Download Homology With Local Coefficients

Annals of Mathematics
Homology With Local Coefficients
Author(s): N. E. Steenrod
Reviewed work(s):
Source: Annals of Mathematics, Second Series, Vol. 44, No. 4 (Oct., 1943), pp. 610-627
Published by: Annals of Mathematics
Stable URL: http://www.jstor.org/stable/1969099 .
Accessed: 28/12/2012 18:40
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .
http://www.jstor.org/page/info/about/policies/terms.jsp
.
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of
content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms
of scholarship. For more information about JSTOR, please contact [email protected].
.
Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of
Mathematics.
http://www.jstor.org
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
ANNALS OF MATHEMATICS
Vol. 44,No. 4, October,1943
HOMOLOGY WITH LOCAL COEFFICIENTS
By N. E.
STEENROD
(Received January 26, 1942)
1. Introduction
In a recentpaper [16] the authorhas had occasion to introduceand use what
he believedto be a new typeof homologytheory,and he named it homology
with
localcoefficients.It provedto be the naturaland fullgeneralizationofthe Whitney notion of locallyisomorphiccomplexes[18]. Whitney,in turn,creditsthe
source of his idea to de Rham's homologygroupsof thesecondkind in a nonorientablemanifold[13]. It has since come to the author's attention that
homologywith local coefficients
is equivalent in a complex to Reidemeister's
Uberdeckung
[10].
Since this new homologytheory(whichincludesthe old) seems to have such
wide applicability,a completereviewof the oldertheoryis needed to determine
to what extentand in what formits theoremsgeneralize. The object of this
paper is to make such a survey. The general conclusion is that all major
parts of the older theorydo extendto the new. In addition the newertheory
fillsin several gaps in the old. The most noteworthyof these is a fullduality
and intersectiontheoryin a non-orientablemanifold(?14).
For the sake of completeness,some of the resultsof Reidemeisterhave been
included. The new approach and new definitionsmake for easier and more
intuitiveproofs. They lead also to resultsnot obtainedby Reidemeister. The
most importantis a proof of the topological invariance of all the homology
groupsobtained.' In additiondevelopmentsare givenof the subjects of multiplicationsof cycles and cocycles,chain mappings,continuouscycles,and Cech
cycles.
Part I containsan abstractdevelopmentof systemsof local groupsin a space
entirelyapart fromtheirapplicationsto homology. Any fibrebundle over a
base space R [18] determinesmany such systemsin R (one foreach homology
group,homotopygroup,etc., of the fibre). These are invariantsof the bundle.
fibrebundles.
They shouldproveto be ofsomehelp in classifying
Part II, which contains the extendedhomologytheory,presupposeson the
part of the readera knowledgeof the classicaltheorysuch as can be foundin the
booksofLefschetz[7]and Alexandroff-Hopf
[1].
I. LOCAL GROUPS IN A SPACE
2. Notations
R.
We shall be dealing throughoutwith an arewiseconnectedtopologicalspace
For any pointx of R, let Fx be the fundamental(Poincar6) groupof R with
1 It is not determined however whether or not the combinatorial invariant called "torsion" by Reidemeister [9] and its generalizations by Franz [41 and de Rham [141are true
topological invariants.
610
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH LOCAL COEFFICIENTS
611
x as initialand terminalpoint. If A is a curve fromx to y, the class of curves
fromx to y homotopicto A withend pointsfixedwe shall denote by a symbol
such as axz. Its inverseis denoted by ax-yor avx. The elementsof Fx are
abbreviated ax , Ox, etc. The class acx determinesan isomorphismFx-F(denotedby ax,) definedby ax,(Ox) = ayx~xa,. In keepingwiththis notation,
the productaCY3xmeans the elementof Fx obtained by traversingfirsta curve
of the class ax then one of Oxz. As is well known,the combinationof two isomorphisms yz(ax (,yx))is the isomorphism(aOy11)((y-z).
3. Local groups
We shall say that we have a systemoflocal groups(rings)in thespace R if (1)
foreach pointx, thereis givena group (ring)Gx, (2) foreach class of paths ar,
thereis givena group (ring)isomorphismGx --+G (denotedby a,), and (3) the
resultof the isomorphisma, followedby Oy,is the isomorphismcorresponding
to the path axyO~z
It followsfromthe transitivitycondition(3) that the identitypath fromx to
x is theidentitytransformation
in Gx. A further
consequenceis that the inverse
of the isomorphisrn
a, is ayx. By (2), a closed path ax e Fx determinesan
automorphismof Gx. From (3) it followsthat Fx is a groupof automorphisms
of Gx. The invariantsubgroupof Fx acting as the identityon Gx is denoted
Fx. Since, by (3),
axv(Ox(ayx('9))) =
(ayx=xaxy)(g),
g e Gy,
it followsthat
(3.1)
axy(Ox(g))= [axy(O,)J(axv(g)),
eGx.
We shall say that the system[Gx is simpleifeveryFx = Fx. If thishappens
forone x, it will be true forall. If EGO}is simple,the isomorphisma, is independentof the path fromx to y. Choosinga fixedpointo as origin,we findthat
each Gxis uniquelyisomorphicto Go. Thus the local systemconsistsof one Go
and as many copies of Goas thereare pointsx z# 0.
Two systemsfGx1,{HT are said to be isomorphicif, foreach x, thereis an
isomorphism
4, of Gxonto Hx such that
= ax~y(Ot:(g)),
Oy(a~xy(g))
9 eGzG.
We shalldeal onlywithpropertiesofsystemswhichare invariantunderisomorphisms. In each case theproofofinvarianceis trivialand willbe omitted.
It was proved in ?2 that the collection {Fx} is a systemof local groups. It
is simpleif and only if it is abelian.
In some instances a system {G1, will consist of topologicalgroups. The
isomorphismsa, will then be continuous. In the followingpages we shall
omit continuityconsiderationswheneversuch are reasonablyobvious.
We shall consistentlyattemptto reducethe studyof a systemto the studyof
what occursat one point of R. As a firststep we have
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
612
N. E. STEENROD
of
1. If G is a group(ring),o a pointofR, and V' a homomorphism
THEOREM
of G, thenthereis one and only one system
F0 into thegroupof automorphisms
IG.} oflocal groups(rings)in R suchthatGois a copy of G and theoperationsof
byV,.
F0 in G, are thosedetermined
For each point x E R, choose a class of paths Xox,choosing Xo, = identity.
Let Gxbe a group (ring)isomorphicto G. Associate thisisomorphismwithXx0.
If axyis a path, we attach to it the isomorphismGx-+ G, definedby
axy(g)= ?o[(XoxaxyXyo)(X20(9))
e Fo by A1.
The second operationis the automorphismof G attached to X0oaxyXyo
holds.
(3)
the transitivitycondition
Since ,6is a homomorphism,
If {GQ} is a local system,and 4 an isomorphismof G' into Go such that ao=
4.a for a e Fo, map G'Gx with the operation ox(g') = Xox(0(Xxo(g'))). It is
easily verifiedthat {fx} establishes an isomorphismbetween {Gig and {Gxl.
This proves the uniquenessand completesthe theorem.
4. Automorphisms
Let {Gx}, {A:} be two systemsof local groups,and suppose that A is a group
of automorphismsof Gxin such a way that, forany a, X we have
(4.1)
axy(a(g)) = ax, (a)(axy(g)),
a
E A:,
g e G,.
of GE}. By (3.1), it follows
Then {AxI is called a systemoflocal automorphisms
that {FZ} is such a systemfor any {Goj.
Let A' be the subgroupof Ax acting as the identityon G,. By (4.1), {Al}
is a systemof local groups. It followsthat, under the natural isomorphisms
Ax/A1 -* AJIA1inducedby Ax -* Ay, {Ax/1A} is a systemof local automorphisms of {Gx}.
of Gxsuch that
If A is a group,and, foreach x, A is a groupof automorphisms
(4.2)
ax,(a(g)) = a(axy(g)),
a
e
A, g e
,
of {Gx . If a system {Ax}
we shall call A a groupof uniformautomorphisms
is simple,thenany one of its groupsis, in a naturalway,
of local automorphisms
a groupof uniformautomorphismsof IGx}. If Fx or Fx/F1is abelian, we have
forany G., .
such a groupof uniformautomorphisms
is obtainAs in Theorem1, completeknowledgeof a systemof automorphisms
able fromknowledgeof what occursat a singlepoint.
2. Let A be a groupof automorphisnsof G withonly theidentity
THEOREM
actingas suchin G (i.e. A' = 1). Let o be a point of R, and letFo be represented
a(a(a'1(g)))
ofG in sucha waythattheautomorphism
as a groupofautomorphisms
ofG is in A for everya e Fo, a e A. Then thereis one and onlyone system{Ax}
of {Gx} such thatthecollection(Fo , Go, Ao) is isomorphic
of local autoniorphisms
A is a groupof uniformautomorphto (Fo, G, A). 'Ax) is simpleand therefore
ofF0 and A in G commute.
ismsof IG } if and onlyif theautomorphisms
By Theorem 1, the system {Gx} is completelydetermined. By assumption
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
613
WITH LOCAL COEFFICIENTS
the automorphismaaa-' is a unique elementof A. Thus each oadeterminesan
automorphismof A, and F0 is a groupof such. By Theorem1, the system {Ax}
of local groupsis completelydetermined. For a E A , g E G., we choose a path
aox and define
a(g)
= aoz(acx(a)(azo(g))).
Clearly (4.1) will hold once we have proved the rightside to be independentof
aox .
This is shown as follows.
9xo(aox(axa)(a
o(g))))
=
(aoxz)
(a>o(a) (azo(g)))
=
[(aozf3>) (azo(a))
(aozB3) i] ((aox2fo)
(azo(g)))
= 0-o (a) (Oxo(g)).
5. Operator rings
If G is an additive abelian group, the set H of all homomorphisms of G into
itself forms a ring under the operations
(a + b)(g) = a(g) + b(g),
(ab)(g) = a(b(g)),
a, 6e H, g e G.
A group A of automorphisms of G forms a multiplicative subgroup of H. It
generates a subring A* of H with unit = identity. If the symbol a(g) be abbreviated by ag, this multiplication of g E G by the scalar a E A* obeys the usual
laws: (a + b)g = ag + bg, (ab)g = a(bg), 1g = g, Og = 0, a(g + g') = ag + ag'.
One may thereforespeak of linear combinations, linear independence, and bases
in G relative to A.
We shall say that the system {Ax} is a systemof operatorringsfor theabelian
system {Gx} if, for each x, Ax is a ring of operators for G, and for each path
axy, we have
a.(ag)
= ac,(a)ax(g),
a E Ax, g eG.G.
The analogue of Theorem 2 is proved with only slight modifications. If a system
of operator rings is simple, we are led naturally to the concept of a uniformoperator ringfor { GZ} .
For any {Gx}, the system {F* } of group rings of { Fx} is a system of operator
rings. If Fx or Fx/F1 is abelian, the group ring of F0 or factor ring thereof is a
uniform operator ring for {G. }.
6. Dual systems
Two abelian systems Qx}, {[Hx of local groups form a pair with respect to a
third {KZ } if, for each x, Gx and Hx form a pair with respect to Kx (i.e. a multiplication gh = k is given which is linear in each factor) in such a way that, for
each path ax, we have a.11(gh)= axy(g)axzv(h). Analogous to Theorem 1, we have
THEOREM 3. Let G, H forma pair withrespectto K, and letF0 be realizedas a
of each of G, H, K in such a way thata(gh) = a(g)a(h)
groupofautomorphisms
E F0, g e G, h e H.
Thenthereis oneand onlyone setofsystems{Gxo, IHx ,
for a
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
614
N. E.
STEENROD
{K:} such thatthefirsttwoforma pair with respectto the third,and Go, Ho,
K0 and theautomorphisins
F0forma setisomorphicto thegivenG, HI,K, F0.
By Theorem 1, thereare unique systems {CG}, {H !, {K1 correspondingto
G, H, K and the operationsof F0 in these groups. In the constructionof all
threesystemslet us use the same collectionofpaths NX,(see proofof Theorem1).
Under the isomorphismof GC, H1, K., with G, H, K attached to 02 the multiplication in the latter groups carriesover into a multiplicationin the former.
The relation a1,,(gh) = ac,,(g)a,.,(h) for g E GC , h f H. is proved by using the
propertya(g'h') = a(g')a(h') of the closed path a = Xo2a1~,Xvo
whereg', h' correspond to g, h under X01. Any other allowable product which agrees with the
constructedproductin GC, Ho will likewiseagree in G., Hz since both products
are invariantunder the translationalong X10Ofparticularinterestis the case ofcharactergroups.
K = realnumbersmod 1, G = a discrete,
THEOREM 4. IJf
or compact,or locally
compactseparablegroup,and Fo is realizedin twoarbitraryways as a groupof
automorphisms
ofKo and GCrespectively,
thenF0 can be realizedin one and only
one way as a groupof automorphisms
of the charactergroupH of G satisfying
a(gh) = a(g)a(h). Thus togivensystems{IK-,, Gx} is attacheda unique system
IH } of charactergroups.
The charactera(h) is definedto be the one withthe value a(a'-(g)h) on any g.
The remainderofthetheoremis readilyverified.
It is to be noted that K admits but one non-trivialautomorphism,namely:
- k. Thus thereare as many charactersystemsof a given system {Gx
k
as thereare factorgroupsof F0 of order2. If it should be desirableto have a
uniquecharactersystem,it would be naturalto choose {'K} to be simple.
7. Local groupsunder mappings
Let R, R' be two arewise connectedspaces and 0 a continuousmap R' - RI.
Let F, F' be theirfundamentalgroups relative to base points o, o' such that
(o') = o. If {IZI is a systemof local groupsin R, then0 inducesin R' a system
{G } as follows. The group G' is chosen isomorphicto G. where x =
is denoted by O. The path a,, in R' maps G' isomorphically
The isonmorphism
on G' accordingto the rule
(7.1)
aYz(g) =
( (avz) (9())),
g eG-
The transitivity
conditionis immediatelyverified.
The existenceof the inducedsystemis apparentin view of Theorem1 and the
F' -, F realizes F' as a group of automorphisms
fact that the homomorphism
of Go.
The induced systemhas numerouspropertieswhich we list withoutproofs.
(a) If {GI} is simple, so is {Gv}.
(b). {G"} is simpleif and only if F' is mapped by O on the subgroupF1 of F
leaving Go fixed.
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
615
WITH LOCAL COEFFICIENTS
(c). If {GX},
HX} are paired relative to {K.}, then likewise the induced
G
svstemsin R'.
(d). If {AZ} is a systemof local automorphismsor operatorringsfor {Gx},
thenlikewisethe inducedsystemsin R'.
(e). If A is a groupof uniformautomorphisms
or a uniformoperatorringfor
{G.}, it is also one forthe inducedsystem.
It followsfrom(b) that,if R' is the coveringspace of R corresponding
to the
subgroupF' of F, the inducedsystemis simple. Thus any systemin R can be
consideredas the continuousimage of a simplesystemin some coveringspace.
It is natural to inquire under what circumstancesa given systemin R' is
inducedby one in R. For this it is necessaryand sufficient
that (1) the kernel
F' ofthehomomorphism
F' -+ F shall act as the identityon G',, and (2) the map
of the subgroupFP/FPof F into the groupof automorphismsof G', shall admit
a homomorphicextensionto F.
8. Special local groups
Because of their importancein the work of Reidemeister,we shall discuss
certainlocal systemsbased on the fundamentalgroup F of R.
Let F' be an invariantsubgroupof F, and let {f = F/F1. Let 9' be an abelian
group. Let G be the set of functionsfrom'Fto 1AJ.If two such functionsare
added by addingfunctionalvalues at each elementof 9X,G becomesan abelian
group. A functionf e G is called a restricted
functioniff(a) = 0 except for a
finitenumberof a E 'F. The restrictedfunctionsforma subgroup G' of G.
The structureof G (G') can be describedas the unrestricted(restricted)direct
sum of as many copies of ( as there are elementsof SF. If g} = integersand
F' = 1, thenG' is the ordinarygroupringof F.
If (tjis a ring,we definea multiplicationof two functionsf, g e G' by
f X g(a) = Epf(af-')g(f),
a, / e
Since the functionsin G' are restricted,the sum is finite;thus G' is a ring. If
= integers,thisproductis the usual one in the groupring.
The groupF can be realizedin threenaturalways as a groupofautomorphisms
of the groups G, G'. For any -yE F, we definethree operationson a function
fEG (G') by
Lyf(a) = f(f'a),
Ryf(a) = f(ay),
Tyf(a) = f(J'a-y),
a eb.
Then Lzf, RJfand T7zfare in G (G'), and are called the lefttranslation,right
off by y, respectively. It is easy to verifythat Lz,
and transform
translation,
RI, T, are automorphismsof G (G'), and that LyLa = Lya, RyRa = Rya, and
7'js = T'a . The subgroupacting as the identityforboth L and R is F'.
In the case that ,j and thereforeG' is a ring,the leftand right translations
are not ring automorphisms. However the transformsare: 7Y'(f X g) =
X (Tfrg)w
(tyfow)
It followsfrontTheorem1 that,corresponding
to each of the threeways that
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
616
N. E.
STEENROD
F can act on G (G'), we have a systemof local groups. These we denote by
{G.', {G.', {Gf , and similarlyforG'. The last of these, {G-T7,is a systemof
local rings,whenever( is a ring.
Since 9 is associative,LRIf = ROLzf. Therefore,by Theorem 2, the left
translationsforma group of uniformautomorphismsof {G--, and likewisefor
the righttranslationsof {G' . Then, as in ?5, the group ringof F is a ringof
uniformoperatorsforthesesystems.
Under the autornorphism
q of G (G') definedby Of(a) = f(a&'), we have
=
by
means of 4), an isomorphismexists between {GL}
Therefore,
RqOf.
4L~f
and {GR.
In the workof Reidemeisteron hornotopychains, S = integers,F' = 1, so
of the homotopychains belong
that G' is the groupringof F. The coefficients
to the local groups{G'RI (these are not rings),and the ringof uniformoperators
is likewiseG' wherelefttranslationsare used. This distinctionbetweenthe two
usages of the group ringof F is necessaryfor a comprehensionof the subject
of homotopychains.
II. THE
COMBINATORIAL
THEORY
9. Chains withlocal coefficients
Let IG-} be a systemof local abelian groupsin a space R whichis decomposed
into a cell complex2 K. A q-cellofK is denotedby a', incidenceby a < a', and
incidencenumbersby [o-': ao]. We suppose as usual that
(9.1)
E
l q-2
q-=][0 q-.
O
In each cell a we choosea representative
pointx(a) and abbreviatethe symbol
f attachingto each orientedq-cella
G.(,) by GQ. A q-chainof K is a function3
E
with
the propertyf(- a) = - f(o). Chains are added by
an elementf(u) G,
values.
functional
They then forma group isomorphicto the direct
adding
for
of
all
the
q-cellsa.
sum
groupsG,
If a' < a, we may choose a path in the closureof a joiningx(a) to x(u') and
an isomorphismG, -+ G,, whichis denotedby hat,. In order
obtain therefrom
that hat,shall be independentof the path, we postulatethat the closureof each
cell is simply connected. A second consequence of this and the transitivity
conditionis
(9.2)
hq'aqhq'j=
For the sake of simplicitywe suppose K is finiteand closed. The extensionof the
subsequentresultsto relativecomplexes,open complexes,and to the finiteand infinite
chains of locally-finitecomplexeswill be obvious.
3We shall abide by the functionalnotationthroughout. This will prove to be as convenientas the classical linearformnotation. We abandon the lattersinceit has algebraic
implicationsmore prejudicial than suggestivein the presentdiscussion.
2
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH LOCAL COEFFICIENTS
617
By means of the isomorphismsh, we can definethe boundary af and coboundary 5fof a q-chainf:
(9.3)
af(ao-1)
=
fo)=
6A0,
= a2[-q: oq+1]ha-sar
2a 2+ i(f (o-q)
q+l)
ZEn
[oq-'
O*qIhcT-lcTQ(f(O*q)),
The sums extendover the q-cellswhich (a) have -q-1as a face, (b) are faces of
oq+. These new functionsare q1 and (q + 1)-chains respectively. It
followsfrom(9.1) and (9.2) that aaf = 0, baf= 0. Thereforecycles,bounding
cycles,homologyand cohomologygroupscan be definedas usual.
A special conventionfor 0-cyclesis necessary. We shall agree that any 0chain is a 0-cycle. Note that the Kroneckerindex (i.e. the sum of the coefficientsof a 0-chain)has no meaningunless the system{GX} is simple.4
Finally it is to be remarkedthat we obtain a completelyisomorphicsituation
ifwe choosenew representatives
is established
y(o) in each a. The isomorphism
by meansof a path x(a) to y(u) in a foreach o.
10. Automorphisms
of chains
Let A be a groupof uniformautomorphismsor a ringof uniformoperatorsof
For any chain f and a e A, define (af)(u) = af(u) for each o. Then
is
and A appears as a group of automorphismsor a ring of opera
chain,
af
ators of the groupof q-chainsforeach q.
Since the operationsof A commutewithtranslationsof the G, along paths, it
followsthat theoperationsofA on the chainscommutewitha and 6. Therefore
A appears as an automorphism
groupor ringofoperatorsofthe groupsof cycles,
cocycles, boundaries, coboundaries, and consequentlyof the homology and
cohomologygroups.
{Go}.
11. Multiplicationof chains
is locally a cycle in the
It should be noted that a cycle withlocal coefficients
ordinarysense; for,in any simplyconnectedopen set U (e.g. the starofa vertex),
of the local groupscan be set up with a fixedgroup (usingpaths
isomorphisms
into an
in U) in such a way as to transformeach chain with local coefficients
ordinarychain mod (K - U) so that the boundaryrelationsare preserved. It
followsthat any operationon ordinarychainswhichis a sum of local operations
can be carriedover to chains withlocal coefficients.We have seen that this is
trueof the boundaryoperator. We shall see that thisis also trueof productsof
cocycles, products of cycles and cocycles, and intersectionsof cycles. The
linkingnumberlike the Kroneckerindex is not of this category.
4Classical
homology with a single coefficientgroup G is isomorphic to homology with
coefficientsin the simple system of local groups determined by G.
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
618
N. E.
STEENROD
paper ofWhitneyon products[17],it is provedthat,correIn a comprehensive
spondingto cells 4l X s,ao +q in K, thereis an integer'"r"I such that
(rI) if aiP,aq are not both facesof o-'+', then "rkI' = 0,
(r2) forall p, q, i, j, k,
is== E r[0.Z P0+llP+lq rk 0jt1~
vM
+ (
emr a P+q:.P+q+llpq
)
Z
>Zm
[ 'a
]
. q+1]P'q-1]FiM
a
m
(I3) for all q and j, Es 0qrj = 1.
Althoughthe last two conditionsappear not to be local in nature,they are so
by virtueof the first.
Let {G-,}be a systemof local ringswithunits. Iffp and fPare p and q-chains
in {G.,}, we definetheir cup product to be the
respectivelywith coefficients
(p + q)-chain
(11 .1)
(11.1)
=
uas aim
k )=
ds
ij am[hpk
i (f (oiP))][hk
q:^(f
(oat))].
Here we heve abbreviatedthe isomorphismof the local group of uaPon that of
The
by h?P+q;. The sum extends over the faces oa, as of 0'+qk
kp+q
relations
(Pl)
fp u fris
zero on any oP+qwhichhas not both a face infP and a face in f,
(P2) 6(fP uf) = afp u fq + (- 1)PfP U af
(P3) I u f = i,
followfromthe relations(F), the transitivityof the h's (9.2), and the preservation of the productsin G. underan h. In (P3), I is the 0-cocyclewhichattaches
itpreservesthe
to each vertexV the unit of G, . Since h is a ringisomorphism,
unit; thereforeA1 = 0. It followsjust as in Whitney [17] that a product is
definableforthe cohomologyclasses withthe usual properties. The associative
law and the commutationrule forthe special productsin a simplicialcomplex
are givenlocal proofs. Once invarianceundersubdivisionhas been established
(see ?16), thesame laws willhold in a generalcomplex.
For the cap product,we shall suppose that {G.,}, {H-,} are paired withrespect
to {Lx}. Given a q-cochainfq (coef. {G,}) and a (p + q)-chain gp+p (coef.
{H_), we definetheircap productto be the p-chain
(11.2)
fn
gP+qP(?(P)
=
jk
Pqrm
(o q))][hpkp+q(gp+q(ofp+q))]
[hmtl(f
in {L-,}. The sum extends over those j, k for which 0', a(q
with coefficients
are faces of oUp+q.Just as before,we obtain
(Q1)frn gp+q is zero on any a' whichwithno aq offq is a face of a ap+qof gp+q
ngp+q + fq n agp+q.
(Q2) a(frng9+q) = (- 1)p
In order to obtain an analogue of (I3), we shall take the special case where
{L:} = {[Hx} and {GX-}is a systemof operatorringsfor {HX } (see ?5). Then
the 0-cocycleI is defined,and we have (QM)I n p = go.
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH LOCAL COEFFICIENTS
619
Underthe same assumption,it can be shownthat
(fq u fi) n g
+q-
if n
(fi n gp 2 )r
for cocycles f, f and a cycle gP+q+r.
The uniquenessof the productsin the followingsense can be proved. Suppose 'pr is anotherset of rFs forwhichthe relations(r) hold. Corresponding
to the two sets of rFs is the Whitneyoperation A (Theorem 8, [17]). Let
PqAV be the coefficient
DefinefA gP+q with
of ok?+ in the producto? A al+.
equationsanalogousto (11.2) usingA in place of r. Then the Whitneyrelations
(R) (loc. cit.) with chains in place of cells can be proved to hold. It follows
that the two sets of P's determinethe same productsamong the homologyand
cohomologyclasses.
12. Duality
Let L be the groupof real numbersmod 1, and let {L-,} be the corresponding
simplesystemin K. Let {G-,}and {HTh}be charactersystemsof one another
in {Lo}
withrespectto L-, . Since {Lo is simple,a 0-cyclef0withcoefficients
in L. It possesses thereforea
may be regardedas a 0-cyclewith coefficients
which we denote by (f). As
Kroneckerindex (= the sum of its coefficients)
0
0. The scalar product of a q-cochainf,
usual (U) = 0 is equivalent to fo
coef. {GXI and a q-chaingqcoef. {H.} is definedto be the Kroneckerindex of
theircap product:
(12.1)
= (fQn gq)
in L.
IffQand gQare zeroexcepton a singlea, then,by 1'3 fg is theproductfq(a)gq(a).
f2,gQis the sum of
Therefore,due to the linearity,the scalar productof arbitrary
coefficients.It followsthat the groupsof q-cochains
productsof corresponding
and q-chainsare charactergroupsof one another.
For any q-cochainf' and (q + 1)-chaing'+' we obtain fromQ2 that
,
(12.2)
fQ. gp+l = Pf ag"l
It followsnow in the usual way (see Whitney [17]) that the qthcohomnology
group coef. {Gzl and the qth homologygroup coef. {Ha} are charactergroups
withthe scalar productas the multiplication.
13. Intersectionin an orientablemanifold
Let K be an orientablesimplicialn-manifoldand let K* be its dual. Denote
by Pa the cell ofK* dual to the orientedsimplexa ofK relativeto a fixedchoice
ofthe fundamentaln-cycleZ' withintegercoefficients.Let {G-} be a systemof
in both K and K*. Let the coefficient
local groupsto be used as coefficients
groupsof a and 9a be the group G. wherex is theircommonpoint. Then for
any chainfofK (cochainf* ofK*), theequation
(13.1)
f*(Oa) = f(a)
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
620
N. E.
STEENROD
definesits dual cochain(chain) ofK* (K) ofthe complementary
dimension. This
isomorphismbetweenthe two groupsof chains has, as usual, the property
(13.2)
(Of)* = (
1)"bf*,
q = dimensionoff.
It followsthat theqthhomologygroupcoef. {GX} and the (n - q)tll cohomology
group coef. {Go,} are 2somorphic.
In case {Go} is a systemof local rings,we have as in ?11 a multiplication
definedforthe cochainsof K*. The isomorphisms
just establishedbetweenthe
chains of K and cochainsof K* enable us to carryover the productin K* into
an intersection
inK. We definethe intersection
oftwo chainsfi ,f2 ofK to be the
dual of the cup productof theirduals:
(13.3)
flOf2 =
(fl
uf
.
(Compare Whitney[17], p. 422, formula(19.9)). It followsthat a systemof
local ringsin an orientable
an intersection
manifolddetermines
ringof cyclesisomorphicto theringofcocycles(same coefficients)
undertheoperationofdual.
As is well known,the ringof integersis a ringof operatorsforany group G
whichcommutewithany automorphismof G. The ringof integersis therefore
a uniformoperatorring (?5) for any {Gal. The equation (11.1) may be interpretedas definingthe cup productof a cochainfP with (simple)integercoefficientsand a cochainf withlocal coefficients
{ G.,. The relations(P) stillhold,
and these togetherwith the associative law lead to the conclusion that the
a ring of operatorsfor
cohomology
ringwithsimpleintegercoefficients
constitutes
thecohornology
groupscoef. {Gal}.
The dual of thislast resultis that theintersection
ringofan orientable
manifold
withsimpleintegercoefficients
is a ringof operators
for thehomology
groupscoef.
{Ga .
14. Intersectionin a non-orientablemanifold
In a non-orientable
manifoldK thereis no n-cyclewithsimpleintegercoefficients. One cannottherefore
determinethe orientationof 93a uniformly
over K
so that (13.2) holds. A customarydevice is to use integersmod 2 as coefficients
so as to restore the basic n-cycle and escape orientationdifficulties.The
resultingdualityand intersectiontheoryis a bit weak due to the inadequacy of
the coefficients.A more ingeniousdevice has been used by de Rham [13].
We shall see that a suitable use of local coefficients
permitsa fulldevelopment
of De Rham's notionand leads to a completeand satisfyingduality and intersectiontheoryin a non-orientable
manifold.
Since K is non-orientable,
the elementsof its fundamentalgroupF divide into
two classes accordingas they do or do not preserveorientation. Those which
do forman invariantsubgroupF1 of index 2. Let T be the group of integers.
For each integert E T and ae F, let a(t) be +t or - t accordingas a is or is not in
FP. In this way F is a group of automorphismsof T. Let { T' be the correspondingsystemof local groupsgivenby Theorem1. We shall say that chains
with coefficients
in I TfI have twistedintegercoefficients.
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH
LOCAL
COEFFICIENTS
621
Let G be an abelian groupand let F be representedas a groupof automorphisms of G. Let G be the directsum of two copies of G (i.e. the group of pairs
IdentifyG withthe subgroupof elementsof the form(g, 0), and call
(91, 92)).
G the real part of G. The subgroupG' of elementsof the form(0, g) we call the
imaginarypart of G. The productof (gl, 92) withthe complexnumbera + ib
(a, b are integers)is definedby
(14.1)
(a + ib)(gi, g2)
=
(ag1- bg2,ag2+ bglq).
It followsimmediatelythat the complexintegersforma ringof operatorsforthe
group G, and that each elementof C can be writtenuniquelyin the formgi +
iq2 (91 ,92 real). If we set
(14.2)
(g9 +
ig2)
=
{(g)
+ ia (g2)
if a e Fl,
ofF in G are extendedto G. For any complexintegera + ib,
theautomorphisms
definea(a + ib) = a + ib accordingas a is or is not in F1. Let T be the ring
{
of complexintegers,and let IT,
be the systemof local ringscorresponding
to
T and theseautomorphisms. It followsthat {lU} is a systemof operatorrings
forthe system {6xI corresponding
to G and the automorphisms(14.2) (see ?5).
We referto {GO} as the complexextensionof {Gx}.
If G is a ring,and the elementsof F are ring automorphisms,we definea
g2q2, q2l
productin G in the usual wvay:(gq, g2) (g , g2) = (gig+ gig')
The equations (14.2) definering automorphismsof G. Furthermorethe elementsof T associate and commutewith the multiplicationsin G. Thus {G }
is a systemof local ringswith {I xI as a systemof operatorrings.
We shall use { GOIand { Tx} as coefficients
forchainsof K and cochainsof K*.
The group of n-cyclesof K coef. { Tx} is infinitecyclic. A generatoris constructedas follows. Choose an oriented n-cell a and let Zn(0_)= ?i in T,
(because of a(i) = ?i, the sign of i has only a local significance). If I 0r'I is
any othern-cell,choosea path a of it-cellsfroma- to a-'I (successivecells having
an (n - 1)-face in common). The path a determinesan orientationa-' of
( a-'I concordantwiththat of o-. Now defineZr(o-r) = a(Z'(o-)).
In words:the
of or'are determinedby translatingalong a path a
orientationand coefficient
of a-. Translatingalong a second path f
both the orientationand coefficient
will eitherproducethe same orientationand the same coefficient
or reversethe
sign of both accordingas ad-1is or is not in FP. Thus the chain Zn is independent of the paths chosen in its construction. It is a cycle since, in any
simplyconnecteddomain, it is a cycle. The usual argumentshows that any
n-cycleof K coef. { Tx is an integralmultipleof ZV. Thus the fundamental
n-cycleof K has pure imaginarycoefficients
(or equally well, twisted integer
coefficients).
If e is an orientedsimply-connected
neighborhoodin K, Zn(e) will be the
with e. Corresponding
coefficient
Zn (a-) of an n-cella- in e orientedconcordantly
to a q-cella e K and an orientedneighborhoode of o-,thereis in the usual way a
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
622
N. E.
STEENROD
unique orientationof the dual cell 0DST in K*. If f is any q-chainof K coef.
{ O}, we defineits dual to be the (n - q)-cochainf*in K* coef. {GOIdefinedby
(14.3)
P(qde)
=
-fA()Z
(f).
It is to be understoodthatf(a) is in G, and Z'(E) is in TXwherex is the point
commonto u and ?IPo. Since reversingthe orientationof e changesthe sign of
both sides of (14.3), if*is independentof the choicesof the E's. The dual of a
chain f* in K* is given by
(14.4)
f**(of)=
f*(9I"0)Zf(E).
=
Since Z'(e)Z'(e)
-1, we have that any chain is thedual of its dual. The
dual of Z' is the unit 0-cocycleI. The dual of a real (imaginary)chain is
imaginary(real). The formula(13.2) now followsforthe dual in a non-orientable manifold;forit is a statementof local properties,and (14.3), (14.4) differ
from(13.1) locallyby a constantfactor. As in ?13,it followsthat theqth homolgroup coef. {IG} are isoogy group coef. {Gx} and the (n - q)th cohoomology
morphic. Using (13.3) to definethe intersection,we obtain a homologyring
ringwheneverthe coef. {Gx} forma systemof
isomorphicto the cohomnology
local rings. In any case, the homologyringcoef. ITtxIformsa ringof operators
forthe homologygroupscoef. {GO
x.
Since G is the directsum of its real and imaginarypart,any chain is uniquely
a sum of a real chain and an imaginarychain. The operationsa and a preserve
the propertyof being real or imaginary. Thereforethe homology and cohomologygroupsdecomposeinto directsums of theirreal and imaginaryparts.
realand imaginary,we obtainthefollowing
Sincepassingto thedual interchanges
results. The qth homologygroup coef. {Gx} (coef. fGx) is isomorphicto the
groupcoef. {G } (coef. {Gx}). The homologyclasses coef.
(n - q)th cohomology
the
}
ringcoef.
{G (i.e.
imaginaryones)forma ringisomorphicto thecohomology
two
The
intersection
The
real
is
intersection
of a real
of
cycles imaginary.
G{x}.
and an imaginarycycleis real.
It is to be noted that the resultsof this sectionapply to an orientablemanifold. The absence of orientationreversingpaths in no wise invalidates the
constructions. We have in this way a single theoryincludingboth types of
manifolds.
is to definedirectlythe intersectionof a
The classicalapproachto intersection
p-chainf of K witha q-chaing of K* to be a chain of the subdivisionof K. We
may do this here as follows. If a is a p-simplex,u' an (n - q)-face of ufand e
ofu,defineo0DE0o' relativeto e in the usual way. Then
an orientedneighborhood
definethe intersectionchainfogto have on ?ooPEo' the value -f(o)g(To')Z'(E).
15. The Poincare duality
If we assumethat {Gx , {Hx are charactersystemsofone anotherwithrespect
to the simplesystem {Lx} of mod 1 groups,we may combinethe resultsof ?12
with those of ?13 and ?14 to obtain: The qth homology
groupof themanifoldK
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH
LOCAL
COEFFICIENTS
623
1
groupcoef. {H'} (coef. {H } )
coef. {GXJ(coef. {G_ ) and the(n- q)th homology
by thescalar
is determined
groupsofone another;themultiplication
are character
of K*.
(n
q)-chain
an
g
is
and
product(f* n g) wheref is a q-chainofK
{Go} are
and
{Go}
that
we
note
The resultssimplifyin the orientablecase if
results
the
case,
the
In
non-orientable
isomorphic,as also are {Hz} and {G'}.
of
simplifyif we note that {H'} ({Hx}) is the charactersystem {G,} ({G'})
with respectto the system {L') of twistedmod 1 groups associated with the
simple system {Lz}.
Beforeleaving the subject of duality,let us observethat the notionof local
has nothingto add to the duality theoremof Alexander. A vital
coefficients
step in the argumentof Alexanderis the following:A cyclein theclosed set R
theboundaryofa chain in Sn.
in then-sphereSn is a cyclein Sn and is therefore
used in R is part
coefficients
of
local
the
if
system
Such a statementis valid only
of local coeffisystem
the
1),
>
(n
Sn
simply-connected
is
of a systemin Sn; as
cientsmustbe simple.
16. Chain mappingand subdivision
of the groups of chains of one
is a homomorphism
A chain transformation
the boundaryor coboundary
with
commutes
which
of
those
another
on
complex
the
of
case
dual). This definitionhas
the
in
them
(as
interchanges
or
operator
All that we need to deterare
used.
local
coefficients
when
of
course
meaning
exist in the
"S" with local coefficients
mine here is that chain transformation
circumstances.
usual
Let a cell mappingK' -* K be givenwhichpreservesthe relationof incidence.
in K, and let {G I be theinducedsystem
Let {Go} be a systemoflocal coefficients
Gal- G, . A chain
in K' (see ?7). If a' -a u, thereis an attachedisomorphism
chain. Let f' be
of K' whichis zero excepton a singleo-'we call an elementary
an elementarychainand f'(o-') ? 0. If the image o-of a-'has a lowerdimension,
we definethe imagef of f' to be zero. If a-, a-' have the same dimension,the
imagef off' is zero except on a-,and f(a) is the image off'(o-') under the isomorphismG, -* G,. An arbitrarychainof K' is uniquelya sum of elementary
chains. Its image is definedas the sum of the images of its elementaryparts.
The resultingchain mappingwe denoteby r. The inversecochain mapping A
attaches to an elementarychain of K the sum of the elementarychains of K'
mapped into it by r; we thenextenda' preservinglinearity. That r (r') commuteswitha (8) is provedby firstestablishingit in the usual wayforelementary
chains (of course (7.1) is used), and then applyingthe linearityof a (3).
It is necessaryto use in K' the inducedsystem {GQJin orderthat r, T' shall
exist in all dimensions. This is seen as follows. Let V' be a vertexof K' and
V its image. AssumingT, H definedforthe elementarychains of V, V1,we aris the systeminduced by
{
rive at an isomorphismG-' -* Gv. ThereforeGJ
Go overthe 0-dimensionalpart of K'. Suppose thisis knownforthe q-dimenthereis theresional part of K'. Any closed (q + 1)-cellis simply-connected,
forejust one systemof local groupsdefinedover it whichagrees with a given
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
624
N. E.
STEENROD
systemon its boundary,and that one is simple. We conclude that the given
systemand the inducedsystemagreeon each closed (q + 1)-cell,and finallyover
the wholeof K'.
If K' is a subdivisionof the simplicialcomplexK, we then have two systems
in K': the given system {G,} forK, and the system {G'} induced by the map
K' -i K definedby mappingeach vertexof K' into a vertexof the simplexof K
containingit. The two systemsare isomorphic. The isomorphism
is set up by
usingthe line segmentswhichjoin each point to its image point. The proofof
the invarianceunder subdivisionof the groupsof K and their multiplications
may now be completedin the standardway (see forexample [17]).
17. Continuouscycles
Let {GaI be a systemof local groupsin a space R. A continuouschain in R
is a collectioncomposedof a complexK, a continuousmap 4 of K in R, and a
chain Z in K with local coefficients
in the system {Go} induced by 4 and {G2}.
If Z is a cycle,thecollection(K, q, Z) is called a continuouscycle. The boundary
of (K, 0, Z) is (K, q, aZ). Two continuouschains (Ki, qi, Zi) (i = 1, 2) are
addedby formingthe abstractsum K1 + K2, definingq5= 4i on Ki, and adding
ifthereexistsa chain (K, X),Z)
ZI to Z2 . Two continuouscyclesare homologous
such that K D K, and K2, 4)= q i on Ki, and AZ = Z1- Z2 . The cyclesof a
fixeddimensiondivide up into homologyclasses. Two classes are added by
adding representative
elements. In this way we definethe homologygroupsof
R based on continuous cycles with local coefficients{G.,}. That they are
topologicalinvariantsof R and the system {G I is an immediateconsequenceof
the definition.
If R is the space of a complex,thesegroupsof R are isomorphicto those of K
withthe same local coefficients.The identitymap q5lof K attaches to a chain
Z of K the continuouschain (K, 01, Z) of R. This chain mapping commutes
with a, and thereforeinduces homomorphisms
of the groupsof K into those of
P. That these are isomorphismsfollowsfromthe lemma: If (K', X, Z') is a
chain withboundaryof the form(K, 01i, Z), then thereis a chain Z, of K such
that a Z1 = Z, and thedifference
(K', O,Z') + (K,
4s , - Z1) boundsa continuous
chain in R. The lemma is proved in the usual way by using the simplicial
approximationtheoremto constructa map of the product complex K' X I
(I = (0, 1)) intoR. The needed chainis foundin K' X I withlocal coefficients
in the induced system.
18. Cech cycles
The only difficulty
in the way of extendinglocal coefficients
to Cech cyclesis
that of constructinga systemof local groups in the nerve K of a finiteopen
coveringwhensuch a systemis givenin R. It is clear that the formermust be
chosen so as to induce the given systemin R under the natural map R -* K.
If R is sufficiently
complicated,it is possible to constructin R a local system
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH LOCAL COEFFICIENTS
625
whichis not inducedby a local systemin any nerve.5 Thereforewe are forced
to restrictourselvesto a system {G.} induced in R by a system {Go} in a fixed
nerve K0. We then admit only those coveringswhich are refinements
of K0,
and we use in them the local groupsinduced by theirnatural projectionsinto
the definitions
of the Cech homologyand cohomolK0. Withthismodification,
ogy groups and theirmultiplicationsproceed as before.
If R is the space of a complexK, a local systemin R is one in K = K0. Using
invarianceunder subdivision,one proves in the customaryway (see [15], ?9)
that the groupsof K and the Cech groupsof R are isomorphic,and the isomorphismspreservethe multiplications. We are thusled to a proofthat the homology theory(coef. tG6}) of a complexK is a topologicalinvariantof the space
K with the local groups EGG}.
19. tberdeckung
In a complexK choose a referencepoint o (preferablya vertex),and foreach
cell a let a,,be a path in K fromo to a point x(r) in a. If a' < a, let a,,,' be a
path in the closureof a fromx(u) to x(u'). The closed path aOaCCYOa,-1is abbreAs elementsof the fundamentalgroup F of K (origino), the
viated by Yao'
ly's have the property
(19.1)
=
fora" < a' < a.
By means of the a's, we map isomorphicallythe chains of K with local coef.
in G0 as follows. The transform
f
JG.} into ordinarychains with coefficients
is definedby f(u)
of the chainf with local coefficients
a-7'(f()).
By means
we defineoperatorsa, 8 forchainsf by: f=
of this isomorphism,
f.
b, e =
From (9.3) we obtain
(19.2)
}(f)-
EOu': uf]^y~'(f(u)),
bf(o")= Z[o:o"]yff't(J(o))
Ad< u,
u < a-"
Thus the systemof ordinarychains (coef.G,) withthe special operatorsa, a are
isomorphicto the system of chains with local coef. OGxEwith the ordinary
operatorsa, &. They determinethereforeisomorphichomologygroups. It is
a corollarythat the homologygroups determinedby c, 3 are independentof
the choices of the &'s.
The systemf, 5, 8 just describedis called an Uberdeckung
of K by Reidemeister
[10; 11]. An advantage of this approach is that it lends itselfmorereadilyto a
computationof the homologygroups. One may attemptto simplifythe incidence matrices[u': u]yO' , with elementsin the group ring of F, by the usual
bases and consolidation(see W. Franz [2]).
methodsof transforming
This is the case if R is not locally simply-connected,
and if JGX,is the system(G'
of local groupringsof ?8.
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
l
626
N. E.
STEENROD
20. Zero and 1-dimensionalgroups
If the fundamentalgroupF of K and the operationsof F in Goare given,then
one may compute the 0 and 1-dimensionalhomologygroups withoutfurther
knowledgeof K. Choose a basis a,, *- *, ah of F and a basis r1, -* *, r8forthe
the unit). Constructa
relationsin F (each r is a productof a's representing
for
each
one
edge
o',
one
vertex
2-complexK' consistingof
ai (likewisedenoted
a
2-cell
by as) withboth end-pointsat o', and, foreach ri,
Ei whose boundary
group of K'.
the
fundamental
is
also
F
a's.
Clearly
the
is the product ri of
to 0 and
K'
leading
in
coefficients
local
F
in
determine
The operationsof
Go
homologyand cohomologygroupswhichwe shall proveisomorphic
1-dimensional
to thoseof K. By the dualityof ?12, it sufficesto prove this forthe homology
groups.
Define a map 0 of K' into K so that 0(o') = o, 0(ai) representsas , and 0 is
continuous. It is readilyseen that a map 4' of a complexK" in K is homotopic
to a map 1' which,on the 1-dimensionalpart K" of K", can be expressedas a
product04" of 0 and a map if" of K',' in K' whichmaps each vertexinto o' and
each edge into a productof a's. Thus everycontinuous0 and 1-cycleis homologous to one of the form(K', X, Z) (see ?17). It is a furtherconsequencethat
the 0-cycleZ bounds in K' if (K', X, Z) bounds in K. This shows that the 0th
homologygroups of K anrd-K'are isomorphic. To prove the same for the 1dimensionalgroups,we must show that a homologyrelationin K of the form
a(E, i', f(E) = g) = (K', 0, Z), whereE is a 2-celland /'(ME) is a productr of
the a's, is a consequenceof the relationsin K'. Let E be regardedas a hemisphereof a 2-sphereS2 and let E' be the otherhemisphere. The map t" of the
equator in K' extendsto a continuousmap A/"of E' in K'; for the productr is
expressiblein termsof the ri. The map JP'of E and 4,6"of E' definea map
4' of 82 in K, and therebya 2-cycle(S2, Jp', f(E) = f(- E) = g). Thus a (E',
Ap',
f(E') = g) = (K', 0, Z), and Vp' factorsinto 44".
Usingthe above resultswe may describethe 0-dimensionalgroupsquite easily.
Let G' be the subgroupof elementsof Go which are fixedunder every automorphisma e F. The oth cohomology groupof K is isomorphicto thegroupGo.
Let G"'be the subgroupof elementsof Goexpressiblein the formEi(ai(gi) - gi)
groupof K is isomorphicto thedifferwhereai e F, gi e Go. The oth homology
ence group Go - Go .
It is worthnotingthat a continuousimage of an n-sphere(n > 1) in K deterfor the sphere is
minesa group of sphericali-cycles for any local coefficients;
simplyconnected. However these cycles may or may not bound accordingto
the structureof the systemof local groups. Consider,as an example,the projective plane P2 and the double coveringof it by the 2-sphereS2. With twisted
(?14), the 2-cycleson P2 forman infinitecyclicgroupand are
integercoefficients
The
even multiplesof the generatorare images of the 2-cycles
nonbounding.
in P2, the
Yet with simple integercoefficients
coefficients.
on S2 with integer
zero).
on
is
(algebraically
S2
bounding
mage of every2-cycle
THE UNIVERSITY OF MICHIGAN
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions
HOMOLOGY
WITH LOCAL COEFFICIENTS
627
BIBLIOGRAPHY
1. P. ALEXANDROFF-H. HOPF, Topologie, Berlin, 1936.
2. W. FRANZ, Uberdeckungentopologischer Komplexe mit hyperkomplexenSystemen, J.
reine angew. Math., 173 (1935), 174-184.
, Uber die Torsion einer Uberdeckung,ibid., 173 (1935), 245-254.
3.
4.
, Torsionideale, Torsionklassen und Torsion, ibid., 176 (1936), 113-124.
5. A. KOMATU, Uber die Dualitdtssatze der Uberdeckingeni,Jap. Journ. Math., 13 (1937),
493-500.
, Uber die Lberdeckungenvon Zellenradmen,Proc. Imp. Acad. Tokyo, 14 (1938),
6.
340-341; 15 (1939), 4-7; 16 (1940), 55-58.
7. S. LEFSCHETZ, Topology, Colloquium lectures, New York, 1930.
8. K. REIDEMEISTER, HomotopiegruppenvonKomplexen, Hamburger Abh., 10 (1934).
, Homotopieringeund Linsenraume, ibid., 11 (1935).
9.
, Uberdeckungenvon Komplexen, J. reine angew. Math., 173 (1935), 164-173.
10.
, Topologie der Polyeder, Leipzig, 1938.
11.
, Durchschnittund Schnitt von Homotopicketteni,
12.
Monats. ffirMath., 48 (1939),
226-239.
13. G. DE RHAM, Sur la theoriedes intersectionsct les idltegralcs multiples,Comment. Math.
Helv., 4 (1932), 151-157.
, Sur les complexesavec automnorphismties,
14.
ibid., 12' (1940), 191-211.
15. N. STEENROD, Universal homologygroups, Amer. Journ. Math., 58 (1936), 661-701.
16.
-, Topological methodsfor the constructionof tensorJunctions, Annals of Math.,
43 (1942).
17. H. WHITNEY, On productsin a complex,Annals of Mathl., 39 (1938), 397-432.
, On thetheoryof sphere-bundles,Proc. Nat. Acad. Sci., 26 (1940), 148-153.
18.
, On the topology of differentiablemanifolds, Lectures in Topology, Univ. of
19.
Michigan, 1940.
This content downloaded on Fri, 28 Dec 2012 18:40:12 PM
All use subject to JSTOR Terms and Conditions