On the Capacity and Depth of Compact Surfaces
aa r X i v : . [ m a t h . A T ] D ec On the Capacity and Depth of Compact Surfaces
Mahboubeh Abbasi , Behrooz Mashayekhy ∗ Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures,Ferdowsi University of Mashhad,P.O.Box 1159-91775, Mashhad, Iran.
Abstract
K. Borsuk in 1979, in the Topological Conference in Moscow, introduced the conceptof capacity and depth of a compactum. In this paper, we compute the capacity anddepth of compact surfaces. We show that the capacity and depth of every compactorientable surface of genus g ≥ g + 2. Also, we prove that the capacityand depth of a compact non-orientable surface of genus g > g ] + 2. Keywords:
Capacity, Compact surface, Homotopy domination, Homotopy type,Eilenberg-MacLane space, Polyhedron, CW-complex, Compactum.
1. Introduction
K. Borsuk in [3], introduced the concept of capacity and depth of a compactum(compact metric space) as follows: the capacity C ( A ) of a compactum A is thecardinality of the set of all shapes of compacta X for which S h ( X ) S h ( A ). Asystem S h ( X ) < S h ( X ) < · · · < S h ( X k ) S h ( A ) is called a chain of length k forcompactum A . The depth D ( A ) of a compactum A is the least upper bound of thelengths of all chains for A . It is clear that D ( A ) ≤ C ( A ) for each compactum A .In the case polyhedra, the notions shape and shape domination in the abovedefinitions can be replaced by the notions homotopy type and homotopy domination,respectively. Indeed, by some known results in shape theory one can conclude thatfor any polyhedron P , there is a 1-1 functorial correspondence between the shapesof compacta shape dominated by P and the homotopy types of CW-complexes (not ∗ Corresponding author
Email addresses: [email protected] (Mahboubeh Abbasi), [email protected] (Behrooz Mashayekhy)
Preprint submitted to May 25, 2018 ecessarily finite) homotopy dominated by P (in both pointed and unpointed cases)[9]. S. Mather in [16] proved that every polyhedron dominates only countably manyof different homotopy types (hence shapes). Since the capacity of a topological spaceis a homotopy invariant, i.e., if topological spaces X and Y have the same homotopytype, then C ( X ) = C ( Y ), it is interesting to know that which topological spaceshave finite capacity and compute the exact capacity of them. Borsuk in [3] asked aquestion: “ Is it true that the capacity of every finite polyhedron is finite? ”. D.Kolodziejczyk in [13] gave a negative answer to this question. Also, in [8] she provedthat there exist polyhedra with infinite capacity and finite depth. Moreover, sheinvestigated some conditions for polyhedra to have finite capacity ([9, 10, 12]). Forinstance, polyhedra with finite fundamental groups and polyhdera P with abelianfundamental groups π ( P ) and finitely generated homology groups H i ( ˜ P ), for i ≥ W k S and S n are equal to k + 1and 2, respectively. Also, M. Mohareri et al. in [17] computed the exact capacityof the Moore space M ( A, n ) and the Eilenberg-MacLane space K ( G, n ) (in finite orinfinite cases). In fact, they showed that the capacity of a Moore space M ( A, n ) andan Eilenberg-MacLane space K ( G, n ) are equal to the number of semidirect factorsof A and G , respectively, up to isomorphism. Also, they computed the capacity ofthe wedge sum of finitely many Moore spaces of different degrees and the capacity ofthe product of finitely many Eilenberg-MacLane spaces of different homotopy types.In particular, they showed that the capacity of W n ∈ I ( ∨ i n S n ) is Q n ∈ I ( i n + 1) where ∨ i n S n denotes the wedge sum of i n copies of S n , I is a finite subset of N and i n ∈ N .In this paper, we compute capacity and depth of compact surfaces. In fact,we show that the capacity and depth of every compact orientable surface of genus g ≥ g + 2. Also, we prove that the capacity and depth of a compactnon-orientable surface of genus g > g ] + 2.
2. Preliminaries
In this paper every topological space is assumed to be connected. We expect thatthe reader is familiar with the basic notions and facts of shape theory (see [5] and[14]) and retract theory (see [4]). We need the following results and definitions forthe rest of the paper.
Definition 2.1. [6]. A topological space X having just one nontrivial homotopygroup π n ( X ) ∼ = G is called an Eilenberg-MacLane space and is denoted by K ( G, n ) . he full subcategory of the category hT op consisting of spaces K ( G, n ) with G ∈ Gp is denoted by K n . Theorem 2.2. [6]. The homotopy type of a CW complex K ( G, n ) is uniquely deter-mined by G and n . Theorem 2.3. [6]. The homology groups of K ( Z n , is Z n for odd n and zero foreven n > . Theorem 2.4. [1]. 1) A connected CW-space X is contractible if and only if all itshomotopy groups π n ( X ) ( n ≥ ) are trivial.2) A simply connected CW-space X is contractible if and only if all its homologygroups H n ( X ) ( n ≥ ) are trivial. Definition 2.5. [20]. Let φ : K −→ X be a cellular map between CW-complexeswith mapping cylinder M = X S φ ( K × I ) . We denote π n ( M, K × { } ) by π n ( φ ) . Themap φ is called n -connected if K and X are connected and π i ( φ ) = 0 for ≤ i ≤ n . Theorem 2.6. [15, Theorem 7.2]. Any compact, orientable surface is homeomorphicto a sphere or a connected sum of tori. Any compact, non-orientable surface ishomeomorphic to the connected sum of either a real projective plane or the Kleinbottle and a compact, orientable surface.
Lemma 2.7. [6, Example 1B.2]. Every (orientable or non-orientable) surface ofgenus g > , is an Eilenberg-MacLane space. Recall that the genus of connected sum of n tori or connected sum of n realprojective planes is n . Definition 2.8. [9]. A homomorphism g : G −→ H of groups is an r -homomorphism if there exists a homomorphism f : H −→ G such that g ◦ f = id H . In this case H is called an r -image of G . In particular, let G be a group with a subgroup H . Then H is called a retract of G if there exists a homomorphism r : G −→ H such that r ◦ i = id H where i : H −→ G is the inclusion homomorphism.Note that if a group H is an r -image of G with an r -homomorphism g and theconverse homomorphism f , then H ∼ = f ( H ) and f ( H ) is a retract of G .3 . The Capacity of Compact Surfaces Borsuk in [3] mentioned that the capacity of W k S and S n are equal to k + 1 and2, respectively. Also, M. Mohareri et al. in [17] showed that the capacity of a Moorespace M ( A, n ) and an Eilenberg-MacLane space K ( G, n ) are equal to the number ofsemidirect factors of A and G , respectively, up to isomorphism. In particular, theyproved that the capacity of W n ∈ I ( ∨ i n S n ) is Q n ∈ I ( i n + 1) where ∨ i n S n denotes thewedge sum of i n copies of S n , I is a finite subset of N and i n ∈ N .In this section, we compute the capacity of compact (orientable or non-orientable)surfaces. Definition 3.1. [20]. Let X be a connected CW-complex. The conditions F i and D i on X are defined as following: F : the group π ( X ) is finitely generated. F : the group π ( X ) is finitely presented, and for any 2-dimensional finite CW-complex K and any map φ : K −→ X inducing an isomorphism of fundamentalgroups, π ( φ ) is a finitely generated module over Z π ( X ) . F n : condition F n − holds, and for any ( n − -dimensional finite CW-complex K and any ( n − -connected map φ : K −→ X , π n ( φ ) is a finitely generated Z π ( X ) -module. D n : H i ( ˜ X ) = 0 for i > n , and H n +1 ( X ; B ) = 0 for all coefficient bundles B . Lemma 3.2. [20, Theorem F]. The CW-complex X is dominated by a finite CW-complex of dimension n if and only if X satisfies D n and F n . Lemma 3.3. [20, Proposition 3.3]. If CW-complex X satisfies D and F and π ( X ) is free, then X has the homotopy type of a finite bouquet of 1-spheres and 2-spheres. Lemma 3.4. [2]. Let X be a topological space which is homotopy dominated by aclosed (compact without boundary) connected topological n -dimensional manifold M .If H n ( X ; Z ) = 0 , then X has the homotopy type of M . Definition 3.5. [21]. The group G is called a surface group if G ∼ = π ( S ) for aclosed (compact without boundary) surface S with χ ( S ) < where χ ( S ) is the Eulercharacteristic of S . Recall that the Euler Characteristic of connected sum of n tori and connectedsum of n real projective planes are 2 − n and 2 − n , respectively. Lemma 3.6. [21, Lemma 4.5]. Let G be a surface group. If K is any proper retractof G , then K is a free group with rank K ≤ rank G where rank G is the minimalnumber of generators of G . heorem 3.7. The capacity of connected sum of n tori is equal to n + 2 .Proof. Suppose that S is connected sum of n tori. First, let n = 1. By [17, Propo-sition 4.6], the capacity of a torus is 3. Now, let n >
1. Then χ ( S ) <
0. Suppose A is homotopy dominated by S . Without loss of generality, we can suppose that H ( A ) = 0. Otherwise, if H ( A ) ∼ = Z , then H ( A ; Z ) = Hom ( H ( A ); Z ) = 0 andso by Lemma 3.4, A has the same homotopy type to S .Put G = π ( S ). We know that G is a finitely presented group with 2 n generatorsand one relation (see [15]) and that π ( A ) is isomorphic to a retract of G . We havetwo following cases:Case One. π ( A ) is isomorphic to a proper retract of G . It is clear that G is a surface group. Using Lemma 3.6, π ( A ) is a free group with rank π ( A ) ≤ rank G = n . Now by Lemma 3.2, A satisfies conditions D and F since S is afinite 2-dimensional CW-complex. Therefore by Lemma 3.3, A has the homotopytype of a bouquet of t copies of 1-spheres since π ( A ) is a free group of rank 0 ≤ t ≤ n (note that H ( A ) = 0). Hence, A has the form as one of following spaces: ∗ , S , S ∨ S , · · · , S ∨ · · · ∨ S | {z } n − folds . On the other hand, it is obvious that S ∨ · · · ∨ S | {z } n − folds is a retract of S . So, each ofspaces above is a retract of S .Case Two. π ( A ) ∼ = G . By Lemma 2.7, S (and also A ) is an Elenberg-MacLancespace. Then by Theorem 2.2, A and S have the same homotopy type. Theorem 3.8.
The capacity of the real projective plane is equal to 2.Proof.
Suppose that A is homotopy dominated by RP . Then π ( A ) ∼ = π ( RP )and so π ( A ) = 1 or π ( A ) ∼ = Z . Let π ( A ) = 1. Since H i ( A ) = 0 for i ≥ A is contractible by Theorem 2.4.(2). Now, let π ( A ) ∼ = Z . Suppose that ˜ A isthe universal covering of A . It is easy to see that ˜ A is homotopy dominated by S .Then, ˜ A has the same homotopy type with either a one-point space or S . If ˜ A hasthe same homotopy type with a one-point space, then π i ( A ) = 0 for i ≥ π i ( A ) ∼ = π i ( ˜ A ) for i ≥
2. Hence, A is an Eilenberg-MacLane space K ( Z , A has the same homotopy type with S .Since H i ( ˜ A ) ∼ = H i ( S ) for i ≥ π ( A ) ∼ = π ( RP ), A and RP have the samehomotopy type by Whitehead Theorem (see [7, Theorem 3.1, p.107]). Remark 3.9. [19]. Let G = H ⋉ ϕ U be the semidirect product of U and H withrespect to ϕ : H → Aut( U ) . For a subgroup Γ of G , put U Γ = Γ ∩ U and Γ H =5 h ∈ H | ( u, h ) ∈ Γ for some u ∈ U } . Group monomorphisms ρ : R −→ H and λ : L −→ U are called a ϕ HU − pair of groups if there exists a short exact sequence ofgroups → L α −→ T β −→ R → with a monomorphism µ : T −→ G such that the following diagram commutes L α / / λ ❍❍❍❍❍❍❍❍❍❍ T µ (cid:15) (cid:15) β / / R ρ z z ✈✈✈✈✈✈✈✈✈✈ H ⋉ ϕ U. We denote a ϕ HU − pair of groups by h L λ , R ρ , ϕ HU i . In 1991, Usenko [19] proved thateach ϕ HU − pair of groups determines a subgroup of the semidirect product G = H ⋉ ϕ U .On the other hand, each subgroup Γ G determines some ϕ HU − pair of groups h U i Γ , Γ jH , ϕ HU i , where i : U Γ −→ U and j : Γ H −→ H are embeddings.As a consequence, suppose G = Z ⋉ Z and Γ G . Since U Γ (and also Γ H )is trivial or Z , up to isomorphism, by the above argument Γ ∼ = Γ H ⋉ U Γ . Henceisomorphism classes of subgroups of G are trivial, Z , Z × Z and Z ⋉ Z . Theorem 3.10.
The capacity of the Klein bottle is equal to 3.Proof.
Suppose that K is the Klein bottle. We know that π ( K ) has a presentationas h x, y | yxy − = x − i ∼ = Z ⋉ Z (see [15]). By Remark 3.9, subgroups of π ( K )are trivial, Z , Z × Z and Z ⋉ Z , up to isomorphism. Then the capacity of K is3 (with retracts ∗ , S and K ). Note that A is not homotopy dominated by K if π ( A ) ∼ = Z × Z .Recall that by the classification theorem for compact surfaces, any compact ori-entable surface is homeomorphic to a sphere, or to the connected sum of n tori, andany compact non-orientable surface is homeomorphic to the connected sum of n realprojective planes. Hence by Theorem 2.6, we have following lemmas: Lemma 3.11.
Let S be a connected sum of m tori and a real projective plane. Then S ∨ · · · ∨ S | {z } m − folds is a retract of S .Proof. Suppose that q : T · · · T | {z } m − folds RP −→ T · · · T | {z } m − folds ∨ RP is the quotientmap. It is obvious that S ∨ · · · ∨ S | {z } m − folds is a retract of T · · · T | {z } m − folds . Also, it is easy to6ee that T · · · T | {z } m − folds is a retract of T · · · T | {z } m − folds ∨ RP . So, S ∨ · · · ∨ S | {z } m − folds is a retractof T · · · T | {z } m − folds ∨ RP . Hence, S ∨ · · · ∨ S | {z } m − folds is a retract of T · · · T | {z } m − folds RP since S ∨ · · · ∨ S | {z } m − folds is invariant under quotient map q (see [18, p.374]). Lemma 3.12.
Let S be connected sum of m tori and a Klein bottle. Then S ∨ · · · ∨ S | {z } ( m +1) − folds is a retract of S .Proof. Suppose that q : T · · · T | {z } m − folds K −→ T · · · T | {z } m − folds ∨K is the quotient map.It is obvious that S ∨ · · · ∨ S | {z } m − folds is a retract of T · · · T | {z } m − folds and S is a retract of K (byTheorem 3.10). Suppose that r : T · · · T | {z } m − folds −→ S ∨ · · · ∨ S | {z } m − folds and r : K −→ S are corresponding retractions.Now, we define map r : T · · · T | {z } m − folds ∨K −→ S ∨ · · · ∨ S | {z } ( m +1) − folds as follows: r ( x ) = r ( x ) q ( x ) ∈ T · · · T | {z } m − folds r ( x ) q ( x ) ∈ K . Let C be the common circle of connected sum of T · · · T | {z } m − folds and K . For all x ∈ C ,we have r i ( q ( x )) = r i ( ∗ ) = ∗ where i = 1 , ∗ is the common point of the wedgesum. It is obvious that r is continuous. It is enough to show that r is a retraction. If x ∈ S ∨ · · · ∨ S | {z } ( m ) − folds ( ⊆ T · · · T | {z } m − folds ), then r ( x ) = r ( x ) = x and if x ∈ S ( ⊆ K ), then r ( x ) = r ( x ) = x . Hence, S ∨ · · · ∨ S | {z } ( m +1) − folds is a retract of T · · · T | {z } m − folds ∨K . Now, one canobtain S ∨ · · · ∨ S | {z } ( m +1) − folds is a retract of T · · · T | {z } m − folds K since S ∨ · · · ∨ S | {z } ( m +1) − folds is invariantunder quotient map q . Theorem 3.13.
The capacity of connected sum of n real projective planes is equalto [ n ] + 2 . roof. Suppose that S is the connected sum of n real projective planes and A ishomotopy dominated by S .First, let n = 1. By Theorem 3.8, the capacity of a real projective plane is 2.Next, let n = 2. In this case, S is a Klein bottle and so by Theorem 3.10, thecapacity of S is 3. Now, suppose that n >
2. Then, χ ( S ) <
0. Put G = π ( S ). Weknow that G is a finitely presented group with n generators and one relation (see[15]). We have two following cases:Case One. Let n = 2 m + 1. Then, S is homeomorphic to connected sum of areal projective plane and m tori. It is obvious that π ( A ) is isomorphic to a retractof G and G is a surface group. If π ( A ) is isomorphic to a proper retract of G , thenby Lemma 3.6, π ( A ) is a free group with rank π ( A ) ≤ rank G = n . Now,by Lemma 3.2, A satisfies conditions D and F since S is a finite 2-dimensionalCW-complex. Therefore by Lemma 3.3, A has the homotopy type of bouquet of tcopies of 1-spheres since π ( A ) is a free group of rank 0 ≤ t ≤ [ n ] = m . Hence, A has the form as one of following spaces: ∗ , S , S ∨ S , · · · , S ∨ · · · ∨ S | {z } m − folds . On the other hand, by Lemma 3.11, S ∨ · · · ∨ S | {z } m − folds is a retract of S . So, each of spacesabove can be a retract of S .Now, if π ( A ) ∼ = G , then by Theorem 2.2, A and S have the same homotopy typesince by Lemma 2.7, S (and also A ) is an Eilenberg-MacLance space. Hence, thecapacity of S is equal to m + 2 = [ n ] + 2.Case Two. Let n = 2 m . Then, S is homeomorphic to connected sum of a Kleinbottle and m − A has the form as oneof following spaces: ∗ , S , S ∨ S , · · · , S ∨ · · · ∨ S | {z } m − folds , S . By Lemma 3.12, S ∨ · · · ∨ S | {z } m − folds is a retract of S . Thus each of the above spaces canbe a retract of S . Hence, the capacity of S is equal to m + 2 = [ n ] + 2.Using Theorems 3.7 and 3.13 the following result can be easily concluded. Corollary 3.14.
The depth of orientable compact surfaces of genus g ≥ is g + 2 .Moreover the depth of non-orientable compact surfaces of genus g ≥ is [ g ] + 2 . eferenceseferences