A parametrized version of the Borsuk Ulam theorem

A parameterized v ersion of the Borsuk-Ulam theorem Thomas Sc hic k Mathematisc hes Institut Georg-August-Unive rsit¨ at Bunsenstr. 3 37073 G¨ ottingen, German y sc hick@uni-math.gwdg.de Rob ert Sam uel Simon Departmen t of Mat hematics LSE Hough ton Street London, W C2A 2AE, UK r.s.simon@lse.a c.uk Stanisla w Spie ˙ z IMP AN ul. Sniadec kic h 8 P .O. Bo x 21 00-956 W arsza w a, Poland spiez@impan.pl Henryk T oru ´ nczyk F aculty of Mathematics Univ ersit y of W arsaw Banac ha 2 02-097 W arsza w a, Poland torunczy@mimu w.edu.pl Octob er 28 , 2018 Abstract W e sho w that for a “con tinuous” family of Borsuk-Ulam situations, parameterized by points of a compact manifol d W , its solution set also dep ends “contin uously” on th e parameter space W . “Con tinuit y” here means that the solution set supp orts a homology class which maps onto the fun damen tal class of W . When W ⊂ R m +1 w e also show how to construct s uch a “contin uous” family starting from a family dep ending in the same wa y contin uously on th e p oints of ∂ W . This solves a problem related to a conjecture whic h is relev an t for the construction of equilibrium strategies in repeated tw o-play er games with incomplete information. A new metho d (of independ en t in terest) u sed in this context is a canonical symmetric squaring construction in ˇ Cec h homology with Z / 2- coefficients. MSC: 91A05, 55N45, 55N05, 55N91 1 In tro duction The classica l Borsuk-Ula m theorem s tates that for every cont inuous map f : S n → R n there e xists v ∈ S n such that f ( v ) = f ( − v ). W e think of { v ∈ S n | f ( v ) = f ( − v ) } as the set o f solutions to the Bo rsuk-Ulam equation. The origi- nal B orsuk-Ulam theorem s tates that this set is non-empty . There are many gener alizations of the B orsuk-Ulam theor em. In this paper, we generaliz e to contin uous families of Borsuk-Ula m situations and study the 1 2 Thomas Schic k, Rob ert S. Simon, Stanislaw Spie ˙ z, Henr y k T o r u´ nczyk global structure of the solution set (and its depe ndence on the parameters ). A t the sa me time, we generalize to cor respondence s, i.e. “multi-v alued functions” . More precisely , our “family version” of the Bors uk-Ulam equa tion starts with a contin uous map F : W × S n → R n and asks ab out prop erties of the s olution set B := { ( w , v ) ∈ W × S n | F ( w, v ) = F ( w, − v ) } . In pa rticular, we establish a suita ble kind of contin uous dep endence of the solution sets B w := { v ∈ S n | F ( w, v ) = F ( w, − v ) } o n w ∈ W . Of cour se, this has to b e made precise in a suitable wa y . Our ma in result, Theo rem 2.4, implies that the solution set B (as subs et of W × S n ) s upports a ( ˇ Cech) homology class whic h is mapped to the fundamental clas s if W is a co mpact manifold. If W = [0 , 1] this means essentially that there is a path s : [0 , 1] → B with s (0 ) ∈ { 0 } × S n and s (1) ∈ { 1 } × S n . The homo logical sta temen t is the c orrect g eneralization if W is an arbitrar y manifold. [3, Theorem 3.1] is a similar family Borsuk- Ulam theor em to ours , with the precur sor [17, Theorem 4 .2]. Note that the a ssumptions in these works are q uite different to ours , and so a re the metho ds. W e show then tha t a special ca se of a co n tinuous family of Bo rsuk-Ulam situations arises as follows. Given is an n -dimensiona l manifold with bo undary , embedded into R n and o n its bo undary a co n tinuous function (or corresp on- dence) with v alues in R n − 1 . Each of its interior p oints can b e considered as a “midp oin t”, g iving rise to a corres ponding Bo rsuk-Ulam equa tion (that the function assumes equal v alues on opp osite p oints o f the bo undary , “ opposite” with res pect to the interior p oin t). W e establish in Theo rem 2.8 that this is indeed a contin uous family o f Borsuk-Ulam situations. As a consequence (The- orem 2 .9), the solution set has the (homological) con tinuit y pro perty alluded to ab o ve. This result is strongly reminiscen t to a (mor e complicated) Borsuk-Ula m t yp e sta temen t (Conjecture 5.5) which is par t of a program in game theor y . This program would establish the existence of equilibria in genera l rep eated t wo-play er games with incomplete information. W e describ e the game theor etic context in some detail in Section 5 . The pro of of our main Theor em 2.4 relies o n a new homo logical co nstruc- tion which we exp ect to b e of indep endent interest, describ ed in Section 3. It is a functorial symmetric sq uare: a transformation a 7→ a s : H k ( X ; Z / 2) → H 2 k ( X s ; Z / 2), where X s = ( X × X/ τ , ∆), τ : X × X → X × X interc hanges the factors, and ∆ is the image o f the diago nal. One s hould think of a s as one half of the K ¨ unneth pro duct a × a (defined on “ half of X × X ”). The ma in prop ert y we use (and establish) states that if X is a manifold and a its fundamental class, then a s is the fundamen tal class of the relative manifold X s . The pap er is orga nized a s follows: in Section 2 we sta te our version of the family Borsuk-Ula m theo rem and introduce the rele v ant notation. In Section 3 we dis cuss the symmetric squaring transformatio n. In Section 4.1 we prov e our Bor suk-Ulam t yp e theorem and in Section 5 we descr ibe the game theor etic background and motiv ation. 2 A parameterized Borsuk-Ulam theorem Below, H deno tes the ˇ Cech ho mology functor with Z / 2 coefficients; see a lso Section 3. A par ameterized version of the Borsuk-Ula m theor em 3 2.1 Definition. Let X b e a compact space and ( W , W 0 ) a compact pair. La ter, ( W , W 0 ) will b e a compact manifold with bo undary . (1) W e say that a map f : X → W is H -essential for ( W , W 0 ) if it induces a surjection H d ( X, f − 1 ( W 0 )) → H d ( W , W 0 ) for d = dim W , co mpare to [1]. (2) Let p : E → W be a fiber bundle, e.g. E = Y × W for some top ological space Y . Assume tha t X ⊂ E . W e say that X has pr op erty S for ( W, W 0 ) if the pro j ection p | X : X → W is H -essential fo r ( W , W 0 ). Here, S stands for “s panning”, compare with [14, 15]. (3) In ca se W is a manifold with b oundary ∂ W (p ossibly empty) or W is a compact subset of R n with topo logical b oundary ∂ W 6 = W in R n , then in the definitions ab ov e we sa y “for W ” in place of “for ( W , ∂ W )”. 2.2 R emark. Suppose W is a connected co mpact manifold with bo undary ∂ W . The top-degree homology group of ( W , ∂ W ) is generated by a single element, which we ca ll the fundamental class of the manifold W and denote [ W ]. Th us f being H -essential for ( W, ∂ W ) is equiv alent to the exis tence of a homolog y class α ∈ H ( X, f − 1 ( ∂ W )) s uc h that f ∗ ( α ) = [ W ]. W e say that such a class α witnesses the H -essentialit y of the map f . W e may also sp eak of an analogo usly defined witness ing of the prop erty S of X for ( W, ∂ W ). 2.3 Definition. Let S = S m denote the m –sphere. W e assume that W is a compact PL–manifold (b oundary admitted), and Z ⊂ W × S × R m is compact. W e asso ciate to Z its Borsuk-Ulam c orr esp ondenc e B ( Z ) ⊂ W × R m defined by ( w, e ) ∈ B ( Z ) ⇐ ⇒ ∃ x ∈ S : ( w , x, e ) ∈ Z and ( w , − x, e ) ∈ Z . The fir st of our tw o results o n corre spondences asser ts the following: 2.4 Theorem. If Z has pr op erty S for W × S , then B ( Z ) has it for W . 2.5 R emark. (1) Note that, if W = { pt } and Z is the g raph of a contin uous function S → R m , then the Borsuk-Ulam theorem sta tes that B ( Z ) is non- empt y , whence the name chosen fo r the Borsuk– Ulam corr espondence . (2) F or the s ame rea son, Theorem 2 .4 is a pa rameterized Bor suk-Ulam theo- rem. Indeed, it yields that theorem when W = { pt } , and, loosely speaking, it asser ts in g eneral the following: for a “contin uous” family of Bor suk- Ulam situations, para meterized b y a manifold W , its solution set dep ends contin uously on the parameters in the sense tha t it supp orts a homology class which hits the fundamental class of W (in par ticular, it surjects on to W ). With pr W denoting the pro jection to W we want to describe the construction of the clas s γ ∈ H ( B ( Z ) , pr − 1 W ( ∂ W ) ∩ B ( Z )) satisfying (pr W ) ∗ ( γ ) = [ W ] more precisely (which will of c ourse a lso be necessa ry to esta blish its prop erties). T o do this, we will use a relative squa ring construction in ˇ Cech ho mology . W e belie v e t hat this constructio n can b e useful in other co n texts and therefore deserves independent interest; it will b e de scribed in the Section 3. 4 Thomas Schic k, Rob ert S. Simon, Stanislaw Spie ˙ z, Henr y k T o r u´ nczyk 2.6 R emark. The theorem above remains tr ue when the manifold pair ( W, ∂ W ) is replaced b y a re lative PL–manifold , b y w hic h we mean a compact pair ( W, W 0 ) such that W \ W 0 is a PL-manifold. W e will sk ip the a rgument , which howev er will b e implicit in the pro of of the result b elow. W e get notable examples of such r elativ e manifo lds by taking W to b e a compact n –dimensional subset o f R n , n < ∞ , and W 0 to b e its top ological b oundary in R n . In our second theorem we assume that W is a compact subset o f R m +1 of co dimension zero, of cours e with a to pologic al b oundary in R m +1 , whic h w e denote ∂ W . W e keep denoting by S = S m the m -sphere. 2.7 Defini tion. The spheric al c orr esp ondenc e Z asso ciated to a compact set Y ⊂ ∂ W × E , where E is an a rbitrary space, is defined by Z := { ( w, x, e ) ∈ W ◦ × S × E | ( w + λx, e ) ∈ Y for some λ > 0 } . Here W ◦ is the interior of the manifold W with b oundary , o r the top ological int erio r of the subset of R n , resp ectiv ely . The ov er-line stands, as usual, for the closure, i.e. Z is the clo sure in W × S × E of the set o f points listed. If we think of Y as a multi-v alued map from ∂ W to E , Z is a cano nical “extension” to W using the con vex s tructure o f R m +1 and keeping track of the directions needed for the extension. It collects a ll Bor suk-Ulam equatio ns arising if the p oints in W ar e treated as “midp oin ts” o f W . 2.8 Theorem . If Y ⊂ ∂ W × E has pr op erty S for ( ∂ W, ∅ ) t hen the spheric al c orr esp ondenc e Z ⊂ W × S × E asso ciate d to Y as a b ove has p r op erty S fo r W × S . As a co nsequence, we immediately obtain fro m Theor em 2.8 and Theo rem 2.4 the following corolla ry . 2.9 Corollary . L et W ⊂ R n b e a c omp act n - dimensio nal emb e dde d manifold with b oundary and assume t hat Y ⊂ ∂ W × R n − 1 has pr op erty S for ( ∂ W , ∅ ) . Then Z := { ( x, v ) ∈ W × R n − 1 | ∃ x 1 , x 2 ∈ ∂ W : x ∈ [ x 1 , x 2 ] , ( x 1 , v ) , ( x 2 , v ) ∈ Y } satisfies pr op erty S for ( W , ∂ W ) . F o r earlier results r elated to Coro llary 2.9 see [6 , 10]. 3 ˇ Cec h homology and symmetric homology squar- ing Throughout this note, a ll manifolds are finite-dimens ional, p ossibly with a nonempty b oundary , a nd all spaces e ncoun tered will b e subspaces of manifolds. The homolo gy g roups we are using will exclusively b e ˇ Ce ch homolo gy groups. Their pr operties c an be found in [5, Chapters IX, X] a nd in [4, VII I,13]. W e list the mos t importa n t pr operties (not all o f them will b e rele v ant to us): (1) ˇ Cech homology is defined for compact pairs. (2) ˇ Cech homology satisfies excisio n in a very strong form: if f : ( X , A ) → ( Y , B ) is a map of compact pa irs such that f | : X \ A → Y \ B is a homeomorphism, then f ∗ : H ( X , A ) → H ( Y , B ) is an isomorphism of ˇ Cech homology g roups. A par ameterized version of the Borsuk-Ula m theor em 5 (3) More g enerally , t he Vietoris t he or em ab out maps with acyclic fibers holds: if f : ( X , A := f − 1 ( B )) → ( Y , B ) is a surjective map of compact pairs such that the r educed ˇ Cech homolo gy groups ˜ H ( f − 1 ( y )) are trivial for all y ∈ Y \ B , then f ∗ : H ( X, A ) → H ( Y , B ) is an isomorphism o f ˇ Cech homology g roups [16] (using [5, Theo rem 5.4]). (4) F or a Euclidea n neighbor hoo d retract (ENR), e.g. for a top ological mani- fold, ˇ Cech homology and sing ular homology a re canonically isomor phic. (5) Con tinuit y: The ˇ Cech homology functor co mm utes with taking an inverse limit. (6) F or ˇ Cech homolo gy , there is a na tural intersection pa iring: If ( X , A ) and ( Y , B ) a re tw o compact pair s ins ide an m - dimensional manifold W (pos - sibly with non-empty b oundary), then there is an intersection pair ing H p ( X, A ) ⊗ H q ( Y , B ) ∋ α ⊗ β 7→ α • β ∈ H p + q − m (( X, A ) ∩ ( Y , B )) which is natura l for inclusions o f pa irs. (Here, ( X, A ) ∩ ( Y , B ) := ( X ∩ Y , A ∩ Y ∪ X ∩ B ).) The reasoning in [4, p. 34 2] not only shows that the pro duct α • β can be computed in any g iv en neig h bo rhoo d of X ∩ Y , but actually that it ca n b e de fined whenever a neighborho o d o f X ∩ Y is a manifold (even if W isn’t). (7) F or c omp act subsets of manifolds, and with coe fficien ts in a field (e.g. Z / 2), ˇ Cech ho mology is a ho mology theory , in particular with a long exact se- quence of a pair. This do es not hold in gene ral for Z -co efficients. The main re ason how ever for which we work with Z / 2 -coefficients is to avoid orientabilit y assumptions. As in [15] these prop erties a llo w to define a certain restric tion op erator , as follows. Suppos e ( X , A ) and ( Y , B ) are compact pairs in a space Z . If X ∩ ( Y \ B ) is a relatively op en subset of X \ A then we say that ( Y , B ) is admissible for ( X, A ). In this ca se we hav e ca nonical homomorphisms H ( X , A ) → H ( X , X \ ( Y \ B )) → H ( X ∩ Y , X ∩ B ) the first of which is induced by the inclusion and the second is the excision of X \ Y from X . The ar ising comp osition H ( X , A ) → H ( X ∩ Y , X ∩ B ) will b e called r estriction and denoted α 7→ α | ( Y , B ) . Note that then an y other compa ct pair ( Y ′ , B ′ ) in Z such that X ∩ ( Y ′ , B ′ ) = X ∩ ( Y , B ) is admiss ible for ( X, A ), to o, and α | ( Y ′ , B ′ ) = α | ( Y , B ). F r om now on w e assume that H denotes the gr aded ˇ Cech homology functor with Z / 2 co efficients. Our a im is to construct a “s ymmetric squar ing”, similar to the squar ing H k ( X, A ) → H 2 k ( X × X , A × X ∪ X × A ) obtained fr om the exterior ho mology pro duct [4], but which takes v alues in the homology of the (slightly mo dified) s ymm et ric square of the pair ( X , A ). W e expla in this below. Let X ⊃ A b e compacta. On X × X we hav e the coo rdinate-switching inv olution τ . W e put ( X, A ) s =  X × X/ τ ,  ∆ ∪ ( A × X ) ∪ ( X × A )  /τ  and X s = ( X, ∅ ) s 6 Thomas Schic k, Rob ert S. Simon, Stanislaw Spie ˙ z, Henr y k T o r u´ nczyk where ∆ = { ( x, x ) : x ∈ X } is the diagonal. Clear ly , a map f : ( X , A ) → ( Y , B ) of compact pairs induces a map ( X, A ) s → ( Y , B ) s , denoted f s . (Here and below the sup erscript s stands for “symmetric square ”.) 3.1 Theorem. F or e ach k ther e is an assignment H k ( X, A ) ∋ α 7→ α s ∈ H 2 k (( X, A ) s ) such that (1) It is n atur al: if f : ( X , A ) → ( Y , B ) is a map of c omp act p airs then ( f ∗ ( α )) s = f s ∗ ( α s ) for α ∈ H k ( X, A ) (2) If ( X , A ) = ( M , ∂ M ) for some c omp act PL– manifold M and α = [ M ] is its fundamental class, then α s is the fundamental class of the r elative PL–manifol d ( M , ∂ M ) s . Before pr o ving this theorem we need also to e xplain wha t we mean by a fundamental class of a r elativ e PL- manifold. When Y is a g en uine co nnected manifold and B = ∂ Y is its bo undary (p ossibly empty), then the fundamental class [ Y ] ∈ H m ( Y , B ) , m = dim Y , has a lready be en defined ab ov e. It is unique, bec ause w e use Z / 2 as th e co efficients. In case Y is disco nnected but r emains a n m -manifold we define [ Y ] ∈ H m ( Y , ∂ Y ) as the sum o f the images of the c lasses [ Y i ] under the homomorphisms induced by inclusions ( Y i , ∂ Y i ) ֒ → ( Y , ∂ Y ), where Y i runs ov er the (finitely many) comp onents of Y . It is clear that when M is a co dimension 0 submanifold of Y , contained in the interior of Y , then [ M ] is the restrictio n o f [ Y ] to ( M , ∂ M ). Thus in the g eneral cas e of a r elativ e PL- manifold ( Y , B ) we may exhaust Y \ B by compact m –ma nifolds M i , i ∈ Z , each contained in the interior of the next, a nd define the fundament al class of ( Y , B ) a s α = lim ← − α i ∈ H m ( Y , B ), where α i ∈ H m ( Y , Y \ M ◦ i ) is the image of [ M i ] under the map H m ( M i , ∂ M i ) → H m ( Y , Y \ M ◦ i ) induced by the inclusion ( M i , ∂ M i ) ֒ → ( Y , Y \ M ◦ i ). This fundamental class α is uniquely characterized by th e pro perty that α | ( M , ∂ M ) 6 = 0 , for each co mpact m –manifold M ⊂ Y \ B . A word s hould b e said ab out the exhaustion o f Y \ A by the manifo lds M i . In the presence of a PL-structur e on Y \ A its exis tence is obvious. Howev er, for m ≥ 6 it ex ists also when Y \ A is a top ological manifold, see [7, p. 108 ]. Pr o of of The or em 3.1 . W e assume first that X and A a re compact polyhedra , in which case ˇ Cech and singular homolog y of ( X , A ) coincide. Let U b e a neighborho o d of the diagonal ∆ in X × X/τ and let ( X, A ) s U = ( X , A ) s ∪ ( X × X / τ , U ) . Let σ = σ U be a repre sen tative of α ∈ H k ( X, A ) by a singular chain σ = P n i =1 σ i such that for a ll i, j the imag e of σ i × σ j under the pro jection p : X × X → X × X/ τ is either contained entirely in U or do es not mee t the dia gonal ∆ at all. Then we define σ s to b e the chain P i 0 (with some mo dification if v k = 0 which leads to the in- tro duction of slack v ariables and the r equirement of sa turation b elo w ). Hence we a re describing a set V whose co n vex hull inc ludes the initial pr obabilit y p 0 with corres ponding vectors ( y v | v ∈ V ) that are all equa l. With some form of contin uit y of these pay off vectors we reco gnize a relation to the B orsuk-Ulam Theorem. This relation beco mes stronger if there are restr ictions o n how we can conv exify members of V, for example that the initial p 0 m ust b e in the c on vex combination of at most tw o members of V . Contin uing with the ga me context, a family L of subsets o f K is intro duced. A member L of L is a set of states for which the fir st play er can confirm that the ch os en state b elongs to L without revealing a n ything more about the chosen A par ameterized version of the Borsuk-Ula m theor em 13 state. A choice of an L in L (which must co n tain the chosen state) b y the first play er determines a kind of sub-g ame, which motiv ates the ma thematics describ ed b elow. Without go ing into further detail, we pres en t t wo statements (of B orsuk- Ulam t yp e): Assertion 5.3 and Conjecture 5.5. T og ether, they imply Conjecture 5.1, as ass erted in [1 3, p. 4 0], compare [12]. This follows rather easily using the metho ds of [15] . T o state them, we first make a definition. 5.2 Definiti on. (1) If A is a subset of an affine spa ce, co( A ) stands for its conv ex hull. (2) If F ⊂ X × Y and y ∈ Y , then F − 1 ( y ) := { x ∈ X | ( x, y ) ∈ F } . (3) If F ⊂ ∆( L ) × Y we define cF ⊂ ∆( L ) × Y by cF := { ( x, y ) | x ∈ co( F − 1 ( y )) } . (4) If F ⊂ ∆( L ) × Y a nd Y ⊂ R L then is F + ⊂ ∆( L ) × Y defined b y F + := { ( p, y ) | ∃ ( p, x ) ∈ F : x l ≤ y l ∀ l ∈ L and x l = y l if p l > 0 } . W e call F + the Y - satur ation of F . If F = F + then F is called sa turated. (5) A constant cor respo ndence ∆( K ) ∋ p 7→ U is denoted U cst (that is, U cst := ∆( K ) × U ). In b oth the Assertion 5.3 and Conjecture 5.5 b elow we assume tha t I = [ a, b ] ⊂ R is a non-triv ial clo sed segment, K is a finite set a nd L is a family of non-empty prop er subsets of K which cov ers K . Eac h s implex ∆( L ) , L ∈ L , is considered as a subset o f ∆( K ). 5.3 Asse rtion. Supp ose fo r every L ∈ L ∪ { K } ther e is given a satur ate d c orr esp ondenc e F L ⊂ ∆( L ) × I L with pr op erty S for ∆( L ) and a close d c onvex set U L ⊂ I K c ontaining the p oint v + = ( b, ..., b ) . Set f F L = F L × I K \ L ⊂ ∆( K ) × I K for L ∈ L (5.4) Also let U := T L ∈L∪{ K } U L , F := U cst ∩ F K and G := U cst ∩ S L ∈L f F L , and define a c orr esp ondenc e Γ ⊂ ∆( K ) × I K so that for y ∈ I K Γ − 1 ( y ) = co( G − 1 ( y )) ∪ [ x ∈ F − 1 ( y ) co( { x } ∪ G − 1 ( y )) The assertion is that if f F L ⊂ U cst L for e ach L ∈ L ∪ { K } , then Γ has pr op erty S for ∆( K ) . The key prop ert y in the definition of Γ is that fo r ev ery y ∈ ∆( K ) we conv exify sets in F − 1 ( y ) ∪ G − 1 ( y ) that have a t most one member x whose pro jection to ∆( K ) b elongs to ∆( K ) \ S L ∈L ∆( L ). 5.5 Conjecture. This time supp ose the satur ate d c orr esp ondenc es F L ⊂ ∆( L ) × I L with pr op ert y S for ∆( L ) ar e given mer ely for those sets L ∈ L which ar e maximal with r esp e ct to inclusion. F or al l L ∈ L let F L := [ L ⊂ J ∈ max L F J ∩ (∆( L ) × I L ) ⊂ ∆( L ) × I L 14 Thomas Schic k, Rob ert S. Simon, Stanislaw Spie ˙ z, Henr y k T o r u´ nczyk and define ˜ F L as in (5.4) and Γ ⊂ ∆( K ) × I K so that for y ∈ I K Γ − 1 ( y ) := [ { co( x L 1 , . . . , x L m ) | x L i ∈ ˜ F − 1 L i ( y ) , L i = L j ∈ max L ⇒ i = j } . The c onje ctur e is that if L 1 ∩ L 2 ∈ L for al l L 1 , L 2 ∈ L , then Γ has pr op erty S for ∆( K ) . Because —in a wa y similar to Assertion 5.3— in Conjecture 5.5 we have a restriction on how we conv exify sets to o btain Γ, its character is close to the original Bors uk-Ulam Theo rem (where one co n vexifies using opp osite po in ts). It sugg ests the s tatemen t of Coro llary 2.9, proved ab ov e. Indeed, this rela tion to Conjecture 5.5 was o ne o f the main motiv a tions for the work presented in this pap er. W e b eliev e that we can prove Asser tion 5.3. W e plan to publish this in a forthcoming pap er. Conjecture 5.5 is still op en. Provided we can e stablish it, as well, we will even tually put all the details together a nd presen t a pro of of the game theor etic Conjecture 5.1. References [1] P . Alexa ndroff and Heinz Hopf. T op ologie. I. Ber ic htigter Reprint. Die Grundlehren der mathematischen Wissenschaften, Band 45 . Springer- V er lag, Berlin-New Y ork, 1974. [2] Robert Aumann and M. Ma sc hler R ep e ate d Games with Inc omplete Infor- mation . 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