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2-polyhedra for which every homotopy domination over itself is a homotopy equivalence
Topology and its Applications 207 (2016) 54–61
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Topology and its Applications www.elsevier.com/locate/topol
2-polyhedra for which every homotopy domination over itself is a homotopy equivalence 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 22 October 2015 Received in revised form 11 April 2016 Accepted 11 April 2016 Available online 4 May 2016 MSC: primary 55P15 secondary 55P10, 55P55 Keywords: Polyhedron CW -complex Homotopy domination Homotopy type Hopfian Elementary amenable group Knot group Hyperbolic group Wall obstruction Projective module Rank
a b s t r a c t We consider an open question: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? The answer is known to be positive for 1-dimensional polyhedra and polyhedra with virtually-polycyclic fundamental groups, so it is natural to ask about 2-dimensional polyhedra, in particular about those with solvable fundamental groups. In this paper we prove that for each 2-dimensional polyhedron P with weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it is not isomorphic to a proper retract of itself. Thus every Hopfian group is weakly Hopfian. The class of Hopfian groups contains: all torsion-free hyperbolic groups, finitely generated linear groups, knot groups, limit groups, and many others. One corollary to the main result is that for 2-dimensional polyhedra with elementary amenable (including virtually-solvable) fundamental groups of finite cohomological dimension, the answer to our question is positive (we show that every elementary amenable group with finite cohomological dimension is Hopfian). The problem in consideration is related in an obvious way to the famous question of K. Borsuk (1967): Is it true that two compact ANR’s homotopy dominating each other have the same homotopy type? © 2016 Elsevier B.V. All rights reserved.
1. Introduction In this paper we study a natural, but still open problem: Is it true that for every polyhedron P , each homotopy domination of P over itself is a homotopy equivalence? [13, Problem (3.15)]. This question is related in an obvious way to the famous problem of K. Borsuk (1967): Is it true that two ANR’s homotopy dominating each other have the same homotopy type? [7, Ch. IX, Problem (12.7)]. Here and subsequently, every polyhedron is finite, and every ANR is compact. (Recall that, by the well-known result of J. West [34], every compact ANR is homotopy equivalent to a polyhedron.)
E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.topol.2016.04.007 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Throughout this paper we assume, without loss of generality, that each polyhedron and ANR are connected. In 1959 I. Bernstein and T. Ganea proved that every homotopy domination of a manifold over itself is a homotopy equivalence [4]. The same is true for 1-dimensional polyhedra. Indeed, every 1-dimensional polyhedron has the homotopy type of a finite wedge of circles S 1, so K(F, 1), where F is a free, finitely generated group. It is well-known that every finitely generated, free group is Hopfian (Nielsen 1921, Hopf 1931; see [28, 6.1.12]). For polyhedra with virtually-polycyclic (i.e., polycyclic-by-finite) fundamental groups, the answers to both questions are also positive (see [23, Theorems 3 and 5]). Every polycyclic group is solvable, so it is natural to ask whether the same is true for polyhedra with solvable fundamental groups (in particular, for 2-dimensional polyhedra with solvable fundamental groups). In this paper we prove that for any polyhedron P with dim P = 2 and weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it cannot be isomorphic to a proper retract of itself (see Definition 1). It should be noted that at present there are no known examples of finitely presented groups that are not weakly Hopfian. Of course, every Hopfian group is weakly Hopfian. Thus, for each 2-dimensional polyhedron whose fundamental group is Hopfian, every homotopy domination over itself is a homotopy equivalence. The class of Hopfian groups contains: all torsion-free hyperbolic groups (in the sense of Gromov), finitely generated linear groups, knot groups, braid groups, small cancellation groups, limit groups, many one-relator groups, and many others (see Remark 5). By the main result, we obtain, in particular, that for any 2-dimensional polyhedron with elementary amenable fundamental group G of finite cohomological dimension cdG, every homotopy domination over itself is a homotopy equivalence. (Note that any virtually-solvable group is elementary amenable; see the proof of Corollary 3.) To this end, we show that every elementary amenable group G with cdG < ∞ is Hopfian. The question in consideration is also closely related to another open problem posed by K. Borsuk: Are the homotopy types (or equivalently, shapes) of two quasi-homeomorphic ANR’s equal? [8, Problem (12.7), p. 233]. We are able to give an affirmative answer to this question, for the same classes of polyhedra (compare [24]). In the paper we use the terms “polyhedron” and “finite CW -complex” interchangeably. Let us note that, without loss of generality, we may study pointed spaces and maps (by [12, Theorem 5.1]). 2. Main theorem with some applications In the sequel X ≤ Y denotes that X is homotopy dominated by Y . We begin by recalling a few definitions: 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 is weakly Hopfian iff every ∼ r-homomorphism r : G → H = G is an isomorphism. In other words, G = K H and H ∼ = G imply 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). Remark 1. Let X be a CW -complex and A be a subcomplex of X. It is well-known that, for k ≥ 2, ˜ can be regarded as Zπ1 (X)-modules (where Zπ1 (X) is the integral group ring of π1 (X)). πk (X) and Hk (X) ˜ A) ˜ can be regarded as Zπ1 (X)-modules Similarly, if k ≥ 3 and π1 (X) ∼ = π1 (A), then πk (X, A) and Hk (X, (see [10, p. 28]). Definition 2. Let K be a CW -complex, X have the homotopy type of one, and ϕ : K → X. We define πk (ϕ), Hk (ϕ) as πk (Mϕ , K × 0), Hk (Mϕ , K × 0) (respectively), where Mϕ is the mapping cylinder of ϕ. We call ϕ
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n-connected, if πk (ϕ) = 0 for 1 ≤ k ≤ n (cf. [33, pp. 56–57]). Thus, if ϕ : K → X is 2-connected, then, for all k ≥ 3, πk (ϕ) and Hk (ϕ) ˜ can be regarded as Zπ1 (X)-modules (cf. Remark 1). Definition 3. A finitely generated projective ZG-module M is stable free iff there exist finitely generated free ZG-modules ZGr and ZGs , where r and s are integers such that M ⊕ ZGr ∼ = ZGs (ZGn denotes the direct sum of n copies of ZG). To every X homotopy dominated by a finite CW -complex, C. T. C. Wall assigned the so called Wall ˜ 0 (Zπ1 (X)). Let us recall the definitions (cf. obstruction of X, denoted by σ(X), an element of the group K [33, pp. 64–65]). Definition 4. Let K 0 (Zπ1 (X)) = F (P)/R(P), where P is the set of isomorphism classes [M ] of all the finitely generated projective Zπ1 (X)-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 (Zπ1 (X)) we denote the quotient of K 0 (Zπ1 (X)) such that two finitely generated projective By K ˜ 0 (Zπ1 (X)) iff there exist finitely generated free Zπ1 (X)-modules Q1 and Q2 have the same class in K Zπ1 (X)-modules R1 and R2 satisfying Q1 ⊕ R1 ∼ = Q2 ⊕ R2 . Then we write {Q1 } = {Q2 }. Definition 5. 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 [33, p. 65]). The Wall obstruction of ˜ 0 (Zπ1 (X)) defined as (−1)n+1 {πn+1 (ϕ)}. X, σ(X), is the element of the group K Remark 2. The Wall obstruction of X, σ(X), is an invariant of the homotopy type of X which vanishes iff X is homotopy equivalent to a finite complex (by [33, Theorem F, p. 66]). It is important that σ(X) does not depend on the choice of Ln and ϕ in the above definition (see [33, Lemma 3.2, p. 65] and [33, Theorem F with the proof]). In the sequel we use two different notions of rank of a projective finitely generated ZG-module, and their properties. The first one is very natural: Definition 6. For a projective finitely generated ZG-module P , let rk(P ) denote the rank over Z of Z ⊗ZG P . Remark 3. If P is a finitely generated projective ZG-module, then it is easy to check that Z ⊗ZG P is indeed a finitely generated free abelian group (cf. [9, Ch. II, p. 34]) and this rank has the following properties: (i) rk(P1 ⊕ P2 ) = rk(P1 ) + rk(P2 ), (ii) rk(ZGn ) = n, where ZGn is a free ZG-module on n generators. However, it is not clear if rk(P) = 0 implies P = 0. The following is a different definition of a rank of a projective finitely generated ZG-module P , denoted by i(P ) (see [14, p. 241], [32]). Definition 7. Given P a finitely generated projective ZG-module, let π : ZGn → P be an epimorphism, and s : P → ZGn a monomorphism such that πs = idP . Then e = sπ is an idempotent. Let M be the n × n matrix with entries in ZG which describes e; i.e, e(xi ) = Σnj=1 Mij xj , where x1 , x2 , . . . , xn are generators of ZGn . Then M = Σg∈G M (g)g, where each M (g) is an n × n matrix with integer values; M (g) = 0 except for finitely many g. We define the rank i(P ) to be i(P ) = traceM (1), where 1 is the identity element of G. Remark 4. The rank i(P ) is an integer and an invariant of the isomorphism class of P which satisfies [14, Prop. 1, p. 241]: (i) i(P1 ⊕ P2 ) = i(P1 ) + i(P2 ), (ii) i(P ) = n iff P = ZGn , (iii) i(P ) = 0 iff P = 0. (It is also known that 0 ≤ i(P ) ≤ n(P ), where n(P ) is the minimum number of elements needed to generate P ; see [14].)
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The basic properties of the rank i(P ) play an important role in the proof of the next theorem. This theorem is a substantial generalization of the main result of [24, Theorem 1] and its proof may be regarded as an extension of the geometric proof of [24, Theorem 1]. Theorem 1. Let P be 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. Proof. (i) Let d : P → P be a homotopy domination, and let G = π1 (P ) be weakly Hopfian. Then d induces an isomorphism on the fundamental groups d∗ : π1 (P ) → π1 (P ). (ii) It is clear that d induces isomorphisms d∗ : Hk (P ) → Hk (P ) on the homology groups for all k (any finitely generated abelian group is well-known to be Hopfian). Therefore Hk (d) = 0, for all k ∈ N . Since d induces an epimorphism d∗ : π2 (P ) → π2 (P ), by the exact sequence of homotopy groups for P , we conclude that π2 (d) = 0 i.e., d is 2-connected. Our goal is to show that π3 (d) = 0. We can consider π3 (d) as a module over ZG (compare Remark 1 and Definition 2). (iii) In the first step we show that π3 (d) is a finitely generated projective ZG-module. Let us observe that from the results of [33] we have that, if X is a CW -complex homotopy dominated by any 2-dimensional finite complex, K 2 is a finite complex with dim K 2 = 2, and ϕ : K 2 → X is a map which is 2-connected, then π3 (ϕ) is a finitely generated, projective module over Zπ1 (X). Indeed, Wall proved that if X satisfies the condition D2 (for the definition, see [33, p. 62]) and ϕ : K 2 → X is 2-connected (where K 2 is a finite complex with dim K 2 = 2), then π3 (ϕ) is a projective Zπ1 (X)-module [33, Complement on p. 63]. From the main results of the Wall’s paper [33, Theorem F, p. 66], if X is dominated by a finite CW -complex of dimension 2, then X satisfies D2. In our situation, P clearly satisfies D2, hence π3 (d) is a projective ZG-module. The module π3 (d) is also finitely generated. To see this, let us observe that P satisfies the condition F3 from [33]. The condition F3 for X implies that for every ϕ : K 2 → X which is 2-connected, where K 2 is a finite complex with dim K 2 = 2, π3 (ϕ) is a finitely generated Zπ1 (X)-module (cf. [33, p. 57]). In general, Wall proved [33, Theorem A, p. 58], that if X is dominated by a finite CW -complex of dimension n, then X satisfies Fn (for the definition of Fn, see [33, p. 57]). Obviously, P which is itself a finite CW -complex of dimension 2 satisfies F3. As a concequence, π3 (d) is a finitely generated ZG-module. (iv) We proceed to show that π3 (d) = 0. Let S = π3 (d). Firstly, we prove that the ZG-module S is stable free. Note that since P is a polyhedron, its Wall obstruction σ(P ) is trivial. It follows that the ZG-module S is stable free. Indeed, according to [33, Lemma 3.2, p. 65], the class of the module π3 (ϕ) in ˜ 0 (ZG) does not depend on the choice of a finite 2-dimensional CW -complex the projective class group K 2 K and a 2-connected map ϕ : K 2 → X (compare Remark 2). Hence, taking just K 2 = P and ϕ = d, we have from σ(P ) = 0 that S is a finitely generated, projective, stable free ZG-module. (v) Next we show that then rk(S) = 0. To see this, we can use a version of the Relative Hurewicz Theorem (see [16, Ch. IV, Sec. 4.2, Theorem 4.37, p. 371]) which implies that if A and X are path-connected spaces and n ≥ 3 is such that πq (X, A, x0 ) = 0 for q < n, then πn (X, A, x0 ) ∼ = Hn (X, A), where πn (X, A, x0 ) is defined as πn (X, A, x0 ) = πn (X, A, x0 ) ⊗ZG Z (compare [16, p. 370]). By the above and (ii), we have 0 = H3 (d) ∼ = π3 (d) ⊗ZG Z. Thus rk(S) = 0. (vi) Then we observe that since S is a finitely generated, projective and stable free ZG-module, rk(S) = 0 implies i(S) = 0. Indeed, if S is a finitely generated projective and stable free ZG-module, then there exist two finitely generated free ZG-modules, say ZGk and ZGn , such that S ⊕ ZGk ∼ = ZGn . Therefore, by the Remarks 3 and 4(i)–(ii), we conclude that 0 = rk(S) = n − k = i(S). From i(S) = 0 we obtain S = 0 (applying Remark 4, (iii)).
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(vii) Since π3 (d) = 0, d induces an isomorphism d∗ : π2 (P ) → π2 (P ). Finally, d induces isomorphisms d∗ : πr (P ) → πr (P ), for r = 1, 2, hence, by the classical Whitehead Theorem, it is a homotopy equivalence. Thus the proof of Theorem 1 is completed. 2 The following is an immediate corollary to Theorem 1: Theorem 2. Let P be a polyhedron with dim P = 2 such that π1 (P ) is weakly Hopfian. If there exist homotopy dominations d1 : P → Q and d2 : Q → P , then P Q. In the sequel we point out some applications. Since every Hopfian group is weakly Hopfian, we have the following corollaries to Theorems 1 and 2 (respectively): Corollary 1. Let P be a polyhedron with dim P = 2 such that π1 (P ) is Hopfian. Then every homotopy domination of P over itself is a homotopy equivalence. Corollary 2. Let P be a polyhedron with dim P = 2 such that π1 (P ) is Hopfian. If there exist homotopy dominations d1 : P → Q and d2 : Q → P , then P Q. Remark 5. The class of Hopfian groups includes the following (among many others): (i) Torsion-free hyperbolic groups. Indeed, by the result of Z. Sela [31], all torsion-free hyperbolic groups are Hopfian. Let us remark that (in statistical meaning) almost every finitely presented group is hyperbolic (see M. Gromov [15], A. Yu. Ol’shanskii [27]). (ii) Knot groups. It is known (by [17], see also [21, p. 15]) that all knot groups are residually finite (that is, their subgroups of finite index have trivial intersection). On the other hand, any finitely generated residually finite group is Hopfian (A. Malcev, see [28, 6.1.11]). (iii) Finitely generated linear groups. All these groups are also residually-finite (A. Malcev, see [29, Theorem 1.3]), hence Hopfian. This class contains, in particular, all virtually-polycyclic groups. (iv) Braid groups. All such groups are also linear (see [5,25]). (v) Limit groups. The Hopficity of limit groups (see [24, Lemma 1]) follows from the proof of [1, Lemma A.1, p. 269]. (vi) In many cases one-relator groups are also known to be Hopfian (see, for example, [35,36,2,11,29,30]). Example 1. The Trefoil Knot group T = a, b | a2 = b3 belongs to both, knot groups and braid groups, simultaneously (hence also to linear groups). The case of 2-dimensional polyhedra with fundamental group isomorphic to T was considered by the author in [22, Theorem 3] and (by the same method) generalized to [24, Theorem 1]. Example 2 (Baumslag–Solitar Groups; see [3]). For each pair of integers 0 < m ≤ |n|, we define BS(m, n) = a, b | abm a−1 = bn . It is known that, if m = 1 or m and n have the same prime divisors, then BS(m, n) is Hopfian (and conversely). Let us remark that all BS(m, n) are non-hyperbolic groups, with cohomological dimension 2 (see [26]). Moreover, for each n > 1, the group BS(1, n) is soluble, but non-polycyclic. 3. An application to elementary amenable fundamental groups In this section we show that Theorems 1 and 2 can be applied to all 2-dimensional polyhedra with elementary amenable (hence, with virtually-solvable) fundamental groups of finite cohomological dimension. Let us recall some definitions: Definition 8. 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 }.
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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 [6, p. 223]). In what follows we prove that every countable, elementary amenable group of finite cohomological dimension is Hopfian. We use the following definition of the Hirsch length for elementary amenable groups [6, Definition I.15, p. 223], which coincides (see [6, p. 223]) with the first one due to J. Hillman (cf., for example, [20]). Definition 10. The Hirsch length, h(G), of an elementary amenable group G 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.) Lemma 1. Let G be an elementary amenable group which is torsion-free. Then h(G) = 0 iff G = 1. Proof. We show the non-trivial implication from the left to the right. According to the definition, there exists a finite series 1 = H0 . . . Hr = G in which all the factors Hi /Hi−1 (for 1 ≤ i ≤ r) are locally finite. Let us take the first one of them which is non-trivial. It is equal to some Hk (where 1 ≤ k ≤ r). Hk is a subgroup of G, so it is torsion-free. On the other hand, Hk is locally finite, hence, in particular, each subgroup of Hk which is generated by one element is finite. Therefore Hk = 1. Consequently, G = 1, and the proof is finished. 2 Lemma 2. Let G be a countable, elementary amenable group with cdG < ∞. Then G is Hopfian. Proof. Suppose that f : G → G is an epimorphism. Let N = kerf . It is known that, if G is a countable, elementary amenable group, then cdG < ∞ implies h(G) < ∞ (by [18, Ch. I, Theorem 5]). Then we use the fact that, if G is an elementary amenable group, H < G is a subgroup of G, and M G is such that H ∼ = G/M , then h(G) = h(M ) + h(H) (see, for example, [19, Theorem 1]). This gives h(N ) = 0. Since G satisfies cdG < ∞, it is torsion-free (see, for example, [9, VIII, 2, Coroll. (2.5)]). Applying Lemma 1, we have N = 1. Thus, f is an isomorphism, which ends the proof. 2 Remark 6. In [18, Ch. I, Theorem 5] the assumption that G is countable can be omitted (see remarks in [18, Ch. I]). Hence, it could also be omitted in the above Lemma 2. Anyway, in our applications we consider only finitely presented groups, which are always countable. Applying Theorem 1 and Lemma 2, we obtain: Theorem 3. Let P be a polyhedron with dim P = 2. If π1 (P ) is an elementary amenable group with finite cohomological dimension, then every homotopy domination of P over itself is a homotopy equivalence. As a consequence, we get in particular: Corollary 3. Let P be a polyhedron with dim P = 2. If π1 (P ) is a virtually-solvable group with finite cohomological dimension, then every homotopy domination of P over itself is a homotopy equivalence. Proof. Any solvable group is well-known to be poly-abelian. Hence, any virtually-solvable group is elementary amenable. Thus, it follows from Theorem 3. 2
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4. Open problems Since so far there is no known example of a finitely presented group that is not weakly Hopfian, let us state the following questions: Problem 1. Does there exist a finitely presented group that is not weakly Hopfian? Problem 2. Is it true that every finitely presented group G with cdG < ∞ is weakly Hopfian? Problem 3. Is it true that every finitely presented solvable group is weakly Hopfian? We may also conjecture that Theorem 1 remains true for polyhedra of higher dimensions. Problem 4. Is it true that for every polyhedron P with weakly Hopfian fundamental group every homotopy domination of P over itself is a homotopy equivalence? Remark 7. It is worth pointing out that the proof of the positive results for polyhedra with polycyclic-byfinite fundamental groups ([23, Theorems 3 and 5]) cannot be extended even to polyhedra with solvable fundamental groups. Indeed, this proof uses the fact that for each polycyclic-by-finite group G, the integral group ring ZG is Noetherian. On the other hand, if G is a solvable group with ZG Noetherian, then G is polycyclic (see [33, remark on p. 61] and [28, 5.4.12]). It should be also noted that the only known examples of finitely presented groups G with Noetherian integral group rings ZG are just polycyclic-by-finite groups (cf. [33, remark on p. 61]). References [1] E. Alibegovic, M. Bestvina, Limit groups are CAT(0), J. Lond. Math. Soc. (2) 74 (2006) 259–272. [2] S. Andreadakis, E. Raptis, D. Varsos, A characterization of residually finite HNN-extensions of finitely generated abelian groups, Arch. Math. 50 (1988) 495–501. [3] G. Baumslag, D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Am. Math. Soc. 68 (1962) 199–201. [4] I. Berstein, T. Ganea, Remark on spaces dominated by manifolds, Fundam. Math. 47 (1959) 45–56. [5] S. Bigelow, Braid groups are linear, J. Am. Math. Soc. 14 (2) (2001) 471–486. [6] M.R. Bridson, P.H. Kropholler, Dimension of elementary amenable groups, J. Reine Angew. Math. 699 (2015) 217–243. [7] K. Borsuk, Theory of Retracts, vol. 44, Polish Scientific Publishers, Warsaw, 1967. [8] K. Borsuk, Theory of Shape, vol. 59, Polish Scientific Publishers, Warsaw, 1975. [9] K. Brown, Cohomology of Groups, Springer-Verlag, Berlin, 1982. [10] M. Cohen, A Course in Simple Homotopy Theory, Grad. Texts in Math., vol. 10, Springer, Berlin, 1970. [11] D.J. Collins, F. Levin, Automorphisms and Hopficity of certain Baumslag–Solitar groups, Arch. Math. 40 (1983) 385–400. [12] J. Dydak, Pointed and unpointed shape and pro-homotopy, Fundam. Math. 107 (1980) 58–69. [13] J. Dydak, A. Kadlof, S. Nowak, Open Problems in Shape Theory, Mimeographed Notes, University of Warsaw, 1981. [14] E. Dyer, A.T. Vasquez, An invariant for finitely generated projectives over Z[G], J. Pure Appl. Algebra 7 (1976) 241–248. [15] M. Gromov, Hyperbolic groups, in: S.M. Gersten (Ed.), Essays in Group Theory, in: M.S.R.I. Pub., vol. 8, Springer-Verlag, Berlin, New York, 1987, pp. 75–263. [16] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [17] J. Hempel, Residual finiteness for 3-manifolds, in: S.M. Gersten, J.R. Stallings (Eds.), Combinatorial Group Theory and Topology, in: Ann. of Math. Stud., vol. 111, Princeton Univ. Press, 1987, pp. 379–396. [18] J.A. Hillman, The Algebraic Characterization of Geometric 4-Manifolds, Cambridge University Press, 1994. [19] J.A. Hillman, Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Aust. Math. Soc. Ser. A 50 (1991) 160–170. [20] 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. [21] R. Kirby, Problems in low dimensional topology, Berkeley, 1995, in: H. Kazez (Ed.), Geometric Topology, in: AMS/IP, vol. 2, International Press, 1997. [22] D. Kołodziejczyk, 2-dimensional polyhedra with infinitely many left neighbors, Topol. Appl. 159 (2012) 1943–1947. [23] D. Kołodziejczyk, Homotopy dominations by polyhedra with polycyclic-by-finite fundamental group, Topol. Appl. 153 (2–3) (2005) 294–302. [24] D. Kołodziejczyk, Polyhedra for which every homotopy domination over itself is a homotopy equivalence, arXiv:1411.1032v1 [math.GT].
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Topology and its Applications www.elsevier.com/locate/topol
2-polyhedra for which every homotopy domination over itself is a homotopy equivalence 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
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Article history: Received 22 October 2015 Received in revised form 11 April 2016 Accepted 11 April 2016 Available online 4 May 2016 MSC: primary 55P15 secondary 55P10, 55P55 Keywords: Polyhedron CW -complex Homotopy domination Homotopy type Hopfian Elementary amenable group Knot group Hyperbolic group Wall obstruction Projective module Rank
a b s t r a c t We consider an open question: Is it true that each homotopy domination of a polyhedron over itself is a homotopy equivalence? The answer is known to be positive for 1-dimensional polyhedra and polyhedra with virtually-polycyclic fundamental groups, so it is natural to ask about 2-dimensional polyhedra, in particular about those with solvable fundamental groups. In this paper we prove that for each 2-dimensional polyhedron P with weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it is not isomorphic to a proper retract of itself. Thus every Hopfian group is weakly Hopfian. The class of Hopfian groups contains: all torsion-free hyperbolic groups, finitely generated linear groups, knot groups, limit groups, and many others. One corollary to the main result is that for 2-dimensional polyhedra with elementary amenable (including virtually-solvable) fundamental groups of finite cohomological dimension, the answer to our question is positive (we show that every elementary amenable group with finite cohomological dimension is Hopfian). The problem in consideration is related in an obvious way to the famous question of K. Borsuk (1967): Is it true that two compact ANR’s homotopy dominating each other have the same homotopy type? © 2016 Elsevier B.V. All rights reserved.
1. Introduction In this paper we study a natural, but still open problem: Is it true that for every polyhedron P , each homotopy domination of P over itself is a homotopy equivalence? [13, Problem (3.15)]. This question is related in an obvious way to the famous problem of K. Borsuk (1967): Is it true that two ANR’s homotopy dominating each other have the same homotopy type? [7, Ch. IX, Problem (12.7)]. Here and subsequently, every polyhedron is finite, and every ANR is compact. (Recall that, by the well-known result of J. West [34], every compact ANR is homotopy equivalent to a polyhedron.)
E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.topol.2016.04.007 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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Throughout this paper we assume, without loss of generality, that each polyhedron and ANR are connected. In 1959 I. Bernstein and T. Ganea proved that every homotopy domination of a manifold over itself is a homotopy equivalence [4]. The same is true for 1-dimensional polyhedra. Indeed, every 1-dimensional polyhedron has the homotopy type of a finite wedge of circles S 1, so K(F, 1), where F is a free, finitely generated group. It is well-known that every finitely generated, free group is Hopfian (Nielsen 1921, Hopf 1931; see [28, 6.1.12]). For polyhedra with virtually-polycyclic (i.e., polycyclic-by-finite) fundamental groups, the answers to both questions are also positive (see [23, Theorems 3 and 5]). Every polycyclic group is solvable, so it is natural to ask whether the same is true for polyhedra with solvable fundamental groups (in particular, for 2-dimensional polyhedra with solvable fundamental groups). In this paper we prove that for any polyhedron P with dim P = 2 and weakly Hopfian fundamental group, every homotopy domination of P over itself is a homotopy equivalence. A group is weakly Hopfian if it cannot be isomorphic to a proper retract of itself (see Definition 1). It should be noted that at present there are no known examples of finitely presented groups that are not weakly Hopfian. Of course, every Hopfian group is weakly Hopfian. Thus, for each 2-dimensional polyhedron whose fundamental group is Hopfian, every homotopy domination over itself is a homotopy equivalence. The class of Hopfian groups contains: all torsion-free hyperbolic groups (in the sense of Gromov), finitely generated linear groups, knot groups, braid groups, small cancellation groups, limit groups, many one-relator groups, and many others (see Remark 5). By the main result, we obtain, in particular, that for any 2-dimensional polyhedron with elementary amenable fundamental group G of finite cohomological dimension cdG, every homotopy domination over itself is a homotopy equivalence. (Note that any virtually-solvable group is elementary amenable; see the proof of Corollary 3.) To this end, we show that every elementary amenable group G with cdG < ∞ is Hopfian. The question in consideration is also closely related to another open problem posed by K. Borsuk: Are the homotopy types (or equivalently, shapes) of two quasi-homeomorphic ANR’s equal? [8, Problem (12.7), p. 233]. We are able to give an affirmative answer to this question, for the same classes of polyhedra (compare [24]). In the paper we use the terms “polyhedron” and “finite CW -complex” interchangeably. Let us note that, without loss of generality, we may study pointed spaces and maps (by [12, Theorem 5.1]). 2. Main theorem with some applications In the sequel X ≤ Y denotes that X is homotopy dominated by Y . We begin by recalling a few definitions: 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 is weakly Hopfian iff every ∼ r-homomorphism r : G → H = G is an isomorphism. In other words, G = K H and H ∼ = G imply 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). Remark 1. Let X be a CW -complex and A be a subcomplex of X. It is well-known that, for k ≥ 2, ˜ can be regarded as Zπ1 (X)-modules (where Zπ1 (X) is the integral group ring of π1 (X)). πk (X) and Hk (X) ˜ A) ˜ can be regarded as Zπ1 (X)-modules Similarly, if k ≥ 3 and π1 (X) ∼ = π1 (A), then πk (X, A) and Hk (X, (see [10, p. 28]). Definition 2. Let K be a CW -complex, X have the homotopy type of one, and ϕ : K → X. We define πk (ϕ), Hk (ϕ) as πk (Mϕ , K × 0), Hk (Mϕ , K × 0) (respectively), where Mϕ is the mapping cylinder of ϕ. We call ϕ
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n-connected, if πk (ϕ) = 0 for 1 ≤ k ≤ n (cf. [33, pp. 56–57]). Thus, if ϕ : K → X is 2-connected, then, for all k ≥ 3, πk (ϕ) and Hk (ϕ) ˜ can be regarded as Zπ1 (X)-modules (cf. Remark 1). Definition 3. A finitely generated projective ZG-module M is stable free iff there exist finitely generated free ZG-modules ZGr and ZGs , where r and s are integers such that M ⊕ ZGr ∼ = ZGs (ZGn denotes the direct sum of n copies of ZG). To every X homotopy dominated by a finite CW -complex, C. T. C. Wall assigned the so called Wall ˜ 0 (Zπ1 (X)). Let us recall the definitions (cf. obstruction of X, denoted by σ(X), an element of the group K [33, pp. 64–65]). Definition 4. Let K 0 (Zπ1 (X)) = F (P)/R(P), where P is the set of isomorphism classes [M ] of all the finitely generated projective Zπ1 (X)-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 (Zπ1 (X)) we denote the quotient of K 0 (Zπ1 (X)) such that two finitely generated projective By K ˜ 0 (Zπ1 (X)) iff there exist finitely generated free Zπ1 (X)-modules Q1 and Q2 have the same class in K Zπ1 (X)-modules R1 and R2 satisfying Q1 ⊕ R1 ∼ = Q2 ⊕ R2 . Then we write {Q1 } = {Q2 }. Definition 5. 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 [33, p. 65]). The Wall obstruction of ˜ 0 (Zπ1 (X)) defined as (−1)n+1 {πn+1 (ϕ)}. X, σ(X), is the element of the group K Remark 2. The Wall obstruction of X, σ(X), is an invariant of the homotopy type of X which vanishes iff X is homotopy equivalent to a finite complex (by [33, Theorem F, p. 66]). It is important that σ(X) does not depend on the choice of Ln and ϕ in the above definition (see [33, Lemma 3.2, p. 65] and [33, Theorem F with the proof]). In the sequel we use two different notions of rank of a projective finitely generated ZG-module, and their properties. The first one is very natural: Definition 6. For a projective finitely generated ZG-module P , let rk(P ) denote the rank over Z of Z ⊗ZG P . Remark 3. If P is a finitely generated projective ZG-module, then it is easy to check that Z ⊗ZG P is indeed a finitely generated free abelian group (cf. [9, Ch. II, p. 34]) and this rank has the following properties: (i) rk(P1 ⊕ P2 ) = rk(P1 ) + rk(P2 ), (ii) rk(ZGn ) = n, where ZGn is a free ZG-module on n generators. However, it is not clear if rk(P) = 0 implies P = 0. The following is a different definition of a rank of a projective finitely generated ZG-module P , denoted by i(P ) (see [14, p. 241], [32]). Definition 7. Given P a finitely generated projective ZG-module, let π : ZGn → P be an epimorphism, and s : P → ZGn a monomorphism such that πs = idP . Then e = sπ is an idempotent. Let M be the n × n matrix with entries in ZG which describes e; i.e, e(xi ) = Σnj=1 Mij xj , where x1 , x2 , . . . , xn are generators of ZGn . Then M = Σg∈G M (g)g, where each M (g) is an n × n matrix with integer values; M (g) = 0 except for finitely many g. We define the rank i(P ) to be i(P ) = traceM (1), where 1 is the identity element of G. Remark 4. The rank i(P ) is an integer and an invariant of the isomorphism class of P which satisfies [14, Prop. 1, p. 241]: (i) i(P1 ⊕ P2 ) = i(P1 ) + i(P2 ), (ii) i(P ) = n iff P = ZGn , (iii) i(P ) = 0 iff P = 0. (It is also known that 0 ≤ i(P ) ≤ n(P ), where n(P ) is the minimum number of elements needed to generate P ; see [14].)
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The basic properties of the rank i(P ) play an important role in the proof of the next theorem. This theorem is a substantial generalization of the main result of [24, Theorem 1] and its proof may be regarded as an extension of the geometric proof of [24, Theorem 1]. Theorem 1. Let P be 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. Proof. (i) Let d : P → P be a homotopy domination, and let G = π1 (P ) be weakly Hopfian. Then d induces an isomorphism on the fundamental groups d∗ : π1 (P ) → π1 (P ). (ii) It is clear that d induces isomorphisms d∗ : Hk (P ) → Hk (P ) on the homology groups for all k (any finitely generated abelian group is well-known to be Hopfian). Therefore Hk (d) = 0, for all k ∈ N . Since d induces an epimorphism d∗ : π2 (P ) → π2 (P ), by the exact sequence of homotopy groups for P , we conclude that π2 (d) = 0 i.e., d is 2-connected. Our goal is to show that π3 (d) = 0. We can consider π3 (d) as a module over ZG (compare Remark 1 and Definition 2). (iii) In the first step we show that π3 (d) is a finitely generated projective ZG-module. Let us observe that from the results of [33] we have that, if X is a CW -complex homotopy dominated by any 2-dimensional finite complex, K 2 is a finite complex with dim K 2 = 2, and ϕ : K 2 → X is a map which is 2-connected, then π3 (ϕ) is a finitely generated, projective module over Zπ1 (X). Indeed, Wall proved that if X satisfies the condition D2 (for the definition, see [33, p. 62]) and ϕ : K 2 → X is 2-connected (where K 2 is a finite complex with dim K 2 = 2), then π3 (ϕ) is a projective Zπ1 (X)-module [33, Complement on p. 63]. From the main results of the Wall’s paper [33, Theorem F, p. 66], if X is dominated by a finite CW -complex of dimension 2, then X satisfies D2. In our situation, P clearly satisfies D2, hence π3 (d) is a projective ZG-module. The module π3 (d) is also finitely generated. To see this, let us observe that P satisfies the condition F3 from [33]. The condition F3 for X implies that for every ϕ : K 2 → X which is 2-connected, where K 2 is a finite complex with dim K 2 = 2, π3 (ϕ) is a finitely generated Zπ1 (X)-module (cf. [33, p. 57]). In general, Wall proved [33, Theorem A, p. 58], that if X is dominated by a finite CW -complex of dimension n, then X satisfies Fn (for the definition of Fn, see [33, p. 57]). Obviously, P which is itself a finite CW -complex of dimension 2 satisfies F3. As a concequence, π3 (d) is a finitely generated ZG-module. (iv) We proceed to show that π3 (d) = 0. Let S = π3 (d). Firstly, we prove that the ZG-module S is stable free. Note that since P is a polyhedron, its Wall obstruction σ(P ) is trivial. It follows that the ZG-module S is stable free. Indeed, according to [33, Lemma 3.2, p. 65], the class of the module π3 (ϕ) in ˜ 0 (ZG) does not depend on the choice of a finite 2-dimensional CW -complex the projective class group K 2 K and a 2-connected map ϕ : K 2 → X (compare Remark 2). Hence, taking just K 2 = P and ϕ = d, we have from σ(P ) = 0 that S is a finitely generated, projective, stable free ZG-module. (v) Next we show that then rk(S) = 0. To see this, we can use a version of the Relative Hurewicz Theorem (see [16, Ch. IV, Sec. 4.2, Theorem 4.37, p. 371]) which implies that if A and X are path-connected spaces and n ≥ 3 is such that πq (X, A, x0 ) = 0 for q < n, then πn (X, A, x0 ) ∼ = Hn (X, A), where πn (X, A, x0 ) is defined as πn (X, A, x0 ) = πn (X, A, x0 ) ⊗ZG Z (compare [16, p. 370]). By the above and (ii), we have 0 = H3 (d) ∼ = π3 (d) ⊗ZG Z. Thus rk(S) = 0. (vi) Then we observe that since S is a finitely generated, projective and stable free ZG-module, rk(S) = 0 implies i(S) = 0. Indeed, if S is a finitely generated projective and stable free ZG-module, then there exist two finitely generated free ZG-modules, say ZGk and ZGn , such that S ⊕ ZGk ∼ = ZGn . Therefore, by the Remarks 3 and 4(i)–(ii), we conclude that 0 = rk(S) = n − k = i(S). From i(S) = 0 we obtain S = 0 (applying Remark 4, (iii)).
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(vii) Since π3 (d) = 0, d induces an isomorphism d∗ : π2 (P ) → π2 (P ). Finally, d induces isomorphisms d∗ : πr (P ) → πr (P ), for r = 1, 2, hence, by the classical Whitehead Theorem, it is a homotopy equivalence. Thus the proof of Theorem 1 is completed. 2 The following is an immediate corollary to Theorem 1: Theorem 2. Let P be a polyhedron with dim P = 2 such that π1 (P ) is weakly Hopfian. If there exist homotopy dominations d1 : P → Q and d2 : Q → P , then P Q. In the sequel we point out some applications. Since every Hopfian group is weakly Hopfian, we have the following corollaries to Theorems 1 and 2 (respectively): Corollary 1. Let P be a polyhedron with dim P = 2 such that π1 (P ) is Hopfian. Then every homotopy domination of P over itself is a homotopy equivalence. Corollary 2. Let P be a polyhedron with dim P = 2 such that π1 (P ) is Hopfian. If there exist homotopy dominations d1 : P → Q and d2 : Q → P , then P Q. Remark 5. The class of Hopfian groups includes the following (among many others): (i) Torsion-free hyperbolic groups. Indeed, by the result of Z. Sela [31], all torsion-free hyperbolic groups are Hopfian. Let us remark that (in statistical meaning) almost every finitely presented group is hyperbolic (see M. Gromov [15], A. Yu. Ol’shanskii [27]). (ii) Knot groups. It is known (by [17], see also [21, p. 15]) that all knot groups are residually finite (that is, their subgroups of finite index have trivial intersection). On the other hand, any finitely generated residually finite group is Hopfian (A. Malcev, see [28, 6.1.11]). (iii) Finitely generated linear groups. All these groups are also residually-finite (A. Malcev, see [29, Theorem 1.3]), hence Hopfian. This class contains, in particular, all virtually-polycyclic groups. (iv) Braid groups. All such groups are also linear (see [5,25]). (v) Limit groups. The Hopficity of limit groups (see [24, Lemma 1]) follows from the proof of [1, Lemma A.1, p. 269]. (vi) In many cases one-relator groups are also known to be Hopfian (see, for example, [35,36,2,11,29,30]). Example 1. The Trefoil Knot group T = a, b | a2 = b3 belongs to both, knot groups and braid groups, simultaneously (hence also to linear groups). The case of 2-dimensional polyhedra with fundamental group isomorphic to T was considered by the author in [22, Theorem 3] and (by the same method) generalized to [24, Theorem 1]. Example 2 (Baumslag–Solitar Groups; see [3]). For each pair of integers 0 < m ≤ |n|, we define BS(m, n) = a, b | abm a−1 = bn . It is known that, if m = 1 or m and n have the same prime divisors, then BS(m, n) is Hopfian (and conversely). Let us remark that all BS(m, n) are non-hyperbolic groups, with cohomological dimension 2 (see [26]). Moreover, for each n > 1, the group BS(1, n) is soluble, but non-polycyclic. 3. An application to elementary amenable fundamental groups In this section we show that Theorems 1 and 2 can be applied to all 2-dimensional polyhedra with elementary amenable (hence, with virtually-solvable) fundamental groups of finite cohomological dimension. Let us recall some definitions: Definition 8. 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 }.
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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 [6, p. 223]). In what follows we prove that every countable, elementary amenable group of finite cohomological dimension is Hopfian. We use the following definition of the Hirsch length for elementary amenable groups [6, Definition I.15, p. 223], which coincides (see [6, p. 223]) with the first one due to J. Hillman (cf., for example, [20]). Definition 10. The Hirsch length, h(G), of an elementary amenable group G 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.) Lemma 1. Let G be an elementary amenable group which is torsion-free. Then h(G) = 0 iff G = 1. Proof. We show the non-trivial implication from the left to the right. According to the definition, there exists a finite series 1 = H0 . . . Hr = G in which all the factors Hi /Hi−1 (for 1 ≤ i ≤ r) are locally finite. Let us take the first one of them which is non-trivial. It is equal to some Hk (where 1 ≤ k ≤ r). Hk is a subgroup of G, so it is torsion-free. On the other hand, Hk is locally finite, hence, in particular, each subgroup of Hk which is generated by one element is finite. Therefore Hk = 1. Consequently, G = 1, and the proof is finished. 2 Lemma 2. Let G be a countable, elementary amenable group with cdG < ∞. Then G is Hopfian. Proof. Suppose that f : G → G is an epimorphism. Let N = kerf . It is known that, if G is a countable, elementary amenable group, then cdG < ∞ implies h(G) < ∞ (by [18, Ch. I, Theorem 5]). Then we use the fact that, if G is an elementary amenable group, H < G is a subgroup of G, and M G is such that H ∼ = G/M , then h(G) = h(M ) + h(H) (see, for example, [19, Theorem 1]). This gives h(N ) = 0. Since G satisfies cdG < ∞, it is torsion-free (see, for example, [9, VIII, 2, Coroll. (2.5)]). Applying Lemma 1, we have N = 1. Thus, f is an isomorphism, which ends the proof. 2 Remark 6. In [18, Ch. I, Theorem 5] the assumption that G is countable can be omitted (see remarks in [18, Ch. I]). Hence, it could also be omitted in the above Lemma 2. Anyway, in our applications we consider only finitely presented groups, which are always countable. Applying Theorem 1 and Lemma 2, we obtain: Theorem 3. Let P be a polyhedron with dim P = 2. If π1 (P ) is an elementary amenable group with finite cohomological dimension, then every homotopy domination of P over itself is a homotopy equivalence. As a consequence, we get in particular: Corollary 3. Let P be a polyhedron with dim P = 2. If π1 (P ) is a virtually-solvable group with finite cohomological dimension, then every homotopy domination of P over itself is a homotopy equivalence. Proof. Any solvable group is well-known to be poly-abelian. Hence, any virtually-solvable group is elementary amenable. Thus, it follows from Theorem 3. 2
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4. Open problems Since so far there is no known example of a finitely presented group that is not weakly Hopfian, let us state the following questions: Problem 1. Does there exist a finitely presented group that is not weakly Hopfian? Problem 2. Is it true that every finitely presented group G with cdG < ∞ is weakly Hopfian? Problem 3. Is it true that every finitely presented solvable group is weakly Hopfian? We may also conjecture that Theorem 1 remains true for polyhedra of higher dimensions. Problem 4. Is it true that for every polyhedron P with weakly Hopfian fundamental group every homotopy domination of P over itself is a homotopy equivalence? Remark 7. It is worth pointing out that the proof of the positive results for polyhedra with polycyclic-byfinite fundamental groups ([23, Theorems 3 and 5]) cannot be extended even to polyhedra with solvable fundamental groups. Indeed, this proof uses the fact that for each polycyclic-by-finite group G, the integral group ring ZG is Noetherian. On the other hand, if G is a solvable group with ZG Noetherian, then G is polycyclic (see [33, remark on p. 61] and [28, 5.4.12]). It should be also noted that the only known examples of finitely presented groups G with Noetherian integral group rings ZG are just polycyclic-by-finite groups (cf. [33, remark on p. 61]). References [1] E. Alibegovic, M. Bestvina, Limit groups are CAT(0), J. Lond. Math. Soc. (2) 74 (2006) 259–272. [2] S. Andreadakis, E. Raptis, D. Varsos, A characterization of residually finite HNN-extensions of finitely generated abelian groups, Arch. Math. 50 (1988) 495–501. [3] G. Baumslag, D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Am. Math. Soc. 68 (1962) 199–201. [4] I. Berstein, T. Ganea, Remark on spaces dominated by manifolds, Fundam. Math. 47 (1959) 45–56. [5] S. Bigelow, Braid groups are linear, J. Am. Math. Soc. 14 (2) (2001) 471–486. [6] M.R. Bridson, P.H. Kropholler, Dimension of elementary amenable groups, J. Reine Angew. Math. 699 (2015) 217–243. [7] K. Borsuk, Theory of Retracts, vol. 44, Polish Scientific Publishers, Warsaw, 1967. [8] K. Borsuk, Theory of Shape, vol. 59, Polish Scientific Publishers, Warsaw, 1975. [9] K. Brown, Cohomology of Groups, Springer-Verlag, Berlin, 1982. [10] M. Cohen, A Course in Simple Homotopy Theory, Grad. Texts in Math., vol. 10, Springer, Berlin, 1970. [11] D.J. Collins, F. Levin, Automorphisms and Hopficity of certain Baumslag–Solitar groups, Arch. Math. 40 (1983) 385–400. [12] J. Dydak, Pointed and unpointed shape and pro-homotopy, Fundam. Math. 107 (1980) 58–69. [13] J. Dydak, A. Kadlof, S. Nowak, Open Problems in Shape Theory, Mimeographed Notes, University of Warsaw, 1981. [14] E. Dyer, A.T. Vasquez, An invariant for finitely generated projectives over Z[G], J. Pure Appl. Algebra 7 (1976) 241–248. [15] M. Gromov, Hyperbolic groups, in: S.M. Gersten (Ed.), Essays in Group Theory, in: M.S.R.I. Pub., vol. 8, Springer-Verlag, Berlin, New York, 1987, pp. 75–263. [16] A. Hatcher, Algebraic Topology, Cambridge University Press, 2002. [17] J. Hempel, Residual finiteness for 3-manifolds, in: S.M. Gersten, J.R. Stallings (Eds.), Combinatorial Group Theory and Topology, in: Ann. of Math. Stud., vol. 111, Princeton Univ. Press, 1987, pp. 379–396. [18] J.A. Hillman, The Algebraic Characterization of Geometric 4-Manifolds, Cambridge University Press, 1994. [19] J.A. Hillman, Elementary amenable groups and 4-manifolds with Euler characteristic 0, J. Aust. Math. Soc. Ser. A 50 (1991) 160–170. [20] 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. [21] R. Kirby, Problems in low dimensional topology, Berkeley, 1995, in: H. Kazez (Ed.), Geometric Topology, in: AMS/IP, vol. 2, International Press, 1997. [22] D. Kołodziejczyk, 2-dimensional polyhedra with infinitely many left neighbors, Topol. Appl. 159 (2012) 1943–1947. [23] D. Kołodziejczyk, Homotopy dominations by polyhedra with polycyclic-by-finite fundamental group, Topol. Appl. 153 (2–3) (2005) 294–302. [24] D. Kołodziejczyk, Polyhedra for which every homotopy domination over itself is a homotopy equivalence, arXiv:1411.1032v1 [math.GT].
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