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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. 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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