On the singular homology of one class of simply-connected cell-like spaces

On the singula r homo logy of one class of simply -connecte d cell-like spaces Katsuy a Eda, Umed H. Karimo v and Du ˇ san Rep o v ˇ s Abstract. In our earlier papers we constructed examples of 2-dimensional nonaspherical simply-connected cell-like P eano contin ua, called Snake space . In the sequ el we in tro duced t he functor S C ( − , − ) defined on the category o f all spaces with base p oints and con tinuous mappings. F or the ci rcle S 1 , the space S C ( S 1 , ∗ ) is a Snake space. In the present pap er w e study t h e higher-dimensional homology and homotopy prop erties of the spaces S C ( Z, ∗ ) for any p ath -connected compact spaces Z . Mathematics Su b ject Classification (2010). Primary: 54G15, 54G20, 54F15; Secondary: 54F35, 55Q52. Keywords. Snake space, T op ologist sine curve, asphericity , simple con- nectivity , cell-likeness, semi-lo cal strong contractibilit y , continuum, free σ -pro duct of groups, v an K amp en th eorem. 1. In tro duction It is well-known that there exist planar noncontractible c ontin ua X all ho- motopy groups π i ( X ) , i ≥ 1 , of whic h are trivial (e.g. the Warsaw cir cle ). Every plana r simply connected Peano contin um is a contractible space, see e.g. [1 0, 14]. Nonco nt ra ctible homology lo cally connected (HLC a nd therefor e Peano) c ontin ua, all homotopy gro ups of which a re trivia l, were constr ucted in [9]. All these examples are infinite-dimensional. The following problem remains op en [6]: Problem 1.1 . Do es ther e exist a finite-dimensional nonc ont r actible Pe ano c ont inuum al l homotopy gr oups of which ar e t rivial? W e co nstructed in [4] the functor S C ( − , − ), defined on the ca tegory of all to p o lo gical s pa ces with ba s e p oints. Roughly s p ea king, for any spa ce Z , one takes the infinite cylinder Z × [0 , ∞ ) a nd attaches it to the squa re [0 , 1 ] × [ − 1 , 1 ] ⊂ R 2 , along the op en T op olo gist sine curve : { ( x, y ) ∈ R 2 | y = sin(1 /x ) , 0 < x ≤ 1 } , 2 K. E da, U. H. Kar imov and D. Repov ˇ s so that its dia meter tends to zero . The space S C ( Z, ∗ ) is called the Snake c one and when Z is the c ir cle S 1 , the space S C ( S 1 , ∗ ) is called the Snake sp ac e . The Snak e s pace w as the firs t candidate for an example of a s imply connected aspherical noncontractible Peano contin uum. How ever, we ha ve discov ered, r ather unexp ectedly , that the group π 2 ( S C ( S 1 )) is nontrivial [6]. It is easy to see that the Sna ke space is a cell-like Peano contin uum (for the verification o f cell-likeness use e.g. [12]). W e hav e alrea dy prov ed the following: Theorem 1.2. [4, Theorem 1.1 ] F or every p ath-c onn e ct e d sp ac e Z , the Snake c one S C ( Z ) is simply-c onne cte d. Our origina l pro of in [4] was quite lo ng and technical. W e s hall give a short pro of o f this result in Section 2 o f the pr esent pa per . W e proved in [6] that whenever π 1 ( Z, z 0 ) is nontrivial, the sing ula r ho- mology gr oup H 2 ( S C ( Z ); Z ) is nontrivial, and since the spaces S C ( Z ) a r e simply connected, it follows by the Hurewicz Theorem, that π 2 ( S C ( Z ) , z 0 ) is isomorphic to H 2 ( S C ( Z ); Z ) , and hence is also nontrivial. The co nverse was prov ed in [7 ], alo ng the lines of the pr o of in [4]. In Section 3 of the present pa- per we sha ll g ive a different a nd sig nificantly sho rter pr o of o f the g e ne r alized result for ( n − 1) − connected spaces, whe n n ≥ 2 . Theorem 1.3 . ([7, Theorem 1.1 ] for n = 2 . ) L et Z b e any ( n − 1) -c onne cte d sp ac e, n ≥ 2 . Then H n ( S C ( Z ); Z ) and π n ( S C ( Z ) , z 0 ) ar e trivial. Since by [6, The o rem 3 .1] the nontrivialit y of π 1 ( Z ) implies that o f H 2 ( S C ( Z )), we obtain the following: Corollary 1 .4 . F or any p ath-c onne cte d sp ac e Z and any p oint z 0 ∈ Z , t he fol lowing statement s ar e e quivalent: (i) π 1 ( Z, z 0 ) is trivial; (ii) H 2 ( S C ( Z ); Z ) is trivial; and (iii) π 2 ( S C ( Z ) , z 0 ) is trivial. Undefined no tions a re the usual ones and w e refer the r eader to [1 5]. 2. Pro of of Theorem 1.2 W e s hall follow the no tations for the Snake cone S C ( Z ) and the pr o jection p : S C ( Z ) → I 2 as in [4]. The p olyg o nal line A 1 B 1 A 2 B 2 · · · on I 2 , together with the limit interv a l AB , is the pie c ewise line ar version of the T opolo g ist sine cur ve in Figur e 1. On the singula r homolo g y 3 ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✆ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✄ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ A B C B 1 B 2 B 3 A 1 A 2 A 3 C 1 C 2 C 3 C 4 C 5 • • • • • • • • • • • • • • Figure 1 F or a pro of o f Theo rem 1 .2, we recall the notion of the free σ -pr o duct of groups and a lemma fro m [2]. An element of × × σ i ∈ I G i is expre s sed by a word W ∈ W σ ( G i : i ∈ I ), wher e G i ∩ G j = { e } for i 6 = j , W : W → S { G i : i ∈ I } , W is a countable linearly ordered set, a nd W − 1 ( G i ) is finite for each i ∈ I (cf. [2 ]). L e t h : × × σ i ∈ I G i → × × σ j ∈ J H j be a homomo rphism. F or a n y W ∈ W σ ( G i : i ∈ I ), express h ( W ( α )) by a reduced w or d V α ∈ W σ ( H j : j ∈ J ). Define V to be { ( α, β ) : α ∈ W , β ∈ V α } with the lexicogra phical ordering and V ( α, β ) = V α ( β ). A homomo rphism h is said to b e standar d , if V as defined ab ov e, is a word in W σ ( H j : j ∈ J ) and h ( W ) = V , for every W ∈ W σ ( G i : i ∈ I ). W e hav e used the sup er script σ in some cases, which means a restr ic tion to the c o unt able ca se. Hence, when an index set is countable, the restrictio n is unnecessar y and w e drop the sup e rscript σ . Let { ( X i , x i ) } i ∈ I be p ointed spa ces. Le t ( e W i ∈ I ( X i , x i ) , x ∗ ) b e a b ouquet of { ( X i , x i ) } i ∈ I . The underlying set ( e W i ∈ I ( X i , x i ) , x ∗ ) is the quotient space of a discrete union of all X i ’s b y the identification of all po int s x i with a singleto n x ∗ and the top olo gy is defined by sp ecifying the neighbo rho o d bases as follows (c.f. [1]): (1) If x ∈ X i \ { x i } , then the neig hborho o d base of x in e W i ∈ I ( X i , x i ) is the one o f X i ; (2) The p oint x ∗ has a neighbor ho o d base , each elemen t o f which is of the form: f _ i ∈ I \ F ( X i , x i ) ∨ _ j ∈ F U j , 4 K. E da, U. H. Kar imov and D. Repov ˇ s where F is a finite s ubs et o f I and each U j is an o pen neig h b orho o d of x j in X j for j ∈ F . Lemma 2.1 . [2, T he o rem A.1] Supp ose that the sp ac e X i is lo c al ly simply- c onn e ct e d and first c ountable at x i , for e ach i ∈ I . Then π 1 ( f _ i ∈ I ( X i , x i ) , x ∗ ) ≃ × × σ i ∈ I π 1 ( X i , x i ) . Lemma 2.2. [3 , Pro po sition 2.1 0] L et X i and Y j b e lo c al ly simply-c onne cte d and first c ountable at x i and y j , r esp e ctively for e ach i ∈ I and j ∈ J . Then for the c ontinuous map f : ( f _ i ∈ I ( X i , x i ) , x ∗ ) → ( f _ j ∈ J ( Y j , y j ) , y ∗ ) , the induc e d homomorphism f ∗ : π 1 ( f _ i ∈ I ( X i , x i ) , x ∗ ) → π 1 ( f _ j ∈ J ( Y j , y j ) , y ∗ ) is st andar d u nder t he natur al identific ations: π 1 ( f _ i ∈ I ( X i , x i ) , x ∗ ) = × × σ i ∈ I π 1 ( X i , x i ) and π 1 ( f _ j ∈ J ( Y j , y j ) , y ∗ ) = × × σ j ∈ J π 1 ( Y j , y j ) . Let Y 0 = p − 1 ( I × [0 , 2 / 3)) and Y 1 = p − 1 ( I × (1 / 3 , 1]). Then S C ( Z ) = Y 0 ∪ Y 1 and Y 0 ∩ Y 1 is op en in S C ( Z ). W e let i 0 : Y 0 ∩ Y 1 → Y 0 , i 1 : Y 0 ∩ Y 1 → Y 1 , j 0 : Y 0 → S C ( X ), j 1 : Y 1 → S C ( X ), a nd i : Y 0 ∩ Y 1 → S C ( X ) b e the inclus io n maps. Pr o of of Theorem 1 .2. W e observe that p − 1 ( I × { 1 / 2 } ) is a stro ng defor- mation r e tract o f Y 0 ∩ Y 1 . L et C n be the p oints on I × { 1 / 2 } such that C 2 n − 1 is on the segment A n B n and C 2 n is on the seg ment B n A n +1 . Let X n be the subspace [ C, C n ] ∪ p − 1 ( { C n } ) of S C ( Z ). Then Y 0 ∩ Y 1 is homotopy equiv a lent to e W n<ω ( X n , C ). Since X n is lo cally simply connected a nd first countable at C and p − 1 ( { C n } ) is homeomorphic to Z , π 1 ( Y 0 ∩ Y 1 ) is is omor- phic to × × n<ω π 1 ( p − 1 ( { C n } ) ∼ = × × n<ω π 1 ( Z ) by Lemmas 2.1 and 2.2. Simlilar ly , π 1 ( Y 0 ) and π 1 ( Y 1 ) are isomor phic to × × n<ω π 1 ( p − 1 ( { A n } ) ∼ = × × n<ω π 1 ( Z ) a nd × × n<ω π 1 ( p − 1 ( { B n } ) ∼ = × × n<ω π 1 ( Z ) resp ectively . Her e we re ma rk tha t i 0 ∗ and i 1 ∗ are standard homomorphisms under these prese ntations of the fundamen- tal gr oups. Since Y 0 , Y 1 and Y 0 ∩ Y 1 are pa th-connected and open in S C ( Z ), w e can a pply the v a n Ka mpen theorem [1 3, Theorem 2.1] for ho momorphisms i 0 ∗ , i 1 ∗ , j 0 ∗ , j 1 ∗ and i ∗ betw een fundamental groups. The diagr a m formed by these five homomo rphisms is a pushout diagra m and hence the range s of j 0 ∗ and j 1 ∗ generate π 1 ( S C ( Z )). Therefore i ∗ is s urjective. F or the simple connectivity of S C ( Z ) it suffices to show that i ∗ is trivial. W e let u n ∈ π 1 ( p − 1 ( { C n } ) b e the copy of u ∈ π 1 ( Z ). Let U n be the word u − 1 n u n +1 · · · u − 1 n +2 k u n +2 k +1 · · · . On the singula r homolo g y 5 Since i 0 ∗ ( u − 1 2 m u 2 m +1 ) = e and i 1 ∗ ( u − 1 2 m − 1 u 2 m ) = e and i 0 ∗ and i 1 ∗ are stan- dard homomorphisms, i 0 ∗ ( U 2 m ) = e , i 1 ∗ ( U 2 m ) = i 1 ∗ ( u − 1 2 m ), i 0 ∗ ( U 2 m − 1 ) = i 0 ∗ ( u − 1 2 m − 1 ), a nd i 1 ∗ ( U 2 m − 1 ) = e . Now, a n ar bitrary element o f π 1 ( Y 0 ∩ Y 1 ) is expressed by a word W ∈ W ( π 1 ( p − 1 ( { C n } )) : n ∈ N ) . F or each letter u n ∈ π 1 ( p − 1 ( { C n } )) for an o dd n app earing in W , we insert U n +1 successively to u n and form W ∗ . Since U n +1 ∈ W ( π 1 ( p − 1 ( { C m } )) : m ≥ n ) , W ∗ is actually a word in W ( π 1 ( p − 1 ( { C n } )) : n ∈ N ). W e let W 0 be the word obtained by deleting all letters in S π 1 ( p − 1 ( { C 2 n − 1 } )) from W . Since i 0 ∗ and i 1 ∗ are standard ho momorphisms, i 0 ∗ ( W ∗ ) = i 0 ∗ ( W ) a nd i 1 ∗ ( W ∗ ) = i 1 ∗ ( W 0 ). Now W 0 ∈ W ( π 1 ( p − 1 ( { C 2 n } )) : n ∈ N ) . W e again inse r t U n +1 for ea ch letter u n app earing in W 0 and for m W ∗ 0 . Then, by the symmetr ic a l a rgument a s ab ov e, we c o nclude tha t i 1 ∗ ( W ∗ 0 ) = i 1 ∗ ( W 0 ) and i 0 ∗ ( W ∗ 0 ) = e . Now, i ∗ ( W ) = j 0 ∗ ◦ i 0 ∗ ( W ∗ ) = j 1 ∗ ◦ i 1 ∗ ( W 0 ) = j 1 ∗ ◦ i 1 ∗ ( W ∗ 0 ) = j 0 ∗ ◦ i 0 ∗ ( W ∗ 0 ) = j 0 ∗ ( e ) = e, which imples that π 1 ( S C ( Z )) is indeed trivial.  3. Pro of of Theorem 1.3 F or every gro up G i , Π σ i ∈ I G i is the subgr oup of Π i ∈ I G i consisting o f elements u such that { i ∈ I : u ( i ) 6 = 0) } is countable. A space X is called semi-lo c al ly str ongly c ontr actible at x ∈ X , if ther e exists an op en neighbor ho o d U ⊂ X of x such that there exists a contraction of U in X to x which fixes x (cf. [8]). Lemma 3 .1. [8, Theor em 1.1] L et n ≥ 2 and let X i b e a sp ac e which is ( n − 1) - c onn e ct e d semi-lo c al ly str ongly c ontr actible at x i for e ach i ∈ I . Then π n ( f _ i ∈ I ( X i , x i ) , x ∗ ) ∼ = Π σ i ∈ I π n ( X i , x i ) . Pr o of of Theorem 1.3. W e sha ll use the May er-Vietoris seque nc e instea d of the v an Kamp en theorem (as in the pre ceding pro o f ). Consider the following May er-Vietoris homology exact seq uence (ov er Z ) for the tr iad ( S C ( Z ); Y 0 , Y 1 ) from Sectio n 2: H n ( Y 0 ∩ Y 1 ) i 0 ∗ + i 1 ∗ − → H n ( Y 0 ) ⊕ H n ( Y 1 ) j 0 ∗ + j 1 ∗ − → H n ( S C ( Z )) ∂ − → H n − 1 ( Y 0 ∩ Y 1 ) . Since Z is ( n − 1 )-connected, Y 0 ∩ Y 1 is also ( n − 1)-co nnected, which implies that H n − 1 ( Y 0 ∩ Y 1 ) = { 0 } . Ther efore it suffices to show that i 0 ∗ + i 1 ∗ is surjective. Note that Y 0 ∩ Y 1 , Y 0 and Y 1 are simply connected and that p − 1 ( I × { 1 / 2 } ), p − 1 ( I × { 0 } ) and p − 1 ( I × { 1 } ) are stro ng defo r mation retracts of Y 0 ∩ Y 1 , Y 0 and Y 1 , r esp ectively . F o r the same r e ason a s explained in the 6 K. E da, U. H. Kar imov and D. Repov ˇ s first parag raph o f the pro o f of Theorem 1.2, the lo cal pro pe r ties required in Lemma 3 .1 for Y 0 ∩ Y 1 , Y 0 and Y 1 are sa tisfied and we hav e: H n ( Y 0 ∩ Y 1 ) = π n ( Y 0 ∩ Y 1 ) = Π ∞ m =1 H n ( p − 1 ( { C m } )) , H n ( Y 0 ) = π n ( Y 0 ) = Π ∞ m =1 H n ( p − 1 ( { A m } )) , and H n ( Y 1 ) = π n ( Y 1 ) = Π ∞ m =1 H n ( p − 1 ( { B m } )) , where A m , B m , C m are the p oints indica ted in Figure 1. Since p − 1 ( { C m } ), p − 1 ( { A m } ) and p − 1 ( { B m } ) ar e homeomor phic to Z , we can identify the ho- mology gro ups of these spac e s with H n ( Z ). Therefore for u ∈ Π ∞ m =1 H n ( p − 1 ( { C m } )) i 0 ∗ ( u )(1) = u (1) , i 0 ∗ ( u )( m ) = u (2 m − 1) + u (2 m − 2) for m ≥ 2 , i 1 ∗ ( u )( m ) = u (2 m − 1) + u (2 m ) for m ≥ 1 . F or any given v ∈ H n ( Y 0 ) , w ∈ H n ( Y 1 ), de fine: u (2 m − 1 ) = Σ m k =1 v ( k ) − Σ m − 1 k =1 w ( k ) and u (2 m ) = Σ m k =1 w ( k ) − Σ m k =1 v ( k ) . Then i 0 ∗ ( u ) = v and i 1 ∗ ( u ) = w and hence ( i 0 ∗ + i 1 ∗ )( u ) = v + w . W e hav e th us shown that H 2 ( S C ( Z ) , Z ) is trivia l a nd consequently π 2 ( S C ( Z )) is also trivial b y the Hurewicz Theo rem a nd Theorem 1.2.  Remark 3 .2. The pro of of Theorem 1 .3 in [7 ] was along the same line as the pro of of [4 , Theo rem 1.1], which contains a pro cedure to av oid p − 1 ((0 , 1] × { 1 } ). The us e of the May er-Vieto r is sequence ab ov e mak es it p oss ible for us to sk ip this pro cedure, as do es the use of the v an Kamp en theorem in the pro of of The o rem 1.2. When Z is not s imply-connected, we cannot av oid p − 1 ((0 , 1] × { 1 } ) for π 2 ( S C ( Z )), which reflects the nontriviality of π 2 ( S C ( Z )) in Theorem 1.3 . W e r emark that the presen tations of the homotopy g roups, i.e. Lem- mas 2.1 , 2.2 and 3.1, are als o useful to ma ke pro o fs shorter. That is, o ur previous pro o fs implicitly contain the pro cedur es us ed in the pr o ofs of the May er-Vietoris sequence , the v an Kamp en theor em and the lemmas. 4. Ac kno wledgemen ts The a uthors thank the refer ee for several co mmen ts a nd sug gestions. This resear ch was suppor ted b y the Slov enian Resea rch Agency gra nts P1 - 0292- 0101, J 1 -9643 -0101 and J1- 2057- 0 101. The first author was als o supp orted by the Gra nt -in-Aid for Scient ific rese a rch (C) of Japan No. 20 5400 9 7. References [1] M. G. Barrat and J. Milnor, A n example of anomalous singular homolo gy , Proc. Amer. Math. So c. 13 (1962), 293–297. On the singula r homolo g y 7 [2] K. Eda, F r e e σ -pr o ducts and nonc omm utatively slender gr oups , J. Algebra 148 (1992), 243–263 . [3] , The fundamental gr oups of one-dimensional wild sp ac es and the Hawai- ian e arring , Pro c. Amer. Math. So c. 130 (2002), 1515–1522 . [4] K. Eda, U. Karimo v, and D. Rep o v ˇ s, A c onstru ction of sim ply c onne cte d non- c ontr actible c el l-li ke two-dimensional Pe ano c ontinua , F und. Math. 195 (2007), 193–203 . [5] , On the f undamental gr oup of R 3 mo dulo the Case-Chamb erin c ontin- uum , Glasnik Mat. 42 (2007), 89–94. [6] , A nonaspheric al c el l-like 2-dimensional simply c onne cte d c ontinuum and r elate d c onstru ctions , T op ology App l. 156 (2009), 515–521. [7] , The se c ond homotopy gr oup of S C ( Z ), Glasnik Mat. 44(64) (2009), 493–498 . [8] K. Eda and K. Kaw amura, Homotopy gr oups an d homolo gy gr oups of the n - dimensional Hawai ian e arring , F und. Math. 165 (2000), 17–28. [9] U. Karimov and D. R ep o v ˇ s On nonc ontr actible c omp acta wi th trivial homolo gy and homotopy gr oups , Proc. Amer. Math. So c. 138 :4 (2010), 1525-1531. [10] U. Karimo v, D. Rep ov ˇ s, W. Rosic ki and A. Zastro w On two dimensional planar c om p acta not homotopy e quivalent to any one-dimensional c omp actum , T op ol. Appl. 153 (2005), 284–293 . [11] K. Kuratowski, T op olo gy II , Academic Press, New Y ork, 1968. [12] S. Marde ˇ si´ c and J. Segal, Shap e the ory , N orth-Holland, A msterdam, 1982 [13] W. Massey , Algeb r aic T op olo gy: An I ntr o duction , Grad. T ex ts Math. 56 , Springer, Berlin 1984. [14] S. B. Nadler, Jr., Continuum The ory. An I ntr o duction , Monogr. and T extb o oks in Pure an d Appl. Math. 158 , Marcel Dekker, Inc., New Y ork, 1992. [15] E. H. Spanier, Algebr aic T op olo gy , McGra w-Hill, New Y ork, 1966. Katsuya Eda School of S cience and Engineering, W aseda Universit y , T okyo 169-8555, Japan e-mail: eda@logic.i nfo.waseda.ac.j p Umed H. K arimo v Institute of Mathematics, Academy of Sciences of T a jikistan, U l. A iny 299 A , D u shanbe 734063, T a jikistan e-mail: umedkarimov @gmail.com Du ˇ san Rep ov ˇ s F aculty of Mathematics and Physics, and F aculty of Education, U niversi ty of Ljubl- jana, P .O.Bo x 2964, Ljubljana 1001, Slov enia e-mail: dusan.repov s@guest.arnes.s i