### 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;Aof topological spaces andwAH^{1}
X;Z2the cobordism
groupWn
X;A;Swwith local coe½cients is introduced. IfXis a CW complex andSw

is a local system overXdetermined byw, then we have an Atiyah-Hirzeburch spectral
sequence Ep;^{2}qHp
X;WqnSw )Wpq
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;j 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}_{q}Hp
X;Wq )Wpq
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 H_{4}
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 wAH^{1}
X;Z2, which reduces to Wn
X;A if w0.

Let w_{1}:BO_{r}!K
Z_{2};1 be the map corresponding to the ®rst Stiefel-Whitney
class. Consider w to be a map of X to K
Z_{2};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 ! X

fr

??

?y

??

?y^{w}
BO_{r} !

w1 K
Z_{2};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 gg 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
gg^{0} S
g S
g^{0} for any gAG
y;x and g^{0}AG
z;y.

By the de®nition we see that S induces a homomorphism Sx:p1
X;x !
AutS
x de®ned by Sx
a S
a
aAp1
X;x for each xAX. Fixx0AX
and choose an element a_{x}AG
x;x_{0} for each xAX. Then we see also that

S
g S
a_{x}^{ÿ1}S_{x}_{0}
a_{x}ga^{ÿ1}_{y} S
a_{y}

for eachgAG
y;x. WhenXis arcwise connected and Gis an abelian group,
any homomorphism r:p_{1}
X;x_{0} !AutG induces one and only one local
system over X such that S
x0 G and Sx0 r [10], which is called a local
system determined by r.

For wAH^{1}
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:p_{1}
X;x_{0}

!H1
X;Z with w considered as a homomorphism from H1
X;Z to
AutZZ_{2}.

We will prove the following theorem.

Theorem 1. Let X be a CW complex and wAH^{1}
X;Z2. Then we have
a spectral sequence

E_{p;q}^{2} H_{p}
X;W_{q}nS_{w} )W_{pq}
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 Hn
N;Nÿu;Z for each uAIntN and SN
u
H_{nÿ1}
qN;qNÿu;Z for each uAqN.

(2.2) S_{N} is determined by the homomorphism r_{N} w_{1}
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 f^{}Sw are
equivalent. Letj fj_{u}g_{u}_{A}_{M} denote the family of isomorphisms j_{u}:S_{M}
u !
f^{}S_{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.
M_{1};f_{1};j_{1}@
M_{2};f_{2};j_{2} means that there exist a
compact
n1-manifold W and a map F :W !X satisfying the following
conditions:

(1) qWM1UM2,
(2) FjM_{j} f_{j}
j1;2,

(3) there exists an equivalence F:S_{W} !F^{}S_{w} such that F_ FjqW :
SqW !F^{}SwjqW satis®es FjM_ 1j_{1} and FjM_ 2 ÿj_{2}.

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;j for the cobordism class in
Wn
X;Sw.

Since j induces an isomorphismj_{}:Hn
M;SM !Hn
M;f^{}Sw, we can
de®ne a homomorphism

m:Wn X;Sw !Hn X;Sw

by m
M;f;j 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 wAH^{1}
X;Z_{2}.

The map m:W4
X;Sw !H4
X;Sw is a surjection and the kernel is W4 if
w0, and W_{4}nZ_{2} if w00.

Let p be a ®nitely presentable group, BpK
p;1 be an Eilenberg-
MacLane complex and w be an element of H^{1}
Bp;Z2. We consider the set
M^{4}_{p;w} consisting of the closed connected 4-manifolds M such that p1
M p
and w_{1}
M w, 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 !
p_{1}
Bp;f
u is isomorphism for any u, and

(3.2) f^{}ww1
MAH^{1}
M;Z2.

By Proposition 15 inO7 M^{4}_{p;}_{w} is not empty. For every MAM^{4}_{p;}_{w} there exists
an element
M;f;jof M4
Bp;Sw by Proposition 11 inO6. For a non-zero

w Proposition 13 in O6 says that M;f;j M;f;ÿj in W_{4}
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]M_{0} and N]N_{0} 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 l^{}ww on H^{1}
Bp;Z_{2}.

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 H^{1}
Bp;Z_{2}. Then, the set of weakly stable equivalence classes
in M_{p;w}^{4} 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
K^{2} of the regular neighborhood of an embedded ®nite 2-complex K^{2} in
RP^{4}R realizing the fundamental group pp1
M and r_{w}_{1}_{
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 H_{4}
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;qMinto
X;A. If
Af then qMf. We denote by f^{}Sw the local system over M induced
from S_{w} by f, that is, f^{}S_{w}
u S_{w}
f
uforuAM and f^{}S_{w}
g S_{w}
f_{}g

for gAG
u^{0};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 !^{j}^{u} T
u

S g

??

?y

??

?y^{T
g}

S v !

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 ! f^{}Sw 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 !
R^{n};0. We put D
r fxAR^{n};jjxjjarg and U
r h^{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 U^{0} of u, a homeomorphism h^{0}, and a
positive number r^{0} we write U^{0}
r^{0} as above. If U^{0}
r^{0}HU
r, then the
homomorphism i_{U}^{U
r}0
r^{0} induced by the inclusion i_{U}^{U
r}0
r^{0}:
M;MÿU
r !
M;MÿU^{0}
r^{0} coincides with the isomorphism
iu^{U}^{0}^{
r}^{0}^{}^{ÿ1}iu^{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_{u} fU
rAB;uAU
rg is a directed set. Therefore fH_{n}
M;Mÿ
U
r;Z;i_{U}^{U
r}0
r^{0};uAU
rg forms an inductive system over Bu and we get a
canonical isomorphism

lim! Hn M;MÿU r;ZGHn 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
rg of [0, 1] and
a division 0t0<t1< <tl1 of [0, 1] such that tjÿtjÿ1 <e. We put
g
t_{j} u_{j}. For each j
1ajal there exists some U
r which contains
g
tjÿ1;tj. Denoting such U
r by Uj
rj, we de®ne a homomorphism
g_{}:H_{n}
M;Mÿv;Z !H_{n}
M;Mÿu;Z by

g_{}i_{u}^{U}^{1}^{
r}^{1}^{}
i_{u}^{U}_{1}^{1}_{}^{
r}^{1}^{}^{ÿ1}i_{u}^{U}_{1}^{2}_{}^{
r}^{2}^{}
i_{u}^{U}_{2}^{2}_{}^{
r}^{2}^{}^{ÿ1} i_{u}^{U}_{lÿ1}^{l}^{
r}_{}^{l}^{}
i_{u}^{U}_{l}_{}^{l}^{
r}^{l}^{}^{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, g^{}w1
M is an
obstruction to the trivialization of g^{}T
M, where T
Mis the tangent bundle
of M. So, SM
g 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 D^{n}, and choose a point v0AVV
IntM and an embbeded path d in V from u to v_{0}. We can assume
dVqM fug. Moreover we put V_{1}VVqM and V_{2}qVÿV_{1}. Let
d:Hn
IntM;IntM ÿv0;Z !Hnÿ1
qM;qMÿu;Z be a composite of the
following maps:

Hn
IntM;IntMÿv0;Z!^{i}^{} Hn
M;Mÿv0;Z

!iVÿ1

Hn
V;qV;Z!^{q}^{} H~nÿ1
qV;Z

!^{j}^{} H_{nÿ1}
qV;V_{2};Z!^{k}^{}^{ÿ1} H_{nÿ1}
V_{1};qV_{1};Z

!^{i}^{V}^{1} Hnÿ1
qM;qMÿu;Z;

where i_{};i_{V};j_{};k_{};i_{V}_{1}_{} are isomorphisms induced by the inclusions. Then
SM
d 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_{M}jqMS_{qM}.

Given an equivalence j:SIntM ! f^{}SwjIntM. Then we can extend it
to an equivalence j:SM ! f^{}Sw by de®ning j_{u} Sw
f_{}d j_{v}_{0}d^{ÿ1}_{} for
uAqM, where d is a path in M from u to v_{0}AIntM. 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_{M}jqMS_{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;cAM_{n}
X;A;S_{w}, we de®ne

ÿ M;f;j M;f;ÿj; M;f;j N;g;c MUN;fUg;jUc:

We say that M;f;jis null cobordant: M;f;j@0, if there exists an element W;F;FAMn1 X;X;Swsuch thatq W;F;F1 M;f;j modA, that is,

(6.1) M is a regular submanifold of qW, (6.2) FjM f and F qWÿMHA, and

(6.3) FjInt_ Mj 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;jAM_{n}
X;A;S_{w} let W MI and de®ne a
map F:W !X by

F u;t f u u;tAMI:

For each v
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
u;st; b_{v}
s
u;1ÿsst
sAI:

Note that identifyinguwith
u;0and
u;1we getj_{u}av ÿj_{u}b_{v} for
v
u;tAIntW and we de®ne F_{v} by this map. Then, identifyingM0 and
M1 with M, it is easy to see

q W;F;F1 M;f;j M;f;ÿj modA:

(2) The re¯ective law is clear.

(3) Assume that

q
W1;F1;F11
M1;f_{1};j_{1}
M2;f_{2};ÿj_{2} modA
q
W_{2};F_{2};F_{2}1
M_{2};f_{2};j_{2}
M_{3};f_{3};ÿj_{3} modA:

We glueW_{1} andW_{2} by identifyingM_{2} 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 F_{i}
v
vAW_{i}fori1;2. For vAIntW_{j}
j1;2 the inclusion ij
v:
IntWj;IntWjÿv !
IntW;IntWÿv induces
an isomorphism

i_{j}
v_{}:H_{n1}
IntW_{j};IntW_{j}ÿv;Z !H_{n1}
IntW;IntWÿv;Z:

If vAIntM_{2}, we take a neighborhood Uofvin IntW such that
U;UVM_{2}is
homeomorphic to
R^{n1};R^{n}. We take further a point v_{j}AUVIntW_{j} and a
path~aj from vto vj in UVWj
j1;2. If we regard~aj as a path in IntW,
we rewrite this a_{j}. Then we have isomorphisms

a~j :Hn1 IntWj;IntWjÿvj;Z !Hn IntM2;IntM2ÿv;Z;

aj :Hn1 IntW;IntWÿvj;Z !Hn1 IntW;IntWÿv;Z:

Since U is simply connected, from the way of the gluing we get
ÿ~a_{1}i_{1}
v_{1}^{ÿ1}_{} a^{ÿ1}_{1} ~a_{2}i_{2}
v_{2}^{ÿ1}_{} a^{ÿ1}_{2}:
So, we de®ne F by

Fv
Fj_{v}ij
v^{ÿ1}_{}
vAIntWj; j1;2

ÿ
j_{2}_{v}~a1i1
v1^{ÿ1}_{} a^{ÿ1}_{1}
vAIntM2.

(

The de®nition is independent of the choice of U;v_{j};~a_{j}. Moreover we
have S_{w}
F_{}a_{j} F_{v}_{j} F_{v}
a_{j}_{} for vAIntM_{2}
j1;2. Let v_{j}AIntW_{j},
vAIntM2 be any point and g_{j} be any path from v to vj in Wj for j1;2.

From the above equality we see that S_{w}
F_{}g_{j} F_{v}_{j}F_{v}
g_{j}_{}. This leads
to Sw
Fg Fv^{0}Fvg_{} for any points v;v^{0}AIntW and any gAG
v^{0};v.

Hence we get an equivalence F:S_{Int}_{W} !^{@} F^{}S_{w}jIntW. Since this can be
extended naturally to F:SW !F^{}Sw as remaked before, we have

q
W;F;F1
M_{1};f_{1};j_{1}
M_{3};f_{3};ÿj_{3} modA: r
We put W_{n}
X;A;S_{w} M_{n}
X;A;S_{w}=@and denote by M;f;j the
equivalence class of
M;f;j. By settingM;f;j N;g;c MUN;fUg;

jUc, 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
w0, 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 hAH^{1}
Y;Z_{2} and a continuous map h:
X;A !
Y;B. For each

M;f;jAWn
X;A;h^{}Sh, we have j:SM !^{@}
h f^{}Sh. Hence we de®ne a
homomorphism h_{}:W_{n}
X;A;h^{}S_{h} !W_{n}
Y;B;S_{h} by

h_{}
M;f;j M;h f;j:

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

q
M;f;j qM;fjqM;j;_
where j_jjS_{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 h^{0} :
Y;B !
Z;C be continuous maps
and zAH^{1}
Z;Z2. Then
h^{0}h_{}:Wn
X;A;
h^{0}h^{}Sz !Wn
Z;C;Sz is a
composite of h_{}:W_{n}
X;A;
h^{0}h^{}S_{z} !W_{n}
Y;B;
h^{0}^{}S_{z} and h_{}^{0} :W_{n}
Y;B;

h^{0}^{}Sz !Wn
Z;C;Sz.

(3) For any hAH^{1}
Y;Z_{2}and any map h:
X;A !
Y;B, the diagram
W_{n}
X;A;h^{}S_{h} !^{q} W_{nÿ1}
A;i^{}h^{}S_{h}

h

??

?y

??

?y^{
hjA}^{}

W_{n}
Y;B;S_{h} !^{q} W_{nÿ1}
B;i^{}S_{h}
is commutative.

(4) For every pair
X;A and every wAH^{1}
X;Z2, the sequence
!Wn
A;i^{}Sw !^{i}^{} Wn
X;Sw !^{j}^{} Wn
X;A;Sw !^{q} Wnÿ1
A;i^{}Sw !
is exact.

(5) If there is a homotopy h_{t}:
X;A !
Y;B, then h_{0}h_{1}:W_{n}
X;A;

Sw !Wn
Y;B;Sh for wh_{0}^{}hh_{1}^{}h, hAH^{1}
Y;Z2.

(6) If UHIntA, then the inclusion i:
XÿU;AÿU !
X;A induces
an isomorphism i_{}:W_{n}
XÿU;AÿU;i^{}S_{w} !W_{n}
X;A;S_{w}.

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

(4) For M;f;jAWn
A;i^{}Sw we put W MI. We de®ne a map
F :W !X by F
u;t f
u
u;tAMI and a path a_{v} from
u;0 to
v
u;t by av
s
u;st. Moreover de®ne F by extending Fvj_{u}av

v
u;tAIntW. Then q
W;F;F1
M;f;j modA. Hence we have
j_{}i0.

Assume that j_{}M;f;j 0 for M;f;jAWn
X;Sw. Then there exists
an element
W;F;FAM_{n1}
X;X;S_{w} such that q
W;F;F1
M;f;j

modA. Now we put

NqWÿM; gFjN; c ÿFjN:_

Then N;g;cAWn
A;i^{}Sw and iN;g;c M;f;j. Hence we have
Kerj_{}HImi_{}.

qj_{} 0 and iq0 are trivially veri®ed. Assume that qM;f;j 0 for

M;f;jAW_{n}
X;A;S_{w}. Then there exists an element
N;g;cAM_{n}
A;A;i^{}S_{w}
such thatq
N;g;c1
qM;fjqM;j. Now we put_

M^{0}MUqMN; f^{0} fUg; j^{0}jUc

and WM^{0}I. De®ne a map F :W !X by F
u;t f^{0}
u. Moreover
de®ne F by extending Fvj_{u}^{0}av
v
u;tAIntW. Then it holds

q
W;F;F1
M^{0};f^{0};j^{0}
M;f;ÿj modA:

This implies j_{}M^{0};f^{0};j^{0} M;f;j. Hence we have KerqHIm j_{}.
Assume that i_{}M;f;j 0 for M;f;jAW_{nÿ1}
A;i^{}S_{w}. Then there
exists an element
W;F;FAMn
X;A;Sw such that q
W;F;F1
M;f;j.

Since W;F;FAW_{n}
X;A;w, we have M;f;jAImq. Hence Keri_{}HImq.

(5) For M;f;jAW_{n}
X;A;S_{w} we put WMI and de®ne a map
F :W !Y byF
u;t ht
f
u
u;tAMI. Sinceh_{t}^{}hwfor anytAI,
we can de®neF just in the same way as in the proof of Proposition 5. Hence
we get

q W;F;F1 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;jAW_{n}
X;A;S_{w}, let Pf^{ÿ1}
XÿIntA andQf^{ÿ1}
U.

Then there exists a compact submanifold NHM such that PHN and
QVNf. We put g fjN and cjIntNjjIntN by identifying H_{n}
IntM;

IntMÿv;Z with Hn
IntN;IntNÿv;Z for any vAIntN. The equiva-
lence c:S_{N}!g^{}S_{w} is de®ned as a natural unique extension. Then we have

N;g;cAWn
XÿU;AÿU;i^{}w and i^{}N;g;c M;f;j. 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;i^{}Sw !^{i}^{} Wn
X;B;Sw !^{j}^{} Wn
X;A;Sw

!^{q} W_{nÿ1}
A;B;i^{}S_{w} !

ForwAH^{1}
X;Z_{2}andhAH^{1}
Y;Z_{2}let xwn11nhAH^{1}
XY;Z_{2}
GH^{1}
X;Z2nH^{0}
Y;Z2lH^{0}
X;Z2nH^{1}
Y;Z2. Then we can choose a
local system S_{x} equivalent to S_{w}nS_{h} on XY. Through this equivalence
for M;f;jAW_{m}
X;A;S_{w} and N;g;cAW_{n}
Y;S_{h} we have

jnc:S_{MN} !^{@}
f g^{}S_{x}:

Then, j_nc:S_{qMN}!^{@}
f g^{}S_{x}jqMN and hence we can de®ne a
homomorphism

Y:W_{m}
X;A;S_{w}nW_{n}
Y;S_{h} !W_{mn}
XY;AY;S_{x}

by Y M;f;jnN;g;c MN;f g;jnc. In particular, if Y pt then we get a homomorphism

Y:Wm X;A;SwnWn!Wmn 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 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;f^{}S_{h} !W_{n}
YU_{f} X;Y;S_{h} is an isomorphism for any
hAH^{1}
YUfX;Z2.

Proof. We put ZYUf 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;f^{}S_{h} ^{i}^{}! W_{n}
X;V;f^{}S_{h} ^{i}^{}^{0} W_{n}
XÿA;VÿA;i^{0}f^{}S_{h}

f

??

?y

??

?y^{f}^{}

??

?y^{f}^{}
Wn
Z;Y;Sh !

j_{} Wn
Z;YUf
V;Sh

j^{0}

Wn
ZÿY;f
VÿA;j^{0}Sh:

For the homotopy ht:V!V between iAr and 1V given by (7.1), h_{t}^{}:
H^{1}
V;Z_{2} !H^{1}
V;Z_{2}is an identity isomorphism for every t. Hence by (1),
(2), (3), (4) and (5) of Proposition 6 we have Wq
V;A;i_{V}^{}f^{}Sh 0 for the
natural inclusion iV:V !X and every q. From the exact sequence of triple
X;V;Awe 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 i_{}^{0} and j_{}^{0} are
isomorphisms for the natural inclusions i^{0} and j^{0}. 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 X`

lD_{l}^{p}, A`

lS_{l}^{pÿ1},
Y X^{pÿ1} and f `

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_{l}^{}i^{}Sw !Wn
X^{p};X^{pÿ1};
i^{}S_{w} is an isomorphism.

Moreover, we have

Corollary 9. The map Y:Wn
X^{n};X^{nÿ1};i^{}SwnWq!Wnq
X^{n};X^{nÿ1};
i^{}S_{w} is an isomorphism.

Proof. Since D_{l}^{p} is simply connected, the local system h_{l}^{}i^{}S_{w} is
equivalent to S0. So, the map

Y_{l}:W_{n}
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}S_{w}nW_{q}!W_{nq}
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}S_{w}

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

P

lWn
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}SwnWq !^{SY}^{l} P

lWnq
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}Sw

Shln1

??

?y

??

?y^{Sh}^{l}
Wn
X^{n};X^{nÿ1};i^{}SwnWq !

Y Wnq
X^{n};X^{nÿ1};i^{}Sw:

Therefore, Corollary 8 implies Corollary 9. r

4. Proof of Theorem 1

For M;f;jAWn
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_{Int}_{M}

!Hn
IntM;f^{}Sw be the isomorphism induced by jjIntM. We know
that there is a natural isomorphism i: H_{n}
IntM;f^{}S_{w} !H_{n}
M;qM;f^{}S_{w}
for any compact manifold M (cf. [6]). We put j_{}ij_{]} and de®ne a
homomorphism

m:Wn X;A;Sw !Hn X;A;Sw

by m
M;f;j f_{}
j_{}
sM, where f_{} is an induced homomorphism
f_{}:H_{n}
M;qM;f^{}S_{w} !H_{n}
X;A;S_{w}

and s_{M} is a fundamental class of H_{n}
IntM;S_{Int}_{M}. Then, for the any CW
complex X we have the following.

Theorem 10. The map m:W_{n}
X^{n};X^{nÿ1};i^{}S_{w} !H_{n}
X^{n};X^{nÿ1};i^{}S_{w} is
an isomorphism for every wAH^{1}
X;Z2.

Proof. We know that the map

m_{l}:W_{n}
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}S_{w} !H_{n}
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}S_{w}

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

P

lWn
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}Sw !^{Sm}^{l} P

lHn
D_{l}^{n};S_{l}^{nÿ1};h_{l}^{}i^{}Sw

Shl

??

?y

??

?y^{Sh}^{l}
Wn
X^{n};X^{nÿ1};i^{}Sw !

m Hn
X^{n};X^{nÿ1};i^{}Sw:

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

implies Theorem 10. r

Proof ofTheorem 1. For wAH^{1}
X;Z_{2} and each pair of integers
p;q

such that ÿyapaqay, we put H p;q P

nWn
X^{ÿ}^{p};X^{ÿq};i^{}Sw. Then
fH
p;qgsatis®es the axioms in the theory of spectral sequences [1, Chap. XV,
p. 334]. Now let H
p;q H
ÿp;ÿq, H
p H
p;ÿy, HH
y;ÿy.

We de®ne a ®ltration F_{p;q}H of H by

Fp;qH Im
Hpq
p !Hpq Im
Wpq
X^{p};i^{}Sw !Wpq
X;Sw:

We de®ne also

Z_{p;q}^{r} Im
Hpq
p;pÿr !Hpq
p;pÿ1

Im
W_{pq}
X^{p};X^{pÿr};i^{}S_{w} !W_{pq}
X^{p};X^{pÿ1};i^{}S_{w}

B_{p;q}^{r} Im
Hpq1
prÿ1;p !Hpq
p;pÿ1

Im
W_{pq1}
X^{prÿ1};X^{p};i^{}S_{w} !W_{pq}
X^{p};X^{pÿ1};i^{}S_{w}

E_{p;q}^{r} Z_{p;q}^{r} =B_{p;q}^{r}

where 1aray,ÿy<p<y. Since H_{n}
p W_{n}
X^{p};i^{}S_{w} 0 for everyn
and paÿ1, F is regular and hence convergent in the sense of [1]. Then we
have particularly

E_{p;q}^{1} Wpq
X^{p};X^{pÿ1};i^{}Sw:

By Corollary 9 and Theorem 10 we get

W_{pq}
X^{p};X^{pÿ1};i^{}S_{w} ^{G} W_{p}
X^{p};X^{pÿ1};i^{}S_{w}nW_{q}!^{G} H_{p}
X^{p};X^{pÿ1};i^{}S_{w}nW_{q}:
By the universal coe½cient theorem for the homology with local coe½cients [6]

we have

Hp
X^{p};X^{pÿ1};i^{}SwnWq G Hp
X^{p};X^{pÿ1};Wqni^{}Sw:

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

W_{pq}
X^{p};X^{pÿ1};i^{}S_{w} !^{G} H_{p}
X^{p};X^{pÿ1};W_{q}ni^{}S_{w}

q

??

?y

??

?y^{q}

W_{pqÿ1}
X^{pÿ1};X^{pÿ2};i^{}S_{w} !_{G} H_{pÿ1}
X^{pÿ1};X^{pÿ2};W_{q}ni^{}S_{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};Wqni^{}Sw !Hpÿ1
X^{pÿ1};X^{pÿ2};Wqni^{}Sw. Hence
we have

E_{p;q}^{2} GH_{p}
X;W_{q}nS_{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 S^{1} and w00. We have an exact sequence
0!E_{0;}^{y}_{n}!Wn
S^{1};Sw !E_{1;}^{y}_{nÿ1}!0

since E_{m;nÿm}^{2} 0 for m00;1. From H_{0}
S^{1};S_{w} Z_{2} and H_{1}
S^{1};S_{w} 0,
we have E_{0;n}^{y} GH_{0}
S^{1};W_{n}nS_{w}GW_{n}nZ_{2} and E_{1;nÿ1}^{y} GH_{1}
S^{1};W_{nÿ1}nS_{w}
GTor
Z2;Wnÿ1. It is known that W0GZ, W1W2W30, W4GZ.

Hence we have W_{n}
S^{1};S_{w}GW_{n}nZ_{2} for na5.

Example 2. LetX be a real projective planeP^{2} andw00. We see that

E_{m;nÿm}^{2} H_{m}
P^{2};W_{nÿm}nS_{w}G

WnnZ2 m0

Tor Z2;Wnÿ1 m1

Wnÿ2 m2

0 mb3

8>

>>

<

>>

>:

since H_{0}
P^{2};S_{w} Z_{2}, H_{1}
P^{2};S_{w} 0,H_{2}
P^{2};S_{w} Z. Hence for na5 we
have an exact sequence

0!E^{y}_{0;n}!W_{n}
P^{2};S_{w} !E_{2;nÿ2}^{y} !0:

Then we have W2
P^{2};SwGW0 and Wn
P^{2};SwGWnnZ2 for n02, na5.

Proof of Crollary 2. Since W0GZ, W1W2W30, 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_{pq}
X;S_{w} to the Atiyah-Hirzeburch spectral sequence fE_{p;q}^{0r} g for
Hpq
X;Sw and we have the following commutative diagram:

W4
X;Sw ! E^{y}_{4;0}H4
X;W0nSw

m

??

?y

??

?y^{m}^{}

H4
X;Sw ^{id}! E_{4;}^{0y}_{0}H4
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;SwGZ if w0, and
H_{0}
X;S_{w}GZ_{2} if w00. Therefore we have E^{y}_{0;4}GW_{4} if w0, and E^{y}_{0;4}G
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 wAH^{1}
X;Z_{2}.

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 f^{}S_{w} if and
only if f^{}ww1
M.

Proof. Assume that j:SM ! f^{}Sw is an equivalence. We regard w
and w_{1}
M as the homomorphisms from H_{1}
X;Z to AutZZ_{2} and
H1
M;Z to AutZZ2 respectively. We put r_{w}wX and r_{M}
w1
M X for the Hurewicz homomorphism X. For every point uAM and
every element gAp_{1}
M;u, the following diagram is commutative:

SM
u !^{j}^{u}
f^{}Sw
u Sw
f
u

SM g

??

?y

??

?y^{S}^{w}^{
}^{f}^{}^{g}

S_{M}
u !^{j}^{u}
f^{}S_{w}
u S_{w}
f
u:

So,
S_{M}_{u}
g j^{ÿ1}_{u}
S_{w}_{f}_{
u}
f_{}g j_{u} as an automorphism of S_{M}
u.

Because j_{u} identi®es SM
u Sw
f
uGZ, this means r_{M} r_{w} f_{}. Since
X is a surjection, we see f^{}ww_{1}
Mby the following commutative diagram:

p_{1}
M;u ^{f}^{}! p_{1}
X;f
u

X

??

?y

??

?y^{X}

H1
M;Z ^{f}^{}! H1
X;Z:

Conversely assume that f^{}ww1
M. Fix a base point u0. Then, the
local systems f^{}Sw and SM have the same associated homomorphism r_{M}
r_{w} f_{}:p_{1}
M;u_{0} !AutZ. We choose an element a_{u}AG
u;u_{0} for each
point uAM. If we choose an isomorphismj_{u}_{0} :SM
u0 !
f^{}Sw
u0for the
base pointu_{0}, the isomorphismj_{u}:S_{M}
u !
f^{}S_{w}
u is determined by j_{u}
Sw
f_{}au^{ÿ1}j_{u}_{0}SM
au. In fact j fj_{u}g satis®es

j_{u}SM
g Sw
f_{}au^{ÿ1}j_{u}_{0}
SM_{u}_{0}
auga^{ÿ1}_{v} SM
av

Sw
f_{}au^{ÿ1}
Sw_{f}_{
u}_{0}_{}
f_{}
auga^{ÿ1}_{v} j_{u}_{0}SM
av

Sw
f_{}g j_{v}

for every gAG
v;u. Hence j is an equivalence. r
Let Mbe a closed connectedn-manifold,pp1
Mand f;f^{0}:
M;u0 !
Bp;y_{0} be two maps which satisfy the conditions (3.1) and (3.2). Moreover
let j:SM ! f^{}Sw and j^{0}:SM ! f^{0}Sw be equivalences. Suppose that f
and f^{0} 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
u;t and
de®ne isomorphisms d_{u}: f^{0}S_{w}
u ! f^{}S_{w}
u and k_{F}
u:S_{M}
u !S_{M}
u by

d_{u}S_{w}
F_{}g_{u} and k_{F}
u j^{ÿ1}_{u} d_{u}j_{u}^{0}:
8:1

Then we have

kF
u SM
a^{ÿ1}kF
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} 1 if kF
u id for anyu
ÿ1 if kF
u ÿidfor anyu.

We have the following proposition.

Proposition 12. Let M be a closed connected n-manifold, pp_{1}
M and
f;f^{0}:
M;u0 !
Bp;y_{0} be two maps which satisfy the conditions (3.1) and
(3.2). Moreover let j:S_{M} ! f^{}S_{w} and j^{0}:S_{M} ! f^{0}S_{w} be equivalences.

Suppose that f and f^{0} are homotopic by a homotopy F. Then it holdsM;f;j

M;f^{0};
sgnk_{F}j^{0} in W_{n}
Bp;S_{w}, where k_{F} is a map de®ned by (8.1).

Proof. We put W MI. For v
u;tAIntW we de®ne F_{v}:
SW
v !F^{}Sw
v by

FvSw
Fav^{ÿ1}j_{u}av;

where a_{v}is a path from
u;0to v
u;tde®ned bya_{v}
s
u;st. Letb_{v} be
a path from
u;1 to v
u;t de®ned by b_{v}
s
u;1ÿsst. By the
de®nitions of Fv and kF
v we see that

Fv ÿSw
Fb_{v}^{ÿ1}j_{u}^{0}kF
u^{ÿ1}b_{v}

ÿSw
Fb_{v}^{ÿ1}
sgnkFj_{u}^{0} b_{v}:
So, F_v:SqW
v !
FjqW^{}Sw
v is written as

F_v j_{u}
v
u;0

ÿ
sgnkFj_{u}^{0}
v
u;1:

Hence we get
W;F;F1
M;f;j
M;f^{0};ÿ
sgnkFj^{0}. r
Let g be an element of orthogonal group O
nÿ1 with detg ÿ1 and
denote by N the quotient space of RD^{nÿ1} gained by identifying
s;v and
s1;gv for each
s;vARD^{nÿ1}. Then N is a non-orientable smooth
O
nÿ1 bundle over S^{1} with ®ber D^{nÿ1}. We denote by s;v the point
represented by
s;v in N.

Let d:0;1 ! 0;1 be a monotone and smooth function such that dj0;e 1 and dj1ÿe;1 0 for a positive number e which is small enough.

For each tAI we de®ne a map Ht:N!N by
H_{t}
s;ru std
r;ru;

where sAR, 0ara1 anduAqD^{nÿ1}. Then H_{t} is a di¨eomorphism such that
HtjqN1qN for each t and H1 is homotopic to H01N.

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 a00. The tubular neighborhood of a is di¨eomorphic to the above bundle N for a some gAO nÿ1with detg 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,