Discreteness and Homogeneity of the Topological Fundamental Group

DISCRETENESS AND HOMOGENE ITY OF THE T OPOLOGICAL FUND AMENT AL GR OUP JACK S. CALCUT AND JOHN D. MCCAR THY Abstract. F or a locally path connect ed topological space, the topological fundamen tal group is discrete if and only if the space is semilocally simply- connect ed. While f unctorialit y of the topological f undamen tal group f or arbi- trary topological spaces r emains an open question, the top ological fundament al group is alwa ys a homogeneous space. 1. Introduction The concept of a natur al topolo gy for the fundamental g r oup appea rs to have originated with Hurewicz [8] in 193 5. It received further attention by Dugundji [2] in 1950 and by Biss [1], F abel [3, 4, 5, 6], and others more recently . The purp ose of this note is to prov e the follo wing folklore theorem. Theorem 1. L et X b e a lo c al ly p ath c onne cte d top olo gic al sp ac e. The top olo gi- c al fundamental gr oup π top 1 ( X ) is discr ete if and only if X is semilo c al ly simply- c onne cte d. Theorem 5.1 of [1] is Theorem 1 without the hypothesis of lo cal path c o nnected- ness. How ever a counterexample of F ab el [4] s hows that this strong er result is fa ls e. F ab el [4] also proves a weak er version of Theor em 1 assuming that X is lo ca lly path connected and a metric space. In this note we r emov e the metr ic hypothesis. Our pr o of pro ceeds from first top olo gical principles making no use o f rigid cov- ering fibrations [1] nor even of c la ssical covering spaces . W e make no use o f the functoriality o f the to po logical fundamental group, a prop er t y which was also a main result in [1, Cor. 3.4] but in fact is unprov en [6 , pp. 188–18 9]. Beware that the misstep in the pro o f of [1, Pro p. 3.1], namely the assumption that the pro duct of quotient ma ps is a quotien t ma p, is rep ea ted in [7, Thm. 2.1]. In genera l the homeomorphism type of the top ologic a l fundamental gro up de- pends on a c hoice of basep oint. W e say that π top 1 ( X ) is discr et e without reference to basep oint provided π top 1 ( X, x ) is discrete for each x ∈ X . If x and y are connected by a path in X , then π top 1 ( X , x ) and π top 1 ( X, y ) are homeo morphic. This fact was prov ed in [1, Prop. 3.2 ] and a detailed pr o of is in Section 4 b elow for co mpleteness. Theorem 1 now immediately implies the following. Corollary . L et X b e a p ath c onn e cte d and lo c al ly p ath c onne cte d top olo gic al sp ac e. The top olo gic al fundamental gr oup π top 1 ( X, x ) is discr ete for some x ∈ X if and only if X is semilo c al ly simply-c onne cte d. Date : March 25, 2009. 1 2 J. CALCUT AND J. MCCAR THY As mentioned a bove it is op e n whether π top 1 is a functor fro m the categor y of po int ed top olo g ical spaces to the catego r y of top olo g ical groups. The unsettled question is whether m ultiplication π top 1 ( X , x ) × π top 1 ( X , x ) µ / / π top 1 ( X, x ) ([ f ] , [ g ])  / / [ f ] · [ g ] is contin uous. By Theorem 1, if X is lo cally path connected and semilo cally simply- connected, then π top 1 ( X, x ), and hence π top 1 ( X, x ) × π top 1 ( X , x ), is discr ete and so µ is trivially c ontin uous. Contin uity of µ in gene r al r emains an interesting ques tio n. Lemma 4 b elow shows that if ( X, x ) is an ar bitrary p ointed topolo gical space, then left a nd right multiplication by any fix ed element in π top 1 ( X, x ) ar e contin uous self maps of π top 1 ( X , x ). Therefore π top 1 ( X, x ) acts o n itself by left and right trans - lation a s a gr oup of self homeomo rphisms. Clearly these actions are b oth transitive. Thu s we obtain the following result. Theorem 2. If ( X , x ) is a p ointe d top olo gic al sp ac e, then π top 1 ( X , x ) is a homo ge- ne ous sp ac e. This note is org anized as follows. Sectio n 2 co n tains definitions and conv e n- tions, Section 3 prov es t w o lemmas and Theorem 1, Section 4 addresses change of basep oint, and Section 5 shows left and r ight transla tion a re homeomor phisms. 2. Definitions and Conventions By conven tion, neig hborho o ds ar e op en. Unless stated otherwise, homomor- phisms are inclusio n induced. Let X b e a top olo gical space and x ∈ X . A neighbo rho o d U of x is r elatively inessential (in X ) provided π 1 ( U, x ) → π 1 ( X, x ) is trivia l. X is semilo c al ly simply- c onne cte d at x provided there exists a relatively inessential neigh bor ho o d U of x . X is semilo c al ly simply-c onne cte d pr ovided it is so at each x ∈ X . A neighborho o d U of x is str ongly r elatively inessent ial (in X ) provided π 1 ( U, y ) → π 1 ( X, y ) is trivial for every y ∈ U . The fundamental gro up is a functor from the catego r y of p o inted to p o lo gical spaces to the categor y of gr oups. Consequently if A a nd B are any subsets of X such that x ∈ A ⊂ B ⊂ X a nd π 1 ( B , x ) → π 1 ( X , x ) is triv ia l, then π 1 ( A, x ) → π 1 ( X, x ) is trivial as w ell. This observ ation justifies the conven tion that neighbo r ho o ds are op en. If X is lo ca lly path connected and semilo ca lly simply-co nnected, then each x ∈ X has a path connected r elatively inessential neig h bo rho o d U . Such a U is necess arily a strongly rela tively inessential neig hborho o d of x as the r eader may verify (see for instance [9, Ex . 5 p. 330]). Let ( X , x ) b e a p ointed top olog ical s pace and let I = [0 , 1] ⊂ R . The space C x ( X ) = { f : ( I , ∂ I ) → ( X , x ) | f is co n tin uous } DISCRETENESS AND HOMOGENE ITY OF THE TOPOLOGICAL FUND AM ENT AL GROUP 3 is endow ed with the compact-o p en top olo gy . The function C x ( X ) q / / π 1 ( X , x ) f  / / [ f ] is surjective s o π 1 ( X, x ) inherits the quotient top olog y and one writes π top 1 ( X, x ) for the resulting t op olo gic al fundamental gr oup . Let e x ∈ C x ( X ) denote the con- stant ma p. If f ∈ C x ( X ), then f − 1 denotes the path defined b y f − 1 ( t ) = f (1 − t ). 3. Proof of Theorem 1 W e prov e tw o le mma s and then Theorem 1. Lemma 1. L et ( X , x ) b e a p ointe d top olo gic al sp ac e. If { [ e x ] } is op en in π top 1 ( X , x ) , then x has a r elatively inessential neighb orho o d in X . Pr o of. The quotient map q is contin uous a nd { [ e x ] } ⊂ π top 1 ( X, x ) is op en, so q − 1 ([ e x ]) = [ e x ] is op en in C x ( X ). Therefore e x has a basic open neigh bo rho o d (1) e x ∈ V = N \ n =1 V ( K n , U n ) ⊂ [ e x ] ⊂ C x ( X ) where each K n ⊂ I is co mpact, each U n ⊂ X is o p en, and each V ( K n , U n ) is a subbasic op en set for the compact- o p e n to po logy on C x ( X ). W e will show that U = N \ n =1 U n is a relatively inesse ntial ne ig hborho o d o f x in X . Clearly U is op en in X and, by (1), x ∈ U . Finally , let f : ( I , ∂ I ) → ( U, x ). F o r each 1 ≤ n ≤ N we hav e f ( K n ) ⊂ U ⊂ U n . Thu s f ∈ [ e x ] by (1) and so [ f ] = [ e x ] is trivia l in π 1 ( X, x ).  Lemma 2. L et ( X, x ) b e a p ointe d top olo gic al sp ac e and let f ∈ C x ( X ) . If X is lo c al ly p ath c onne cte d and semilo c al ly simply-c onne cte d, then { [ f ] } is op en in π top 1 ( X, x ) . Pr o of. As q is a q uo tient map, we m ust show that q − 1 ([ f ]) = [ f ] is op en in C x ( X ). So let g ∈ [ f ]. F or each t ∈ I let U t be a path connected r elatively inessential neighborho o d o f g ( t ) in X . The sets g − 1 ( U t ), t ∈ I , for m an op en co ver of I . Let λ > 0 be a Leb esgue num b er for this cov er. Cho ose N ∈ N so that 1 / N < λ . F or each 1 ≤ n ≤ N let I n =  n − 1 N , n N  ⊂ I . Reindex the U t s so that g ( I n ) ⊂ U n for each 1 ≤ n ≤ N . The U n s ar e not necessarily distinct, nor do es the pro of req uire this condition. F or each 1 ≤ n ≤ N let W n denote the pa th co mp onent of U n ∩ U n +1 containing g ( n/ N ), so (2) g  n N  ∈ W n ⊂ ( U n ∩ U n +1 ) ⊂ X . 4 J. CALCUT AND J. MCCAR THY Consider the basic op en set (3) V = N \ n =1 V ( I n , U n ) ! ∩ N − 1 \ n =1 V n n N o , W n  ! ⊂ C x ( X ) . By constructio n, g ∈ V . It r emains to show that V ⊂ [ f ]. So let h ∈ V . As [ g ] = [ f ], it suffices to show that [ h ] = [ g ]. By (3) we hav e h ( I n ) ⊂ U n for each 1 ≤ n ≤ N a nd h  n N  ∈ W n for each 1 ≤ n ≤ N − 1 . (4) F or each 1 ≤ n ≤ N − 1 let γ n : I → W n be a contin uous path such that γ n (0) = h  n N  and γ n (1) = g  n N  , which exists by (2) and (4). Let γ 0 = e x and γ N = e x . F or each 1 ≤ n ≤ N define I s n / / I n t  / / 1 N t + n − 1 N and let g n = g ◦ s n and h n = h ◦ s n . So g n and h n are affine r e parameteriza tions of g | I n and h | I n resp ectively . F or ea ch 1 ≤ n ≤ N δ n = g n ∗ γ − 1 n ∗ h − 1 n ∗ γ n − 1 is a lo o p in U n based a t g n (0) (se e Figure 1). As U n is a strong ly relatively inessential h n g n γ n γ n − 1 U n U n − 1 U n + 1 Figure 1. Lo o p δ n = g n ∗ γ − 1 n ∗ h − 1 n ∗ γ n − 1 in U n based at g n (0). neighborho o d, [ δ n ] = 1 ∈ π 1 ( X, g n (0)). Ther efore g n and γ − 1 n − 1 ∗ h n ∗ γ n are path DISCRETENESS AND HOMOGENE ITY OF THE TOPOLOGICAL FUND AM ENT AL GROUP 5 homotopic. In π 1 ( X, x ) we hav e [ h ] = [ h 1 ∗ h 2 ∗ · · · ∗ h N ] =  γ − 1 0 ∗ h 1 ∗ γ 1 ∗ γ − 1 1 ∗ h 2 ∗ γ 2 ∗ · · · ∗ γ − 1 N − 1 ∗ h N ∗ γ N  = [ g 1 ∗ g 2 ∗ · · · ∗ g N ] = [ g ] proving the le mma.  In the previo us pro o f, the second collection o f s ubba sic op en sets in (3) are essential. Figure 2 shows t wo lo ops g and h bas ed at x in the a nnu lus X = S 1 × I . All conditions in the pro of a re satis fie d except g (1 / N ) and h (1 / N ) fail to lie in the x h g g ( ) U 1 U 2 1 N h ( ) 1 N Figure 2. Lo o ps g and h based a t x in the ann ulus X . same connected co mp onent of U 1 ∩ U 2 . Clear ly g and h are no t homotopic lo ops. Pr o of of The or em 1. First assume π top 1 ( X ) is discrete a nd let x ∈ X . By definition π top 1 ( X, x ) is discrete and so { [ e x ] } is op en in π top 1 ( X, x ). By Lemma 1, x has a relatively inessent ial neighborho o d in X . The choice of x ∈ X was arbitrary and so X is semiloca lly simply-connected. Next assume X is semilo cally s imply-connected and let x ∈ X . Poin ts in π top 1 ( X, x ) are op en by Lemma 2 and so π top 1 ( X, x ) is discr ete. The choice o f x ∈ X was arbitra ry and so π top 1 ( X ) is discr e te.  4. Basepoint change Lemma 3. L et X b e a top olo gic al sp ac e and x, y ∈ X . If x and y lie in the same p ath c omp onent of X , then π top 1 ( X , x ) and π top 1 ( X, y ) ar e home omorphi c. Pr o of. Let γ : I → X b e a contin uous path with γ (0) = y and γ (1) = x . Define the function C y ( X ) Γ / / C x ( X ) f  / /  γ − 1 ∗ f  ∗ γ 6 J. CALCUT AND J. MCCAR THY First we show that Γ is contin uo us. Let I 1 = [0 , 1 / 4], I 2 = [1 / 4 , 1 / 2], and I 3 = [1 / 2 , 1]. Define the affine ho meomorphisms I 1 s 1 / / I I 2 s 2 / / I I 3 s 3 / / I t  / / 4 t t  / / 4 t − 1 t  / / 2 t − 1 and note that I Γ( f ) / / X t  / / γ − 1 ◦ s 1 ( t ) 0 ≤ t ≤ 1 4 t  / / f ◦ s 2 ( t ) 1 4 ≤ t ≤ 1 2 t  / / γ ◦ s 3 ( t ) 1 2 ≤ t ≤ 1 Consider an a rbitrary subbasic op en set V = V ( K, U ) ⊂ C x ( X ) . Observe that Γ( f ) ∈ V if and only if γ − 1 ◦ s 1 ( K ∩ I 1 ) ⊂ U , (5) f ◦ s 2 ( K ∩ I 2 ) ⊂ U , and (6) γ ◦ s 3 ( K ∩ I 3 ) ⊂ U . (7) Define the subba sic op en set V ′ = V ( s 2 ( K ∩ I 2 ) , U ) ⊂ C y ( X ) . Observe that f ∈ V ′ if and only if (6) holds. As conditions (5) a nd (7) ar e inde- pendent of f , either Γ − 1 ( V ) = ∅ or Γ − 1 ( V ) = V ′ . Thus Γ is contin uo us. Next consider the diag ram C y ( X ) Γ / / q y   C x ( X ) q x   π top 1 ( X, y ) π (Γ) / / _ _ _ π top 1 ( X, x ) The c o mpo sition q x ◦ Γ is constant on each fib er of q y so there is a unique s et function making the diagr a m c o mm ute, na mely π (Γ) : [ f ] 7→ [Γ( f )]. As q y is a quotient map, the universal pr op erty of quo tien t maps [9, Thm. 11.1 p. 139] implies that π (Γ) is contin uous. It is well known that π (Γ) is a bijection [9 , Thm. 2.1 p. 327]. Repea ting the a bove a rgument w ith the roles o f x and y interc ha nged and the ro les of γ and γ − 1 int erchanged, w e see that π (Γ) − 1 is contin uous. Thus π (Γ) is a homeomo rphism as desired.  5. Transl a tion Lemma 4. L et ( X , x ) b e a p ointe d t op olo gic al sp ac e. If [ f ] ∈ π top 1 ( X , x ) , then left and right t r anslation by [ f ] ar e self home omorphisms of π top 1 ( X , x ) . Pr o of. Fix [ f ] ∈ π top 1 ( X, x ) a nd consider left tr anslation by [ f ] on π top 1 ( X, x ) π top 1 ( X, x ) L [ f ] / / π top 1 ( X, x ) [ g ]  / / [ f ] · [ g ] DISCRETENESS AND HOMOGENE ITY OF THE TOPOLOGICAL FUND AM ENT AL GROUP 7 Plainly L [ f ] is a bijectio n of sets. Co nsider the co mm utative diagr am (8) C x ( X ) L f / / q   C x ( X ) q   π top 1 ( X, x ) L [ f ] / / π top 1 ( X, x ) where L f is defined by C x ( X ) L f / / C x ( X ) g  / / f ∗ g First w e show L f is cont inuous. Let I 1 = [0 , 1 / 2] a nd I 2 = [1 / 2 , 1]. Define the affine homeomor phisms I 1 s 1 / / I I 2 s 2 / / I t  / / 2 t t  / / 2 t − 1 and note that I f ∗ g / / X t  / / f ◦ s 1 ( t ) 0 ≤ t ≤ 1 2 t  / / g ◦ s 2 ( t ) 1 2 ≤ t ≤ 1 Consider an a rbitrary subbasic op en set V = V ( K, U ) ⊂ C x ( X ) . Observe that f ∗ g ∈ V if a nd only if f ◦ s 1 ( K ∩ I 1 ) ⊂ U and (9) g ◦ s 2 ( K ∩ I 2 ) ⊂ U . (10) Define the subba sic op en set V ′ = V ( s 2 ( K ∩ I 2 ) , U ) ⊂ C x ( X ) . Observe that g ∈ V ′ if and o nly if (10) holds. As co ndition (9) is independent o f g , either L − 1 f ( V ) = ∅ or L − 1 f ( V ) = V ′ . Thus L f is contin uous. The comp ositio n q ◦ L f is co nstant o n ea ch fib er of the quotient map q and (8) commutes, so the universal prop erty of q uotient maps [9, Thm. 11.1 p. 1 39] implies that L [ f ] is contin uous . Applying the pr evious argument to f − 1 we get L − 1 [ f ] = L [ f − 1 ] is con tin uous and L [ f ] is a ho meomorphism. The pr o of for rig ht translation is a lmost identical.  References [1] D. K. Biss, The topological fundamental group and generalized co vering spaces, T op olo gy Appl. 124 (2002) 355–371. [2] J. Dugundji, A topologized fundament al group, Pr o c. Nat. A c ad. Sci. USA 36 (1950) 141– 143. [3] P . 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Dep ar tment of Ma thema tics, Michigan St a te University, East Lansing, MI 48824-102 7 E-mail addr ess : jack@math.msu. edu URL : http://www.math .msu.edu /~jack/ Dep ar tment of Ma thema tics, Michigan St a te University, East Lansing, MI 48824-102 7 E-mail addr ess : mccarthy@math. msu.edu URL : http://www.math .msu.edu /~mccarthy/