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Cobordism group with local coe½cients and its application to 4-manifolds

Ichiji Kurazono

(Received July 28, 2000) (Revised January 16, 2001)

Abstract. For a pair…X;of topological spaces andwAH1…X;Z2†the cobordism groupWn…X;A;Sw†with local coe½cients is introduced. IfXis a CW complex andSw

is a local system overXdetermined byw, then we have an Atiyah-Hirzeburch spectral sequence Ep;2qˆHp…X;WqnSw† )Wp‡q…X;Sw† which is regular and hence conver- gent. For a connected CW complexXthe mapm:W4…X;Sw† !H4…X;Sw†, de®ned by m…‰M;f;jŠ† ˆ f…j…s††, is a surjection and its kernel isW4nZ2 ifw00, wheresis a fundamental homology class with respect to the orientation sheaf of a manifoldMandj is a local orientation. The closed 4-manifolds with ®nitely presentable fundamental grouppand the ®rst Stiefel-Whitney class induced fromware almost classi®ed modulo connected sums with simply connected manifolds by the quotientH4…Bp;Sw†=…Autw, and precisely in the case that p is abelian.

1. Introduction

The oriented cobordism functor fW…X;A†;j;qg satis®es the ®rst six Eilenberg-Steenrod axioms for the category of pairs of topological spaces and maps [2]. So, for any CW complexXthe Atiyah-Hirzeburch spectral sequence

Ep;2qˆHp…X;Wq† )Wp‡q…X†

is regular and hence convergent in the sense of [1]. Using this spectral se- quence, the classi®cation of oriented closed 4-manifolds having the ®nitely presentable fundamental group p modulo connected sums with simply con- nected manifolds is given by the quotient H4…Bp;Z†=…Autp† [4], [7].

Our goal of this paper is to extend the above result to the non-orientable case. We introduce a cobordism group Wn…X;A;Sw† for a pair …X;A† of topological spaces and wAH1…X;Z2†, which reduces to Wn…X;A† if wˆ0.

Let w1:BOr!K…Z2;1† be the map corresponding to the ®rst Stiefel-Whitney class. Consider w to be a map of X to K…Z2;1†, and let

2000 Mathematics Subject Classi®cation. 55M30, 55N25, 55T25, 57M50, 57N13

Key words and phrases. cobordism group with local coe½cients, Atiyah-Hirzeburch spectral sequence, weakly stable classi®cation of closed 4-manifolds, Lusternik-Schnirelmann p1-category

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Br ƒƒƒ! X

fr

??

?y

??

?yw BOr ƒƒƒ!

w1 K…Z2;1†

be the pull-back. Then Wn…X;Sw† coincides with Wn…B;f† given by Stong in [12, p. 17]. We show that this cobordism group has the properties similar to the oriented cobordism group.

For a pair of points x;yAX we denote by G…y;x† the set of relative homotopy classes of paths from x to y. Let S be a family fS…x†;S…g†g satisfying the following conditions, which will be called a local system (of abelian groups) over X:

(1) for each xAX, S…x† is an abelian group,

(2) for each gAG…y;x†, S…g† is an isomorphism of S…y† to S…x† and (3) S…gg0† ˆS…g† S…g0† for any gAG…y;x† and g0AG…z;y†.

By the de®nition we see that S induces a homomorphism Sx:p1…X;x† ! AutS…x† de®ned by Sx…a† ˆS…a† …aAp1…X;x†† for each xAX. Fixx0AX and choose an element axAG…x;x0† for each xAX. Then we see also that

S…g† ˆS…ax†ÿ1Sx0…axgaÿ1y † S…ay†

for eachgAG…y;x†. WhenXis arcwise connected and Gis an abelian group, any homomorphism r:p1…X;x0† !AutG induces one and only one local system over X such that S…x0† ˆG and Sx0 ˆr [10], which is called a local system determined by r.

For wAH1…X;Z2† let Sw be a local system over X which satis®es the following conditions.

(1.1) For each xAX, Sw…x† is isomorphic to the group Z of integers.

(1.2) Sw is determined by the homomorphism rw:p1…X;x0† !AutZ.

Here rw is a composite of the Hurewicz homomorphism X:p1…X;x0†

!H1…X;Z† with w considered as a homomorphism from H1…X;Z† to AutZˆZ2.

We will prove the following theorem.

Theorem 1. Let X be a CW complex and wAH1…X;Z2†. Then we have a spectral sequence

Ep;q2 ˆHp…X;WqnSw† )Wp‡q…X;Sw† which is regular and hence convergent.

For an n-manifold N the orientation sheaf SN is de®ned as follows.

(2.1) SN…u† ˆHn…N;Nÿu;Z† for each uAIntN and SN…u† ˆ Hnÿ1…qN;qNÿu;Z† for each uAqN.

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(2.2) SN is determined by the homomorphism rN ˆw1…N† X, where X is the Hurewicz homomorphism and w1…N† is the ®rst Stiefel-Whitney class of N.

Now we de®ne Wn…X;Sw† assuming the notion of equivalence between local systems. We consider a pair of a closedn-manifoldM and a continuous map f :M!X such that SM and the induced local system fSw are equivalent. Letjˆ fjuguAM denote the family of isomorphisms ju:SM…u† ! fSw…u† which gives this equivalence (See O2). Let Mn…X;Sw† be the set which consists of such triples …M;f;j†. We de®ne the equivalence relation in Mn…X;Sw† as follows. …M1;f1;j1†@…M2;f2;j2† means that there exist a compact …n‡1†-manifold W and a map F :W !X satisfying the following conditions:

(1) qWˆM1UM2, (2) FjMjˆ fj …jˆ1;2†,

(3) there exists an equivalence F:SW !FSw such that F_ ˆFjqW : SqW !FSwjqW satis®es FjM_ 1ˆj1 and FjM_ 2ˆ ÿj2.

The set of equivalence classes Mn…X;Sw†=@has a natural group structure and is denoted by Wn…X;Sw† and called a cobordism group with local coe½cients. We use the notation ‰M;f;jŠ for the cobordism class in Wn…X;Sw†.

Since j induces an isomorphismj:Hn…M;SM† !Hn…M;fSw†, we can de®ne a homomorphism

m:Wn…X;Sw† !Hn…X;Sw†

by m…‰M;f;jŠ† ˆ f…j…s††, where s is the fundamental class in Hn…M;SM†.

We may call ja local orientation of Massociated with f. We have only two local orientationsGj associated with f provided that M is connected.

Using Theorem 1 we will get the following corollary.

Corollary 2. Let X be a connected CW complex and wAH1…X;Z2†.

The map m:W4…X;Sw† !H4…X;Sw† is a surjection and the kernel is W4 if wˆ0, and W4nZ2 if w00.

Let p be a ®nitely presentable group, BpˆK…p;1† be an Eilenberg- MacLane complex and w be an element of H1…Bp;Z2†. We consider the set M4p;w consisting of the closed connected 4-manifolds M such that p1…M† ˆp and w1…M† ˆw, or more precisely, there is a map f :M!Bp satisfying

(3.1) f induces an isomorphism on p1, that is, f:p1…M;u† ! p1…Bp;f…u†† is isomorphism for any u, and

(3.2) fwˆw1…M†AH1…M;Z2†.

By Proposition 15 inO7 M4p;w is not empty. For every MAM4p;w there exists an element …M;f;j†of M4…Bp;Sw† by Proposition 11 inO6. For a non-zero

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w Proposition 13 in O6 says that ‰M;f;jŠ ˆ ‰M;f;ÿjŠ in W4…Bp;Sw† under some condition which is automaticaly satis®ed when p is abelian.

We will say that closed connected 4-manifoldsM andNare weakly stably equivalent, if there exist closed simply connected 4-manifolds M0 and N0 such that M]M0 and N]N0 are di¨eomorphic to each other. Let …Autp†w be the subgroup of Autp consisting of the elements whose corresponding classifying base point preserving maps l:Bp!Bp satisfy lwˆw on H1…Bp;Z2†.

Then we can extend Theorem 1 in [7] to the non-orientable case at least in the case of abelian fundamental groups.

Theorem 3. Let p be a ®nitely generated abelian group and w be a non- trivial element of H1…Bp;Z2†. Then, the set of weakly stable equivalence classes in Mp;w4 is in one-to-one correspondence with the quotient H4…Bp;Sw†=…Autp†w by the correspondence…M;f;j† 7! f…j…s††, where s is the fundamental homol- ogy class of M with local coe½cients SM.

A more general form of Theorem 3 (Theorem 20 in O7) implies the following theorem which characterizes the Lusternik-Schnirelmann p1-category of closed connected 4-manifolds including both the orientable and non- orientable cases.

Theorem 4. If the Lusternik-Schnirelmann p1-category of a connected closed4-manifold M is not 4, then M is weakly stably equivalent to the boundary qN…K2† of the regular neighborhood of an embedded ®nite 2-complex K2 in RP4R realizing the fundamental group pˆp1…M† and rw1…M†:p!AutZ.

We recall the notion of equivalence between local systems and de®ne the relative cobordism group with local coe½cients in O2, and we describe the properties of cobordism group with local coe½ciens inO3. We prove Theorem 1 in O4 and then we compute some cobordism groups with local coe½cients and prove Corollary 2 inO5. We discuss the relation of local orientations and cobordism classes inO6 and we prove Theorem 3, its generalized form Theorem 20, and Theorem 4 in O7. Finally we give some calculations of H4…Bp;Sw†=

…Autp†w in O8.

The auther would like to thank Prof. Takao Matumoto for his advice and suggestions.

2. Cobordism group with local coe½cients

LetMbe a compactn-manifold, and f a map of …M;qM†into…X;A†. If Aˆf then qMˆf. We denote by fSw the local system over M induced from Sw by f, that is, fSw…u† ˆSw…f…u††foruAM and fSw…g† ˆSw…f

for gAG…u0;u†.

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If the following conditions are satis®ed, two local systems S;T over M are called equivalent, and denoted by j:S!@ T.

(4.1) For every uAM, there exists an isomorphism ju:S…u† !T…u†.

(4.2) For every pair of points u;vAM and every homotopy class g of path from v to u, the following diagram is commutative.

S…u† ƒƒƒ!ju T…u†

S…g†

??

?y

??

?yT…g†

S…v† ƒƒƒ!

jv T…v†

Now we de®ne a w-singular manifold …M;f;j† of dimension n in …X;A†

by the following three conditions.

(5.1) M is a compact n-manifold.

(5.2) f is a continuous map from …M;qM† into …X;A†.

(5.3) j:SM ! fSw is an equivalence.

We recall here the de®nition of the isomorphism SM…a† for the relative homotopy class a of any path from u to v. For each point uAIntM there exists an open neighborhood U of u with a homeomorphism h:…U;u† ! …Rn;0†. We put D…r† ˆ fxARn;jjxjjarg and U…r† ˆhÿ1…IntD…r†† for a positive number r. Then the inclusion iuU…r†:…M;MÿU…r†† ! …M;Mÿu†

induces an isomorphism iuU…r†:Hn…M;MÿU…r†;Z† !Hn…M;Mÿu;Z†. For another choice of open neghborhood U0 of u, a homeomorphism h0, and a positive number r0 we write U0…r0† as above. If U0…r0†HU…r†, then the homomorphism iUU…r†0…r0† induced by the inclusion iUU…r†0…r0†:…M;MÿU…r†† ! …M;MÿU0…r0†† coincides with the isomorphism …iuU0…r0††ÿ1iuU…r†. The set B consisting of all U…r†'s obtained by changing u;U;h;r forms an open basis of M and Buˆ fU…r†AB;uAU…r†g is a directed set. Therefore fHn…M;Mÿ U…r†;Z†;iUU…r†0…r0†;uAU…r†g forms an inductive system over Bu and we get a canonical isomorphism

lim! Hn…M;MÿU…r†;Z†GHn…M;Mÿu;Z†:

For any two points u;v of IntM and any embbeded path g from u to v, we take a Lebesgue number e of an open covering fgÿ1…U…r††g of [0, 1] and a division 0ˆt0<t1< <tlˆ1 of [0, 1] such that tjÿtjÿ1 <e. We put g…tj† ˆuj. For each j …1ajal† there exists some U…r† which contains g…‰tjÿ1;tjŠ†. Denoting such U…r† by Uj…rj†, we de®ne a homomorphism g:Hn…M;Mÿv;Z† !Hn…M;Mÿu;Z† by

gˆiuU1…r1† …iuU11…r1††ÿ1iuU12…r2† …iuU22…r2††ÿ1 iuUlÿ1l…rl† …iuUll…rl††ÿ1:

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It is known that the homomorphismg depends only on the homotopy class of g keeping the boundary ®xed [6]. When g is a closed path, gw1…M† is an obstruction to the trivialization of gT…M†, where T…M†is the tangent bundle of M. So, SM…‰gŠ† is given by g for any path g connecting two points of IntM. If vAIntM and uAqM, we choose a closed neighborhood V of u in M homeomorphic to a closed disk Dn, and choose a point v0AVV IntM and an embbeded path d in V from u to v0. We can assume dVqMˆ fug. Moreover we put V1ˆVVqM and V2ˆqVÿV1. Let d:Hn…IntM;IntM ÿv0;Z† !Hnÿ1…qM;qMÿu;Z† be a composite of the following maps:

Hn…IntM;IntMÿv0;Z†ƒ!i Hn…M;Mÿv0;Z†

ƒ!iVÿ1

Hn…V;qV;Z†ƒ!q H~nÿ1…qV;Z†

ƒ!j Hnÿ1…qV;V2;Z†ƒ!kÿ1 Hnÿ1…V1;qV1;Z†

ƒ!iV1 Hnÿ1…qM;qMÿu;Z†;

where i;iV;j;k;iV1 are isomorphisms induced by the inclusions. Then SM…‰dŠ† is given by d. The composition of g and d's gives the iso- morphism SM…a† for the relative homotopy class a of any path from u to v with u;vAqM. Note that SMjqM is also determined by w1…qM† and SMjqMˆSqM.

Given an equivalence j:SIntM ! fSwjIntM. Then we can extend it to an equivalence j:SM ! fSw by de®ning ju ˆSw…f‰dŠ† jv0dÿ1 for uAqM, where d is a path in M from u to v0AIntM. This remark is very useful, especially in the proof of Propositions 5 (3) and 6. We will use the notation j_:SqM ! …fjqM†Sw as a restriction of j on SMjqMˆSqM hereafter.

Let Mn…X;A;Sw†be the set of all w-singular manifolds of dimension n in …X;A†. For …M;f;j†, …N;g;c†AMn…X;A;Sw†, we de®ne

ÿ…M;f;j† ˆ …M;f;ÿj†; …M;f;j† ‡ …N;g;c† ˆ …MUN;fUg;jUc†:

We say that…M;f;j†is null cobordant:…M;f;j†@0, if there exists an element …W;F;F†AMn‡1…X;X;Sw†such thatq…W;F;F†1…M;f;j† modA, that is,

(6.1) M is a regular submanifold of qW, (6.2) FjMˆ f and F…qWÿM†HA, and

(6.3) FjInt_ Mˆj by identifying Hn…qW;qWÿv;Z† with Hn…IntM;

IntMÿv;Z† for any vAIntM.

We de®ne …M;f;j†@…N;g;c† when …M;f;j† ‡ …N;g;ÿc†@0. Then we have the following proposition.

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Proposition5. The relation@inMn…X;A;Sw† is an equivalence relation.

Proof. (1) For …M;f;j†AMn…X;A;Sw† let W ˆMI and de®ne a map F:W !X by

F…u;t† ˆ f…u† ……u;t†AMI†:

For each vˆ …u;t† we de®ne a path av from …u;0† to v and a path bv from …u;1† to v by

av…s† ˆ …u;st†; bv…s† ˆ …u;1ÿs‡st† …sAI†:

Note that identifyinguwith…u;0†and…u;1†we getjuavˆ ÿjubv for vˆ …u;t†AIntW and we de®ne Fv by this map. Then, identifyingM0 and M1 with M, it is easy to see

q…W;F;F†1…M;f;j† ‡ …M;f;ÿj† modA:

(2) The re¯ective law is clear.

(3) Assume that

q…W1;F1;F1†1…M1;f1;j1† ‡ …M2;f2;ÿj2† modA q…W2;F2;F2†1…M2;f2;j2† ‡ …M3;f3;ÿj3† modA:

We glueW1 andW2 by identifyingM2 by a di¨eomorphism which reverses the local orientation at each point, and denote the resulting manifold by W. We de®ne a map F:W!X byF…v† ˆFi…v† …vAWi†foriˆ1;2. For vAIntWj …jˆ1;2† the inclusion ij…v†:…IntWj;IntWjÿv† ! …IntW;IntWÿv† induces an isomorphism

ij…v†:Hn‡1…IntWj;IntWjÿv;Z† !Hn‡1…IntW;IntWÿv;Z†:

If vAIntM2, we take a neighborhood Uofvin IntW such that…U;UVM2†is homeomorphic to …Rn‡1;Rn†. We take further a point vjAUVIntWj and a path~aj from vto vj in UVWj …jˆ1;2†. If we regard~aj as a path in IntW, we rewrite this aj. Then we have isomorphisms

a~j :Hn‡1…IntWj;IntWjÿvj;Z† !Hn…IntM2;IntM2ÿv;Z†;

aj :Hn‡1…IntW;IntWÿvj;Z† !Hn‡1…IntW;IntWÿv;Z†:

Since U is simply connected, from the way of the gluing we get ÿ~a1i1…v1†ÿ1 aÿ11 ˆ~a2i2…v2†ÿ1 aÿ12: So, we de®ne F by

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Fvˆ …Fj†vij…v†ÿ1 …vAIntWj; jˆ1;2†

ÿ…j2†v~a1i1…v1†ÿ1 aÿ11 …vAIntM2†.

(

The de®nition is independent of the choice of U;vj;~aj. Moreover we have Sw…F‰ajŠ† Fvj ˆFv …aj† for vAIntM2 …jˆ1;2†. Let vjAIntWj, vAIntM2 be any point and gj be any path from v to vj in Wj for jˆ1;2.

From the above equality we see that Sw…F‰gjŠ† FvjˆFv …gj†. This leads to Sw…Fg† Fv0ˆFvg for any points v;v0AIntW and any gAG…v0;v†.

Hence we get an equivalence F:SIntW !@ FSwjIntW. Since this can be extended naturally to F:SW !FSw as remaked before, we have

q…W;F;F†1…M1;f1;j1† ‡ …M3;f3;ÿj3† modA: r We put Wn…X;A;Sw† ˆMn…X;A;Sw†=@and denote by ‰M;f;jŠ the equivalence class of …M;f;j†. By setting‰M;f;jŠ ‡ ‰N;g;cŠ ˆ ‰MUN;fUg;

jUcŠ, Wn…X;A;Sw† has a structure of an abelian group. We call this group an n-dimensional cobordism group with local coe½cients Sw of …X;A†. If wˆ0, thenMandWare orientable;jandFgive the orientation ofMandW respectively. Therefore Wn…X;A;S0† coincides with Wn…X;A†.

The relative cobordism group may be also de®ned by the method of [12, p. 43], but our method makes clear the representatives and able to prove Theorems 1 and 3.

3. Properties of cobordism group with local coe½cients

In this section, we study the properties of cobordism group with local coe½cients needed to construct the Atiyah-Hirzeburch spectral sequence.

Cobordism groups with local coe½cients have properties similar to the Eilenberg-Steenrod axioms for the homology theory.

Fix hAH1…Y;Z2† and a continuous map h:…X;A† ! …Y;B†. For each

‰M;f;jŠAWn…X;A;hSh†, we have j:SM !@ …h f†Sh. Hence we de®ne a homomorphism h:Wn…X;A;hSh† !Wn…Y;B;Sh† by

h…‰M;f;jŠ† ˆ ‰M;h f;jŠ:

Let i:A!X be the inclusion map. We de®ne a boundary operator q:Wn…X;A;Sw† !Wnÿ1…A;iSw† by

q…‰M;f;jŠ† ˆ ‰qM;fjqM;jŠ;_ where j_ˆjjSqM.

Proposition6. Cobordism groups with local coe½cients have the following properties.

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(1) If id:…X;A† ! …X;A† is the identity map, then id:Wn…X;A;Sw† ! Wn…X;A;Sw† is the identity map.

(2) Let h:…X;A† ! …Y;B† and h0 :…Y;B† ! …Z;C† be continuous maps and zAH1…Z;Z2†. Then …h0:Wn…X;A;…h0Sz† !Wn…Z;C;Sz† is a composite of h:Wn…X;A;…h0Sz† !Wn…Y;B;…h0†Sz† and h0 :Wn…Y;B;

…h0†Sz† !Wn…Z;C;Sz†.

(3) For any hAH1…Y;Z2†and any map h:…X;A† ! …Y;B†, the diagram Wn…X;A;hSh† ƒƒƒ!q Wnÿ1…A;ihSh†

h

??

?y

??

?y…hjA†

Wn…Y;B;Sh† ƒƒƒ!q Wnÿ1…B;iSh† is commutative.

(4) For every pair …X;A† and every wAH1…X;Z2†, the sequence !Wn…A;iSw† !i Wn…X;Sw† !j Wn…X;A;Sw† !q Wnÿ1…A;iSw† ! is exact.

(5) If there is a homotopy ht:…X;A† ! …Y;B†, then h0ˆh1:Wn…X;A;

Sw† !Wn…Y;B;Sh† for wˆh0hˆh1h, hAH1…Y;Z2†.

(6) If UHIntA, then the inclusion i:…XÿU;AÿU† ! …X;A† induces an isomorphism i:Wn…XÿU;AÿU;iSw† !Wn…X;A;Sw†.

Proof. (1), (2) and (3) are trivial.

(4) For ‰M;f;jŠAWn…A;iSw† we put W ˆMI. We de®ne a map F :W !X by F…u;t† ˆ f…u† ……u;t†AMI† and a path av from …u;0† to vˆ …u;t† by av…s† ˆ …u;st†. Moreover de®ne F by extending Fvˆjuav

…vˆ …u;t†AIntW†. Then q…W;F;F†1…M;f;j† modA. Hence we have jiˆ0.

Assume that j‰M;f;jŠ ˆ0 for ‰M;f;jŠAWn…X;Sw†. Then there exists an element …W;F;F†AMn‡1…X;X;Sw† such that q…W;F;F†1…M;f;j†

modA. Now we put

NˆqWÿM; gˆFjN; cˆ ÿFjN:_

Then ‰N;g;cŠAWn…A;iSw† and i‰N;g;cŠ ˆ ‰M;f;jŠ. Hence we have KerjHImi.

qj ˆ0 and iqˆ0 are trivially veri®ed. Assume that q‰M;f;jŠ ˆ0 for

‰M;f;jŠAWn…X;A;Sw†. Then there exists an element…N;g;c†AMn…A;A;iSw† such thatq…N;g;c†1…qM;fjqM;j†. Now we put_

M0ˆMUqMN; f0ˆ fUg; j0ˆjUc

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and WˆM0I. De®ne a map F :W !X by F…u;t† ˆ f0…u†. Moreover de®ne F by extending Fvˆju0av …vˆ …u;t†AIntW†. Then it holds

q…W;F;F†1…M0;f0;j0† ‡ …M;f;ÿj† modA:

This implies j‰M0;f0;j0Š ˆ ‰M;f;jŠ. Hence we have KerqHIm j. Assume that i‰M;f;jŠ ˆ0 for ‰M;f;jŠAWnÿ1…A;iSw†. Then there exists an element …W;F;F†AMn…X;A;Sw† such that q…W;F;F†1…M;f;j†.

Since ‰W;F;FŠAWn…X;A;w†, we have ‰M;f;jŠAImq. Hence KeriHImq.

(5) For ‰M;f;jŠAWn…X;A;Sw† we put WˆMI and de®ne a map F :W !Y byF…u;t† ˆht…f…u†† ……u;t†AMI†. Sincehthˆwfor anytAI, we can de®neF just in the same way as in the proof of Proposition 5. Hence we get

q…W;F;F†1…M;h0 f;j† ‡ …M;h1 f;ÿj† modA:

(6) We will show that i is surjective; the remainder of argument is similar. For ‰M;f;jŠAWn…X;A;Sw†, let Pˆfÿ1…XÿIntA† andQˆfÿ1…U†.

Then there exists a compact submanifold NHM such that PHN and QVNˆf. We put gˆ fjN and cjIntNˆjjIntN by identifying Hn…IntM;

IntMÿv;Z† with Hn…IntN;IntNÿv;Z† for any vAIntN. The equiva- lence c:SN!gSw is de®ned as a natural unique extension. Then we have

‰N;g;cŠAWn…XÿU;AÿU;iw† and i‰N;g;cŠ ˆ ‰M;f;jŠ. r From (1), (2), (3) and (4) of Proposition 6 we see that the following sequence is exact for any triple …X;A;B† and w according to [3].

!Wn…A;B;iSw† !i Wn…X;B;Sw† !j Wn…X;A;Sw†

!q Wnÿ1…A;B;iSw† !

ForwAH1…X;Z2†andhAH1…Y;Z2†let xˆwn1‡1nhAH1…XY;Z2† GH1…X;Z2†nH0…Y;Z2†lH0…X;Z2†nH1…Y;Z2†. Then we can choose a local system Sx equivalent to SwnSh on XY. Through this equivalence for ‰M;f;jŠAWm…X;A;Sw† and ‰N;g;cŠAWn…Y;Sh† we have

jnc:SMN !@ …f g†Sx:

Then, j_nc:SqMN!@ …f g†SxjqMN and hence we can de®ne a homomorphism

Y:Wm…X;A;Sw†nWn…Y;Sh† !Wm‡n…XY;AY;Sx†

by Y…‰M;f;jŠn‰N;g;cŠ† ˆ ‰MN;f g;jncŠ. In particular, if Y ˆpt then we get a homomorphism

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Y:Wm…X;A;Sw†nWn!Wm‡n…X;A;Sw†;

where Wn is the Thom group ([2], [13]).

Let A be a closed subset of X. We want to use an open subset V of X which contains A and

(7.1) A is a deformation retract of V by a retraction r:V !A, that is, iAr:V!V is homotopic to the identity 1V:V !V for the natural in- clusion iA:A!V.

For a continuous map f :A!Y, let f :…X;A† ! …YUf X;Y† be a map de®ned by

f…x† ˆ f…x† …xAA†

x …xAXÿA†:

We have the following theorem.

Theorem 7 (Cf. [6]). Let A be a closed subeset of X and f :A!Y be a continuous map. If there exists an open subset VIA satisfying (7.1), then f:Wn…X;A;fSh† !Wn…YUf X;Y;Sh† is an isomorphism for any hAH1…YUfX;Z2†.

Proof. We put ZˆYUf X and let i:…X;A† ! …X;V†; j:…Z;Y† ! …Z;YUf…V†† be inclusion maps. Consider the left part of the following commutative diagram:

Wn…X;A;fSh† ƒƒƒi! Wn…X;V;fSh† ƒƒƒi0 Wn…XÿA;VÿA;i0fSh†

f

??

?y

??

?yf

??

?yf Wn…Z;Y;Sh† ƒƒƒ!

j Wn…Z;YUf…V†;Sh† ƒƒƒ

j0

Wn…ZÿY;f…VÿA†;j0Sh†:

For the homotopy ht:V!V between iAr and 1V given by (7.1), ht: H1…V;Z2† !H1…V;Z2†is an identity isomorphism for every t. Hence by (1), (2), (3), (4) and (5) of Proposition 6 we have Wq…V;A;iVfSh† ˆ0 for the natural inclusion iV:V !X and every q. From the exact sequence of triple …X;V;A†we see thati is an isomorphism. By a similar argument we see that j is also an isomorphism. Next we consider the right part of the above commutative diagram. From (6) of Proposition 6 we see that i0 and j0 are isomorphisms for the natural inclusions i0 and j0. Since the map f :…XÿA;

VÿA† ! …ZÿY;f…VÿA†† is a homeomorphism, f on the right-hand side is an isomorphism. Hence so is f on the center. Consequently f on the

left-hand side is an isomorphism. r

Let X be a CW complex and Xp its p-skeleton. Hereafter until the end of O5, i:Xp!X denotes the natural inclusion. For each p-cell el of X,

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hl:…Dlp;Slpÿ1† ! …el;e_l† denotes its characteristic map. Then we have the following corollary applying Proposition 7 to Xˆ`

lDlp, Aˆ`

lSlpÿ1, Y ˆXpÿ1 and f ˆ`

lhl, because a CW complex has the homotopy extension property.

Corollary 8. The map Shl:P

lWn…Dlp;Slpÿ1;hliSw† !Wn…Xp;Xpÿ1; iSw† is an isomorphism.

Moreover, we have

Corollary 9. The map Y:Wn…Xn;Xnÿ1;iSw†nWq!Wn‡q…Xn;Xnÿ1; iSw† is an isomorphism.

Proof. Since Dlp is simply connected, the local system hliSw is equivalent to S0. So, the map

Yl:Wn…Dln;Slnÿ1;hliSw†nWq!Wn‡q…Dln;Slnÿ1;hliSw†

is an isomorphism for every l by [2]. Furthermore, the following diagram is commutative:

P

lWn…Dln;Slnÿ1;hliSw†nWq ƒƒƒ!SYl P

lWn‡q…Dln;Slnÿ1;hliSw†

Shln1

??

?y

??

?yShl Wn…Xn;Xnÿ1;iSw†nWq ƒƒƒ!

Y Wn‡q…Xn;Xnÿ1;iSw†:

Therefore, Corollary 8 implies Corollary 9. r

4. Proof of Theorem 1

For ‰M;f;jŠAWn…X;A;Sw† let Hn…IntM;SIntM† be a homology group of in®nite chains with local coe½cients SIntM and j]: Hn…IntM;SIntM†

!Hn…IntM;fSw† be the isomorphism induced by jjIntM. We know that there is a natural isomorphism i: Hn…IntM;fSw† !Hn…M;qM;fSw† for any compact manifold M (cf. [6]). We put jˆij] and de®ne a homomorphism

m:Wn…X;A;Sw† !Hn…X;A;Sw†

by m…‰M;f;jŠ† ˆ f…j…sM††, where f is an induced homomorphism f:Hn…M;qM;fSw† !Hn…X;A;Sw†

and sM is a fundamental class of Hn…IntM;SIntM†. Then, for the any CW complex X we have the following.

(13)

Theorem 10. The map m:Wn…Xn;Xnÿ1;iSw† !Hn…Xn;Xnÿ1;iSw† is an isomorphism for every wAH1…X;Z2†.

Proof. We know that the map

ml:Wn…Dln;Slnÿ1;hliSw† !Hn…Dln;Slnÿ1;hliSw†

is an isomorphism for every l by [2], and the following diagram is commutative:

P

lWn…Dln;Slnÿ1;hliSw† ƒƒƒ!Sml P

lHn…Dln;Slnÿ1;hliSw†

Shl

??

?y

??

?yShl Wn…Xn;Xnÿ1;iSw† ƒƒƒ!

m Hn…Xn;Xnÿ1;iSw†:

Since the vertical map at the right-hand side is an isomorphism, Corollary 8

implies Theorem 10. r

Proof ofTheorem 1. For wAH1…X;Z2† and each pair of integers…p;q†

such that ÿyapaqay, we put H…p;q† ˆP

nWn…Xÿp;Xÿq;iSw†. Then fH…p;q†gsatis®es the axioms in the theory of spectral sequences [1, Chap. XV, p. 334]. Now let H…p;q† ˆH…ÿp;ÿq†, H…p† ˆH…p;ÿy†, HˆH…y;ÿy†.

We de®ne a ®ltration Fp;qH of H by

Fp;qH ˆIm…Hp‡q…p† !Hp‡q† ˆIm…Wp‡q…Xp;iSw† !Wp‡q…X;Sw††:

We de®ne also

Zp;qr ˆIm…Hp‡q…p;pÿr† !Hp‡q…p;pÿ1††

ˆIm…Wp‡q…Xp;Xpÿr;iSw† !Wp‡q…Xp;Xpÿ1;iSw††

Bp;qr ˆIm…Hp‡q‡1…p‡rÿ1;p† !Hp‡q…p;pÿ1††

ˆIm…Wp‡q‡1…Xp‡rÿ1;Xp;iSw† !Wp‡q…Xp;Xpÿ1;iSw††

Ep;qr ˆZp;qr =Bp;qr

where 1aray,ÿy<p<y. Since Hn…p† ˆWn…Xp;iSw† ˆ0 for everyn and paÿ1, F is regular and hence convergent in the sense of [1]. Then we have particularly

Ep;q1 ˆWp‡q…Xp;Xpÿ1;iSw†:

By Corollary 9 and Theorem 10 we get

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Wp‡q…Xp;Xpÿ1;iSw† G Wp…Xp;Xpÿ1;iSw†nWq!G Hp…Xp;Xpÿ1;iSw†nWq: By the universal coe½cient theorem for the homology with local coe½cients [6]

we have

Hp…Xp;Xpÿ1;iSw†nWq G Hp…Xp;Xpÿ1;WqniSw†:

Moreover, through these isomorphisms, we have the following commutative diagram:

Wp‡q…Xp;Xpÿ1;iSw† ƒƒƒ!G Hp…Xp;Xpÿ1;WqniSw†

q

??

?y

??

?yq

Wp‡qÿ1…Xpÿ1;Xpÿ2;iSw† ƒƒƒ!G Hpÿ1…Xpÿ1;Xpÿ2;WqniSw†:

Therefore the di¨erential dp;1q:Ep;1q!Epÿ1;1 q is identi®ed with the boundary operator q:Hp…Xp;Xpÿ1;WqniSw† !Hpÿ1…Xpÿ1;Xpÿ2;WqniSw†. Hence we have

Ep;q2 GHp…X;WqnSw†:

Thus we proved Theorem 1.

5. Some calculations and proof of Corollary 2

Using Theorem 1 we will calculate the cobordism group with local coe½cients for some examples and prove Corollary 2.

Example 1. Let X ˆS1 and w00. We have an exact sequence 0!E0;yn!Wn…S1;Sw† !E1;ynÿ1!0

since Em;nÿm2 ˆ0 for m00;1. From H0…S1;Sw† ˆZ2 and H1…S1;Sw† ˆ0, we have E0;ny GH0…S1;WnnSw†GWnnZ2 and E1;nÿ1y GH1…S1;Wnÿ1nSw† GTor…Z2;Wnÿ1†. It is known that W0GZ, W1ˆW2ˆW3ˆ0, W4GZ.

Hence we have Wn…S1;Sw†GWnnZ2 for na5.

Example 2. LetX be a real projective planeP2 andw00. We see that

Em;nÿm2 ˆHm…P2;WnÿmnSw†G

WnnZ2 …mˆ0†

Tor…Z2;Wnÿ1† …mˆ1†

Wnÿ2 …mˆ2†

0 …mb3†

8>

>>

<

>>

>:

since H0…P2;Sw† ˆZ2, H1…P2;Sw† ˆ0,H2…P2;Sw† ˆZ. Hence for na5 we have an exact sequence

(15)

0!Ey0;n!Wn…P2;Sw† !E2;nÿ2y !0:

Then we have W2…P2;Sw†GW0 and Wn…P2;Sw†GWnnZ2 for n02, na5.

Proof of Crollary 2. Since W0GZ, W1ˆW2ˆW3ˆ0, W4GZ, we have an exact sequence

0!E0;4y !W4…X;Sw† !E4;0y !0:

The map m induces a map m from the Atiyah-Hirzeburch spectral sequence for Wp‡q…X;Sw† to the Atiyah-Hirzeburch spectral sequence fEp;q0r g for Hp‡q…X;Sw† and we have the following commutative diagram:

W4…X;Sw† ƒƒƒ! Ey4;0ˆH4…X;W0nSw†

m

??

?y

??

?ym

H4…X;Sw† ƒƒƒid! E4;0y0ˆH4…X;Sw†:

Since m is an isomorphism, we may identify the map m with the above map W4…X;Sw† !Ey4;0. Since X is connected, H0…X;Sw†GZ if wˆ0, and H0…X;Sw†GZ2 if w00. Therefore we have Ey0;4GW4 if wˆ0, and Ey0;4G W4nZ2 if w00. Hence we get the conclusion.

6. Local orientations of non-orientable manifolds At ®rst we prove the following Proposition.

Proposition 11. Let X be an arcwise connected space and wAH1…X;Z2†.

Suppose that M is a connected manifold without boundary. Then for any continuous map f :M!X the local system SM is equivalent to fSw if and only if fwˆw1…M†.

Proof. Assume that j:SM ! fSw is an equivalence. We regard w and w1…M† as the homomorphisms from H1…X;Z† to AutZˆZ2 and H1…M;Z† to AutZˆZ2 respectively. We put rwˆwX and rM ˆ w1…M† X for the Hurewicz homomorphism X. For every point uAM and every element gAp1…M;u†, the following diagram is commutative:

SM…u† ƒƒƒ!ju …fSw†…u† ˆSw…f…u††

SM…g†

??

?y

??

?ySw…f

SM…u† ƒƒƒ!ju …fSw†…u† ˆSw…f…u††:

(16)

So, …SM†u…g† ˆjÿ1u …Sw†f…u†…fg† ju as an automorphism of SM…u†.

Because ju identi®es SM…u† ˆSw…f…u††GZ, this means rM ˆrw f. Since X is a surjection, we see fwˆw1…M†by the following commutative diagram:

p1…M;u† ƒƒƒf! p1…X;f…u††

X

??

?y

??

?yX

H1…M;Z† ƒƒƒf! H1…X;Z†:

Conversely assume that fwˆw1…M†. Fix a base point u0. Then, the local systems fSw and SM have the same associated homomorphism rM ˆ rw f:p1…M;u0† !AutZ. We choose an element auAG…u;u0† for each point uAM. If we choose an isomorphismju0 :SM…u0† ! …fSw†…u0†for the base pointu0, the isomorphismju:SM…u† ! …fSw†…u† is determined by juˆ Sw…fau†ÿ1ju0SM…au†. In fact jˆ fjug satis®es

juSM…g† ˆSw…fau†ÿ1ju0 …SM†u0…augaÿ1v † SM…av†

ˆSw…fau†ÿ1 …Sw†f…u0†…f…augaÿ1v †† ju0SM…av†

ˆSw…fg† jv

for every gAG…v;u†. Hence j is an equivalence. r Let Mbe a closed connectedn-manifold,pˆp1…M†and f;f0:…M;u0† ! …Bp;y0† be two maps which satisfy the conditions (3.1) and (3.2). Moreover let j:SM ! fSw and j0:SM ! f0Sw be equivalences. Suppose that f and f0 are homotopic by a homotopyF :MI!Bp. For each point uAM let gu be a path from …u;0† to …u;1† in MI de®ned by gu…t† ˆ …u;t† and de®ne isomorphisms du: f0Sw…u† ! fSw…u† and kF…u†:SM…u† !SM…u† by

duˆSw…F‰guŠ† and kF…u† ˆjÿ1u duju0: …8:1†

Then we have

kF…u† ˆSM…a†ÿ1kF…u0† SM…a†

for every relative homotopy class a of paths from u0 to u in M. We may regard kF as a map from M to AutZ. From the above equation we see that kF is continuous. We de®ne sgnkF by

sgnkF ˆ 1 if kF…u† ˆid for anyu ÿ1 if kF…u† ˆ ÿidfor anyu.

We have the following proposition.

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Proposition 12. Let M be a closed connected n-manifold, pˆp1…M† and f;f0:…M;u0† ! …Bp;y0† be two maps which satisfy the conditions (3.1) and (3.2). Moreover let j:SM ! fSw and j0:SM ! f0Sw be equivalences.

Suppose that f and f0 are homotopic by a homotopy F. Then it holds‰M;f;jŠ

ˆ ‰M;f0;…sgnkF†j0Š in Wn…Bp;Sw†, where kF is a map de®ned by (8.1).

Proof. We put W ˆMI. For vˆ …u;t†AIntW we de®ne Fv: SW…v† !FSw…v† by

FvˆSw…F‰avŠ†ÿ1juav;

where avis a path from …u;0†to vˆ …u;t†de®ned byav…s† ˆ …u;st†. Letbv be a path from …u;1† to vˆ …u;t† de®ned by bv…s† ˆ …u;1ÿs‡st†. By the de®nitions of Fv and kF…v† we see that

Fvˆ ÿSw…F‰bvŠ†ÿ1ju0kF…u†ÿ1bv

ˆ ÿSw…F‰bvŠ†ÿ1 …sgnkF†ju0 bv: So, F_v:SqW…v† ! …FjqW†Sw…v† is written as

F_vˆ ju …vˆ …u;0††

ÿ…sgnkF†ju0 …vˆ …u;1††:

Hence we get …W;F;F†1…M;f;j† ‡ …M;f0;ÿ…sgnkF†j0†. r Let g be an element of orthogonal group O…nÿ1† with detgˆ ÿ1 and denote by N the quotient space of RDnÿ1 gained by identifying …s;v† and …s‡1;gv† for each …s;v†ARDnÿ1. Then N is a non-orientable smooth O…nÿ1† bundle over S1 with ®ber Dnÿ1. We denote by ‰s;vŠ the point represented by …s;v† in N.

Let d:‰0;1Š ! ‰0;1Š be a monotone and smooth function such that dj‰0;eŠ ˆ1 and dj‰1ÿe;1Š ˆ0 for a positive number e which is small enough.

For each tAI we de®ne a map Ht:N!N by Ht…‰s;ruŠ† ˆ ‰s‡td…r†;ruŠ;

where sAR, 0ara1 anduAqDnÿ1. Then Ht is a di¨eomorphism such that HtjqNˆ1qN for each t and H1 is homotopic to H0ˆ1N.

Let M be a closed, connected and non-orientable n-manifold and a be a simple closed arc with based point u0 such that w1…M†…‰aŠ†00. The tubular neighborhood of a is di¨eomorphic to the above bundle N for a some gAO…nÿ1†with detgˆ ÿ1. Hence we have a di¨eomorphismh:…M;u0† ! …M;u0† which satis®es the conditions

(9.1) h is the identity map out of a tubular neighborhood N…a†,

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