 2W^q.) Thus to each …₁;. . .; _n† 2 …A_q†ⁿ and 2K we can ¢nd an index with _qⁿ _q¹…† 2W^q. Since _qⁿ _q¹ is con- tinuous Lˆ_qⁿ _q¹…K† is a compact subset of Sin_q…X†; actually LW^qˆS

2IW^q. The support of ˆ …_qⁿ _q¹†…†is contained in L. Choose 1;. . .; m2I such that LW^q₁[. . .[W^q_m and let VjˆW^q_j\L. Since Vj2b…Sinq…X†† it follows that b…Vj† b…Sinq…X††.

measure homology 215

The restriction _j of to b…V_j† de¢nes a real Borel measure on V_j, jˆ1;2;. . .;m.fV_jg^m_jˆ1 is an open covering ofLandLis normal since it is a closed subset of the normal spaceSinq…X†. We can therefore choose a parti- tion of unity fjg^m_jˆ1 subordinated to the covering fVjg^m_jˆ1. The maps _j :V_j!Rare continuous and therefore Borel measurable. Now let

_j…B† ˆ Z

B_jd…_j†; B2b…V_j†; jˆ1;2;. . .;m:

Since_j…L† ‰0;1Šit follows that_j2L¹…_j†. This implies that_j is a signed Borel measure onV_j of bounded total variation, and we can de¢ne_j by

j…B† ˆj…ij†_#…B† \L

; B2b…Sinq…Uj††; jˆ1;2;. . .;m:

SinceV_j 2b…W^q_j†we have

b…Vj† ˆnB\VjjB2b…W^q_j†o

ˆnB\LjB2b…W^q_j†o :

Now …i_j†_#: Sin_q…U_j† !W^q_j is a homeomorphism, so induces a bijection p…ij†_#

:b…Sinq…Uj†† !b…W^q_j†. Thus j is a well de¢ned real valued Borel measure onSinq…Uj†. If we putLjˆSupp_L…j† VjW^q_j then

j…B† ˆ Z

Bjd…j† ˆ Z

Vj

Bjd…j† ˆ Z

Vj

BLjjd…j†

ˆ Z

Vj

B\Ljjd…j† ˆ Z

B\Lj

jd…j† ˆj…B\Lj†

forB2b…Vj†, whereBis the characteristic function ofBetc. Observe here thatL_j is a closed subset ofLhence of Sin_q…X†, so L_j2b…Sin_q…X††. More- over B2b…V_j† b…Sin_q…X†† so B\L_jb…Sin_q…X††. But then B\Ljb…Vj† since B\LjVj2b…Sinq…X††. This shows that the above calculations are allowed. Also Mjˆ …ij†^ÿ1_#…Lj† is a compact subset of Sinq…Uj†and we have

j…D† ˆj…ij†_#…D† \L

ˆj …ij†_#…D† \L\Lj

ˆ_j…i_j†_#…D† \L\ …i_j†_#…M_j†

ˆj…ij†_#…D\Mj† \L

ˆj…D\Mj†

for allD2b…Sin_q…U_j††, i.e._j 2c_q…U_j†. IfB2b…Sin_q…X††we have

X^m

jˆ1

i_j…j†…B† ˆX^m

jˆ1

_j…i_j†^ÿ1_#…B\W^q_j†

ˆX^m

jˆ1

_j…B\W^q_j\L†

ˆX^m

jˆ1

Z

B\Vj

jd…j† ˆX^m

jˆ1

Z

Vj

B\Vjjd…j†

ˆX^m

jˆ1

Z

Sinq…X†_B\Vjjd;

where _j: Sin_q…X† !R is the Borel function de¢ned by _j…x† ˆ_jL…x†, x2Sinq…X†. Now L\BˆB\Vj‡…L\B†nVj and jˆ0 on …L\B† nVj so B\VjjˆL\Bj. SinceP

j1 onL, X^m

jˆ1

ij…j†…B† ˆX^m

jˆ1

Z

Sinq…X†B\Ljd…† ˆ Z

Sinq…X†B\Ld…†

ˆ…B\L† ˆ…B†:

It follows thatˆP_m

jˆ1ij…j†and we conclude that …_q†ⁿ…† ˆ X

12Aq

X

n2Aq

r_1...n…qⁿ _q¹†…† 2c^u_q…X†:

For qˆ0 the result easily follows by the preceding. Observe that W⁰ ˆS

2IW⁰ˆSin₀…X†and skip the ¢rst part of the proof and let ˆ andLˆK in the last part the proof.

Proof of theorem 3.1. Set c…X;u† ˆc…X†=c^u…X†, giving the exact sequence

0ÿ!c^u…X† ÿ!ⁱ c…X† ÿ! c…X;u† ÿ!0:

We must show thatH…c…X;u†† ˆ0. Let‡c^u_q…X† 2Zq…c…X;u††. Then

@… ‡c^u_q…X†† ˆ@‡c^u_qÿ1…X† ˆ0‡c^u_qÿ1…X† ,@2c^u_qÿ1…X†:

By lemma 4.4 we can choosen2Nsuch that…q†ⁿ…† 2c^u_q…X†. Letcbe the chain homotopy between ⁿ and id_c…X†, see lemma 4.3. Set yˆ …_q†ⁿ…† ÿc_qÿ1…@†andxˆ ÿc_q…†. Since@2c^u_qÿ1…X†it follows from lemma 4.3 that c_qÿ1…@† 2c^u_q…X†which implies that y2c^u_q…X†. Moreover we have that x2c_q‡1…X† and ˆid…† ˆ …_q†ⁿ…† ÿc_qÿ1…@†ÿ

@…cq…†† ˆy‡@x. This implies that

ÿ@x2c^u_q…X† ,‡c^u_q…X† ˆ@x‡c^u_q…X† ˆ@…x ‡c^u_q‡1…X††:

But then‡c^u_q…X† 2Bq…c…X;u††.

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5. Measure homology and CW-complexes.

In this section we show that the measure homology groups are isomorphic to the singular homology groups on the category of CW-complexes. The pro- blem is that not all CW-complexes are metrizable so we need the following result. LetXbe a topological space and letfXg_2I be the family of compact subsets ofXpartially ordered by inclusion. Leti:X!Xbe the inclusion whenXX. ThenH…X†

2I forms a direct system of real vector spaces with the linear maps fˆH…i†:H…X† !H…X† induced by the in- clusion maps. We now have

Proposition5.1.

H…X† lim

ÿ! H…X†

Proof. Let i:X!X be the inclusion maps, fˆH…i†:H…X† ! H…X† and let f ˆL

2If:L

2IH…X† !H…X†. If P_n

iˆ1x_i 2L H…X† and 2I is such that X_iX for iˆ1;2;. . .;n and2I

P_n

iˆ1f_i…x_i† ˆ0, then 0ˆf P_n

iˆ1f_i…x_i†

ˆP_n

iˆ1f_i…x_i† ˆf P_n

iˆ1x_i

ÿ

. Thusf induces a linear map

f~: lim

ÿ! H…X† !H…X†:

Now let ‰Š 2H_q…X† and choose a compact subset KSin_q…X† with Supp…†K. The evaluation map !: Sin_q…X† ^q!X de¢ned by

!…;x† ˆ…x† is continuous since ^q is compact so Aˆ!…K^q† is a compact subset of X. Now if 2K we have that …^q†A, and it follows thatK j_#…Sin_q…A††where j:A!X is the inclusion. But then Lˆj^ÿ1_#…K†

is a compact subset of Sin_q…A†. The homeomorphism j_#:L!K induces a bijection Qˆp…j_#†:b…L† !b…K†. Since K2b…Sin_q…X†† we have that b…K†b…Sinq…X††. Moreover b…L† ˆL\DjD2b…Sinq…A††

, so we can de¢ne a signed Borel measure on Sinq…A† of bounded total variation by …D† ˆQ…L\D†. By de¢nition Supp…†L so 2cq…A†. If B2b…Sin_q…X††we have

j…†…B† ˆ…j_#^ÿ1…B†† ˆQ…L\j^ÿ1_# …B†† ˆQ…j_#^ÿ1…K\B††

ˆ…K\B† ˆ…B†:

Now 0ˆ@ˆj…@† and by lemma 2.1 j:c…A† !c…X† is injective so

@ˆ0. All in all we see that‰Š ˆH_q…j†…‰Š†sof and thus~f is surjective.

Suppose thatP_n

iˆ1‰iŠ 2L

2IH…X†withf P_n

iˆ1‰iŠ

ÿ

ˆ0. Since

f Xⁿ

iˆ1

‰_iŠ

!

ˆXⁿ

iˆ1

f_i…‰_iŠ† ˆXⁿ

iˆ1

i_i…i†

‰ Š ˆ Xⁿ

iˆ1

i_i…i†

" #

we can choose2cq‡1…X†with @ˆP_n

iˆ1ii…i†. LetKSinq‡1…X†be a compact support of and let Aˆ!…K^q‡1† where

!: Sinq‡1…X† ^q‡1!X is the continuous evaluation map. Then XˆA[S_n

iˆ1X_i is a compact subset ofX. For2K,…^q‡1†AX, so we can choose2c_q‡1…X†withi…† ˆ. Since

i…@† ˆ@i…† ˆ@ˆXⁿ

iˆ1

ii…i† ˆi

Xⁿ

iˆ1

i_i…i†

!

and since i:c_q…X† !c_q…X† is injective, P_n

iˆ1i_i…i† is a boundary in cq…X†. HenceP_n

iˆ1f_i…‰iŠ† ˆP_n

iˆ1hi_i…i†i

ˆ0.

Corollary5.2. The measure homology groups and the singular homology groups are isomorphic on the category of CW-complexes.

Proof. This follows by the fact that a compact subset of a CW-complex is metrizable.

REFERENCES

[D] A. Dold,Lectures on Algebraic Topology, New York: Springer 1972.

[S] A. H. Stone, A note on paracompactness and normality of mapping spaces, Proc. Amer.

Math. Soc. 14 (1963), 81^83.

[T] W. Thurston, The Geometry and Topology of 3-manifolds, Princeton University, Lecture notes, 1978.

MATEMATISK INSTITUT AARHUS UNIVERSITET BYGNING 530 8000 AARHUS C DENMARK

email:kold@mi.aau.dk

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