Symmetric continuous cohomology of topological groups

SYMMETRIC CONTI NUOUS COHOMOLOGY OF TOPOLOGICAL GR OUPS MAHENDER SINGH Abstra ct. In this pap er, w e introduce a sy mm etric continuous cohomol ogy of topological groups. This is obtained by top ologizing a recent construction due to Staic [23], where a sym- metric cohomology of abstract groups is constructed. W e give a chara cterization of top ological group ex tensions th at corresp ond to elements of the second symmetric continuous cohomology . W e also show that the symmetric con tinuous cohomology of a p rofinite group with co efficients in a discrete mo du le is equal to the direct limit of the symmetric cohomology of fin ite group s. In the end, w e also define symm etric smooth cohomology of Lie groups and p rove similar results. 1. Introduction The cohomology of abstr act groups came in to b eing with the f u ndamenta l work of Eilen b erg and MacLane [6, 7]. T he theory dev elop ed rapidly with the wo rks of E ilen b erg, MacLane, Hopf , Ec kmann, Segal, Ser r e and man y other authors. The cohomology of groups has b een a p opular researc h su b ject and has b een studied from different p ers p ectiv es with applications in algebraic n umb er theory , algebraic top ology a nd Lie algebras, to name a few. A detaile d accoun t of the history of the sub ject app ears in [27]. When th e group u n der consideration is equipp ed with a top ology , then it is natural to look for a cohomolog y theory which also tak es the top ology into accoun t. This lead to many new cohomology theories of top ological groups and the top ology w as first inser ted in the formal definition of group cohomology in th e works of Heller [10], Hu [13] and v an Est [26]. In [8], Fiedoro wicz and Lo day defined a homology theory of crossed simplicial grou p s. Moti- v ated b y their construction, Staic [23] in tro duced the notio n of the ∆-group Γ( X ) for a topo- logica l space X . Giv en a group G and a G -mo du le A , for eac h n ≥ 0, Staic defi n ed an action of the symmetric g roup Σ n +1 on the sta nd ard n th coc hain group C n ( G, A ) used to co mp ute the usual group cohomology an d pro ved it to b e compatible with the standard cob oundary op erators ∂ n . Th us , the s u b complex { C n ( G, A ) Σ n +1 , ∂ n } n ≥ 0 of in v arian t elemen ts of { C n ( G, A ) , ∂ n } n ≥ 0 giv es a new cohomology theory H S ∗ ( G, A ) called the symmetric cohomolog y . Staic show ed that the ∆-group Γ( X ) is determined by the action of π 1 ( X ) on π 2 ( X ) and an elemen t of H S 3 ( π 1 ( X ) , π 2 ( X )). The inclusion of th e co chain groups C n ( G, A ) Σ n +1 ֒ → C n ( G, A ) ind uces a h omomorphism from H S n ( G, A ) → H n ( G, A ). In [23], it is sho wn that H S 2 ( G, A ) → H 2 ( G, A ) is in jectiv e. 2010 Mathematics Subje ct Classific ation. Primary 20J06; S econdary 54H11,57T10. Key wor ds and phr ases. Contin uous cohomolo gy , group extension, Lie group, profinite group, symmetric co- homology , top ological group. 1 2 MAHENDER SINGH It is we ll known that, if A is a G -mo du le, then there is a bijection b et w een H 2 ( G, A ) and the set of equiv alence classes of group extensions of G by A with the giv en G -modu le structure. Therefore, it is n atural to ask what kind of group extensions corresp ond to element s of the second symmetric cohomology . I n [24], Staic prov ed that H S 2 ( G, A ) is in bijection with the set of equiv alence classes of group extensions 0 → A → E → G → 1 admitting a section s : G → E with the prop ert y th at s ( g − 1 ) = s ( g ) − 1 for all g ∈ G . Note that the condition is slight ly w eak er than s b eing a homomorphism. W e sh all see that there are examples of non-split extensions of groups admitting su c h a section. The purp ose of this pap er is to top olog ize this construction and in tro duce a sy m metric con- tin uous cohomology of top ological groups. As for the discrete ca se, we giv e a c h aracterizati on of top ological group extensions that corresp ond to elemen ts of the second symm etric con tinuous cohomology . W e also show that the s y m metric con tinuous cohomolog y of a profin ite group with co efficien ts in a discrete m o dule is equal to the direct limit of the symmetric cohomology of finite groups. W e similarly defin e symmetric smo oth cohomolo gy of Lie groups. The pap er is organized as follo ws. In S ection 2, we fix some notatio n a nd recall some kn o wn definitions and results that will b e used in the pap er. In Section 3, w e introd u ce the symmetric con tin uous cohomology of top ological groups. In Section 4, we giv e some examples to illustrate the p rop osed cohomology theory . In S ection 5, we prov e some prop erties of the symmetric con tin uous cohomology of top ological groups. In S ection 6, we discuss th e symmetric contin- uous cohomology of profinite groups. Finally , in Section 7, we define the symmetric smo oth cohomology of Lie groups and pr o v e some of its prop erties. Ac kno wledgemen t. The author w ould lik e to thank the referee for commen ts w hic h impro ve d the presenta tion of the pap er. The author is grateful to the MathOv erflo w comm un it y w hic h w as helpful in clarifying some examples. The author would also lik e to thank the Departmen t of Science and T ec h nology of India for supp ort via the INSPIRE Sc heme IF A -11MA-01/2 011 and the S ER C F ast T rac k Sc heme SR/FTP/MS-027/20 10. 2. Not a tion and terminology In this section, w e fix some notation and recall some kn o wn d efinitions and results. W e refer the reader to Brown [3] for b asic material on the cohomology of groups. F or an y extension 0 → A → E → G → 1 of groups (abstract, top ological or Lie), the group A is written additiv ely and the groups E and G are written multi plicativ ely , u nless otherwise stated or it is clea r fr om the conte xt. Cohomology of abstract groups. Let us r ecall the constru ction of th e co c hain complex defining the cohomology of abstract groups (groups without an y other structure). Let G b e a group and A b e a G -mo dule. More precisely , there is a group action G × A → A b y automorph isms. As usual A is written additively and G is written multiplicat ive ly , un less otherwise stated or it is clear from the con text. F or eac h n ≥ 0, the group of n -co c hains C n ( G, A ) SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 3 is th e group of all maps σ : G n → A . Th e cob oundary ∂ n : C n ( G, A ) → C n +1 ( G, A ) is given b y ∂ n ( σ )( g 1 , ..., g n +1 ) = g 1 σ ( g 2 , ..., g n +1 ) + n X i =1 ( − 1) i +1 σ ( g 1 , ..., g i g i +1 , ..., g n +1 ) + σ ( g 1 , ..., g n ) , (1) for all σ ∈ C n ( G, A ) an d ( g 1 , ..., g n +1 ) ∈ G n +1 . It is straigh tforw ard to v erify that ∂ n +1 ∂ n = 0 and hence w e obtain a co chain complex. Let Z n ( G, A ) = Ker( ∂ n ) b e the group of n -co cycles and B n ( G, A ) = Im( ∂ n − 1 ) b e the group of n -cob ound aries. Th en the n th cohomolog y group is giv en b y H n ( G, A ) = Z n ( G, A ) /B n ( G, A ) . If σ ∈ Z n ( G, A ) is a n -co cyle, w e denote b y [ σ ] ∈ H n ( G, A ) th e corresp ondin g cohomology class. Symmetric cohomolog y of abstract groups. F or eac h n ≥ 0, let Σ n +1 b e the symmetric group on n + 1 sym b ols. In [23 ], Staic defined an action of th e symmetric group Σ n +1 on the n th cochain group C n ( G, A ). Since the transp ositions of adjacen t elemen ts form a generating set for Σ n +1 , it is enough to defin e the action of these transp ositions τ i = ( i, i + 1 ) for 1 ≤ i ≤ n . F or σ ∈ C n ( G, A ) and ( g 1 , ..., g n ) ∈ G n , d efine ( τ 1 σ )( g 1 , g 2 , g 3 , ..., g n ) = − g 1 σ  ( g 1 ) − 1 , g 1 g 2 , g 3 , ..., g n  , ( τ i σ )( g 1 , g 2 , g 3 , ..., g n ) = − σ  g 1 , ..., g i − 2 , g i − 1 g i , ( g i ) − 1 , g i g i +1 , g i +2 , ..., g n  for 1 < i < n, ( τ n σ )( g 1 , g 2 , g 3 , ..., g n ) = − σ  g 1 , g 2 , g 3 , ..., g n − 1 g n , ( g n ) − 1  . (2) It is shown in [23] that the ab o v e action is compatible with the cob oundary op erators ∂ n and hence yields the sub complex of in v ariants { C S n ( G, A ) , ∂ n } n ≥ 0 = { C n ( G, A ) Σ n +1 , ∂ n } n ≥ 0 . The cohomology of this co c hain complex is called the symmetric cohomology of G with co- efficien ts in A and is denoted H S n ( G, A ). The co cycles and the cob oun daries are called the symmetric cocycles and the symmetric cob oundaries, r esp ectiv ely . Con tinuo us c ohomology o f topological groups. W e assume that all top ological group s under consideration satisfy the T 0 separation axiom. Let G b e a top ological group and A b e an ab elian top ological group . W e s a y that A is a top ologic al G -modu le if there is a con tinuous action of G on A . The contin u ous cohomolog y of top ological groups w as defi ned ind ep endently b y Hu [13], v an Est [26] and Heller [10] as follo ws. F or eac h n ≥ 0, let C n c ( G, A ) b e th e group of all conti nuous maps f rom G n → A , where G n is the pro duct top ologica l group. The cob oundary maps giv en by the s tandard form ula as in (1), giv es the co c hain complex { C n c ( G, A ) , ∂ n } n ≥ 0 . The con tin uous cohomology of G with co efficients in A is defined to b e the cohomolog y of this co c hain complex and is denoted b y H ∗ c ( G, A ). 4 MAHENDER SINGH Clearly , this cohomology theory coincides with the abstract cohomology theory wh en the groups under consid eration are discrete (in particular finite). The lo w dimensional cohomology groups are as exp ected. More p r ecisely , H 0 c ( G, A ) = A G and Z 1 c ( G, A ) = the group of contin u ous crossed h omomorphisms from G to A . An extension of top ologica l groups 0 → A i → E π → G → 1 is an algebraically exact sequence of top ologica l groups with the additional prop ert y that i is closed conti nuous and π is op en contin u ou s . Note th at, if w e assu me that i and π are only con tin uous, then A view ed as a sub group of E m ay not ha ve the relativ e top ology and the isomorphism E /i ( A ) ∼ = G may not b e a homeomorphism. A section to the giv en extension is a map s : G → E suc h that π s ( g ) = g for all g ∈ G . Since A is closed and i is closed cont inuous, it follo ws that i ( A ) = π − 1 ( { 1 } ) is a clo sed subgroup of E and i : A → E is an em b edding of A on to a closed sub group of E . Let G b e a top ological group and A a top ological G -mo du le. Let 0 → A i → E π → G → 1 b e a top ological group extension and let s : G → E b e a section to π . Since A is ab elian, for a ∈ A and g ∈ G , one can see that the elemen t i − 1  s ( g ) i ( a ) s ( g ) − 1  do es not dep end on the choic e of the section. The extension 0 → A i → E π → G → 1 is said to corresp ond to the give n wa y in whic h G acts on A if g a = i − 1  s ( g ) i ( a ) s ( g ) − 1  for all a ∈ A and g ∈ G. Consider the s et of all top ological group extensions of G b y A corresp ond ing to the giv en wa y in whic h G acts on A . Two s uc h extensions 0 → A i → E π → G → 1 and 0 → A i ′ → E ′ π ′ → G → 1 are said to b e equiv alen t if there exists an op en cont inuous isomorphism φ : E → E ′ suc h th at the follo wing diag ram comm ute 0 / / A i / / E π / / φ   G / / 1 0 / / A i ′ / / E ′ π ′ / / G / / 1 . F or br evit y , E ∼ = E ′ denotes the equiv alence of extensions. Heller [10] and Hu [13, Th eorem 5.3] indep endently pro v ed the follo wing r esult. Theorem 2.1. L et G b e a top olo gic al gr oup and A a top olo gic al G -mo dule. Then H 2 c ( G, A ) is in bije ction with the set of e quivalenc e classes of top olo gic al gr oup extensions of G by A admitting a (glob al) c ontinuous se c tion and the gi v en G -mo dule structur e. W e sh all pr o v e similar theorems usin g symmetric con tinuous cohomology of top ological groups (Theorem 3.3) and symmetric smo oth cohomology of Lie group s (Th eorem 7.2) in the follo wing sections. An extension of top ological group s is said to b e top ologicall y s plit if E is A × G as a top ological space. Note that if an extension of top ological group s admits a contin uou s sectio n, then the SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 5 extension is top ologic ally split. Extensions of top ological groups admitting a contin u ou s s ection are assured by the follo w ing theorem. Theorem 2.2. [22, T heorem 2] L et G b e a c onne c te d lo c al ly c omp act gr oup. Then any top olo gic al gr oup extension of G by a simply c onne cte d Lie gr oup admits a c ontinuous se ction. 3. Symmetric continuous cohomology o f topological groups In this section, we define th e symmetric con tinuous cohomology of top ological groups, ha ving the exp ected cohomolo gy groups in lo w d imension. F rom no w on, G is a top ologica l group and A a top ological G -mo d ule. Let n ≥ 0. Since G is a top ologica l group, for ( g 1 , ..., g n ) ∈ G n , ( g 1 , g 2 , g 3 , ..., g n ) 7→  ( g 1 ) − 1 , g 1 g 2 , g 3 , ..., g n  , ( g 1 , g 2 , g 3 , ..., g n ) 7→  g 1 , ..., g i − 2 , g i − 1 g i , ( g i ) − 1 , g i g i +1 , g i +2 , ..., g n  for 1 < i < n, ( g 1 , g 2 , g 3 , ..., g n ) 7→  g 1 , g 2 , g 3 , ..., g n − 1 g n , ( g n ) − 1  , (3) are all con tinuous maps G n → G n . The con tin uity of the action G × A → A and the maps give n by (3) sh o ws that τ σ ∈ C n c ( G, A ) for eac h τ ∈ Σ n +1 and σ ∈ C n c ( G, A ). By [23, Prop osition 5.1], the formulas giv en b y (2) define an action compatible with the cob ou n dary op erators ∂ n giv en by (1) . This give s the sub complex of inv arian ts { C S n c ( G, A ) , ∂ n } n ≥ 0 = { C n c ( G, A ) Σ n +1 , ∂ n } n ≥ 0 . W e call the cohomology of this co c hain complex the symmetric con tinuous cohomology of G with coefficien ts in A and den ote it b y H S n c ( G, A ). Clearly , when the group s under consideration are discrete, then H S n c ( G, A ) = H S n ( G, A ). When G is a connected top ological group and A is a discrete G-mod ule, then G acts trivially on A and the con tin uous co chains are only constant m aps. Th en it follo ws th at H S 0 c ( G, A ) = A and H S n c ( G, A ) = 0 for eac h n ≥ 1. Observe that a 1-cocycle λ : G → A is symmetric if λ ( g ) = − g λ ( g − 1 ) for all g ∈ G and a 2-cocycle σ : G × G → A is symm etric if σ ( g , k ) = − g σ ( g − 1 , g k ) = − σ ( g k , k − 1 ) for all g , k ∈ G. (4) It is easy to establish the follo wing prop erties. Prop osition 3.1. Let G b e a top ologica l group and A b e a top ological G -mod ule. Then (1) H S 0 c ( G, A ) = A G = H 0 c ( G, A ) (2) Z S 1 c ( G, A ) = the group of symmetric con tinuous crossed h omomorphisms from G to A . 6 MAHENDER SINGH Pr o of. (1) is straigh tforward. By definition Z S 1 c ( G, A ) = Ker { ∂ 1 : C S 1 c ( G, A ) → C S 2 c ( G, A ) } . Therefore, λ ∈ Z S 1 c ( G, A ) if and only if λ is contin u ou s and satisfy λ ( g ) = − g λ ( g − 1 ) and λ ( g k ) = g λ ( k ) + λ ( g ) for all g , k ∈ G . In other w ords, λ is a symmetric con tinuous crossed homomorphism. This pro v es (2).  As in the discrete case, the inclusion of th e su b complex C S ∗ c ( G, A ) ֒ → C ∗ c ( G, A ) induces a homomorph ism h ∗ : H S ∗ c ( G, A ) → H ∗ c ( G, A ) . Clearly h ∗ : H S 1 c ( G, A ) → H 1 c ( G, A ) is injectiv e. In dimension t wo , w e ha ve the follo wing prop osition, whic h is essen tially [24, Lemma 3.1]. W e pr o vide a pr o of here for the sak e of completeness. Prop osition 3.2. The map h ∗ : H S 2 c ( G, A ) → H 2 c ( G, A ) is in jectiv e. Pr o of. Let σ represen t an elemen t in K er( h ∗ ). In other words, σ ∈ Z S 2 c ( G, A ) ∩ B 2 c ( G, A ). This implies th at σ is symmetric and there exists λ ∈ C 1 c ( G, A ) such that σ ( g , k ) = ∂ 1 λ ( g , k ) = g λ ( k ) − λ ( g k ) + λ ( g ) for all g , k ∈ G . The s y m metry of σ giv es, σ ( g , k ) = − g σ ( g − 1 , g k ) = − λ ( g k ) + g λ ( k ) − g λ ( g − 1 ) and σ ( g , k ) = − σ ( g k , k − 1 ) = − g k λ ( k − 1 ) + λ ( g ) − λ ( g k ) for all g, k ∈ G. By taking g = 1 and equating the ab o ve t w o equations, we get λ ( k ) = − k λ ( k − 1 ) for all k ∈ G . This sh o ws th at λ ∈ C S 1 c ( G, A ) and h en ce σ ∈ B S 2 c ( G, A ). Th us σ repr esen ts the trivial elemen t in H S 2 c ( G, A ) and the m ap h ∗ is in jectiv e.  Note that the map h ∗ need not b e surjectiv e in general. Th is will b e illustrated by examples in Sectio n 4. By Theorem 2.1 , H 2 c ( G, A ) classifies equiv alence classes of top olog ical group extensions of G b y A ad m itting a contin u ou s section. In view o f Prop osition 3.2, we w ould lik e to kno w wh ic h of these extensions corresp ond to H S 2 c ( G, A ). Let 0 → A → E → G → 1 b e an extension of top ological groups. W e sa y that a section s : G → E is symmetric if s ( g − 1 ) = s ( g ) − 1 for all g ∈ G. F or simplicit y , we assume that s satisfies the n ormalization condition s (1) = 1. Let C ( G, A ) denote the set of equiv alence classes of top ologi cal group extensions of G by A admitting a symmetric cont inuous section and b eing equipp ed with the giv en G -modu le structure. With these definitions, we p ro v e the f ollo win g theorem. Theorem 3.3. L e t G b e a top olo gic al gr oup and A b e a top olo gic al G -mo dule. Then ther e is a bije ction Φ : C ( G, A ) → H S 2 c ( G, A ) . SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 7 Pr o of. Let 0 → A i − → E π − → G → 1 be a top ological group extension of G b y A admitting a symmetric con tinuous sectio n s : G → E and corresp onding to the giv en wa y in whic h G acts on A . Ev ery elemen t of E can b e written u n iquely as i ( a ) s ( g ) for some a ∈ A and g ∈ G . Since the extension corresp onds to the giv en w ay in whic h G acts on A , w e ha ve that g a = i − 1  s ( g ) i ( a ) s ( g ) − 1  for all a ∈ A and g ∈ G. As i and s are con tinuous, w e see that the acti on is con tinuous. Also, we hav e π ( s ( g h )) = g h = π ( s ( g )) π ( s ( h )) = π ( s ( g ) s ( h )) for all g, h ∈ G. Th us , there exists a u nique element (say) σ ( g, h ) in A s u c h that σ ( g , h ) = i − 1  s ( g ) s ( h ) s ( g h ) − 1  . Observe that σ : G × G → A s atisfies the condition g σ ( h, k ) − σ ( g h, k ) + σ ( g , hk ) − σ ( g , h ) = 0 for all g, h, k ∈ G. (5) In ot her w ords , σ is a 2-cocycle. Moreo v er, con tinuit y of i and s implies that σ is con tinuous. Finally , using the symmetry of s and the action of G on A , we sh o w th at σ is in fact symmetric. That is, for all g, h ∈ G , w e ha v e − g σ ( g − 1 , g h ) = − g i − 1  s ( g − 1 ) s ( g h ) s ( h ) − 1  = − g i − 1  s ( g ) − 1 s ( g h ) s ( h ) − 1  = − i − 1  s ( g ) s ( g ) − 1 s ( g h ) s ( h ) − 1 s ( g ) − 1  = − i − 1  s ( g ) s ( h ) s ( g h ) − 1  − 1  = i − 1  s ( g ) s ( h ) s ( g h ) − 1  = σ ( g , h ) and − σ ( g h, h − 1 ) = − i − 1  s ( g h ) s ( h − 1 ) s ( g ) − 1  = − i − 1  s ( g h ) s ( h ) − 1 s ( g ) − 1  = − i − 1  s ( g ) s ( h ) s ( g h ) − 1  − 1  = i − 1  s ( g ) s ( h ) s ( g h ) − 1  = σ ( g , h ) . Th us , σ giv es an elemen t in H S 2 c ( G, A ). W e need to sh o w that the cohomology cl ass of σ is indep end en t of the c hoice of the section. Let s, t : G → E b e t wo symmetric con tin uous sections. As ab o v e w e get sy m metric con tinuous 2-cocycles σ, µ : G × G → A suc h that for all g , h ∈ G , we h a v e σ ( g , h ) = i − 1  s ( g ) s ( h ) s ( g h ) − 1  and µ ( g , h ) = i − 1  t ( g ) t ( h ) t ( gh ) − 1  . 8 MAHENDER SINGH Since s ( g ) and t ( g ) satisfy π ( s ( g )) = g = π ( t ( g )), there exists a u nique elemen t (sa y) λ ( g ) ∈ A suc h that λ ( g ) = i − 1  s ( g ) t ( g ) − 1  . This yields a 1-cochain λ : G → A whic h is con tinuous and symmetric, as τ 1 λ ( g ) = − g λ ( g − 1 ) = − g i − 1  s ( g − 1 ) t ( g − 1 ) − 1  = − g i − 1  s ( g ) − 1 t ( g )  = − i − 1  s ( g ) s ( g ) − 1 t ( g ) s ( g ) − 1  = − i − 1  ( s ( g ) t ( g ) − 1 ) − 1  = i − 1  s ( g ) t ( g ) − 1  = λ ( g ) . Th us , we ha ve that σ ( g , h ) − µ ( g , h ) = g λ ( h ) − λ ( g h ) + λ ( g ) . In other wo rd s, σ − µ ∈ B S 2 c ( G, A ) and hence [ σ ] = [ µ ] in H S 2 c ( G, A ). Let 0 → A i ′ → E ′ π ′ → G → 1 b e an extension equiv alen t to 0 → A i → E π → G → 1 via an op en con tin uous isomorphism φ : E → E ′ . T hen s ′ = φs : G → E ′ is a symmetric con tin uous section. It is clear that the 2-co cycle corresp onding to s ′ is same as the one corresp onding to s . Hence equiv alen t extensions giv es the same eleme nt in H S 2 c ( G, A ). No w w e can defi n e Φ : C ( G, A ) → H S 2 c ( G, A ) b y mappin g an equiv alence class of extensions to the corresp ond ing cohomology class as obtained ab o ve . T he abov e argument s show that Φ is well defin ed . W e first p r o v e that Φ is sur jectiv e. Let σ ∈ Z S 2 c ( G, A ) b e a sy m metric c ontin u ou s 2-co cyle represent ing an elemen t in H S 2 c ( G, A ). The sym metry of the 2-co cyle giv es the equation (4). Let E σ := A × G b e equipp ed with the pro du ct topology . Define a binary op eration on E σ b y ( a, g )( b, h ) =  a + g b + σ ( g , h ) , g h  for all a, b ∈ A and g , h ∈ G. It is routine to c heck that this binary op eration giv es a group structur e on E σ . S ince A is a top ological G -mo dule and the 2-co cyle σ is con tin uous , the group op eration from E σ × E σ → E σ and the inv erting op eration from E σ → E σ are con tin uous w ith r esp ect to the pr o duct top ology on E σ . Hence, E σ is a top ological group. Clearly , the map π : E σ → G giv en b y π ( a, g ) = g is an op en contin uou s h omomorphism; and the map i : A → E σ giv en by i ( a ) = ( a, 1) is an em b edding of A onto the closed sub group i ( A ) of E σ . This giv es the follo wing extension of top olog ical group s 0 → A i − → E σ π − → G → 1 . SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 9 The extension has an obvious con tin uous sec tion s : G → E σ , giv en by s ( g ) = (0 , g ). Using the group op eration and the symmetry of σ , we get s ( g )  s ( g − 1 ) s ( g h )  =  g σ ( g − 1 , g h ) + σ ( g, h ) , g h  = (0 , g h ) = s ( g h ) and  s ( g h ) s ( h − 1 )  s ( h ) =  σ ( g , h ) + σ ( g h, h − 1 ) , g h  = (0 , g h ) = s ( g h ) . This give s s ( g − 1 ) = s ( g ) − 1 and hence th e section s is s y m metric. Note that by (5) and the normalizatio n o f the section, we hav e σ ( g , 1) = σ (1 , g ) = σ (1 , 1) = 0 for all g ∈ G. F or all a ∈ A and g ∈ G , w e ha ve i − 1  s ( g ) i ( a ) s ( g ) − 1  = i − 1  (0 , g )( a, 1)(0 , g ) − 1  = i − 1  ( g a + σ ( g , 1) , g )(0 , g ) − 1  = i − 1  ( g a + σ ( g , 1) , g )( − g − 1 σ ( g , g − 1 ) , g − 1 )  = i − 1  ( g a + σ ( g , 1) + g ( − g − 1 σ ( g , g − 1 )) + σ ( g, g − 1 ) , 1)  = i − 1  ( g a + σ ( g , 1) , 1)  = i − 1  ( g a, 1)  = g a. Th us , the extension 0 → A → E σ → G → 1 corresp onds to the giv en G -module structure on A . Next, f or all g , h ∈ G , w e ha ve σ ( g , h ) = i − 1  ( σ ( g , h ) , 1)  = i − 1  ( σ ( g , h ) − σ (1 , g h ) , 1)  = i − 1  ( σ ( g , h ) − σ (1 , g h ) , 1)(0 , g h )(0 , g h ) − 1  = i − 1  ( σ ( g , h ) , g h )(0 , g h ) − 1  = i − 1  (0 , g )(0 , h )(0 , g h ) − 1  = i − 1  s ( g ) s ( h ) s ( g h ) − 1  . Th us , σ is the 2-cocycle co rr esp onding to the section s . Hence Φ is surjectiv e. Finally , we pro ve that Φ is injectiv e. Let 0 → A i → E π → G → 1 and 0 → A i ′ → E ′ π ′ → G → 1 b e t w o top ological group extensions admitting symmetric con tin uous sections s and s ′ , resp ectiv ely . Let σ and σ ′ b e the 2-co cyles asso ciated to s and s ′ , resp ectiv ely . Supp ose that σ an d σ ′ represent the same elemen t in H S 2 c ( G, A ). In other w ords, σ ′ − σ = ∂ 1 ( λ ) for some λ ∈ C S 1 c ( G, A ). Define t : G → E by t ( g ) = iλ ( g ) s ( g ) f or all g ∈ G. 10 MAHENDER SINGH W e can see that t is a con tinuous sectio n to π and also giv es rise to the 2-cocycle σ ′ as i − 1  t ( g ) t ( h ) t ( gh ) − 1  = i − 1  iλ ( g ) s ( g ) iλ ( h ) s ( h ) s ( gh ) − 1 iλ ( g h ) − 1  = i − 1  iλ ( g ) s ( g ) iλ ( h ) s ( g ) − 1 s ( g ) s ( h ) s ( g h ) − 1 iλ ( g h ) − 1  = i − 1  iλ ( g ) s ( g ) iλ ( h ) s ( g ) − 1 i ( σ ( g , h )) iλ ( g h ) − 1  = λ ( g ) + i − 1  s ( g ) iλ ( h ) s ( g ) − 1  + σ ( g , h ) − λ ( g h ) = λ ( g ) + g λ ( h ) + σ ( g, h ) − λ ( g h ) = σ ′ ( g , h ) . Let 0 → A → E σ ′ → G → 1 b e the extension asso ciated to σ ′ . Define φ t : E σ ′ → E b y φ t ( a, g ) = i ( a ) t ( g ) for all a ∈ A and g ∈ G. Clearly , φ t is contin u ou s ; and it is a h omomorphism b ecause φ t  ( a, g )( b, h )  = φ t ( a + g b + σ ′ ( g , h ) , g h ) = i ( a + g b + σ ′ ( g , h )) t ( g h ) = i ( a ) i ( gb ) i ( σ ′ ( g , h )) t ( g h ) = i ( a ) t ( g ) i ( b ) t ( g ) − 1 t ( g ) t ( h ) t ( gh ) − 1 t ( g h ) = i ( a ) t ( g ) i ( b ) t ( h ) = φ t ( a, g ) φ t ( b, h ) . It is easy to s ee th at φ t is bijectiv e with inv ers e i ( a ) t ( g ) 7→ ( a, g ). As b oth E σ ′ and E ha ve th e pro du ct top ology , th e in v erse homomorph ism is also con tin uous and hence φ t is an equiv alence of extensions E σ ′ ∼ = E . Similarly , define φ s ′ : E σ ′ → E ′ b y φ s ′ ( a, g ) = i ′ ( a ) s ′ ( g ) for all a ∈ A and g ∈ G. Just as ab o v e, we c an see that φ s ′ is an equiv alence of extensions E σ ′ ∼ = E ′ . Hence w e get an equiv alence of extensions E ∼ = E ′ pro ving that Φ is injectiv e. The pro of of the theorem is no w complete.  W e conclude this section with the follo wing remarks. Remark 3.4. Recall that a top ologi cal group G is said to b e a free top ological grou p if there exists a completely regular space X su c h that: (i) X is top ologica lly em b eddable in G ; (ii) when em b edd ed X generates G ; (iii) ev ery conti nuous map from X to a top ological group H can b e extended to a unique con tin uous homomorph ism fr om G to H . If G is a free top ological group , then H 2 c ( G, A ) = 0 by [13, Prop osition 5.6]. Thus, if G is a free top olog ical group and A is a top ological G -mo d ule, then H S 2 c ( G, A ) = 0 b y Prop osition 3.2. SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 11 Remark 3.5. Let G b e a topological group and A b e a top ological G -mod ule. Restriction to the u n derlying abstract group structure gives, for eac h n ≥ 0, th e homomorphisms (with the same notatio n) r ∗ : H n c ( G, A ) → H n ( G, A ) and r ∗ : H S n c ( G, A ) → H S n ( G, A ) . It is easy to see that the follo wing d iagram is comm u tativ e H S 2 c ( G, A ) h ∗   r ∗ / / H S 2 ( G, A ) h ∗   H 2 c ( G, A ) r ∗ / / H 2 ( G, A ) . Note that both the v ertical maps are injectiv e by P rop osition 3.2 and [24, Lemma 3.1]. When G an d A are locally compact groups, then Mo ore [19] defined a cohomology theory H ∗ m ( G, A ) with measurable co c hains. He sho w ed that if G is p erfect, then the restriction map H 2 m ( G, A ) → H 2 ( G, A ) is injectiv e [19, Theorem 2.3]. F u rther, he sh o w ed that, if G is a profinite group and A a d iscrete G -mo d ule or G is a Lie group and A a fi nite dimensional G -v ector sp ace, then the restriction map H 2 c ( G, A ) ∼ = → H 2 m ( G, A ) is an isomorphism [18, p.32]. When the group G is p erfect, com bining the ab ov e t wo results of Mo ore sho ws that the restriction map r ∗ : H 2 c ( G, A ) → H 2 ( G, A ) is injectiv e. This to gether with the injectivit y of the v ertical maps h ∗ in the abov e commuta- tiv e diagram prov es the follo wing: L et G b e a p erfect group satisfying either of the follo wing conditions: (i) G is a pr ofinite group and A a discrete G -module; (ii) G is a Lie group and A a fi n ite dimensional G -v ector sp ace. Then the restriction map r ∗ : H S 2 c ( G, A ) → H S 2 ( G, A ) is injectiv e. W e s h all sho w in Example 4.4 that this map is not injectiv e in general. 4. Some examples In this section, we giv e some examples to illustrate the s y m metric con tinuous cohomology in tro du ced in the previous section. Example 4.1. W e giv e an example of a top ological group and a top olog ical mod u le for whic h the t w o cohomology theories H S ∗ c ( − , − ) and H ∗ c ( − , − ) are different. More p r ecisely , we sho w that, the map h ∗ : H S 2 c ( − , − ) → H 2 c ( − , − ) is not su rjectiv e in general. 12 MAHENDER SINGH First consider the extension 0 → Z i → Z × Z / 2 π → Z / 4 → 0 , where i ( n ) = (2 n, n ) and π ( n, m ) = n + 2 m . Here n denotes the class of n mo d ulo 2 or 4 dep end ing on the context. L et s : Z / 4 → Z × Z / 2 b e the section giv en by s ( 0) = (0 , 0 ) , s (1 ) = ( − 1 , 1 ) , s (2) = (0 , 1) and s (3) = (1 , 1) . Equippin g eac h group with the discrete top ology , w e can consider this as an extension of topo- logica l groups. Then s is clearly a symm etic con tin uous section. L et G b e a non-discrete ab elian top ological group. C onsider the extension 0 → Z × G i ′ → Z × Z / 2 × G × G π ′ → Z / 4 × G → 0 , (6) where i ′ ( n, g ) = ( i ( n ) , g , 0) and π ′ ( n, m, g , h ) = ( π ( n, m ) , h ). T h is is a non-split extension of top ologi cal group s. Note that the extension also admits a symmetric con tinuous section s ′ : Z / 4 × G → Z × Z / 2 × G × G giv en b y s ′ ( n, h ) = ( s ( n ) , 0 , h ) . Therefore, (6) r epresen ts a u nique n on-trivial element in H S 2 c ( Z / 4 × G, Z × G ). Next consider the extension 0 → Z j → Z ν → Z / 4 → 0 , where j ( n ) = 4 n and ν ( n ) = n . With the discrete top olog y , w e can consider this as an extension of top olog ical group s. This extension do es not admit any symmetric con tin uous section. F or an y non-discrete ab elian top ological group G , we get the follo wing non-split extension of top ologic al groups 0 → Z × G j ′ → Z × G × G ν ′ → Z / 4 × G → 0 , (7) where j ′ ( n, g ) = ( j ( n ) , g , 0) and ν ′ ( n, g , h ) = ( ν ( n ) , h ). Let s : Z / 4 → Z b e any cont inuous section, w h ic h exists as the top ologies are discrete. Then the sectio n s ′ : Z / 4 × G → Z × G × G giv en b y s ′ ( n, h ) = ( s ( n ) , 0 , h ) is con tin uous . Ho wev er, there do es not exist any symmetric con tin uous section. Th erefore, (7) represent s a u nique non-trivial elemen t in H 2 c ( Z / 4 × G, Z × G ), b ut do es n ot represent an element in H S 2 c ( Z / 4 × G, Z × G ). Example 4.2. W e no w giv e an example wh ic h is sp ecific to the con tinuous case . Let H 3 ( R ) =      1 x z 0 1 y 0 0 1         x, y , z ∈ R    SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 13 b e the 3-dimensional real Heisen b erg grou p . Note that H 3 ( R ) is a non-ab elian top ological group (in f act a Lie group) with resp ect to matrix m ultiplication. Let A =      1 0 z 0 1 0 0 0 1         z ∈ R    b e the cen ter of H 3 ( R ). Th en A ∼ = R and H 3 ( R ) / A ∼ = R 2 as a top ological group . T his giv es an extension of top ological groups 0 → R → H 3 ( R ) → R 2 → 0 . (8) The extension is n on-split as an extension of to p ologic al groups as this w ould mak e the group H 3 ( R ) to b e ab elian. How eve r, the extension admits a section s : R 2 → H 3 ( R ) give n b y s ( x, y ) =   1 x xy 2 0 1 y 0 0 1   whic h is contin u ou s and symmetric, since s ( − x, − y ) =   1 − x xy 2 0 1 − y 0 0 1   = s ( x, y ) − 1 . Therefore, (8) r epresen ts a u nique n on-trivial element in H S 2 c ( R 2 , R ). Example 4.3. W e giv e an example of a top ological group and a top olog ical mod u le for whic h H S 2 c ( − , − ) ∼ = H 2 c ( − , − ). Examples of (ab elian) top ological group extensions admitting a symmetric cont inuous section are guaran teed b y a w ell kno wn result of Mic h ael [15, Prop osition 7.2], whic h states that: If X and Y are real or complex Banac h spaces rega rd ed as topological groups with resp ect to their addition and π : X → Y is a surjectiv e con tinuous linear transformation, th en there exists a con tin uous map s : Y → X suc h that π s ( y ) = y and s ( − y ) = − s ( y ) for all y ∈ Y . Let Y and A b e infinite dimensional real or complex Banac h spaces. C onsider A as a trivial Y -mo d ule. If 0 → A → X → Y → 0 is any extension of Banac h spaces regarded as an extension of topological group s, then the map X → Y alw a ys admits a symmetric co ntin u ous section by the ab o v e men tioned r esult of Mic hael. Th is together with Prop osition 3.2 s h o ws that H S 2 c ( Y , A ) ∼ = H 2 c ( Y , A ). Example 4.4. Let G b e a top olog ical group and A b e a top olog ical G -mo du le. As announced in the p revious section, w e giv e an example to sho w that the h omomorphism r ∗ : H S 2 c ( G, A ) → H S 2 ( G, A ) is n ot injectiv e in general. Let X b e an in finite d im en sional real or complex Banac h space and let A b e a non-complemented subspace of X . Th e q u otien t map X → X/ A ad m its a symmetric contin u ous section by the ab o v e men tioned result of Michae l. Since A is n on-complemen ted in X , the extension is non-split as an 14 MAHENDER SINGH extension of top ological groups. But, X is isomorphic to A × X/ A as an ab elian group and the extension is split as an extension of abstract groups. Hence r ∗ : H S 2 c ( X/ A, A ) → H S 2 ( X/ A, A ) is n ot injectiv e. Let us consider a p articular example. Let k b e the fi eld of real or complex num b ers and ℓ ∞ b e the space of all b ounded sequences x = ( x n ) ∞ n =1 , wh ere x n ∈ k for eac h n ≥ 1 . Note that ℓ ∞ is a Banac h space with resp ect to the n orm k x k ∞ = sup n | x n | . Let c 0 b e the su bspace of ℓ ∞ consisting of all sequences whose limit is zero. This is a closed subs pace of ℓ ∞ and hence a Banac h space. By a w ell-known resu lt due to Phillips [5, p.33, Corolla ry 4], c 0 is a non-complemen ted subspace of ℓ ∞ and hence 0 → c 0 → ℓ ∞ → ℓ ∞ /c 0 → 0 is a n on-split extension o f top ological groups. W e would lik e to mentio n that there is a general metho d of constructing non-split extensions of Banac h spaces due to K alton and Pec k [14]. Example 4.5. W e no w g ive an example to s h o w that the restriction homomorphism r ∗ is not surjectiv e in general. Consider the 3-dimens ional real Heisen b erg group H 3 ( R ) as an abstract group. Consider the cen ter A ∼ = R of H 3 ( R ) as a top ological group with the discrete top ology and consider H 3 ( R ) / A ∼ = R 2 as a top ological group with the usu al top ology . Then r egarding 0 → R → H 3 ( R ) → R 2 → 0 (9) as an extension of abstr act groups, we see that it is non-split and admits a symmetric section s : R 2 → H 3 ( R ). Th us, the extension (9) represents a non-trivial element in H S 2 ( R 2 , R ). Supp ose that, there is a top ology on H 3 ( R ) making (9) int o an extension of top ological groups admitting a symmetric con tinuous section and ind ucing the underlyin g abstract group extension. Then H 3 ( R ) is a topological group with the pro du ct top ology R 2 × R . In particular, the map I : H 3 ( R ) → H 3 ( R ) sending eac h m atrix to its inv erse m ust b e con tinuous. Bu t this is not tr u e. Consider th e op en s et U =      1 x 0 0 1 y 0 0 1         x, y ∈ ( − 1 , 1)    in H 3 ( R ). Then I − 1 ( U ) =      1 x z 0 1 y 0 0 1           1 − x xy − z 0 1 − y 0 0 1   ∈ U    =      1 x z 0 1 y 0 0 1         − x, − y ∈ ( − 1 , 1) and xy = z    =      1 x xy 0 1 y 0 0 1         x, y ∈ ( − 1 , 1)    SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 15 is not op en in H 3 ( R ). Hence, the element repr esen ted b y (9) in H S 2 ( R 2 , R ) has no pre-image in H S 2 c ( R 2 , R ). 5. Pr op er ties of s ymmetric continuous co homology Let G a nd G ′ b e to p ologic al groups . Let A b e a topological G -module and A ′ a top ologic al G ′ -mo dule. W e sa y that a pair ( α, β ) of con tinuous group homomorphisms α : G ′ → G and β : A → A ′ is compatible if the follo win g diag ram comm utes G × A − − − − → A α x     y β   y β G ′ × A ′ − − − − → A ′ . In other w ords, g ′ β ( a ) = β ( α ( g ′ ) a ) for all a ∈ A and g ′ ∈ G . F or simplicit y , we write ψ = ( α, β ). Under th ese conditions, w e ha ve the follo w ing prop osition. Prop osition 5.1. There is a h omomorphism of cohomology groups ψ n : H S n c ( G, A ) → H S n c ( G ′ , A ′ ) for eac h n ≥ 0. Pr o of. Fix n ≥ 0. F or eac h σ ∈ C S n c ( G, A ), define σ ′ : G ′ n → A ′ b y σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) = β  σ ( α ( g ′ 1 ) , α ( g ′ 2 ) , α ( g ′ 3 ) , ..., α ( g ′ n ))  for all ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) ∈ G ′ n . Clearly σ ′ is con tin uous b eing comp osite of con tin uous maps . Next w e sho w that σ ′ is symmetric. W e ha ve τ 1 σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) = − g ′ 1 σ ′  g ′ 1 − 1 , g ′ 1 g ′ 2 , g ′ 3 , ..., g ′ n  = − g ′ 1 β  σ ( α ( g ′ 1 − 1 ) , α ( g ′ 1 g ′ 2 ) , α ( g ′ 3 ) , ..., α ( g ′ n ))  = β  − α ( g ′ 1 ) σ ( α ( g ′ 1 ) − 1 , α ( g ′ 1 ) α ( g ′ 2 ) , α ( g ′ 3 ) , ..., α ( g ′ n ))  b y compatibilit y = β  σ ( α ( g ′ 1 ) , α ( g ′ 2 ) , α ( g ′ 3 ) , ..., α ( g ′ n ))  b y τ 1 σ = σ = σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) . F or 2 ≤ i ≤ n − 1, w e ha ve τ i σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) = − σ ′  g ′ 1 , ..., g ′ i − 2 , g ′ i − 1 g ′ i , g ′ i − 1 , g ′ i g ′ i +1 , g ′ i +2 , ..., g ′ n  = − β  σ ( α ( g ′ 1 ) , ..., α ( g ′ i − 2 ) , α ( g ′ i − 1 g ′ i ) , α ( g ′ i − 1 ) , α ( g ′ i g ′ i +1 ) , α ( g ′ i +2 ) , ..., α ( g ′ n ))  = β  − σ ( α ( g ′ 1 ) , ..., α ( g ′ i − 2 ) , α ( g ′ i − 1 ) α ( g ′ i ) , α ( g ′ i ) − 1 , α ( g ′ i ) α ( g ′ i +1 ) , α ( g ′ i +2 ) , ..., α ( g ′ n ))  = β  σ ( α ( g ′ 1 ) , α ( g ′ 2 ) , α ( g ′ 3 ) , ..., α ( g ′ n ))  b y τ i σ = σ = σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) . Similarly , we can see that τ n σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ) = σ ′ ( g ′ 1 , g ′ 2 , g ′ 3 , ..., g ′ n ). Th is sho ws that σ ′ is symmetric, and h ence an elemen t of C S n c ( G ′ , A ′ ). 16 MAHENDER SINGH Define ψ n : C S n c ( G, A ) → C S n c ( G ′ , A ′ ) b y ψ n ( σ ) = σ ′ . It it routine to c hec k that ψ n is a homomorphism comm uting with the cob ound ary op erators, that is, the follo w ing d iagram comm utes C S n c ( G, A ) ∂ n   ψ n / / C S n c ( G ′ , A ′ ) ∂ n   C S n +1 c ( G, A ) ψ n +1 / / C S n +1 c ( G ′ , A ′ ) . This sh o ws that ψ n preserve s b oth cycles and b oun daries and hence defines a map ψ n : H S n c ( G, A ) → H S n c ( G ′ , A ′ ) giv en b y ψ n ([ σ ]) = [ σ ′ ] . It is again routine to chec k that ψ n is a h omomorphism. This completes the pro of.  In particular, for G = G ′ and α = id G , we ha ve a homomorph ism β ∗ : H S ∗ c ( G, A ) → H S ∗ c ( G, A ′ ). The f ollo win g is an immediate consequence of Prop osition 5.1. Corollary 5.2. The follo wing statement s h old: (1) Let H b e a sub group of G . Then th e compatible pair of homomorphisms, the inclu- sion map H ֒ → G and the iden tit y map A → A , gives the restriction homomorphism H S ∗ c ( G, A ) → H S ∗ c ( H , A ). (2) Let H b e a normal subgroup of G . Then the compatible pair of homomorphisms, the quotien t map G → G/H and the inclus ion map A H ֒ → A , giv es the inflation homomor- phism H S ∗ c ( G/H, A H ) → H S ∗ c ( G, A ). W e also ha v e, a long exact sequence in cohomolo gy asso ciated to a short exact sequence of top ological G -mo dules admitting a sym m etric con tin uous sec tion whic h is co mp atible with th e actions. Prop osition 5 .3. Let 0 → A ′ i → A j → A ′′ → 0 b e a short exact sequ en ce of top ological G - mo dules admitting a symmetric con tinuous section which is compatible with the actions. Then there is a long exact sequ ence of symmetric con tinuous cohomology groups, · · · → H S n c ( G, A ′ ) i n → H S n c ( G, A ) j n → H S n c ( G, A ′′ ) δ → H S n +1 c ( G, A ′ ) → · · · . Pr o of. W e fi rst sho w th at, for eac h n ≥ 0, there is a short exact sequence of s y m metric con tinuous co c hain groups 0 → C S n c ( G, A ′ ) i n → C S n c ( G, A ) j n → C S n c ( G, A ′′ ) → 0 . Let σ ∈ C S n c ( G, A ′ ) b e suc h that i n ( σ ) = 0, th at is, i ( σ ( g 1 , ..., g n )) = 0 for all ( g 1 , ..., g n ) ∈ G n . But injectivit y of i implies that σ ( g 1 , ..., g n ) = 0 for all ( g 1 , ..., g n ) ∈ G n . Hence σ = 0 and i n is injectiv e. Since j i = 0, we h av e j n i n = 0 and hence Im ( i n ) ⊆ Ker( j n ). Su pp ose σ ∈ Ker( j n ), that is, j ( σ ( g 1 , ..., g n )) = 0. Th is implies σ ( g 1 , ..., g n ) ∈ Ker( j ) = I m ( i ). But i : A ′ → Im( i ) is a SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 17 homeomorphism and hence has a con tinuous inv ers e i − 1 : Im( i ) → A ′ . T aking µ = i − 1 σ , we ha v e i n ( µ ) = σ and hence Ker( j n ) ⊆ Im( i n ). Next w e show that j n is surjectiv e. Let σ ∈ C S n c ( G, A ′′ ) and let s : A ′′ → A b e a s y m metric con tin uous section wh ic h is compatible with the actions. T aking µ = sσ , we see that µ is con tin uous and j n ( µ ) = j ( sσ ) = σ . It remains to chec k th at µ is symmetric. W e ha v e τ 1 µ ( g 1 , g 2 , g 3 , ..., g n ) = − g 1 µ ( g − 1 1 , g 1 g 2 , g 3 , ..., g n ) = − g 1 s  σ ( g − 1 1 , g 1 g 2 , g 3 , ..., g n )  = − s  g 1 σ ( g − 1 1 , g 1 g 2 , g 3 , ..., g n )  b y compatibilit y of s = − s  − ( − g 1 σ ( g − 1 1 , g 1 g 2 , g 3 , ..., g n ))  = − s  − σ ( g 1 , g 2 , g 3 , ..., g n )  b y τ 1 σ = σ = s  σ ( g 1 , g 2 , g 3 , ..., g n )  b y symmetry of s = µ ( g 1 , g 2 , g 3 , ..., g n ) . F or 2 ≤ i ≤ n − 1, w e ha ve τ i µ ( g 1 , g 2 , g 3 , ..., g n ) = − µ ( g 1 , ..., g i − 2 , g i − 1 g i , g − 1 i , g i g i +1 , g i +2 , ..., g n ) = − s  σ ( g 1 , ..., g i − 2 , g i − 1 g i , g − 1 i , g i g i +1 , g i +2 , ..., g n )  = − s  − ( − σ ( g 1 , ..., g i − 2 , g i − 1 g i , g − 1 i , g i g i +1 , g i +2 , ..., g n ))  = − s  − σ ( g 1 , g 2 , g 3 , ..., g n )  ) by τ i σ = σ = s  σ ( g 1 , g 2 , g 3 , ..., g n )  b y symmetry of s = µ ( g 1 , g 2 , g 3 , ..., g n ) . Similarly , one can sh o w that τ n µ ( g 1 , g 2 , g 3 , ..., g n ) = µ ( g 1 , g 2 , g 3 , ..., g n ) . Hence j n is surjectiv e. Note that only th e surjectivit y of j n dep end s on the c hoice of th e section. As in the previous prop osition, the maps i ∗ and j ∗ comm ute with the cob oundary op erators and hence we get th e follo w ing short exact sequence of symmetric con tinuous co chain complexes 0 → C S ∗ c ( G, A ′ ) i ∗ → C S ∗ c ( G, A ) j ∗ → C S ∗ c ( G, A ′′ ) → 0 . It is no w routine to obtain the d esired long exact sequence of symmetric con tin uous cohomol- ogy grou p s b y a diagram c hase. This completes the pro of.  6. Symmetric continuous cohomology o f p rofinite groups Profinite groups form a sp ecial class of top ological groups . W e r efer the r eader to [21] f or basic defin itions and r esu lts regarding profinite groups . Th e contin u ous cohomology of a profinite group with co efficien ts in a discrete mo du le is w ell studied and equals the dir ect limit of the cohomology of finite grou p s. W e prov e the follo win g similar result for symmetric con tinuous cohomology . 18 MAHENDER SINGH Theorem 6.1. The symmetric c ontinuous c ohomolo gy of a pr ofinite gr oup with c o efficients in a discr ete mo dule e quals the dir e ct limit of the symmetric c ohomolo gy of finite gr oups. W e n o w set notations for th e pro of of Theorem 6.1. Let G b e a profinite group and A a discrete G -mo dule. Let U b e the set of all op en normal subgroup s of G . It can be pr ov ed that for eac h U ∈ U , the qu otien t group G/U is finite. Also, for eac h U ∈ U , the group of in v ariants A U = { a ∈ A | ua = a for all u ∈ U } is a G/U -mo du le b y means of the action ( g U, a ) 7→ g a for g U ∈ G/U and a ∈ A. F or elements U, V ∈ U , w e sa y that V ≤ U if U is a subgroup of V . This makes U a directed p oset. F or V ≤ U , there are canonical homomorphisms α U V : G/U → G/V and β V U : A V → A U whic h form compatible p airs and giv es rise to an in v erse s y s tem of finite groups { G/U } U ∈U and a direct system of ab elian group s { A U } U ∈U . I t is then w ell known that G = lim ← − G/U and A = lim − → A U . F urther, for eac h n ≥ 0 and eac h V ≤ U , as in Prop osition 5.1, the compatible pair of homomorphisms ( α U V , β V U ) ind uces a homomorph ism ψ n V U : C S n ( G/V , A V ) → C S n ( G/U, A U ) . Th us , we obtain in a natural w a y the follo win g direct systems of ab elian groups o ver U : { C S n ( G/U, A U ) } U ∈U and { H S n ( G/U, A U ) } U ∈U . Note that, for eac h n ≥ 0, the cob oundary op erator ∂ n U : C S n ( G/U, A U ) → C n +1 ( G/U, A U ) comm utes with the b onding maps ψ n V U and hence giv es a cob oundary op erator ∂ n : lim − → C S n ( G/U, A U ) → lim − → C S n +1 ( G/U, A U ) making { lim − → C S n ( G/U, A U ) , ∂ n } into a co c hain complex. T o pr o v e Theorem 6.1, it suffices to prov e the follo wing lemma, w hic h is essen tially [21, Lemma 6.5.4]. Lemma 6.2. L et G b e a pr ofinite gr oup and A a discr e te G -mo dule. Then for e ach n ≥ 0 , ther e is an isomo rphism lim − → C S n ( G/U, A U ) ∼ = C S n c ( G, A ) c ommuting with the c orr esp onding c ob oundary op er ators. SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 19 Pr o of. The p ro of is s ame as that of [21, Lemma 6.5.4] and we outline it b riefly for the con v enience of the readers. Fix n ≥ 0. F or eac h U ∈ U , let α U : G → G/U and β U : A U → A b e the obvious homomorphisms. Note that β U ( α U ( g ) a ) = β U ( g a ) = g a = g β U ( a ) for all g ∈ G and a ∈ A U . Th us, the p air ( α U , β U ) is compatible. Let ψ n U : C S n c ( G/U, A U ) → C S n c ( G, A ) b e the homomorph ism in duced b y the compatible pair ( α U , β U ) as in Prop osition 5.1. Note that this also comm utes with the cob oun dary op erators. Considering b oth G/U and A U equipp ed with discrete top ology , we hav e C S n c ( G/U, A U ) = C S n ( G/U, A U ). Therefore ψ n U : C S n ( G/U, A U ) → C S n c ( G, A ) . F or elemen ts U, V ∈ U with V ≤ U , by definitions, the follo win g diagram comm utes C S n ( G/V , A V ) ψ n V U   ψ n V / / C S n c ( G, A ) C S n ( G/U, A U ) . ψ n U 6 6 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ Hence, th ere is a homomorphism ψ n : lim − → C S n ( G/U, A U ) → C S n c ( G, A ) giv en b y ψ n ([ σ U ]) = ψ n U ( σ U ) for σ U ∈ C S n ( G/U, A U ) . The pro of of the bijectivit y of ψ n is routine as in [21, Lemma 6.5.4] . The comm utativit y of the homomorphisms ψ n with the co b oundary operators is immediate from the definitions and the form ula (1). This pro v es the lemma.  Note that lim − → is an exact functor and hence we obtain H S n c ( G, A ) = H n ( C S ∗ c ( G, A )) = H n (lim − → C S ∗ ( G/U, A U )) = lim − → H n ( C S ∗ ( G/U, A U )) = lim − → H S n ( G/U, A U ) . This completes the pro of of th e Th eorem 6.1. 20 MAHENDER SINGH 7. Symmetric smooth cohomology of Lie groups The theory of Lie group s, p articularly cohomology of Lie groups, has b een studied fr om differen t p oints of view. V arious cohomology th eories of Lie groups ha ve b een constructed in the literature [11, 12, 13, 26, 20]. T here is a r ic h in terpla y b et w een the con tinuous cohomology of a Lie group, the cohomology of its Lie algebra and the de Rham cohomology of its asso ciated symmetric space [2, 25]. There is a w ell kno w n theo ry of smooth cohomology of a Lie group G with coefficien ts in a top ological ve ctor space V on wh ic h G acts smo othly . This th eory w as d efined b y Blanc [1 ] and was later extended by Brylinski [4] to coefficien ts in an arbitrary ab elian Lie group. In this section, we d efine the sym metric smo oth cohomology of a Lie group and pro ve some basic prop erties as we did for top ologic al groups. Let G b e a Lie group and A b e a smo oth G -mo dule. W e can defin e an analogous cohomolog y theory by imp osing the condition that the standard co c hains are symmetric and smo oth. More precisely , for eac h n ≥ 0, let C n s ( G, A ) b e the group of all smooth maps from the pr o duct Lie group G n → A and let the cob oundary b e giv en b y the s tand ard form ula as in (1). Analogous to the constru ction in the con tinuous case, f or eac h n ≥ 0, consider the action of the symmetric group Σ n +1 on C n s ( G, A ) as giv en b y equations (2). Th e smo othness of the action of G on A implies th at the actio n is w ell-defined. As in the con tin uou s case, the action is compatible with the stand ard coboun dary op erators ∂ n and hence gives the sub complex of in v ariants { C S n s ( G, A ) , ∂ n } n ≥ 0 = { C n s ( G, A ) Σ n +1 , ∂ n } n ≥ 0 . W e d efine the symmetric smooth cohomol ogy H S n s ( G, A ) to b e the cohomology groups o f this new coc hain complex. W e obtain some basic prop erties of this cohomology theory as follo ws. Prop osition 7.1. Let G b e a Lie group and A b e a smo oth G -modu le. Th en we hav e the follo w ing: (1) H S 0 s ( G, A ) = A G . (2) Z S 1 s ( G, A ) = the group of symmetric smooth crossed homomorphisms from G to A . (3) The map h ∗ : H S 2 s ( G, A ) → H 2 s ( G, A ) is injectiv e. (4) Let A b e a G -mo dule and A ′ b e a G ′ -mo dule suc h that the actions are compatible. Then there is a h omomorphism of cohomology groups H S n s ( G, A ) → H S n s ( G ′ , A ′ ) for eac h n ≥ 0. (5) Let 0 → A ′ i → A j → A ′′ → 0 b e a sh ort exact sequence of smo oth G - mo d ules admitting a symmetric s mo oth secti on w hic h is compatible with the actions. Then there is a long exact sequence of symmetric smo oth cohomol ogy groups · · · → H S n s ( G, A ′ ) i n → H S n s ( G, A ) j n → H S n s ( G, A ′′ ) δ → H S n +1 s ( G, A ′ ) → · · · . Pr o of. W e lea ve the p ro ofs to the reader as th ey are similar to those of the con tin uous case.  SYMMETRIC CONTINUOUS COHOMOLOGY OF TOPOLOGICAL GROUPS 21 As in the con tinuous case, we w ould like to ha v e an in terpretation of the symmetric smo oth cohomology in dimension t w o. F or that purp ose, w e recall that, an extension of Lie group s 0 → A i → E π → G → 1 is an algebraic sh ort exact sequence of Lie groups w ith th e additional prop ert y that b oth i and π are smooth homomorphisms and π admits a smo oth lo cal section s : U → E , where U ⊂ G is an op en neig hb ourho o d of iden tity . T he existence o f a smo oth lo cal sectio n means that E is a principal A -bundle o v er G w ith resp ect to the left action of A on E giv en by ( a, e ) 7→ i ( a ) e for a ∈ A and e ∈ E . Sin ce, an extension of Lie groups is a principal bund le, it follo ws th at it is a trivial bundle ( E is A × G as a smo oth manifold) if and only if it admits a smo oth sectio n. Tw o extensions of Lie groups 0 → A i → E π → G → 1 and 0 → A i ′ → E ′ π ′ → G → 1 are s aid to b e equiv alent if there exists a smo oth isomorphism φ : E → E ′ with s m o oth in v erse suc h th at the follo wing diag ram comm ute 0 / / A i / / E π / / φ   G / / 1 0 / / A i ′ / / E ′ π ′ / / G / / 1 . Let S ( G, A ) denote the set of equiv alence classes of Lie group extensions of G b y A admitting a symmetric smooth section and corresp ond ing to the given w a y in whic h G acts on A . Suc h extensions are classified b y the second symmetric smo oth cohomology as follo ws. Theorem 7.2. L et G b e a Lie gr oup and A b e a smo oth G -mo dule. Then ther e is a bije ction Ψ : S ( G, A ) → H S 2 s ( G, A ) Pr o of. W e lea ve the pro of to the reader as it is similar to that of the con tinuous case.  Note that countable grou p s with the discrete top ology are 0-dimens ional Lie groups. T aking G to b e an ab elian Lie group of p ositiv e dimension in the Example 4.1, we ha v e an example of a p ositiv e dimensional Lie group an d a smo oth mo d ule for wh ic h the tw o cohomology theories H S ∗ s ( − , − ) and H ∗ s ( − , − ) are different. Similarly , Example 4.2 also serv es as an example for the Lie group case. Note that an extension of Lie group s 0 → A i → E π → G → 1 can b e thought of as an extension of top ologica l groups b y considering only the u nderlying top ological group structure ( i b ecomes closed con tinuous and π b ecomes op en con tinuous). Th is giv es the restriction homomorphism r ∗ : H S n s ( G, A ) → H S n c ( G, A ) for eac h n ≥ 0 . W e inv estigate this homomorphism in d im en sion tw o. Before th at, w e recall Hilb ert’s fi fth problem, which ask ed: Is ev ery locally E uclidean top ological group necessarily a Lie group ? It is well kno wn that Hilb ert’s fifth problem has a p ositiv e solution [9, 17, 28]. W e use this in the follo w ing concluding theorem. Theorem 7.3. L et G b e a Lie gr oup and A b e a smo oth G -mo dule. Then the natur al hom o- morphism r ∗ : H S 2 s ( G, A ) → H S 2 c ( G, A ) 22 MAHENDER SINGH is an isomo rphism. Pr o of. Let [ σ ] ∈ H S 2 s ( G, A ) and let 0 → A → E → G → 1 b e an extension of Lie groups corresp ondin g to [ σ ] by Theorem 7.2, wh ic h is unique u p to equiv alence of extensions. Su pp ose that r ∗ ([ σ ]) is trivial in H S 2 c ( G, A ). Then there exists a con tin uous section s : G → E which is a group homomorphism. This giv es a con tinuous isomorphism b et we en th e Lie groups E and A ⋊ G . A con tinuous homomorp hism betw een Lie groups is smo oth [16, Theorem 4.21]. As a consequence this isomorphism is smo oth. Hence the cohomology class [ σ ] is trivial in H S 2 s ( G, A ) and the homomorph ism r ∗ is in jectiv e. Let [ σ ] ∈ H S 2 c ( G, A ) and let 0 → A i → E σ π → G → 1 b e the extension of top ologica l group s defined using the 2-co cycle σ as in the pr o of of Theorem 3.3. Th e extension admits a symm etric con tin uous section s : G → E σ giv en by s ( g ) = (0 , g ) for all g ∈ G . By constru ction, E σ is A × G as a top ologica l sp ace. Also, E σ is locally Euclidean as b oth A and G are Lie groups. Hence, w e conclude that E σ is a Lie group b y the p ositiv e solution to Hilb ert’s fifth problem. Since a contin uou s homomorphism b et ween Lie group s is smo oth, w e hav e that b oth i and π are smo oth homomorphisms . Ap plying the implicit function theorem, we can fin d a smo oth section of π defined in a neighbour ho o d of iden tit y in G . This sho ws that 0 → A → E σ → G → 1 is an extension of Lie groups. F urther, the section s : G → E σ b ecomes smo oth as E σ has the pro du ct smo oth structure. 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