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2-dimensional polyhedra with finite depth
Topology and its Applications 241 (2018) 50–61
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Topology and its Applications www.elsevier.com/locate/topol
Virtual Special Issue – In memory of Professor Sibe Mardešić
2-dimensional polyhedra with finite depth Danuta Kołodziejczyk Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 27 February 2017 Received in revised form 27 December 2017 Available online 6 March 2018 Dedicated to the memory of Prof. Sibe Mardešić MSC: primary 55P15 secondary 55P10, 55P55 Keywords: Polyhedron CW -complex AN R F AN R Homotopy type Homotopy domination Shape Depth Elementary amenable group Virtually-solvable group Limit group
a b s t r a c t In this paper every polyhedron is finite and every AN R is compact. Let P ≥ X1 ≥ X2 ≥ . . ., be a sequence, in which P is a polyhedron and ≥ are homotopy dominations. One may ask, if each sequence of this form contains only finitely many homotopy dominations that are not homotopy equivalences, or if there exists an integer lP (independent of the sequence) that each sequence contains only ≤ lP homotopy dominations that are not homotopy equivalences. Closely related open questions are: Does there exist a polyhedron P homotopy dominating an infinite sequence of polyhedra {Pi }, where Pi homotopy dominates Pi+1 but Pi and Pi+1 have different homotopy types, for every i ∈ N ? (M. Moron) [30, Problem 1436], and the famous problem of K. Borsuk (1967): Is it true that two AN R s homotopy dominating each other have the same homotopy type? Here we prove that, if dim P = 2, then the answers to all these questions depend only on the properties of the fundamental group of P (for 1-dimensional polyhedra, the answers are obvious). Furthermore, if sequences in consideration contain only polyhedra, then, for the positive answer it suffices to answer positively the analog of our topological question for finitely presented groups (the fundamental groups) with retractions. Applying these results, we prove that for each polyhedron P with dim P ≤ 2 and elementary amenable fundamental group G with cdG < ∞, there is a bound lP on the lengths of all descending sequences P ≥ X1 ≥ X2 . . . of homotopy dominations that are not homotopy equivalences. The same holds if the fundamental group of P is a limit group. It means that such polyhedra P have finite depth. © 2018 Elsevier B.V. All rights reserved.
1. Introduction In this paper every polyhedron is finite and every AN R is compact. We assume (without loss of generality) that every polyhedron, AN R, F AN R and CW -complex is connected. Recall that a domination in a given category C is a morphism f : X → Y , X, Y ∈ ObC, for which there exists a morphism g : Y → X of C such that f g = idY . Then we say that Y is dominated by X, and we write Y ≤ X, or X ≥ Y . E-mail addresses: [email protected], [email protected]. URL: https://www.mini.pw.edu.pl/~dkolodz. https://doi.org/10.1016/j.topol.2018.02.016 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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We write that X < Y iff X ≤ Y holds but Y ≤ X fails. In 1968 K. Borsuk asked (in the shape-theoretical language): Is it true that every polyhedron homotopy dominates only finitely many different homotopy types? [4]. We showed in [24] that the answer to this question is negative — there exists a finite polyhedron (even 2-dimensional) dominating infinitely many different homotopy types (of 2-dimensional polyhedra). Furthermore, counterexamples to Borsuk’s question are possible even with polycyclic fundamental groups [25,22]. Now, one can consider sequences of spaces homotopy dominated by a given polyhedron P with a quasiorder determined by the relation of homotopy domination P ≥ X1 ≥ X2 ≥ . . . . Recall that, by the classical results of J.H.C. Whitehead and C.T.C. Wall, each topological space homotopy dominated by a polyhedron has the homotopy type of a CW -complex, but not necessarily finite [31]. Let us state the following question: Problem 1. Does there exist a sequence P ≥ X1 ≥ X2 ≥ . . ., where P is a polyhedron, containing infinitely many homotopy dominations that are not homotopy equivalences? This problem is closely related to other natural open questions published in the literature. One of them is the famous problem of K. Borsuk posed in 1967, in his monograph “Theory of Retracts” [6, Ch. IX, Problem (12.7)]: Problem 2. Is it true that two AN R s homotopy dominating each other have the same homotopy type? Since, by the result of J. West [33], every AN R has the homotopy type of a polyhedron, this question is equivalent to its analog for polyhedra. In [13] the same problem was stated for F AN R s, compacta which are generalizations of ANR’s, and correspond in the shape category to CW -complexes dominated by polyhedra in the homotopy category (see below). Recall that on AN R’s shape and homotopy theory coincide (see [29], [14], [5]). An other problem of the similar nature was posed by M. Moron and published in [30, Problem 1436 (or Problem 9), p. 672]: Problem 3. Is there a polyhedron P dominating a sequence of polyhedra {Pi } where Pi homotopy dominates Pi+1 , but Pi and Pi+1 have different homotopy types, for every i ∈ N ? In the 1970’s K. Borsuk introduced the following notion of depth (in the shape category of compacta, see [4]): A system X1 < X2 < . . . < Xk ≤ A, where Xi ∈ ObC, for i = 1, 2, . . . , k, is called a chain of length k for A ∈ ObC. The depth D(A) of A is the least upper bound of the lengths of all chains for A. If this upper bound is infinite, we write D(A) = ℵ0 . We will consider depth in the homotopy category of CW -complexes. However, there is a 1–1 functorial correspondence between the shapes of compacta shape dominated by a given polyhedron and the homotopy types of CW -complexes homotopy dominated by it (see [14, Theorem 2.2.6]; [15]). Moreover, the depth of a given polyhedron is the same in both, pointed and unpointed, cases (it follows from [12, Theorem 5.1]). Clearly, the following problem is also closely related to Problems 1, 2 and 3: Problem 4. Does there exist a polyhedron P with infinite depth D(P )? In this paper we study 2-dimensional polyhedra. In dimension 1 the answers to all the above questions are obvious. This follows from the fact that every 1-dimensional polyhedron has the homotopy type of a finite wedge of the circles S 1 , hence (as a K(G, 1)) space homotopy dominates just over the polyhedra of the same kind (with not greater Betti numbers).
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For all polyhedra with virtually-polycyclic (i.e., polycyclic-by-finite) fundamental groups the answers to Problems 1, 3, 4 are also negative (hence Problem 2 has positive answer), by the results of the author published in [21, Theorem 5] and in [23, Theorem 1]. In particular, every polyhedron with virtually-polycyclic fundamental group has finite depth. We proved in fact that for such a polyhedron there is a bound on the lengths of all descending sequences P ≥ X1 ≥ X2 . . . of homotopy dominations that are not homotopy equivalences [23, Theorem 1]. In this paper we show the same for all 2-dimensional polyhedra whose fundamental groups are elementary amenable with finite cohomological dimension or limit groups. The class of finitely presented elementary amenable groups is larger than the class of virtually-polycyclic groups (which are always finitely presented and finitely dimensional) and includes all the virtually-solvable groups. (Recall that every polycyclic group is virtually solvable.) Limit groups (introduced by Z. Sela) are generalizations of free groups (and are nonsolvable except of the case of free abelian groups). Thus, for 2-dimensional polyhedra with these fundamental groups, Problems 1, 3, 4 have negative answers, and the answer to Problem 2 is positive. In general, we prove that in the case of 2-dimensional polyhedra the answers to the questions in consideration depend only on properties of the fundamental group. One of the main theorems states that if we consider sequences containing only polyhedra (as in Problem 3), it suffices to answer (negatively) the analog of the topological question for finitely presented groups, with retractions instead of homotopy dominations (compare Section 6). We do it for the classes of fundamental groups pointed out above. Recall that a group is weakly Hopfian if it cannot be isomorphic to a proper retract of itself. Applying main results we show here that, for two AN R’s of dimension 2 with weakly Hopfian fundamental groups, the answer to Problem 2 is positive. Obviously, every Hopfian group is weakly Hopfian. There is not known an example of a finitely presented group that is not weakly Hopfian. In the paper we use the terms “polyhedron” and “finite CW -complex” interchangeably. Without loss of generality, we may study pointed spaces and maps (see [12, Theorem 5.1]). For the basic notions and results of shape theory we refer the reader to [14], [29], [5]. 2. Algebraic preliminaries Definition 1. (i) A group G is Hopfian iff every epimorphism f : G → G is an automorphism (equivalently, N = 1 is the only normal subgroup for which G/N ∼ = G). (ii) A group G weakly Hopfian iff every r-homomorphism (see [6]) r : G → G ∼ = G imply = G is an isomorphism. In other words, G = K H and H ∼ K = 1 (where G = K H means that G = KH, K G, H < G, K ∩ H = 1; that is, H is a retract of G). Definition 2. (i) By a cohomological dimension of a group G we mean cdG = sup{n such that H n (G, M ) = 0, for some ZG-module M } (see [8], [3]). (ii) A geometrical dimension, gdG, of a group G is the smallest dimension of a CW -complex K(G, 1) (see, for example, [8]). Definition 3. (i) A group G is said to be geometrically finite if there exists a finite CW -complex K(G, 1). (ii) If cdG < ∞, then G is called finitely dimensional. (Recall that the condition cdG < ∞ is equivalent to the condition gdG < ∞ [8].) Definition 4. Let P, S be classes of groups. A group G is said to be P-by-S if it has a normal subgroup A G, A ∈ P such that G/A ∈ S. Definition 5. A finitely generated projective Λ-module M is called stable free iff there exist finitely generated free Λ-modules Λr and Λs , where r and s are integers, such that M ⊕ Λr ∼ = Λs (Λn denotes the direct sum of n copies of Λ).
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Definition 6. Let Λ be a ring. (i) Let K 0 (Λ) = F (P)/R(P), where P is the set of isomorphism classes [M ] of all the finitely generated projective Λ-modules M , F (P) is the free abelian group with the base P, and R(P) is the subgroup of F (P) generated by elements of the form [M1 ⊕ M2 ] − [M1 ] − [M2 ]. ˜ 0 (Λ) we denote the quotient of K 0 (Λ), such that two finitely generated projective Λ-modules Q1 (ii) By K ˜ 0 (Λ) iff there exist finitely generated free Λ-modules R1 and R2 satisfying and Q2 have the same class in K Q1 ⊕ R1 ∼ = Q2 ⊕ R2 . Then we write {Q1 } = {Q2 } (see [31, p. 64]). 3. Main results In what follows we prove that for 2-dimensional polyhedra, the answers to all the problems in consideration depend only on the properties of the fundamental groups. For more, if we examine sequences containing only polyhedra (as in Problems 2 and 3), then sometimes it suffices to study the analogous questions for the fundamental groups with retractions. We begin with the definition and the main property of the Wall obstruction. Let ϕ : K → X, where K be a CW -complex and X have the homotopy type of one. If ϕ : K → X is 2-connected, then, for all k ≥ 3, πk (ϕ) and Hk (ϕ) ˜ can be regarded as Zπ1 (X)-modules (see, for example, [11, pp. 28–29]). To every X homotopy dominated by a finite CW -complex (such an X has always the homotopy type of ˜ 0 (Zπ1 (X)), where a CW -complex), Wall assigned the so called Wall obstruction of X, denoted by σ(X) ∈ K Zπ1 (X) is the integral group ring of π1 (X). The obstruction σ(X) is an invariant of the homotopy type of X which vanishes iff X is homotopy equivalent to a finite complex (compare [31, pp. 64–65]). Definition 7. If X ≤ P , where P is a polyhedron and dim P = n ≥ 2, then one can find an n-connected map ϕ : Ln → X of a finite CW -complex Ln with dim Ln = n into X (see [31, p. 65]). The Wall obstruction of ˜ 0 (Zπ1 (X)) defined as (−1)n {πn+1 (ϕ)}. X, σ(X), is the element of the group K (In this situation {πn+1 (ϕ)} is a finitely generated, projective Zπ1 (X)-module.) Remark 1. Precisely, Wall proved that if X is homotopy dominated by a polyhedron P with dim P = m ≥ 2 ˜ 0 (Zπ1 (X)) = 0, then X is homotopy equivalent to a polyhedron of dimension ≤ max(m, 3) (see [31, and K Theorem F, p. 66]). In some previous paper we showed that if P is a polyhedron with dim P = 2 such that π1 (P ) is weakly Hopfian, then every homotopy domination of P over itself is a homotopy equivalence [20, Theorem 1]. The same proof gives in fact the following: Theorem 1. Let P be a polyhedron with dim P = 2. Then every homotopy domination of P over itself which induces an isomorphism on the fundamental group is a homotopy equivalence. Proof. See the proof of [20, Theorem 1], steps (ii)–(vii).
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In the sequel we will use the following theorem, known as the Whitehead Theorem on Trees ([35, Theorem 14; 34]): Theorem. Let X and Y be finite polyhedra with dim X = 2 = dim Y , π1 (X) ∼ = π1 (Y ). Then there exist 1 integers mX and mY such that 1
k
S 2 , where k ∈ N , denotes the wedge of k spheres S 2 .
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X∨
mX
S2 Y ∨
S2.
mY
Applying Theorem 1, and the Whitehead Theorem on Trees, we will obtain: Theorem 2. Let P and Q be polyhedra homotopy dominated by 2-dimensional polyhedra, π1 (P ) ∼ = π1 (Q) and χ(P ) = χ(Q). Then every homotopy domination between P and Q that induces an isomorphism on the fundamental groups is a homotopy equivalence. Proof. By the main results of [31] (compare Remark 1) we may assume, that dim P ≤ 3, dim Q ≤ 3. In the proof we will use the following [10, Cor. 2, p. 412]: (∗) If X is a finite complex dominated by a 2-complex, then there is a finite wedge of 2-spheres W such that X ∨ W X (2) , where X (2) is the 2-sceleton of X. Let P and Q be polyhedra homotopy dominated by 2-dimensional polyhedra, dim P ≤ 3, dim Q ≤ 3, π1 (P ) ∼ = π1 (Q) and χ(P ) = χ(Q). By (∗), there exist two finite wedges of spheres S 2 , WP and WQ , such that P ∨ WP P (2) and Q ∨ WQ (2) Q (respectively), where P (2) and Q(2) are the 2-sceleta of P and Q. By the Whitehead Theorem on Trees, there exist integers mP and mQ such that P (2) ∨ WP Q(2) ∨ WQ , 2 2 where WP = S , WQ = S . mP
mQ
Therefore, P ∨ WP ∨ WP Q ∨ WQ ∨ WQ . Let us observe that since χ(P ) = χ(Q), then WP ∨ WP = WQ ∨ WQ . Thus, we have P¯ P ∨ W Q ∨ W , where W = WP ∨ WP is a finite wedge of spheres S 2 and P¯ = P (2) ∨ WP is a polyhedron with dim P¯ ≤ 2. Now, let d : P → Q be a homotopy domination which induces an isomorphism d∗1 : π1 (P ) → π1 (Q) on the fundamental groups. As a homotopy domination, d induces an epimorphism d∗2 : π2 (P ) → π2 (Q) on the second homotopy groups. By the classical Whitehead Theorem (see [19]), if P and Q are homotopy dominated by 2-dimensional polyhedra, then d : P → Q is a homotopy equivalence iff it induces isomorphisms d∗r : πr (P ) → πr (Q), for r ≤ 2. Suppose, on the contrary, that d is not a homotopy equivalence. It means that d∗2 : π2 (P ) → π2 (Q) is not a monomorphism. Since d : P → Q is a homotopy domination, it is easily seen that the map d¯ : P ∨ W → Q ∨ W defined ¯ = d(x), for x ∈ P and d(x) ¯ = x, for x ∈ W , is also a homotopy domination. as d(x) ¯ Observe that d induces an isomorphism d¯∗1 : π1 (P ∨W ) → π1 (Q ∨W ) on the fundamental groups. On the other hand, d¯ does not induce a monomorphism d¯∗2 : π2 (P ∨W ) → π2 (Q ∨W ). Indeed, d∗2 : π2 (P ) → π2 (Q) is not a monomorphism, and the homomorphism i∗2 : π2 (P ) → π2 (P ∨ W ), induced by the inclusion i : P → P ∨ W , is a monomorphism (P is a retract of P ∨ W ). Therefore, there exists a homotopy domination d¯¯ : P¯ → P¯ with the same properties, i.e., such that d¯∗1 : π1 (P¯ ) → π1 (P¯ ) is an isomorphism on the fundamental group, and d¯∗2 : π2 (P¯ ) → π2 (P¯ ) is not a monomorphism. Thus d¯¯ is not a homotopy equivalence. (Indeed, a composition of a homotopy domination with a homotopy equivalence, on the left or on the right, is again a homotopy domination.) Since dim P¯ ≤ 2, this contradicts Theorem 1, and the proof is finished. 2 As a corollary to the above theorem we obtain the following: Corollary 1. Let G be weakly Hopfian. Let P and Q be polyhedra with dim P = dim Q = 2 (or polyhedra homotopy dominated by 2-polyhedra), π1 (P ) ∼ = π1 (Q) ∼ = G and χ(P ) = χ(Q). Then every homotopy domination d : P → Q is a homotopy equivalence.
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Proof. It follows directly from the previous theorem — if G is weakly Hopfian and π1 (P ) ∼ = π1 (Q) ∼ = G, then the epimorphism d∗1 : π1 (P ) → π1 (Q) induced on the fundamental groups by the homotopy domination d is clearly an isomorphism. 2 Applying Theorem 2, we can prove: Theorem 3. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the condition: (Gf) Each sequence of groups and r-homomorphisms G → G1 → G2 → . . . contains only finitely many homomorphisms that are not isomorphisms. Then each sequence P ≥ P1 ≥ P2 ≥ . . . of polyhedra with homotopy dominations contains only finitely many dominations that are not homotopy equivalences. Proof. Assume that P is a polyhedron with dim P ≤ 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G. Let P = P0 → P1 → P2 → . . . be a sequence of polyhedra with homotopy dominations di : Pi−1 → Pi , for i ∈ N . Since G satisfies (Gf), there exists an integer k such that, for i ≥ k, all the epimorphisms di ∗1 : π1 (Pi−1 ) → π1 (Pi ) induced by the maps di on the fundamental groups are isomorphisms. Similarly, there exists an integer l such that for i ≥ l, χ(Pi−1 ) = χ(Pi ). Indeed, the homology groups H1 (P ) and H2 (P ), are finitely generated Abelian, thus also satisfy the condition (Gf) (all of their subgroups are also finitely generated Abelian, hence Hopfian). Clearly, Hr (P ) = 0, for r ≥ 3. By Theorem 2, there exists an integer m = max(k, l) such that for i ≥ m, all the homotopy dominations di : Pi−1 → Pi , are homotopy equivalences, which ends the proof. 2 Therefore, for 2-dimensional polyhedra (and polyhedra homotopy dominated by 2-dimensional polyhedra) the answer to Problem 3 depends only on the similar property of the fundamental group: Corollary 2. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the condition: (Gf) Each sequence of groups and r-homomorphisms G → G1 → G2 → . . . contains only finitely many homomorphisms that are not isomorphisms. Then there is no infinite sequence of polyhedra {Pi } where P = P1 , such that Pi ≥ Pi+1 but Pi and Pi+1 have different homotopy types for every i ∈ N . Proof. This follows immediately from the previous theorem. 2 Using the results of Wall, we obtain: Theorem 4. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the conditions: (Gf) Each sequence of groups and r-homomorphisms G → G1 → G2 → . . . contains only finitely many homomorphisms that are not isomorphisms, ˜ 0 (ZH) = 0, for each retract H of G. (W) K
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Then each sequence P ≥ X1 ≥ X2 ≥ . . . of homotopy dominations contains only finitely many dominations that are not homotopy equivalences. Proof. Assume that P is a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G. Let P = X0 → X1 → X2 → . . . be a sequence of topological spaces with homotopy dominations ˜ 0 (ZH) = 0, for each retract H of G, then, by the main result of [31] di : Xi−1 → Xi , for i ∈ N . Since K (compare Remark 1), every Xi in the above sequence has the homotopy type of a polyhedron of dimension ≤ 3. Thus, the assertion follows from Theorem 3. 2 With a stronger condition on G we conclude that 2-dimensional polyhedra with π1 (P ) ∼ = G have finite depth. Theorem 5. Let P be a polyhedron with dim P = 2 (or P is homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the conditions: (Gd) There exists an integer dG such that each sequence of groups and r-homomorphisms G → G1 → G2 → . . . → Gk contains at most dG r-homomorphisms that are not isomorphisms, ˜ 0 (ZH) = 0, for each retract H of G. (W) K Then there exists an integer kP such that each sequence P ≥ X1 ≥ X2 ≥ . . . of homotopy dominations contains only at most kP dominations that are not homotopy equivalences. Thus D(P ) < ∞. ˜ 0 (ZH) = 0, for each retract H of G, then all the Xi in the Proof. As in the proof of Theorem 4, since K above sequence may be assumed to be polyhedra of dimension ≤ 3. Then the proof is the same as the proof of Theorem 3. The only one difference is that now, the integer k from the proof of Theorem 3 can be chosen independently of the sequence P → P1 → P2 → . . . . (It is clear that the integer l in the proof of Theorem 3 can be always chosen independently of the sequence). 2 In [20] we proved the following theorem (see [20, Theorem 2]): Theorem 6. Let P be a polyhedron with dim P = 2 such that π1 (P ) is weakly Hopfian. If there exist homotopy dominations P ≥ Q and Q ≥ P , then P Q. Now we will show that the same holds for 2-dimensional ANR’s. Theorem 7. Let X, Y ∈ AN R s, dim X = 2 and π1 (X) is weakly Hopfian. If there exist homotopy dominations X ≥ Y and Y ≥ X, then X Y . Proof. Assume that X, Y ∈ AN R s, dim X = 2 and π1 (X) is weakly Hopfian. By [14; 29, p. 318], every topological space X which is homotopy dominated by some AN R of dimension ≤ n is homotopy dominated by a polyhedron of dimension ≤ n. In particular, if X ∈ AN R and dim X = 2, then X is homotopy dominated by a polyhedron of dimension 2 (clearly, the same holds for Y ). By the result of West [33], every AN R has the homotopy type of a polyhedron. Thus, X and Y have both the homotopy types of some polyhedra, so the Wall obstructions of X and Y vanish. By the Wall’s result (see Remark 1), both X and Y have the homotopy types of polyhedra of dimension ≤ 3. Thus there exist two polyhedra P and Q (where P X and Q Y ) homotopy dominated by a 2-dimensional polyhedron, such that P ≤ Q and Q ≤ P . Since all the homology groups of P and Q are isomorphic, then χ(P ) = χ(Q). We apply Corollary 1, and the proof is finished. 2
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4. Corollaries to limit groups as fundamental groups We will apply our results to the class of limit groups (introduced by Z. Sela) which is a generalization of the class of free groups: Definition 8 (Limit groups). A finitely generated group G is called a limit group if, for any finite subset S ⊂ G, there exists a homomorphism f : G → F (where F is a free group of finite rank) so that the restriction of f to S is injective. Example 1. Examples of limit groups include: finite-rank free abelian groups, fundamental groups of closed surfaces with the Euler characteristic less then −1, some doubles. Remark 2. (i) Limit groups are always finitely presented [1]. (ii) All the limit groups are geometrically finite and torsion-free [1]. (iii) Finitely generated subgroups of limit groups are limit groups [1]. (iv) Limit groups are non-solvable except of free abelian groups. In the sequel we will need the following two lemmas: Lemma 1. Let G be a limit group. Then there exists an integer kG such that any descending sequence of subgroups of G with proper retractions ri : Gi−1 → Gi , G = G0 ⊇ G1 ⊇ . . . Gi ⊇ Gi+1 ⊇ . . . , for i = 1, 2, . . . , contains no more than kG distinct subgroups. Proof. To see this, we may use [27, Theorem 1.1] stating that there is a function D(n) such that if Fn → L1 → . . . → Lk is a sequence of proper epimorphisms of limit groups, where Fn is a free group with n-generators, then k ≤ D(n). Thus, in particular, take L1 equal to G which is a finitely presented group (as a limit group, compare Remark 2), then there exists an integer kG such that for every sequence with proper retractions G = L1 → G2 . . . → Gk , k ≤ kG . Indeed, every retract Gi of a limit group G is a finitely generated subgroup of G [31, Lemma 1.3]. Thus, as a finitely generated subgroup of a limit group, it is also a limit group (see Remark 2), and we can apply [27, Theorem 1.1]. 2 ˜ 0 (ZG) = 0, and the same holds for all the retracts of G. Lemma 2. Let G be a limit group. Then K Proof. Observe that each retract of a limit group is also a limit group (see the proof of Lemma 1). It is known that all the limit groups are CAT (0) (see [1]; [2, p. 636]). According to [2], every finitedimensional CAT (0) group belongs to the class B of groups satisfying the Borel Conjecture. Moreover, for ˜ 0 (RG) = 0, if R is a principal ideal domain (see [2, p. 633]). every torsion-free group G from B, K Every limit group is finitely-dimensional and torsion-free (see Remark 2). Thus, for every limit group G, ˜ 0 (ZG) = 0, and the same holds for all the retracts of G. This ends the proof. 2 we have K As applications of the main results, we obtain Corollary 3. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) such that π1 (X) is a limit group. Then there exists an integer lP such that each sequence P ≥ X1 ≥ X2 ≥ . . . Xk contains at most lP homotopy dominations that are not homotopy equivalences. Thus, D(P ) < ∞. Proof. It follows from Theorem 5 and Lemmas 1 and 2. 2
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Corollary 4. Let P be a polyhedron P with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) such that π1 (P ) is a limit group. Then there is no infinite sequence of polyhedra {Pi } homotopy dominated by P , such that Pi ≥ Pi+1 but Pi and Pi+1 have different homotopy types for every i ∈ N . Proof. This is a consequence of Corollary 3.
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5. Corollaries to elementary amenable fundamental groups Definition 9. Elementary amenable groups is the smallest class of groups that contains all abelian and all finite groups, and is closed under extensions and directed unions (see [26], [7, p. 223]). Remark 3. Every virtually-solvable group is elementary amenable, because each solvable group is well known to be a poly-abelian group. In the sequel we will use properties of the Hirsch length of elementary amenable groups. Definition 10 (see [7, Definition I.15, p. 223]). The Hirsch length, h(G), of an elementary amenable group is finite and equal to n ≥ 0, if G has a series 1 = H0 . . . Hr = G in which the factors are either locally finite or infinite cyclic, and exactly n factors are infinite cyclic. In all other cases, h(G) = ∞. (Recall that a group is locally finite if all its finitely generated subgroups are finite). In the next proof we will also use the following lemma [20, Lemma 1]: Lemma 3. Let G be a countable elementary amenable group with cdG < ∞. Then h(G) = 0 iff G = 1. Applying properties of the Hirsch length of elementary amenable groups, we will prove: Lemma 4. Let G be an elementary amenable group with cdG < ∞. Then exists an integer kG such that any descending sequence G = G0 ⊇ G1 ⊇ . . . Gi ⊇ Gi+1 ⊇ . . . of subgroups of G with retractions ri : Gi−1 → Gi , for i = 1, 2, . . . , contains no more than kG distinct subgroups. Proof. Let G be a countable elementary amenable group. Then each retract of G is also countable, elementary amenable (the class of elementary amenable groups is closed under taking subgroups [9]). Since cdG < ∞, then, h(G) < ∞ [16, Theorem 5] (cf. also [20, Remark 6]). It is known that H ⊆ G implies that cdH < cdG [8]. Thus, we have cdH < ∞, hence h(H) < ∞, also for each retract H of G. If h(G) is a Hirsch number of a given elementary amenable group G with finite cdG, H ⊆ G and N G is such that H = G/N , then h(G) = h(N ) + h(H) [17, Theorem 1]. Let Mi be the kernel of the retraction ri : Gi−1 → Gi , i.e. Mi Gi−1 satisfies Gi−1 = Gi Mi and Gi ∩ Mi = 1, for i = 1, 2, . . . . Then we have h(Gi−1 ) = h(Mi ) + h(Gi ), for i = 1, 2, . . . , thus
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h(G) = h(M1 ) + h(M2 ) + . . . h(Mi ) + h(Gi ). Hence, there exists an integer k such that h(Mi ) = 0 for all i except no more than k. As a subgroup of a countable elementary amenable group with finite cohomological dimension, Mi is also elementary amenable, countable and finite dimensional. Then, by Lemma 3, Mi = 1 for such i. Therefore we get immediately that Gi = Gi−1 for these i. Thus, every descending sequence of subgroups of G, G = G0 ⊇ G1 ⊇ . . . ⊇ Gi ⊇ Gi+1 ⊇ . . . , with retractions ri : Gi−1 → Gi , for i = 1, 2, . . . , contains no more than k different groups, where k−1 = h(G) is the Hirsch number of G. This finishes the proof. 2 ˜ 0 (ZG) = 0, and the Lemma 5. Let G be a countable elementary amenable group with cdG < ∞. Then K same holds for all the retracts of G. Proof. Let G be a countable elementary amenable group with cdG < ∞. Then, in particular, G is virtually solvable [16, Theorem 5; 18]. Recently it was shown in [32, Theorem 1.1] that all virtually-solvable groups belong to the class FJ of groups satisfying the Full Farrell–Jones Conjecture (see [28, 11.5], [28, 14.2, Theorem 14.1]). Moreover, every subgroup of a group from the class FJ also belongs to FJ (see, for example, [28, Theorem 11.21, p. 337]). ˜ 0 (ZG) = 0 (see [28, p. 352]). Since It is also known that for each torsion-free group G from FJ , K G is finitely-dimensional, G is torsion-free [8]. Thus each subgroup of G is also torsion-free. Therefore, ˜ 0 (ZG) = 0, and the same holds for all the retracts of G, and the proof is finished. 2 K As corollaries to the main result, we get: Corollary 5. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron), with elementary amenable fundamental group of finite cohomological dimension. Then there exists an integer lP such that each sequence P ≥ X1 ≥ X2 ≥ . . . Xk contains at most lP homotopy dominations that are not homotopy equivalences. Thus, D(P ) < ∞. Proof. This is a corollary to Theorem 5 and Lemmas 4 and 5. 2 Corollary 6. Let P be a polyhedron P with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron), with elementary amenable fundamental group of finite cohomological dimension. Then there is no infinite sequence of polyhedra {Pi } homotopy dominated by P , such that Pi ≥ Pi+1 but Pi and Pi+1 have different homotopy types for every i ∈ N . Proof. We get this immediately from Corollary 5.
2
Remark 4. Since every virtually-polycyclic group is known to be virtually-solvable and always has finite cohomological dimension, these results generalize in dimension 2 the results of [21] and [23] for polyhedra with virtually-polycyclic fundamental groups. 6. Final remarks and open problems We will finish with some questions. Since there is no example of a finitely presented group that is not weakly Hopfian, let us state the following:
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Problem 5. Does there exist a finitely presented group that is not weakly Hopfian? Problem 6. Is it true that each finitely presented group with finite cohomological dimension is weakly Hopfian? Remark 5. Obviously, every Hopfian group is weakly Hopfian. The class of Hopfian groups includes the following and many others: torsion-free hyperbolic groups, Knot groups, finitely generated linear groups, Braid groups, Limit groups, countable elementary amenable groups of finite cohomological dimension (including virtually-solvable groups of finite cohomological dimension), many one-relator groups (see [20, Remark 5, Lemma 2]). The next questions are closely related to Problems 2 and 3. Problem 7. Do there exist two finitely presented groups G and H such that H G with r-homomorphisms r1 : G → H and r2 : H → G? Problem 8. Does there exist a finitely presented group G with an infinite sequence of groups Gi and rhomomorphisms G → G1 → G2 → . . . that are not isomorphisms? Problem 9. Does there exist a finitely presented group G of finite cohomological dimension with an infinite sequence of groups Gi and r-homomorphisms G → G1 → G2 → . . . that are not isomorphisms? It should be noted that an answer to the following related question is also unknown: Problem 10. Does there exist a finitely presented group G with infinitely many different retracts up to isomorphism? Remark 6. A famous conjecture due to Hsiang (see [28]) states that for any torsion-free group G, we have ˜ 0 (ZG) = 0. Let us also remark that it is known that if a group G is finitely dimensional, then it is K torsion-free (see [8]). The results of this paper are generalizations of my results on this subject included into my lecture “Homotopy dominations within 2-dimensional polyhedra” at the “Dubrovnik VII Topology Conference”, Croatia, 2011. At this conference I met Prof. Sibe Mardešić for the last time. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
E. Alibegovic, M. Bestvina, Limit groups are CAT(0), J. Lond. Math. Soc. (2) 74 (2006) 259–272. A. Bartels, W. Lück, The Borel conjecture for hyperbolic and CAT(0)-groups, Ann. Math. 175 (2) (2012) 631–689. D.J. Benson, P.H. Kropholler, Cohomology of groups, in: Handbook of Algebraic Topology, Elsevier Science, 1995. K. Borsuk, Some problems in the theory of shape of compacta, Russ. Math. Surv. 34 (6) (1979) 24–26. K. Borsuk, Theory of Shape, vol. 59, Polish Scientific Publishers, Warsaw, 1975. K. Borsuk, Theory of Retracts, vol. 44, Polish Scientific Publishers, Warsaw, 1967. M.R. Bridson, P.H. Kropholler, Dimension of elementary amenable groups, J. Reine Angew. Math. 699 (2015) 217–243. K. Brown, Cohomology of Groups, Springer-Verlag, Berlin, 1982. Ch. Chou, Elementary amenable groups, Ill. J. Math. 24 (3) (1980) 396–407. J.M. Cohen, Complexes dominated by a 2-complex, Topology 16 (1977) 409–415. M. Cohen, A Course in Simple Homotopy Theory, Grad. Texts in Math., vol. 10, Springer, Berlin, 1970. J. Dydak, Pointed and unpointed shape and pro-homotopy, Fundam. Math. 107 (1980) 58–69. J. Dydak, A. Kadlof, S. Nowak, Open Problems in Shape Theory, Mimeographed notes, University of Warsaw, 1981. J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Math., vol. 688, Springer, Berlin, 1978. H.M. Hastings, A. Heller, Homotopy idempotents on finite-dimensional complexes split, Proc. Am. Math. Soc. 85 (4) (1982) 619–622. [16] J.A. Hillman, The Algebraic Characterization of Geometric 4-Manifolds, Cambridge University Press, 1994.
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[17] J.A. Hillman, Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Aust. Math. Soc. (Ser. A) 50 (1991) 160–170. [18] J.A. Hillman, P.A. Linnell, Elementary amenable groups of finite Hirsch length are locally-finite by virtually-solvable, J. Aust. Math. Soc. (Ser. A) 52 (1992) 237–241. [19] P. Hilton, An Introduction to Homotopy Theory, Cambridge University Press, 1961. [20] D. Kołodziejczyk, 2-polyhedra for which every homotopy domination over itself is a homotopy equivalence, Topol. Appl. 207 (2016) 54–61. [21] D. Kołodziejczyk, Homotopy dominations by polyhedra with polycyclic-by-finite fundamental group, Topol. Appl. 153 (2–3) (2005) 294–302. [22] D. Kołodziejczyk, Homotopy dominations within polyhedra, Fundam. Math. 178 (2003) 189–202. [23] D. Kołodziejczyk, Polyhedra with virtually polycyclic fundamental groups have finite depth, Fundam. Math. 197 (2007) 229–238. [24] D. Kołodziejczyk, There exists a polyhedron dominating infinitely many different homotopy types, Fundam. Math. 151 (1996) 39–46. [25] D. Kołodziejczyk, There exists a polyhedron with infinitely many left neighbors, Proc. Am. Math. Soc. 129 (2001) 303–309. [26] P. Kropholler, P. Linnell, W. Lück, Groups of small homological dimension and the Atijah conjecture, in: Geometric and Cohomological Methods in Group Theory, in: Lond. Math. Soc. Lect. Note Ser., vol. 358, 2009, Chapter 7. [27] L. Louder, Krull dimension for limit groups, Geom. Topol. 16 (2012) 219–299. [28] W. Lück, Isomorphism Conjectures in K and L-Theory, SPIN Springer’s Internal Project, Mathematics – Monograph (English), 2015. [29] S. Mardešić, J. Segal, Shape Theory. The Inverse System Approach, North-Holland Math. Library, vol. 26, Amsterdam, 1982. [30] J.M. Sanjurjo, Problems from the Madrid department of geometry and topology, in: Elliott Pearl (Ed.), Open Problems in Topology II, Toronto, Canada, 2007, pp. 669–673. [31] C.T.C. Wall, Finiteness conditions for CW-complexes, Ann. Math. 81 (1965) 56–69. [32] Ch. Wegner, The Farrell–Jones conjecture for virtually solvable groups, J. Topol. 8 (4) (2015) 975–1016. [33] J. West, Compact AN R’s have finite type, Bull. Am. Math. Soc. (1975) 163–165. [34] J.H.C. Whitehead, Combinatorial homotopy I (and II), Bull. Am. Math. Soc. 55 (1949) 213–245, 453–496. [35] J.H.C. Whitehead, Simple homotopy types, Am. J. Math. 72 (1952) 1–57.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Virtual Special Issue – In memory of Professor Sibe Mardešić
2-dimensional polyhedra with finite depth Danuta Kołodziejczyk Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 27 February 2017 Received in revised form 27 December 2017 Available online 6 March 2018 Dedicated to the memory of Prof. Sibe Mardešić MSC: primary 55P15 secondary 55P10, 55P55 Keywords: Polyhedron CW -complex AN R F AN R Homotopy type Homotopy domination Shape Depth Elementary amenable group Virtually-solvable group Limit group
a b s t r a c t In this paper every polyhedron is finite and every AN R is compact. Let P ≥ X1 ≥ X2 ≥ . . ., be a sequence, in which P is a polyhedron and ≥ are homotopy dominations. One may ask, if each sequence of this form contains only finitely many homotopy dominations that are not homotopy equivalences, or if there exists an integer lP (independent of the sequence) that each sequence contains only ≤ lP homotopy dominations that are not homotopy equivalences. Closely related open questions are: Does there exist a polyhedron P homotopy dominating an infinite sequence of polyhedra {Pi }, where Pi homotopy dominates Pi+1 but Pi and Pi+1 have different homotopy types, for every i ∈ N ? (M. Moron) [30, Problem 1436], and the famous problem of K. Borsuk (1967): Is it true that two AN R s homotopy dominating each other have the same homotopy type? Here we prove that, if dim P = 2, then the answers to all these questions depend only on the properties of the fundamental group of P (for 1-dimensional polyhedra, the answers are obvious). Furthermore, if sequences in consideration contain only polyhedra, then, for the positive answer it suffices to answer positively the analog of our topological question for finitely presented groups (the fundamental groups) with retractions. Applying these results, we prove that for each polyhedron P with dim P ≤ 2 and elementary amenable fundamental group G with cdG < ∞, there is a bound lP on the lengths of all descending sequences P ≥ X1 ≥ X2 . . . of homotopy dominations that are not homotopy equivalences. The same holds if the fundamental group of P is a limit group. It means that such polyhedra P have finite depth. © 2018 Elsevier B.V. All rights reserved.
1. Introduction In this paper every polyhedron is finite and every AN R is compact. We assume (without loss of generality) that every polyhedron, AN R, F AN R and CW -complex is connected. Recall that a domination in a given category C is a morphism f : X → Y , X, Y ∈ ObC, for which there exists a morphism g : Y → X of C such that f g = idY . Then we say that Y is dominated by X, and we write Y ≤ X, or X ≥ Y . E-mail addresses: [email protected], [email protected]. URL: https://www.mini.pw.edu.pl/~dkolodz. https://doi.org/10.1016/j.topol.2018.02.016 0166-8641/© 2018 Elsevier B.V. All rights reserved.
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We write that X < Y iff X ≤ Y holds but Y ≤ X fails. In 1968 K. Borsuk asked (in the shape-theoretical language): Is it true that every polyhedron homotopy dominates only finitely many different homotopy types? [4]. We showed in [24] that the answer to this question is negative — there exists a finite polyhedron (even 2-dimensional) dominating infinitely many different homotopy types (of 2-dimensional polyhedra). Furthermore, counterexamples to Borsuk’s question are possible even with polycyclic fundamental groups [25,22]. Now, one can consider sequences of spaces homotopy dominated by a given polyhedron P with a quasiorder determined by the relation of homotopy domination P ≥ X1 ≥ X2 ≥ . . . . Recall that, by the classical results of J.H.C. Whitehead and C.T.C. Wall, each topological space homotopy dominated by a polyhedron has the homotopy type of a CW -complex, but not necessarily finite [31]. Let us state the following question: Problem 1. Does there exist a sequence P ≥ X1 ≥ X2 ≥ . . ., where P is a polyhedron, containing infinitely many homotopy dominations that are not homotopy equivalences? This problem is closely related to other natural open questions published in the literature. One of them is the famous problem of K. Borsuk posed in 1967, in his monograph “Theory of Retracts” [6, Ch. IX, Problem (12.7)]: Problem 2. Is it true that two AN R s homotopy dominating each other have the same homotopy type? Since, by the result of J. West [33], every AN R has the homotopy type of a polyhedron, this question is equivalent to its analog for polyhedra. In [13] the same problem was stated for F AN R s, compacta which are generalizations of ANR’s, and correspond in the shape category to CW -complexes dominated by polyhedra in the homotopy category (see below). Recall that on AN R’s shape and homotopy theory coincide (see [29], [14], [5]). An other problem of the similar nature was posed by M. Moron and published in [30, Problem 1436 (or Problem 9), p. 672]: Problem 3. Is there a polyhedron P dominating a sequence of polyhedra {Pi } where Pi homotopy dominates Pi+1 , but Pi and Pi+1 have different homotopy types, for every i ∈ N ? In the 1970’s K. Borsuk introduced the following notion of depth (in the shape category of compacta, see [4]): A system X1 < X2 < . . . < Xk ≤ A, where Xi ∈ ObC, for i = 1, 2, . . . , k, is called a chain of length k for A ∈ ObC. The depth D(A) of A is the least upper bound of the lengths of all chains for A. If this upper bound is infinite, we write D(A) = ℵ0 . We will consider depth in the homotopy category of CW -complexes. However, there is a 1–1 functorial correspondence between the shapes of compacta shape dominated by a given polyhedron and the homotopy types of CW -complexes homotopy dominated by it (see [14, Theorem 2.2.6]; [15]). Moreover, the depth of a given polyhedron is the same in both, pointed and unpointed, cases (it follows from [12, Theorem 5.1]). Clearly, the following problem is also closely related to Problems 1, 2 and 3: Problem 4. Does there exist a polyhedron P with infinite depth D(P )? In this paper we study 2-dimensional polyhedra. In dimension 1 the answers to all the above questions are obvious. This follows from the fact that every 1-dimensional polyhedron has the homotopy type of a finite wedge of the circles S 1 , hence (as a K(G, 1)) space homotopy dominates just over the polyhedra of the same kind (with not greater Betti numbers).
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For all polyhedra with virtually-polycyclic (i.e., polycyclic-by-finite) fundamental groups the answers to Problems 1, 3, 4 are also negative (hence Problem 2 has positive answer), by the results of the author published in [21, Theorem 5] and in [23, Theorem 1]. In particular, every polyhedron with virtually-polycyclic fundamental group has finite depth. We proved in fact that for such a polyhedron there is a bound on the lengths of all descending sequences P ≥ X1 ≥ X2 . . . of homotopy dominations that are not homotopy equivalences [23, Theorem 1]. In this paper we show the same for all 2-dimensional polyhedra whose fundamental groups are elementary amenable with finite cohomological dimension or limit groups. The class of finitely presented elementary amenable groups is larger than the class of virtually-polycyclic groups (which are always finitely presented and finitely dimensional) and includes all the virtually-solvable groups. (Recall that every polycyclic group is virtually solvable.) Limit groups (introduced by Z. Sela) are generalizations of free groups (and are nonsolvable except of the case of free abelian groups). Thus, for 2-dimensional polyhedra with these fundamental groups, Problems 1, 3, 4 have negative answers, and the answer to Problem 2 is positive. In general, we prove that in the case of 2-dimensional polyhedra the answers to the questions in consideration depend only on properties of the fundamental group. One of the main theorems states that if we consider sequences containing only polyhedra (as in Problem 3), it suffices to answer (negatively) the analog of the topological question for finitely presented groups, with retractions instead of homotopy dominations (compare Section 6). We do it for the classes of fundamental groups pointed out above. Recall that a group is weakly Hopfian if it cannot be isomorphic to a proper retract of itself. Applying main results we show here that, for two AN R’s of dimension 2 with weakly Hopfian fundamental groups, the answer to Problem 2 is positive. Obviously, every Hopfian group is weakly Hopfian. There is not known an example of a finitely presented group that is not weakly Hopfian. In the paper we use the terms “polyhedron” and “finite CW -complex” interchangeably. Without loss of generality, we may study pointed spaces and maps (see [12, Theorem 5.1]). For the basic notions and results of shape theory we refer the reader to [14], [29], [5]. 2. Algebraic preliminaries Definition 1. (i) A group G is Hopfian iff every epimorphism f : G → G is an automorphism (equivalently, N = 1 is the only normal subgroup for which G/N ∼ = G). (ii) A group G weakly Hopfian iff every r-homomorphism (see [6]) r : G → G ∼ = G imply = G is an isomorphism. In other words, G = K H and H ∼ K = 1 (where G = K H means that G = KH, K G, H < G, K ∩ H = 1; that is, H is a retract of G). Definition 2. (i) By a cohomological dimension of a group G we mean cdG = sup{n such that H n (G, M ) = 0, for some ZG-module M } (see [8], [3]). (ii) A geometrical dimension, gdG, of a group G is the smallest dimension of a CW -complex K(G, 1) (see, for example, [8]). Definition 3. (i) A group G is said to be geometrically finite if there exists a finite CW -complex K(G, 1). (ii) If cdG < ∞, then G is called finitely dimensional. (Recall that the condition cdG < ∞ is equivalent to the condition gdG < ∞ [8].) Definition 4. Let P, S be classes of groups. A group G is said to be P-by-S if it has a normal subgroup A G, A ∈ P such that G/A ∈ S. Definition 5. A finitely generated projective Λ-module M is called stable free iff there exist finitely generated free Λ-modules Λr and Λs , where r and s are integers, such that M ⊕ Λr ∼ = Λs (Λn denotes the direct sum of n copies of Λ).
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Definition 6. Let Λ be a ring. (i) Let K 0 (Λ) = F (P)/R(P), where P is the set of isomorphism classes [M ] of all the finitely generated projective Λ-modules M , F (P) is the free abelian group with the base P, and R(P) is the subgroup of F (P) generated by elements of the form [M1 ⊕ M2 ] − [M1 ] − [M2 ]. ˜ 0 (Λ) we denote the quotient of K 0 (Λ), such that two finitely generated projective Λ-modules Q1 (ii) By K ˜ 0 (Λ) iff there exist finitely generated free Λ-modules R1 and R2 satisfying and Q2 have the same class in K Q1 ⊕ R1 ∼ = Q2 ⊕ R2 . Then we write {Q1 } = {Q2 } (see [31, p. 64]). 3. Main results In what follows we prove that for 2-dimensional polyhedra, the answers to all the problems in consideration depend only on the properties of the fundamental groups. For more, if we examine sequences containing only polyhedra (as in Problems 2 and 3), then sometimes it suffices to study the analogous questions for the fundamental groups with retractions. We begin with the definition and the main property of the Wall obstruction. Let ϕ : K → X, where K be a CW -complex and X have the homotopy type of one. If ϕ : K → X is 2-connected, then, for all k ≥ 3, πk (ϕ) and Hk (ϕ) ˜ can be regarded as Zπ1 (X)-modules (see, for example, [11, pp. 28–29]). To every X homotopy dominated by a finite CW -complex (such an X has always the homotopy type of ˜ 0 (Zπ1 (X)), where a CW -complex), Wall assigned the so called Wall obstruction of X, denoted by σ(X) ∈ K Zπ1 (X) is the integral group ring of π1 (X). The obstruction σ(X) is an invariant of the homotopy type of X which vanishes iff X is homotopy equivalent to a finite complex (compare [31, pp. 64–65]). Definition 7. If X ≤ P , where P is a polyhedron and dim P = n ≥ 2, then one can find an n-connected map ϕ : Ln → X of a finite CW -complex Ln with dim Ln = n into X (see [31, p. 65]). The Wall obstruction of ˜ 0 (Zπ1 (X)) defined as (−1)n {πn+1 (ϕ)}. X, σ(X), is the element of the group K (In this situation {πn+1 (ϕ)} is a finitely generated, projective Zπ1 (X)-module.) Remark 1. Precisely, Wall proved that if X is homotopy dominated by a polyhedron P with dim P = m ≥ 2 ˜ 0 (Zπ1 (X)) = 0, then X is homotopy equivalent to a polyhedron of dimension ≤ max(m, 3) (see [31, and K Theorem F, p. 66]). In some previous paper we showed that if P is a polyhedron with dim P = 2 such that π1 (P ) is weakly Hopfian, then every homotopy domination of P over itself is a homotopy equivalence [20, Theorem 1]. The same proof gives in fact the following: Theorem 1. Let P be a polyhedron with dim P = 2. Then every homotopy domination of P over itself which induces an isomorphism on the fundamental group is a homotopy equivalence. Proof. See the proof of [20, Theorem 1], steps (ii)–(vii).
2
In the sequel we will use the following theorem, known as the Whitehead Theorem on Trees ([35, Theorem 14; 34]): Theorem. Let X and Y be finite polyhedra with dim X = 2 = dim Y , π1 (X) ∼ = π1 (Y ). Then there exist 1 integers mX and mY such that 1
k
S 2 , where k ∈ N , denotes the wedge of k spheres S 2 .
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X∨
mX
S2 Y ∨
S2.
mY
Applying Theorem 1, and the Whitehead Theorem on Trees, we will obtain: Theorem 2. Let P and Q be polyhedra homotopy dominated by 2-dimensional polyhedra, π1 (P ) ∼ = π1 (Q) and χ(P ) = χ(Q). Then every homotopy domination between P and Q that induces an isomorphism on the fundamental groups is a homotopy equivalence. Proof. By the main results of [31] (compare Remark 1) we may assume, that dim P ≤ 3, dim Q ≤ 3. In the proof we will use the following [10, Cor. 2, p. 412]: (∗) If X is a finite complex dominated by a 2-complex, then there is a finite wedge of 2-spheres W such that X ∨ W X (2) , where X (2) is the 2-sceleton of X. Let P and Q be polyhedra homotopy dominated by 2-dimensional polyhedra, dim P ≤ 3, dim Q ≤ 3, π1 (P ) ∼ = π1 (Q) and χ(P ) = χ(Q). By (∗), there exist two finite wedges of spheres S 2 , WP and WQ , such that P ∨ WP P (2) and Q ∨ WQ (2) Q (respectively), where P (2) and Q(2) are the 2-sceleta of P and Q. By the Whitehead Theorem on Trees, there exist integers mP and mQ such that P (2) ∨ WP Q(2) ∨ WQ , 2 2 where WP = S , WQ = S . mP
mQ
Therefore, P ∨ WP ∨ WP Q ∨ WQ ∨ WQ . Let us observe that since χ(P ) = χ(Q), then WP ∨ WP = WQ ∨ WQ . Thus, we have P¯ P ∨ W Q ∨ W , where W = WP ∨ WP is a finite wedge of spheres S 2 and P¯ = P (2) ∨ WP is a polyhedron with dim P¯ ≤ 2. Now, let d : P → Q be a homotopy domination which induces an isomorphism d∗1 : π1 (P ) → π1 (Q) on the fundamental groups. As a homotopy domination, d induces an epimorphism d∗2 : π2 (P ) → π2 (Q) on the second homotopy groups. By the classical Whitehead Theorem (see [19]), if P and Q are homotopy dominated by 2-dimensional polyhedra, then d : P → Q is a homotopy equivalence iff it induces isomorphisms d∗r : πr (P ) → πr (Q), for r ≤ 2. Suppose, on the contrary, that d is not a homotopy equivalence. It means that d∗2 : π2 (P ) → π2 (Q) is not a monomorphism. Since d : P → Q is a homotopy domination, it is easily seen that the map d¯ : P ∨ W → Q ∨ W defined ¯ = d(x), for x ∈ P and d(x) ¯ = x, for x ∈ W , is also a homotopy domination. as d(x) ¯ Observe that d induces an isomorphism d¯∗1 : π1 (P ∨W ) → π1 (Q ∨W ) on the fundamental groups. On the other hand, d¯ does not induce a monomorphism d¯∗2 : π2 (P ∨W ) → π2 (Q ∨W ). Indeed, d∗2 : π2 (P ) → π2 (Q) is not a monomorphism, and the homomorphism i∗2 : π2 (P ) → π2 (P ∨ W ), induced by the inclusion i : P → P ∨ W , is a monomorphism (P is a retract of P ∨ W ). Therefore, there exists a homotopy domination d¯¯ : P¯ → P¯ with the same properties, i.e., such that d¯∗1 : π1 (P¯ ) → π1 (P¯ ) is an isomorphism on the fundamental group, and d¯∗2 : π2 (P¯ ) → π2 (P¯ ) is not a monomorphism. Thus d¯¯ is not a homotopy equivalence. (Indeed, a composition of a homotopy domination with a homotopy equivalence, on the left or on the right, is again a homotopy domination.) Since dim P¯ ≤ 2, this contradicts Theorem 1, and the proof is finished. 2 As a corollary to the above theorem we obtain the following: Corollary 1. Let G be weakly Hopfian. Let P and Q be polyhedra with dim P = dim Q = 2 (or polyhedra homotopy dominated by 2-polyhedra), π1 (P ) ∼ = π1 (Q) ∼ = G and χ(P ) = χ(Q). Then every homotopy domination d : P → Q is a homotopy equivalence.
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Proof. It follows directly from the previous theorem — if G is weakly Hopfian and π1 (P ) ∼ = π1 (Q) ∼ = G, then the epimorphism d∗1 : π1 (P ) → π1 (Q) induced on the fundamental groups by the homotopy domination d is clearly an isomorphism. 2 Applying Theorem 2, we can prove: Theorem 3. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the condition: (Gf) Each sequence of groups and r-homomorphisms G → G1 → G2 → . . . contains only finitely many homomorphisms that are not isomorphisms. Then each sequence P ≥ P1 ≥ P2 ≥ . . . of polyhedra with homotopy dominations contains only finitely many dominations that are not homotopy equivalences. Proof. Assume that P is a polyhedron with dim P ≤ 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G. Let P = P0 → P1 → P2 → . . . be a sequence of polyhedra with homotopy dominations di : Pi−1 → Pi , for i ∈ N . Since G satisfies (Gf), there exists an integer k such that, for i ≥ k, all the epimorphisms di ∗1 : π1 (Pi−1 ) → π1 (Pi ) induced by the maps di on the fundamental groups are isomorphisms. Similarly, there exists an integer l such that for i ≥ l, χ(Pi−1 ) = χ(Pi ). Indeed, the homology groups H1 (P ) and H2 (P ), are finitely generated Abelian, thus also satisfy the condition (Gf) (all of their subgroups are also finitely generated Abelian, hence Hopfian). Clearly, Hr (P ) = 0, for r ≥ 3. By Theorem 2, there exists an integer m = max(k, l) such that for i ≥ m, all the homotopy dominations di : Pi−1 → Pi , are homotopy equivalences, which ends the proof. 2 Therefore, for 2-dimensional polyhedra (and polyhedra homotopy dominated by 2-dimensional polyhedra) the answer to Problem 3 depends only on the similar property of the fundamental group: Corollary 2. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the condition: (Gf) Each sequence of groups and r-homomorphisms G → G1 → G2 → . . . contains only finitely many homomorphisms that are not isomorphisms. Then there is no infinite sequence of polyhedra {Pi } where P = P1 , such that Pi ≥ Pi+1 but Pi and Pi+1 have different homotopy types for every i ∈ N . Proof. This follows immediately from the previous theorem. 2 Using the results of Wall, we obtain: Theorem 4. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the conditions: (Gf) Each sequence of groups and r-homomorphisms G → G1 → G2 → . . . contains only finitely many homomorphisms that are not isomorphisms, ˜ 0 (ZH) = 0, for each retract H of G. (W) K
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Then each sequence P ≥ X1 ≥ X2 ≥ . . . of homotopy dominations contains only finitely many dominations that are not homotopy equivalences. Proof. Assume that P is a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G. Let P = X0 → X1 → X2 → . . . be a sequence of topological spaces with homotopy dominations ˜ 0 (ZH) = 0, for each retract H of G, then, by the main result of [31] di : Xi−1 → Xi , for i ∈ N . Since K (compare Remark 1), every Xi in the above sequence has the homotopy type of a polyhedron of dimension ≤ 3. Thus, the assertion follows from Theorem 3. 2 With a stronger condition on G we conclude that 2-dimensional polyhedra with π1 (P ) ∼ = G have finite depth. Theorem 5. Let P be a polyhedron with dim P = 2 (or P is homotopy dominated by a 2-dimensional polyhedron) and π1 (P ) ∼ = G satisfies the conditions: (Gd) There exists an integer dG such that each sequence of groups and r-homomorphisms G → G1 → G2 → . . . → Gk contains at most dG r-homomorphisms that are not isomorphisms, ˜ 0 (ZH) = 0, for each retract H of G. (W) K Then there exists an integer kP such that each sequence P ≥ X1 ≥ X2 ≥ . . . of homotopy dominations contains only at most kP dominations that are not homotopy equivalences. Thus D(P ) < ∞. ˜ 0 (ZH) = 0, for each retract H of G, then all the Xi in the Proof. As in the proof of Theorem 4, since K above sequence may be assumed to be polyhedra of dimension ≤ 3. Then the proof is the same as the proof of Theorem 3. The only one difference is that now, the integer k from the proof of Theorem 3 can be chosen independently of the sequence P → P1 → P2 → . . . . (It is clear that the integer l in the proof of Theorem 3 can be always chosen independently of the sequence). 2 In [20] we proved the following theorem (see [20, Theorem 2]): Theorem 6. Let P be a polyhedron with dim P = 2 such that π1 (P ) is weakly Hopfian. If there exist homotopy dominations P ≥ Q and Q ≥ P , then P Q. Now we will show that the same holds for 2-dimensional ANR’s. Theorem 7. Let X, Y ∈ AN R s, dim X = 2 and π1 (X) is weakly Hopfian. If there exist homotopy dominations X ≥ Y and Y ≥ X, then X Y . Proof. Assume that X, Y ∈ AN R s, dim X = 2 and π1 (X) is weakly Hopfian. By [14; 29, p. 318], every topological space X which is homotopy dominated by some AN R of dimension ≤ n is homotopy dominated by a polyhedron of dimension ≤ n. In particular, if X ∈ AN R and dim X = 2, then X is homotopy dominated by a polyhedron of dimension 2 (clearly, the same holds for Y ). By the result of West [33], every AN R has the homotopy type of a polyhedron. Thus, X and Y have both the homotopy types of some polyhedra, so the Wall obstructions of X and Y vanish. By the Wall’s result (see Remark 1), both X and Y have the homotopy types of polyhedra of dimension ≤ 3. Thus there exist two polyhedra P and Q (where P X and Q Y ) homotopy dominated by a 2-dimensional polyhedron, such that P ≤ Q and Q ≤ P . Since all the homology groups of P and Q are isomorphic, then χ(P ) = χ(Q). We apply Corollary 1, and the proof is finished. 2
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4. Corollaries to limit groups as fundamental groups We will apply our results to the class of limit groups (introduced by Z. Sela) which is a generalization of the class of free groups: Definition 8 (Limit groups). A finitely generated group G is called a limit group if, for any finite subset S ⊂ G, there exists a homomorphism f : G → F (where F is a free group of finite rank) so that the restriction of f to S is injective. Example 1. Examples of limit groups include: finite-rank free abelian groups, fundamental groups of closed surfaces with the Euler characteristic less then −1, some doubles. Remark 2. (i) Limit groups are always finitely presented [1]. (ii) All the limit groups are geometrically finite and torsion-free [1]. (iii) Finitely generated subgroups of limit groups are limit groups [1]. (iv) Limit groups are non-solvable except of free abelian groups. In the sequel we will need the following two lemmas: Lemma 1. Let G be a limit group. Then there exists an integer kG such that any descending sequence of subgroups of G with proper retractions ri : Gi−1 → Gi , G = G0 ⊇ G1 ⊇ . . . Gi ⊇ Gi+1 ⊇ . . . , for i = 1, 2, . . . , contains no more than kG distinct subgroups. Proof. To see this, we may use [27, Theorem 1.1] stating that there is a function D(n) such that if Fn → L1 → . . . → Lk is a sequence of proper epimorphisms of limit groups, where Fn is a free group with n-generators, then k ≤ D(n). Thus, in particular, take L1 equal to G which is a finitely presented group (as a limit group, compare Remark 2), then there exists an integer kG such that for every sequence with proper retractions G = L1 → G2 . . . → Gk , k ≤ kG . Indeed, every retract Gi of a limit group G is a finitely generated subgroup of G [31, Lemma 1.3]. Thus, as a finitely generated subgroup of a limit group, it is also a limit group (see Remark 2), and we can apply [27, Theorem 1.1]. 2 ˜ 0 (ZG) = 0, and the same holds for all the retracts of G. Lemma 2. Let G be a limit group. Then K Proof. Observe that each retract of a limit group is also a limit group (see the proof of Lemma 1). It is known that all the limit groups are CAT (0) (see [1]; [2, p. 636]). According to [2], every finitedimensional CAT (0) group belongs to the class B of groups satisfying the Borel Conjecture. Moreover, for ˜ 0 (RG) = 0, if R is a principal ideal domain (see [2, p. 633]). every torsion-free group G from B, K Every limit group is finitely-dimensional and torsion-free (see Remark 2). Thus, for every limit group G, ˜ 0 (ZG) = 0, and the same holds for all the retracts of G. This ends the proof. 2 we have K As applications of the main results, we obtain Corollary 3. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) such that π1 (X) is a limit group. Then there exists an integer lP such that each sequence P ≥ X1 ≥ X2 ≥ . . . Xk contains at most lP homotopy dominations that are not homotopy equivalences. Thus, D(P ) < ∞. Proof. It follows from Theorem 5 and Lemmas 1 and 2. 2
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Corollary 4. Let P be a polyhedron P with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron) such that π1 (P ) is a limit group. Then there is no infinite sequence of polyhedra {Pi } homotopy dominated by P , such that Pi ≥ Pi+1 but Pi and Pi+1 have different homotopy types for every i ∈ N . Proof. This is a consequence of Corollary 3.
2
5. Corollaries to elementary amenable fundamental groups Definition 9. Elementary amenable groups is the smallest class of groups that contains all abelian and all finite groups, and is closed under extensions and directed unions (see [26], [7, p. 223]). Remark 3. Every virtually-solvable group is elementary amenable, because each solvable group is well known to be a poly-abelian group. In the sequel we will use properties of the Hirsch length of elementary amenable groups. Definition 10 (see [7, Definition I.15, p. 223]). The Hirsch length, h(G), of an elementary amenable group is finite and equal to n ≥ 0, if G has a series 1 = H0 . . . Hr = G in which the factors are either locally finite or infinite cyclic, and exactly n factors are infinite cyclic. In all other cases, h(G) = ∞. (Recall that a group is locally finite if all its finitely generated subgroups are finite). In the next proof we will also use the following lemma [20, Lemma 1]: Lemma 3. Let G be a countable elementary amenable group with cdG < ∞. Then h(G) = 0 iff G = 1. Applying properties of the Hirsch length of elementary amenable groups, we will prove: Lemma 4. Let G be an elementary amenable group with cdG < ∞. Then exists an integer kG such that any descending sequence G = G0 ⊇ G1 ⊇ . . . Gi ⊇ Gi+1 ⊇ . . . of subgroups of G with retractions ri : Gi−1 → Gi , for i = 1, 2, . . . , contains no more than kG distinct subgroups. Proof. Let G be a countable elementary amenable group. Then each retract of G is also countable, elementary amenable (the class of elementary amenable groups is closed under taking subgroups [9]). Since cdG < ∞, then, h(G) < ∞ [16, Theorem 5] (cf. also [20, Remark 6]). It is known that H ⊆ G implies that cdH < cdG [8]. Thus, we have cdH < ∞, hence h(H) < ∞, also for each retract H of G. If h(G) is a Hirsch number of a given elementary amenable group G with finite cdG, H ⊆ G and N G is such that H = G/N , then h(G) = h(N ) + h(H) [17, Theorem 1]. Let Mi be the kernel of the retraction ri : Gi−1 → Gi , i.e. Mi Gi−1 satisfies Gi−1 = Gi Mi and Gi ∩ Mi = 1, for i = 1, 2, . . . . Then we have h(Gi−1 ) = h(Mi ) + h(Gi ), for i = 1, 2, . . . , thus
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h(G) = h(M1 ) + h(M2 ) + . . . h(Mi ) + h(Gi ). Hence, there exists an integer k such that h(Mi ) = 0 for all i except no more than k. As a subgroup of a countable elementary amenable group with finite cohomological dimension, Mi is also elementary amenable, countable and finite dimensional. Then, by Lemma 3, Mi = 1 for such i. Therefore we get immediately that Gi = Gi−1 for these i. Thus, every descending sequence of subgroups of G, G = G0 ⊇ G1 ⊇ . . . ⊇ Gi ⊇ Gi+1 ⊇ . . . , with retractions ri : Gi−1 → Gi , for i = 1, 2, . . . , contains no more than k different groups, where k−1 = h(G) is the Hirsch number of G. This finishes the proof. 2 ˜ 0 (ZG) = 0, and the Lemma 5. Let G be a countable elementary amenable group with cdG < ∞. Then K same holds for all the retracts of G. Proof. Let G be a countable elementary amenable group with cdG < ∞. Then, in particular, G is virtually solvable [16, Theorem 5; 18]. Recently it was shown in [32, Theorem 1.1] that all virtually-solvable groups belong to the class FJ of groups satisfying the Full Farrell–Jones Conjecture (see [28, 11.5], [28, 14.2, Theorem 14.1]). Moreover, every subgroup of a group from the class FJ also belongs to FJ (see, for example, [28, Theorem 11.21, p. 337]). ˜ 0 (ZG) = 0 (see [28, p. 352]). Since It is also known that for each torsion-free group G from FJ , K G is finitely-dimensional, G is torsion-free [8]. Thus each subgroup of G is also torsion-free. Therefore, ˜ 0 (ZG) = 0, and the same holds for all the retracts of G, and the proof is finished. 2 K As corollaries to the main result, we get: Corollary 5. Let P be a polyhedron with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron), with elementary amenable fundamental group of finite cohomological dimension. Then there exists an integer lP such that each sequence P ≥ X1 ≥ X2 ≥ . . . Xk contains at most lP homotopy dominations that are not homotopy equivalences. Thus, D(P ) < ∞. Proof. This is a corollary to Theorem 5 and Lemmas 4 and 5. 2 Corollary 6. Let P be a polyhedron P with dim P = 2 (or a polyhedron homotopy dominated by a 2-dimensional polyhedron), with elementary amenable fundamental group of finite cohomological dimension. Then there is no infinite sequence of polyhedra {Pi } homotopy dominated by P , such that Pi ≥ Pi+1 but Pi and Pi+1 have different homotopy types for every i ∈ N . Proof. We get this immediately from Corollary 5.
2
Remark 4. Since every virtually-polycyclic group is known to be virtually-solvable and always has finite cohomological dimension, these results generalize in dimension 2 the results of [21] and [23] for polyhedra with virtually-polycyclic fundamental groups. 6. Final remarks and open problems We will finish with some questions. Since there is no example of a finitely presented group that is not weakly Hopfian, let us state the following:
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Problem 5. Does there exist a finitely presented group that is not weakly Hopfian? Problem 6. Is it true that each finitely presented group with finite cohomological dimension is weakly Hopfian? Remark 5. Obviously, every Hopfian group is weakly Hopfian. The class of Hopfian groups includes the following and many others: torsion-free hyperbolic groups, Knot groups, finitely generated linear groups, Braid groups, Limit groups, countable elementary amenable groups of finite cohomological dimension (including virtually-solvable groups of finite cohomological dimension), many one-relator groups (see [20, Remark 5, Lemma 2]). The next questions are closely related to Problems 2 and 3. Problem 7. Do there exist two finitely presented groups G and H such that H G with r-homomorphisms r1 : G → H and r2 : H → G? Problem 8. Does there exist a finitely presented group G with an infinite sequence of groups Gi and rhomomorphisms G → G1 → G2 → . . . that are not isomorphisms? Problem 9. Does there exist a finitely presented group G of finite cohomological dimension with an infinite sequence of groups Gi and r-homomorphisms G → G1 → G2 → . . . that are not isomorphisms? It should be noted that an answer to the following related question is also unknown: Problem 10. Does there exist a finitely presented group G with infinitely many different retracts up to isomorphism? Remark 6. A famous conjecture due to Hsiang (see [28]) states that for any torsion-free group G, we have ˜ 0 (ZG) = 0. Let us also remark that it is known that if a group G is finitely dimensional, then it is K torsion-free (see [8]). The results of this paper are generalizations of my results on this subject included into my lecture “Homotopy dominations within 2-dimensional polyhedra” at the “Dubrovnik VII Topology Conference”, Croatia, 2011. At this conference I met Prof. Sibe Mardešić for the last time. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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