On the second homotopy group of $SC(Z)$

ON THE SECOND HOMOTOPY GR OUP OF S C ( Z ) KA TSUY A EDA, UMED H. KARIMOV, AND DU ˇ SAN REPOV ˇ S Abstract. In our earlier paper we in troduced a cone-like space S C ( Z ). In the present note w e establish some new algebraic prop e rties of S C ( Z ). 1. Introduction In our earlier pap er [1] we introduce d a new, cone-like construction o f a space S C ( Z ) using the topo logist sine curve and pr oved that S C ( Z ) is simply connected for every path-connected s pace Z . In another pa p er [3] we prov ed that the sing ular homology H 2 ( S C ( Z ); Z ) is non-trivial if π 1 ( Z ) is non-trivia l. In the present pa pe r we prov e its conv erse, that is: Theorem 1.1. L et Z b e any p ath-c onne cte d sp ac e, z 0 ∈ Z . If π 1 ( Z, z 0 ) is t rivial, then π 2 ( S C ( Z ) , z 0 ) is also trivial. Consequently , we ge t the following: Corollary 1.2. F or any p ath-c onne cte d sp ac e Z , z 0 ∈ Z , the fol lowing statements ar e e quivalent: (1) π 1 ( Z, z 0 ) is trivial; (2) π 2 ( S C ( Z ) , z 0 ) is trivial; and (3) H 2 ( S C ( Z ); Z ) is trivial. W e also use this o ppo rtunity to cor rect the pro of of Lemma 3.2 in [2] (see Sectio n 3). 2. Proof of Theorem 1.1 In order to des crib e the ho motopies we sha ll need to intro duce some nota tions. The unit interv al [0 , 1] is denoted by I . F or a map f : [ a 0 , b 0 ] × [ c, d ] → X , define f − : [ a 0 , b 0 ] × [ c, d ] → X by: f − ( x, y ) = f ( a 0 + b 0 − x, y ) and f [ a,b ] : [ a, b ] × [ c, d ] → X by: f [ a,b ] ( x, y ) = f ( a 0 + ( b 0 − a 0 )( x − a ) / ( b − a ) , y ) . W e follow the nota tion in [1] for the spa ce S C ( Z ) a nd the pro jection p : S C ( Z ) → I 2 . In par ticular, a p oint of the subspace I 2 of S C ( Z ) is denoted by ( x ; y ). The fol- lowing figure denotes the part I 2 of S C ( Z ), where the po ly gonal line A 1 B 1 A 2 B 2 · · · AB is the piecewis e linea r v ersion of the top olo gists s ine cur ve in Figure 1. Date : April 27, 2022. 2000 Mathematics Subje ct Classific ation. Primary: 54G15, 54G20, 54F15; Secondary: 54F35, 55Q52. Key wor ds and phr ases. Cone-li ke space, si mple connectivit y , local strong con tractibility , sec- ond homotop y group. 1 2 KA TSUY A EDA, UMED H. KARIMOV, AND D U ˇ SAN REPOV ˇ S Definition 2. 1. [4, Definition 2.1] A c ontinuous map f : I 2 → p − 1 ( I × { 0 } ) with f ( ∂ I 2 ) = { A } is said to b e standar d, if f ([1 / ( m + 1) , 1 /m ] × I ) ⊂ [ A, A m ] ∪ p − 1 ( { A m } ) , and f ( ∂ ([1 / ( m + 1) , 1 /m ] × I )) = { A } for e ach m < ω . ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ A B B 1 B 2 B 3 A 1 A 2 A 3 • • • • • • • • (Figure 1) Pr o of of Theorem 1.1. Recall that p : S C ( Z ) → I 2 and p ( z ) = ( p 1 ( z ); p 2 ( z )). Let f : I 2 → S C ( Z ) with f ( ∂ I 2 ) = A = (0 , 0). First we c laim that f is homoto pic to a map f 1 in p − 1 ( I × { 0 } ) relative to ∂ I 2 . Since p − 1 ( I × { 0 } ) is a retraction of S C ( Z ) \ S { p − 1 ( { B n } ) : n ∈ N } , it suffices to show that f is homotopic to a ma p in S C ( Z ) \ S { p − 1 ( { B n } ) : n ∈ N } relative to ∂ I 2 . The idea to remove p − 1 ( { B n } ) from the image of f is basically the sa me as tha t in the pr o of of the simple connectivity of S C ( Z ). Since the b oundar y o f an op en connected subset of the sq uare is co mplicated in comparis on with those o f the in terv al, some more ca re is needed. Here we also use the s imple connectivity of Z . Us ing this prop erty we get the simple connectivity of a certa in small op en subset containing p − 1 ( { B n } ) in S C ( Z ), i.e. p − 1 ( U n ) a ccording to the fo llowing no tation. Let U n be a s q uare neighborho o d of B n in I 2 which do es not cont ain any B i for i 6 = n , and choose a point u n ∈ U n ∩ T satisfying u n 6 = B n . W e hav e finitely many p olygo nal connected o p en s ets O i in I 2 such that ( p ◦ f ) − 1 ( { B n } ) ⊆ S i O i ⊆ ( p ◦ f ) − 1 ( U n ), where O i may fail to b e simply connected. Observe that p − 1 ( U n ) is homotopy equiv a lent to Z and so it is simply connected. Hence, working in ea ch O i , w e have g 1 : I 2 → S C ( Z ) satisfying: (1) g 1 is ho motopic to f r elative to I 2 \ O i ; a nd (2) ther e exist finitely many simply connected p olyg onal pairwise disjoint s ub- sets P ij of O i such that ( p ◦ g 1 ) − 1 ( { B n } )) ⊆ S j P ij , a nd (3) g ( ∂ P ij ) = { u n } . W e r emark that the r ange of the homotopy betw een f and g 1 is contained in p − 1 ( U n ). Since p − 1 ( { u n } ) is a strong deformation retrac t of p − 1 ( U n ), we hav e ON THE SE COND HOMOTOPY GROUP OF S C ( Z ) 3 g 2 : I 2 → S C ( Z ) such that g 2 ( S j P ij ) ⊆ p − 1 ( { u n } ) ∪ I × I , g 2 is homotopic g 1 relative to I 2 \ S j P ij , and the ra nge of the homotopy is contained in p − 1 ( U n ). Observe that there are only finitely many O i for each B n and that p − 1 ( U n ) conv erge to B . W o r king o n each B n successively , w e o btain ma ps ho motopic to f . Let f 0 be the limit of these maps. Since f ( O i ) ⊆ p − 1 ( U n ) and p − 1 ( U n ) conv erge to B , f 0 is contin uous and f 0 is ho- motopic to f relative to ∂ I 2 and also f 0 ( I 2 ) do es not in tersect w ith any p − 1 ( { B n } ), as desired. Hence w e hav e f 1 : I 2 → p − 1 ( I × { 0 } ) which is homoto pic to f relative to ∂ I 2 . The nex t pro cedure is similar to the pro o f o f [4, L e mma 2.2 ], which is rather long but each s tep is simple. Using the co mmut ativity of π 2 and the s imple connectivity of Z ag ain and also using the fa c t that p − 1 ( I × { 0 } ) is lo cally strong ly contractible at p o ints ( x ; 0) with ( x ; 0) / ∈ { A n : n ∈ N } ∪ { A } , w e g et a standar d ma p f 2 : I 2 → p − 1 ( I × { 0 } ) which is homoto pic to f 1 relative to ∂ I 2 . The pro o f that f 2 is null-homotopic is the 2- dimensional version of pro cedures II and I I I in the pro o f of the simple connectivit y o f S C ( Z ) [1, Theore m 1.1 ]. W e outline these pro cedure s. W e concentrate on description of the null homo topy of f 2 | [1 / ( k + 1) , 1 /k ] × I . Fix k ∈ N . F or m ∈ N , define h m : [1 / ( k + 1) , 1 /k ] × I → S C ( Z ) by: h m ( x, y ) =  ( k u/ ( k + m − 1); 0) if f 2 ( x, y ) = ( u ; 0) ( A k + m − 1 , z ) if f 2 ( x, y ) = ( A k , z ) . Next define g k,m : [1 / ( k + 1 ) + 1 / (( m + 1) k ( k + 1)) , 1 / ( k + 1 ) + 1 / ( mk ( k + 1))] × I → S C ( Z ) by: g k, 2 m − 1 = ( h m ) [1 / ( k +1)+1 / (2 mk ( k +1)) , 1 / ( k +1)+1 / ((2 m − 1) k ( k +1))] g k, 2 m = ( h − m +1 ) [1 / ( k +1)+1 / ((2 m +1) k ( k +1)) , 1 / ( k +1)+1 / (2 mk ( k +1))] Let g k : [1 / ( k + 1) , 1 /k ] × I → S C ( Z ) b e the unique contin uo us extensio n o f [ m ∈ N g k,m , i.e. g k (1 / ( k + 1) , y ) = A . Since the ima ges of g k,m conv erge to A and g 2 m +1 is the homotopy inverse of g 2 m in p − 1 ( I × { 0 } ) for ea ch m , g k is c o ntin uous and is ho motopic re la tive to the bo undary to the restriction f 2 | [1 / ( k + 1) , 1 /k ] × I , and the homotopy can b e taken in p − 1 ([ A, A k ]). Hence f 2 is homotopic relative to the bounda r y to g : I × I → S C ( Z ) which is the unique contin uous extensio n o f S { g k : k ∈ N } , i.e. g (0 , y ) = A . F or the next step we do not ca r e for the bounda ry for a while. W e push up the ranges of g k, 2 m − 1 along A k + m B k + m for m ≥ 0 and g − k, 2 m along A k + m B k + m − 1 so that the y - co ordinates o f p ( u ) for u in each of the ra nges are the sa me. Then the resulting map is defined in p − 1 ( I × { 1 } ) and we couple g k, 1 and g k, 2 , a nd generally g k, 2 m − 1 and g k, 2 m . Since these homotopies of co uplings conv erge to B , we see that the resulting map is n ull-homotopic. W e ca n p e rform these pr o cedures uniformly in k , and we ha ve a homotopy fro m f 2 to the co ns tant ma p B . In order to obtain the des ired homotopy to the constant A relative to the bo und- ary , we can mo dify the homotopy ab ov e to the des ired o ne, b eca use we hav e homo- topies in the pushing up pro cedure above, so that the y -co ordina tes of p ( u ) for u in the ra ng es ar e the sa me, even unifor mly in k .  4 KA TSUY A EDA, UMED H. KARIMOV, AND D U ˇ SAN REPOV ˇ S 3. Correction of the proof of Lemma 3. 2 of [2] In our earlier pa pe r [2, Lemma 3.2] we used the following auxiliary r esult: Lemma 3.1. L et p 0 , p 1 , p ∗ b e distinct p oints in a Hausdorff sp ac e X and let f b e a lo op in X with the b ase p oint p 1 such t hat f − 1 ( { p 0 } ) = ∅ and f − 1 ( { p ∗ } ) is a singleton. If f is nul l- homotopic r elative to end p oints, then ther e exists a lo op f ′ in X with the b ase p oint p 1 in X \ { p 0 , p ∗ } such that f and f ′ ar e homotopic r elative t o end p oints in X \ { p 0 } . W e use this oppo rtunity to correct our original pro of. The assertion “ G − 1 ( { p ∗ , p 0 } ) ∩ O is compa ct” in [2 , p.92 l.5 from the b ottom] is wro ng . W e b egin by the follo wing well-kno wn result - see e .g . [5, p.169]: Lemma 3.2. L et X b e a c omp act sp ac e and C a close d c omp onent of X . Then C is the interse ction of clop en sets c ontaining C .  Pr o of of Lemma 3.1. Since f is null-homotopic, we have a homotopy F : I × I → X from f to the consta nt mapping to p 1 , i.e. F ( s, 0) = f ( s ) , F ( s, 1 ) = F (0 , t ) = F (1 , t ) = p 1 for s, t ∈ I . Let { s 0 } be the s ingleton f − 1 ( { p ∗ } ). Let M 0 be the co nnectedness comp o nent of F − 1 ( { p ∗ } ) containing ( s 0 , 0), and O the connectedness comp onent of I × I \ M 0 containing I × { 1 } . Define G : I × I → P ∗ by: G ( s, t ) =  F ( s, t ) if ( s, t ) ∈ O , p ∗ otherwise. Then G is also a homoto py from f to the co nstant ma pping to p 1 and G − 1 ( { p 0 } ) is c o ntained in O . (Observe that ∂ ( I × I ) \ { ( s 0 , 0) } ⊆ O .) Put M 1 = ( I × I \ O ) ∪ I × { 0 } . Since G − 1 ( { p 0 } ) ∩ O is co mpact a nd dis joint from M 1 , we have a p olygo nal neighbo rho o d U of M 1 whose closur e is disjoint from G − 1 ( { p 0 } ) and also I × { 1 } . The b oundary o f U is a piecewise linear ar c connecting a p oint in { 0 } × (0 , 1 ) and a p oint in { 1 } × (0 , 1). W e wan t to g et a piecewise linear injectiv e path g : I → I × I s uch tha t Im( G ◦ g ) ⊆ X \ { p 0 , p ∗ } , g (0) ∈ { 0 } × I , and g (1) ∈ { 1 } × I and Im( g ) divides I × I into tw o co mp o ne nts, o ne of which contains G − 1 ( { p 0 } ) and the other con tains M 1 . If the b oundary of U is disjoint fro m G − 1 ( { p ∗ } ), then we hav e such a path g tracing the bo undary . Other wise, let C 0 be the intersection of the b oundary of U and G − 1 ( { p ∗ } ). Since M 0 is contained in a co nnected c o mp o nent of G − 1 ( { p ∗ } ) which is disjoin t with C 0 , w e hav e clopen s e ts C 1 and C 2 in G − 1 ( { p ∗ } ) ∩ U such that C 1 ∩ C 2 = ∅ , C 0 ⊆ C 1 and M 0 ⊆ C 2 by Lemma 3 .2. Observe that C 1 and C 2 are closed subset of U . Then we ca n see that { 0 } × (0 , 1) ∩ U and { 1 } × (0 , 1) ∩ U b elong to the same comp onent of U \ G − 1 ( { p ∗ } ). W e hav e a piecewise linear ar c b e tween a p oint in { 0 } × (0 , 1 ) a nd a po int in { 1 } × (0 , 1) in U which do es not intersect with G − 1 ( { p ∗ } ) and hav e a path g with the req uired prop erties. W e now se e that G ◦ g is the desired lo op f ′ .  ON THE SE COND HOMOTOPY GROUP OF S C ( Z ) 5 4. Ackno wledgements This resea rch was suppo rted by the Slov e nia n Resear ch Agency gra nt s P1 -029 2 - 0101- 04 and J1-9 6 43-0 1 01-0 7. The first autho r was supp orted by the Gra nt -in- Aid for Scientific r e search (C) of Japan No. 2054 0 097. W e thank the referee for comments and suggestions. References [1] K. Eda, U . Kari mov , and D. Repov ˇ s, A c onstruction of simply c onne cte d nonc ontr actible c e l l-like two-dimensional Pe ano c ontinua , F und. Math. 1 95 (2007) , 193–203. [2] , On the f undamental gr oup of R 3 mo dulo the Case-Chamb erin c ontinuum , Gl asni k Mat. 42 (2007), 89–94. [3] , A nonaspheric al c el l-like 2-dimensional simply c onne cted c ontinuum and some r elate d c onstructi ons , T opol ogy Appl., 1 56 (2009) , 515–521. [4] K. Eda and K. Kaw amura, Homotopy gr oups and homolo gy gr oups of the n -dimensional Hawaiian e arring , F und. Math. 165 (2000), 17–28. [5] K. Kuratowski, T op olo gy II , Academic Press, New Y ork, 1968. School of S cience a nd Eng ineering, W aseda University, Tokyo 169-855 5, Japan E-mail addr ess : eda@logic .info.was eda.ac.jp Institute of Ma thema tics, Academy of Sciences of T ajikist an, Ul. Ainy 299 A , Dushan be 734063, T ajikist a n E-mail addr ess : umed-kari mov@mail. ru F acul ty of Ma thematic s an d Physics, and F acul ty of Educa tion, University of Ljubl- jana, P.O.Box 2964, Ljubljana 1 001, Slovenia E-mail addr ess : dusan.rep ovs@guest .arnes.si