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

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;A†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†=…Autp†^w, 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

E_p;²_qˆHp…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 W_n…X;A;S_w† for a pair …X;A† of topological spaces and wAH¹…X;Z2†, which reduces to Wn…X;A† if wˆ0.

Let w₁:BO_r!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

Br ƒƒƒ! X

fr

??

?y

??

?y^w BO_r ƒƒƒ!

w1 K…Z₂;1†

be the pull-back. Then W_n…X;S_w† coincides with W_n…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 a_xAG…x;x₀† for each xAX. Then we see also that

S…g† ˆS…a_x†^ÿ1S_x0…a_xga^ÿ1_y † S…a_y†

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, S_w…x† is isomorphic to the group Z of integers.

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

Here r_w 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

E_p;q² ˆH_p…X;W_qnS_w† )W_p‡q…X;S_w† 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 H_nÿ1…qN;qNÿu;Z† for each uAqN.

(2.2) S_N is determined by the homomorphism r_N 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 fj_ug_uAM denote the family of isomorphisms j_u:S_M…u† ! fS_w…u† which gives this equivalence (See O2). Let M_n…X;S_w† be the set which consists of such triples …M;f;j†. We de®ne the equivalence relation in M_n…X;S_w† 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) FjM_jˆ f_j …jˆ1;2†,

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

The set of equivalence classes M_n…X;S_w†=@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 H_n…M;S_M†.

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 M⁴_p;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 M⁴_p;w is not empty. For every MAM⁴_p;w there exists an element …M;f;j†of M4…Bp;Sw† by Proposition 11 inO6. For a non-zero

w Proposition 13 in O6 says that ‰M;f;jŠ ˆ ‰M;f;ÿjŠ in W₄…Bp;S_w† 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 M_p;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 S_M.

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 r_w1…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;S_w†=

…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 S_w by f, that is, fS_w…u† ˆS_w…f…u††foruAM and fS_w…g† ˆS_w…fg†

for gAG…u⁰;u†.

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 j_u: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

??

?y^T…g†

S…v† ƒƒƒ!

j_v 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 S_M…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† ˆh^ÿ1…IntD…r†† for a positive number r. Then the inclusion iu^U…r†:…M;MÿU…r†† ! …M;Mÿu†

induces an isomorphism i_u^U…r†:H_n…M;MÿU…r†;Z† !H_n…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 i_U^U…r†0…r⁰† induced by the inclusion i_U^U…r†0…r⁰†:…M;MÿU…r†† ! …M;MÿU⁰…r⁰†† coincides with the isomorphism …iu^U0…r0††^ÿ1iu^U…r†. The set B consisting of all U…r†'s obtained by changing u;U;h;r forms an open basis of M and B_uˆ fU…r†AB;uAU…r†g is a directed set. Therefore fH_n…M;Mÿ U…r†;Z†;i_U^U…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 fg^ÿ1…U…r††g of [0, 1] and a division 0ˆt0<t1< <tlˆ1 of [0, 1] such that tjÿtjÿ1 <e. We put g…t_j† ˆu_j. 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:H_n…M;Mÿv;Z† !H_n…M;Mÿu;Z† by

gˆi_u^U1…r1† …i_u^U₁^1…r1††^ÿ1i_u^U₁^2…r2† …i_u^U₂^2…r2††^ÿ1 i_u^U_lÿ1^l…rl† …i_u^U_l^l…rl††^ÿ1:

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 H_nÿ1…qV;V₂;Z†ƒ!^kÿ1 H_nÿ1…V₁;qV₁;Z†

ƒ!^iV1 Hnÿ1…qM;qMÿu;Z†;

where i;i_V;j;k;i_V1 are isomorphisms induced by the inclusions. Then SM…‰dŠ† is given by d. The composition of g and d's gives the iso- morphism S_M…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 S_MjqMˆS_qM.

Given an equivalence j:SIntM ! fSwjIntM. Then we can extend it to an equivalence j:SM ! fSw by de®ning j_u ˆSw…f‰dŠ† j_v0d^ÿ1 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_:S_qM ! …fjqM†S_w as a restriction of j on S_MjqMˆS_qM hereafter.

Let M_n…X;A;S_w†be the set of all w-singular manifolds of dimension n in …X;A†. For …M;f;j†, …N;g;c†AM_n…X;A;S_w†, 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.

Proposition5. The relation@inM_n…X;A;S_w† is an equivalence relation.

Proof. (1) For …M;f;j†AM_n…X;A;S_w† 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 b_v from …u;1† to v by

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

Note that identifyinguwith…u;0†and…u;1†we getj_uavˆ ÿj_ub_v for vˆ …u;t†AIntW and we de®ne F_v 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† ˆF_i…v† …vAW_i†foriˆ1;2. For vAIntW_j …jˆ1;2† the inclusion ij…v†:…IntWj;IntWjÿv† ! …IntW;IntWÿv† induces an isomorphism

i_j…v†:H_n‡1…IntW_j;IntW_jÿv;Z† !H_n‡1…IntW;IntWÿv;Z†:

If vAIntM₂, we take a neighborhood Uofvin IntW such that…U;UVM₂†is homeomorphic to …R^n‡1;Rⁿ†. We take further a point v_jAUVIntW_j and a path~aj from vto vj in UVWj …jˆ1;2†. If we regard~aj as a path in IntW, we rewrite this a_j. 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 a^ÿ1₁ ˆ~a₂i₂…v₂†^ÿ1 a^ÿ1₂: So, we de®ne F by

Fvˆ …Fj†_vij…v†^ÿ1 …vAIntWj; jˆ1;2†

ÿ…j₂†_v~a1i1…v1†^ÿ1 a^ÿ1₁ …vAIntM2†.

(

The de®nition is independent of the choice of U;v_j;~a_j. Moreover we have S_w…F‰a_jŠ† F_vj ˆF_v …a_j† for vAIntM₂ …jˆ1;2†. Let v_jAIntW_j, vAIntM2 be any point and g_j be any path from v to vj in Wj for jˆ1;2.

From the above equality we see that S_w…F‰g_jŠ† F_vjˆF_v …g_j†. This leads to Sw…Fg† Fv⁰ˆFvg for any points v;v⁰AIntW and any gAG…v⁰;v†.

Hence we get an equivalence F:S_IntW !^@ FS_wjIntW. 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 W_n…X;A;S_w† ˆM_n…X;A;S_w†=@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, W_n…X;A;S_w† 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:W_n…X;A;hS_h† !W_n…Y;B;S_h† 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_ˆjjS_qM.

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

(1) If id:…X;A† ! …X;A† is the identity map, then id:W_n…X;A;S_w† ! 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 …h⁰h†:Wn…X;A;…h⁰h†Sz† !Wn…Z;C;Sz† is a composite of h:W_n…X;A;…h⁰h†S_z† !W_n…Y;B;…h⁰†S_z† and h⁰ :W_n…Y;B;

…h⁰†Sz† !Wn…Z;C;Sz†.

(3) For any hAH¹…Y;Z₂†and any map h:…X;A† ! …Y;B†, the diagram W_n…X;A;hS_h† ƒƒƒ!^q W_nÿ1…A;ihS_h†

h

??

?y

??

?y^hjA†

W_n…Y;B;S_h† ƒƒƒ!^q W_nÿ1…B;iS_h† 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 h_t:…X;A† ! …Y;B†, then h₀ˆh₁:W_n…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:W_n…XÿU;AÿU;iS_w† !W_n…X;A;S_w†.

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 a_v from …u;0† to vˆ …u;t† by av…s† ˆ …u;st†. Moreover de®ne F by extending Fvˆj_uav

…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†AM_n‡1…X;X;S_w† 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;jŠAW_n…X;A;S_w†. Then there exists an element…N;g;c†AM_n…A;A;iS_w† such thatq…N;g;c†1…qM;fjqM;j†. Now we put_

M⁰ˆMUqMN; f⁰ˆ fUg; j⁰ˆjUc

and WˆM⁰I. De®ne a map F :W !X by F…u;t† ˆ f⁰…u†. Moreover de®ne F by extending Fvˆj_u⁰av …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;jŠAW_nÿ1…A;iS_w†. 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;FŠAW_n…X;A;w†, we have ‰M;f;jŠAImq. Hence KeriHImq.

(5) For ‰M;f;jŠAW_n…X;A;S_w† we put WˆMI and de®ne a map F :W !Y byF…u;t† ˆht…f…u†† ……u;t†AMI†. Sinceh_thˆwfor 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;jŠAW_n…X;A;S_w†, let Pˆf^ÿ1…XÿIntA† andQˆf^ÿ1…U†.

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

IntMÿv;Z† with Hn…IntN;IntNÿv;Z† for any vAIntN. The equiva- lence c:S_N!gS_w 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 W_nÿ1…A;B;iS_w† !

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 S_x equivalent to S_wnS_h on XY. Through this equivalence for ‰M;f;jŠAW_m…X;A;S_w† and ‰N;g;cŠAW_n…Y;S_h† we have

jnc:S_MN !^@ …f g†S_x:

Then, j_nc:S_qMN!^@ …f g†S_xjqMN and hence we can de®ne a homomorphism

Y:W_m…X;A;S_w†nW_n…Y;S_h† !W_m‡n…XY;AY;S_x†

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

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† ! …YU_f 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:W_n…X;A;fS_h† !W_n…YU_f X;Y;S_h† 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:

W_n…X;A;fS_h† ƒƒƒⁱ! W_n…X;V;fS_h† ƒƒƒⁱ⁰ W_n…XÿA;VÿA;i⁰fS_h†

f

??

?y

??

?y^f

??

?y^f 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), h_t: 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;i_VfSh† 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 X^p its p-skeleton. Hereafter until the end of O5, i:X^p!X denotes the natural inclusion. For each p-cell e_l of X,

h_l:…D_l^p;S_l^pÿ1† ! …e_l;e__l† denotes its characteristic map. Then we have the following corollary applying Proposition 7 to Xˆ`

lD_l^p, Aˆ`

lS_l^pÿ1, Y ˆX^pÿ1 and f ˆ`

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

Corollary 8. The map Shl:P

lWn…D_l^p;S_l^pÿ1;h_liSw† !Wn…X^p;X^pÿ1; iS_w† is an isomorphism.

Moreover, we have

Corollary 9. The map Y:Wn…Xⁿ;X^nÿ1;iSw†nWq!Wn‡q…Xⁿ;X^nÿ1; iS_w† is an isomorphism.

Proof. Since D_l^p is simply connected, the local system h_liS_w is equivalent to S0. So, the map

Y_l:W_n…D_lⁿ;S_l^nÿ1;h_liS_w†nW_q!W_n‡q…D_lⁿ;S_l^nÿ1;h_liS_w†

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

P

lWn…D_lⁿ;S_l^nÿ1;h_liSw†nWq ƒƒƒ!^SYl P

lWn‡q…D_lⁿ;S_l^nÿ1;h_liSw†

Shln1

??

?y

??

?y^Shl Wn…Xⁿ;X^nÿ1;iSw†nWq ƒƒƒ!

Y Wn‡q…Xⁿ;X^nÿ1;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 S_IntM and j_]: H_n…IntM;S_IntM†

!Hn…IntM;fSw† be the isomorphism induced by jjIntM. We know that there is a natural isomorphism i: H_n…IntM;fS_w† !H_n…M;qM;fS_w† 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:H_n…M;qM;fS_w† !H_n…X;A;S_w†

and s_M is a fundamental class of H_n…IntM;S_IntM†. Then, for the any CW complex X we have the following.

Theorem 10. The map m:W_n…Xⁿ;X^nÿ1;iS_w† !H_n…Xⁿ;X^nÿ1;iS_w† is an isomorphism for every wAH¹…X;Z2†.

Proof. We know that the map

m_l:W_n…D_lⁿ;S_l^nÿ1;h_liS_w† !H_n…D_lⁿ;S_l^nÿ1;h_liS_w†

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

P

lWn…D_lⁿ;S_l^nÿ1;h_liSw† ƒƒƒ!^Sml P

lHn…D_lⁿ;S_l^nÿ1;h_liSw†

Shl

??

?y

??

?y^Shl Wn…Xⁿ;X^nÿ1;iSw† ƒƒƒ!

m Hn…Xⁿ;X^nÿ1;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…X^ÿp;X^ÿq;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 F_p;qH of H by

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

We de®ne also

Z_p;q^r ˆIm…Hp‡q…p;pÿr† !Hp‡q…p;pÿ1††

ˆIm…W_p‡q…X^p;X^pÿr;iS_w† !W_p‡q…X^p;X^pÿ1;iS_w††

B_p;q^r ˆIm…Hp‡q‡1…p‡rÿ1;p† !Hp‡q…p;pÿ1††

ˆIm…W_p‡q‡1…X^p‡rÿ1;X^p;iS_w† !W_p‡q…X^p;X^pÿ1;iS_w††

E_p;q^r ˆZ_p;q^r =B_p;q^r

where 1aray,ÿy<p<y. Since H_n…p† ˆW_n…X^p;iS_w† ˆ0 for everyn and paÿ1, F is regular and hence convergent in the sense of [1]. Then we have particularly

E_p;q¹ ˆWp‡q…X^p;X^pÿ1;iSw†:

By Corollary 9 and Theorem 10 we get

W_p‡q…X^p;X^pÿ1;iS_w† ^G W_p…X^p;X^pÿ1;iS_w†nW_q!^G H_p…X^p;X^pÿ1;iS_w†nW_q: By the universal coe½cient theorem for the homology with local coe½cients [6]

we have

Hp…X^p;X^pÿ1;iSw†nWq G Hp…X^p;X^pÿ1;WqniSw†:

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

W_p‡q…X^p;X^pÿ1;iS_w† ƒƒƒ!^G H_p…X^p;X^pÿ1;W_qniS_w†

q

??

?y

??

?y^q

W_p‡qÿ1…X^pÿ1;X^pÿ2;iS_w† ƒƒƒ!_G H_pÿ1…X^pÿ1;X^pÿ2;W_qniS_w†:

Therefore the di¨erential d_p;¹_q:E_p;¹_q!E_pÿ1;¹ _q is identi®ed with the boundary operator q:Hp…X^p;X^pÿ1;WqniSw† !Hpÿ1…X^pÿ1;X^pÿ2;WqniSw†. Hence we have

E_p;q² GH_p…X;W_qnS_w†:

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!E_0;^y_n!Wn…S¹;Sw† !E_1;^y_nÿ1!0

since E_m;nÿm² ˆ0 for m00;1. From H₀…S¹;S_w† ˆZ₂ and H₁…S¹;S_w† ˆ0, we have E_0;n^y GH₀…S¹;W_nnS_w†GW_nnZ₂ and E_1;nÿ1^y GH₁…S¹;W_nÿ1nS_w† GTor…Z2;Wnÿ1†. It is known that W0GZ, W1ˆW2ˆW3ˆ0, W4GZ.

Hence we have W_n…S¹;S_w†GW_nnZ₂ for na5.

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

E_m;nÿm² ˆH_m…P²;W_nÿmnS_w†G

WnnZ2 …mˆ0†

Tor…Z2;Wnÿ1† …mˆ1†

Wnÿ2 …mˆ2†

0 …mb3†

8>

>>

<

>>

>:

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

0!E^y_0;n!W_n…P²;S_w† !E_2;nÿ2^y !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!E_0;4^y !W4…X;Sw† !E_4;0^y !0:

The map m induces a map m from the Atiyah-Hirzeburch spectral sequence for W_p‡q…X;S_w† to the Atiyah-Hirzeburch spectral sequence fE_p;q^0r g for Hp‡q…X;Sw† and we have the following commutative diagram:

W4…X;Sw† ƒƒƒ! E^y_4;0ˆH4…X;W0nSw†

m

??

?y

??

?y^m

H4…X;Sw† ƒƒƒ^id! E_4;^0y₀ˆH4…X;Sw†:

Since m is an isomorphism, we may identify the map m with the above map W4…X;Sw† !E^y_4;0. Since X is connected, H0…X;Sw†GZ if wˆ0, and H₀…X;S_w†GZ₂ if w00. Therefore we have E^y_0;4GW₄ if wˆ0, and E^y_0;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 S_M is equivalent to fS_w 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 r_wˆwX and r_M 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

??

?y^Sw…fg†

S_M…u† ƒƒƒ!^ju …fS_w†…u† ˆS_w…f…u††:

So, …S_M†_u…g† ˆj^ÿ1_u …S_w†_f…u†…fg† j_u as an automorphism of S_M…u†.

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

p₁…M;u† ƒƒƒ^f! p₁…X;f…u††

X

??

?y

??

?y^X

H1…M;Z† ƒƒƒ^f! 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 r_M ˆ r_w f:p₁…M;u₀† !AutZ. We choose an element a_uAG…u;u₀† for each point uAM. If we choose an isomorphismj_u0 :SM…u0† ! …fSw†…u0†for the base pointu₀, the isomorphismj_u:S_M…u† ! …fS_w†…u† is determined by j_uˆ Sw…fau†^ÿ1j_u0SM…au†. In fact jˆ fj_ug satis®es

j_uSM…g† ˆSw…fau†^ÿ1j_u0 …SM†_u0…auga^ÿ1_v † SM…av†

ˆSw…fau†^ÿ1 …Sw†_f…u0†…f…auga^ÿ1_v †† j_u0SM…av†

ˆSw…fg† j_v

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 g_u be a path from …u;0† to …u;1† in MI de®ned by g_u…t† ˆ …u;t† and de®ne isomorphisms d_u: f⁰S_w…u† ! fS_w…u† and k_F…u†:S_M…u† !S_M…u† by

d_uˆS_w…F‰g_uŠ† and k_F…u† ˆj^ÿ1_u d_uj_u⁰: …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 k_F as a map from M to AutZ. From the above equation we see that kF is continuous. We de®ne sgnkF by

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

We have the following proposition.

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:S_M ! fS_w and j⁰:S_M ! f⁰S_w be equivalences.

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

ˆ ‰M;f⁰;…sgnk_F†j⁰Š in W_n…Bp;S_w†, where k_F is a map de®ned by (8.1).

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

FvˆSw…F‰avŠ†^ÿ1j_uav;

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

Fvˆ ÿSw…F‰b_vŠ†^ÿ1j_u⁰kF…u†^ÿ1b_v

ˆ ÿSw…F‰b_vŠ†^ÿ1 …sgnkF†j_u⁰ b_v: So, F_v:SqW…v† ! …FjqW†Sw…v† is written as

F_vˆ j_u …vˆ …u;0††

ÿ…sgnkF†j_u⁰ …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 RD^nÿ1 gained by identifying …s;v† and …s‡1;gv† for each …s;v†ARD^nÿ1. Then N is a non-orientable smooth O…nÿ1† bundle over S¹ with ®ber D^nÿ1. 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 H_t…‰s;ruŠ† ˆ ‰s‡td…r†;ruŠ;

where sAR, 0ara1 anduAqD^nÿ1. Then H_t 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†,