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Homotopy dominations by polyhedra with polycyclic-by-finite fundamental groups
Topology and its Applications 153 (2005) 294–302 www.elsevier.com/locate/topol
Homotopy dominations by polyhedra with polycyclic-by-finite fundamental groups Danuta Kołodziejczyk 1 Faculty of Mathematics and Informational Sciences, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warsaw, Poland Received 1 February 2003; received in revised form 22 September 2003; accepted 22 September 2003
Abstract In the previous papers, in connection with a question of K. Borsuk, we proved that there exist polyhedra with polycyclic fundamental groups homotopy dominating infinitely many different homotopy types. Here we consider a few problems of K. Borsuk concerning infinite chains of polyhedra or FANR’s ordered by the relation of domination (in homotopy or shape category) and obtain that for polyhedra with polycyclic-by-finite fundamental groups, there are no pathology similar to the above. 2004 Elsevier B.V. All rights reserved. MSC: 55P55; 55P15; 55P10 Keywords: Polyhedron; ANR; CW-complex; FANR; Compactum; Homotopy domination; Homotopy type; Shape domination; Shape
1. Introduction In this paper every polyhedron, ANR, FANR and CW-complex is assumed to be connected (only for convenience). We expect that the reader is familiar with the basic notions and facts of shape theory (see [5,9,25]) and of retract theory (see [4]).
E-mail address: [email protected] (D. Kołodziejczyk). 1 Address for correspondence: ul. Jasna 8/18, 00-013 Warsaw, Poland.
0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2003.09.016
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Define the capacity C(A) of a compactum A as the cardinality of the class of the shapes of all compacta X such that Sh(X) Sh(A). Let us say that a system consisting of k compacta X1 , X2 , . . . , Xk is a chain of length k for a compactum A if Sh(X1 ) < Sh(X2 ) < · · · < Sh(Xk ) Sh(A) (Sh(X) < Sh(Y ) if and only if Sh(X) Sh(Y ) holds but Sh(Y ) Sh(X) fails). The depth D(A) of a compactum 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 . The above notions were introduced by K. Borsuk in 1979, at the Topological Conference in Moscow, together with some relevant problems (see [3]). In [19], answering the most interesting question: Is the capacity of each polyhedron finite? (see also [8]), we showed an example of a polyhedron dominating infinitely many different homotopy types of polyhedra (see also [20]). Such phenomena are even frequent: in [16] we proved that, for every non-Abelian poly-Z-group G, there exists a polyhedron P with π1 (P ) ∼ = G dominating infinitely many polyhedra of different homotopy types. Thus, there exist polyhedra with polycyclic or nilpotent fundamental groups with this property. Using those examples, we gave also a positive answer to the other question from [3] (Question (8), see also [8, Problem (6.1)]) stating that there exist polyhedra (with polycyclic fundamental groups) with infinite capacity and finite depth. We conjecture that while there exist polyhedra with polycyclic fundamental groups and infinite capacity, every polyhedron with such a fundamental group has finite depth. The problem, if there exists a polyhedron with infinite depth, remains unsolved. It is an obvious observation that finite capacity of a given polyhedron implies finite depth. Let us mention that we showed that all the polyhedra with finite fundamental groups, nilpotent polyhedra, and some others, dominate only finitely many different shapes (or homotopy types), see [18] and [17]. It is also known, by the classical results of shape theory, that the capacity of each 1-dimensional polyhedron is finite. Observe that a polyhedron P has finite depth if and only if: there exists an integer k such that each sequence · · · Sh(Xi ) · · · Sh(X1 ) Sh(P ) contains at most k different shapes (compare [8, Problem (6.1)]). Here we show that for polyhedra with polycyclic-by-finite fundamental groups there are no infinite sequences of this kind. This implies the solutions of some questions of K. Borsuk included in his books [4,5] concerning infinite sequences of shape or homotopy dominations between FANR’s or ANR’s, for FANR’s with polycyclic-by-finite first shape groups and ANR’s with such fundamental groups (recall that by the well-known result of J. West [30], every ANR has the homotopy type of a polyhedron). In the sequel X Y will denote that X is homotopy dominated by Y . Let us remark that by the classical results in shape theory (see [11] or [12]; [9, Theorem 2.2.6]; [10]) 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 (not necessarily finite) homotopy dominated by this polyhedron (in both, pointed and unpointed cases). Hence, if X and Y are two compacta shape dominated by a polyhedron, and Sh(X) Sh(Y ), then X Y , for corresponding CW-complexes X and Y (respectively), and conversely. Thus considering questions on shape dominations between FANR’s we may work in the homotopy category of CW-complexes and homotopy classes of cellular maps between them. Also in the definitions of the capacity and the depth of a polyhedron
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one can investigate dominations in the homotopy category of CW-complexes. Moreover, there is no difference which case, pointed or unpointed, is studied (see [7]).
2. Algebraic preliminaries Let us recall some definitions (see [27]): A group G is polycyclic if it has a finite normal series G = G0 ⊇ G1 ⊇ · · · ⊇ Gl = 1, (Gi G, for i = 1, . . . , l) for which each factor Gi−1 /Gi is infinite cyclic or finite cyclic (this is true for all the finitely generated nilpotent groups, see, for instance, [28, 15.4, p. 92]). G is a poly-Z-group if it has a finite normal series with factors Z. A group G is called polycyclic-by-finite if it is an extension of a polycyclic group by a finite group, i.e., there exists a polycyclic group H G such that G/H is finite. It is said that a group G satisfies ascending chain condition (or max-condition) if and only if every ascending sequence of subgroups of G, G1 ⊆ G2 ⊆ · · · , where Gi = Gi+1 , is finite. A group G is called residually finite if and only if their subgroups of finite index have trivial intersection. Recall that a group G is Hopfian if every epimorphism f : G → G is an automorphism (equivalently, N = 1 is the only normal subgroup for which G/N ∼ = G). Let R be a ring. A right R-module M is said to be noetherian if every R-submodule of M is finitely generated, equivalently, M satisfies ascending chain condition, i.e., every ascending sequence of submodules of M: M1 ⊆ M2 ⊆ · · · such that Mi = Mi+1 is finite (compare [26, p. 419], or [22, Chapter VI,1]). We call a ring R noetherian if it is a noetherian module as a left module over itself. A module M is called Hopfian (see [14]) if every epimorphism f : M → M is an isomorphism (equivalently M is not isomorphic to a proper quotient of itself).
3. Main results Let us begin with the following: Lemma 1. Assume that G is a polycyclic-by-finite group. Let G = G0 → G1 → · · · → Gi → Gi+1 → · · · be a sequence of groups with epimorphisms fi : Gi−1 → Gi , for i = 1, 2, . . . , then there exists an integer i0 such that for i > i0 , fi is an isomorphism. Proof. Assign to each group Gi a normal divisor Ni G such that Ni = ker si , where si = fi · · · f2 f1 : G → Gi (i = 1, 2, . . .). Consider the following sequence of subgroups of G G ⊇ · · · ⊇ Ni+1 ⊇ Ni ⊇ · · · ⊇ N1 .
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As a polycyclic-by-finite group, G satisfies the max-condition (see [27, Chapter 3, p. 69 or (5.4.12), p. 152], hence there exists an integer i0 such that Ni+1 = Ni , for i i0 . Then for i i0 , we have Gi+1 ∼ = Gi . Every subgroup of a polycyclic-by-finite group is a group of this kind. Moreover, it is known that every polycyclic-by-finite group is Hopfian. Indeed, by the result of Hirsch, every polycyclic-by-finite group is residually finite (see [27, (5.4.17)]), and by the result of Malcev, if a finitely generated group G is residually finite, then it is Hopfian (see [23,15]). Obviously, an epimorphism between isomorphic Hopfian groups is an isomorphism. Therefore, for every i > i0 , fi : Gi−1 → Gi is an isomorphism. This ends the proof of the lemma. 2 In what follows we will also use the following: Lemma 2 (See [29, the proof of Theorem A, implication (ii) ⇒ (i), p. 60]). Let X P , where P is a finite CW-complex, then there exists a finite CW-complex P such that X P and π1 (P ) ∼ = π1 (X). Proof. Assume that X P and P is a finite CW-complex. Then, by attaching finitely many 2-cells, we may change P into a finite CW-complex P such that X P and π1 (P ) ∼ = π1 (X). Indeed, without loss of generality one can assume P to have just one 0-cell. Let P(P ) = gi | rj be a presentation of π1 (P ) in which the generators correspond to the 1-cells of P and the relations are given by the attaching maps of the 2-cells of P . Let P = gi | rj , gi−1 ωi be a presentation which is obtained from P(P ) by adding one new relation gi−1 ωi for each generator gi of P(P ), where ωi is the following word on generators of P: If j : π1 (X) → π1 (P ) and r : π1 (P ) → π1 (X) are homomorphisms of fundamental groups induced by a map converse to the domination P X and this domination (respectively) and p is a natural epimorphism of a free group on generators {gi } on the group π1 (P ), then for each i, there exists a word ωi on the generators {gi } such that j rp(gi ) = p(ωi ). One can see that P is a presentation of π1 (X) (see [29, the proof of Lemma 1.3, p. 60]). Let P be a CW-complex naturally corresponding to the presentation P . Since all of the relations gi−1 ωi hold in π1 (X), a given domination of P over X can be extended to P . Thus X P . 2 will denote, as usually, the universal covering space of X. In what follows, X Let f : X → Y be a cellular map of CW-complexes such that f (x) = y, for vertices x ∈ X, y ∈ Y . Choose x˜ ∈ p −1 (x), y˜ ∈ p −1 (y) (where p denote the projections). Then →Y such that p f˜ = fp and f˜(x) there exists a unique map f˜ : X ˜ = y. ˜ The map f˜ induces → Hk (Y ), for k = 1, 2, . . . , and we say that f˜k is induced by homomorphisms f˜k : Hk (X) f (see [13, p. 107]). Theorem 1. Assume that P is a polyhedron with polycyclic-by-finite fundamental group π1 (P ). Let P = X0 , X1 , X2 , . . . be a sequence of CW-complexes with maps
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fi : Xi−1 → Xi , for i = 1, 2, . . . , inducing epimorphisms π1 (Xi−1 ) → π1 (Xi ) and epii−1 ) → Hk (X i ), for k = 2, 3, . . . . Then there exists an integer i0 such morphisms Hk (X that for i > i0 , fi : Xi−1 → Xi are all homotopy equivalences. Proof. Let π1 (P ) ∼ = G. By Lemma 1, without loss of generality, we may consider a sequence in which homomorphisms fi : Xi−1 → Xi , for i = 2, 3, . . . , induce isomorphisms of fundamental groups π1 (Xi−1 ) → π1 (Xi ). Let π1 (Xi ) ∼ = H , for i = 1, 2, . . . , where H is a subgroup of G. Since H is a polycyclic-by-finite group, ZH is a noetherian ring (P. Hall proved that if H is a polycyclic-by-finite group and R is right noetherian ring with identity, then the group ring RG is right noetherian (see [27, p. 446, (15.3.3)]). Hence every finitely generated ZH module is noetherian (see [22, Chapter VI]). Then any submodule of a finitely generated ZH -module is also finitely generated. Recall that for every finite CW-complex L, the kth chains in the cellular complex of have a structure of a finitely generated module over Zπ1 (L) Ck (L), the universal cover L, with the basis corresponding to the k-cells of L (Whitehead [31], or see, for example, [6, Chapter 2, p. 28]). is also a finitely generated Zπ1 (L)-module. Indeed, since Zk (L) is a Then Hk (L) submodule of Ck (L), then Zk (L) is a finitely generated Zπ1 (L)-module. Thus Hk (L), a quotient module of Zk (L), is also a finitely generated Zπ1 (L)-module. i ), for i = 1, 2, . . . , are all finitely generated as ZH -modules. Therefore the Hk (X Indeed, by Lemma 2, we may replace P by a polyhedron P with π1 (P ) ∼ = H. This is also a corollary to Theorems A and B in [29] which imply that if ZH is a noetherian ring, then for every CW-complex X with π1 (X) ∼ = H which is dominated by a (for k 2) are finitely generated. polyhedron, all the ZH -modules Hk (X) i−1 → X i be the lifting to the universal covers of fi : Xi−1 → Xi . Let f˜i : X i ) and consider a sequence of ZH -modules For i = 1, 2, . . . , take Ai = Hk (X A1 → · · · → Ai → Ai+1 → · · · i−1 → X i , for i = 2, 3, . . . . We will with epimorphisms f˜i∗ : Ai−1 → Ai induced by f˜i : X ˜ show that there exists an integer i0 such that, for i > i0 , fi∗ : Ai−1 → Ai are all isomorphisms. To do this, assign to each ZH -module Ai , for i = 2, 3, . . . , a ZH -module Mi = ker ki∗ , where ki∗ = f˜i∗ · · · f˜3∗ f˜2∗ : A1 → Ai . Then · · · ⊇ Mi+1 ⊇ Mi ⊇ · · · ⊇ M2 constitute an ascending sequence of ZH -submodules of A1 . The ZH -module A1 is noetherian, hence there exists an integer i0 , such that Mi+1 = Mi , for i i0 . Therefore also Ai+1 ∼ = Ai , for i i0 . Since every finitely generated noetherian module is Hopfian (see, for example, [14, Proposition 6]), f˜(i+1)∗ : Ai → Ai+1 are all isomorphisms, for i i0 . Thus, for each sequence of CW-complexes P = X0 , X1 , X2 . . . , where P is a polyhedron and π1 (P ) is polycyclic-by-finite, with homomorphisms fi : Xi−1 → Xi , for i = 1, 2, . . . , satisfying the assumptions of the theorem, there exists an integer i0 such that
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i−1 ) → Hk (X i ), for for i > i0 , fi induce isomorphisms π1 (Xi−1 ) → π1 (Xi ) and Hk (X k = 2, 3, . . . . By the Whitehead Theorem (see, for example, [13, Theorem 3.1, p. 107]), if X and Y are CW-complexes and there exists a map f : X → Y inducing isomorphisms π1 (X) → π1 (Y ) → Hk (Y ), for all k = 2, 3, . . . , then f is a homotopy equivalence. Thus the and Hk (X) proof is complete. 2 Therefore we obtain: Corollary 1. Assume that P is a polyhedron and P = X0 , X1 , X2 , . . . is a sequence of spaces with dominations di : Xi−1 → Xi and converse maps ui : Xi → Xi−1 (i.e., di ui idXi ), for i = 1, 2, . . . . If π1 (Xi ) of some Xi in this sequence is polycyclic-by-finite, then there exists an integer i0 such that for i > i0 , di and ui are all homotopy equivalences. Proof. Without loss of generality, each Xi may be assumed to be a CW-complex, not necessarily finite (by the known results of J.H.C. Whitehead, each space homotopy dominated by a polyhedron has the homotopy type of some CW-complex, see also [29]). Applying Theorem 1 and Lemma 2 we obtain that there exists an integer i0 such that for i > i0 , di : Xi−1 → Xi are all homotopy equivalences. Then for i > i0 , ui : Xi → Xi−1 induce also isomorphisms π1 (Xi ) → π1 (Xi−1 ) and i ) → Hk (X i−1 ), for k = 2, 3, . . . . Hence by the Whitehead Theorem, for i > i0 , ui Hk (X are also homotopy equivalences and the proof is finished. 2 Remark. Let X and Y be FANR’s, x ∈ X, y ∈ Y , and f : X → Y be shape equivalence (domination). Then there exists a pointed shape equivalence (domination) f : (X, x) → (Y, y) equal to f if the base points are omitted (by [7, Theorem 5.1]). Corollary 2. Assume that (X1 , x1 ), (X2 , x2 ), . . . is a sequence of FANR’s with shape dominations d i : Xi−1 → Xi and converse shape morphisms u i : Xi → Xi−1 (i.e., d i u i = id Xi ), for i = 2, 3, . . . . If πˇ 1 (Xi , xi ) of some (Xi , xi ) in this sequence is polycyclic-byfinite, then there exists an integer i0 such that for i > i0 , d i and u i are all shape equivalences. Proof. According to the remark in the Introduction, we may consider CW-complexes (not necessarily finite) corresponding to the FANR’s Xi . Then we apply Corollary 1. 2
4. On some problems of K. Borsuk on infinite sequences of retracts In his monograph [5] K. Borsuk stated the following questions (see also [8]): Problem 1 [5, Problem (8.2), p. 285]. Is it true that if X1 , X2 , . . . ∈ ANR’s and Xi+1 is a retract of Xi , for i = 1, 2, . . . , then Xi ∈ FANR?
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Problem 2 [5, Problem (8.3), p. 285]. Is it true that for every sequence X1 , X2 , . . . of ANR’s such that Xi+1 is a retract of Xi , there exists an index i0 such that Xi+1 is a deformation retract of Xi , for i i0 ? With the assumption that π1 (Xi ) is polycyclic-by-finite for some Xi , the answer to any of these two questions is affirmative. In fact, applying Corollary 2, we obtain more: Corollary 3. Assume that (X1 , ∗), (X2 , ∗), . . . is a sequence of FANR’s such that Xi+1 is a fundamental retract of Xi and πˇ 1 (Xi , ∗) of some (Xi , ∗) in this sequence is polycyclicby-finite. Then there exists an index i0 such that Xi+1 is a fundamental deformation retract of Xi , for i i0 . Proof. Let (X1 , ∗), (X2 , ∗), . . . be a sequence of FANR’s such that Xi+1 is a fundamental retract of Xi and πˇ 1 (Xi , ∗) of some (Xi , ∗) in this sequence is polycyclic-by-finite. By Corollary 2, there exists i0 such that for i i0 , the fundamental retraction between Xi and Xi+1 and the converse shape morphism induced by the inclusion Xi+1 ⊂ Xi are both shape equivalences. Thus, for i i0 , Xi+1 is a fundamental deformation retract of Xi . 2 Note that a fundamental deformation retraction between two ANR’s is a deformation retraction. K. Borsuk [5, Chapter 8, Theorem (8.1), p. 285] stated that if X1 , X2 , . . . is a sequence of ANR’s such that Xi+1 is a deformation retract of Xi , for every i = 1, 2, . . . , then Xi ∈ FANR. As a consequence of Corollary 3 we also obtain: Corollary 4. If (X1 , ∗), (X2 , ∗), . . . ∈ FANR’s such that Xi+1is a fundamental retract of Xi and πˇ 1 (Xi , ∗) of some (Xi , ∗) is polycyclic-by-finite, then Xi ∈ FANR. Proof. By Corollary 3, there exists an index i0 such that, for i i0 , Xi+1 is a fundamental deformation retract of Xi . K. Borsuk showed thatif X1 , X2 , . . . ∈ FANR’s such that Xi+1 is a fundamental deformation retract of Xi , then Xi ∈ FANR [2, Corollary (4.26)]. This ends the proof. 2 5. On some problems of K. Borsuk concerning two ANR’s and FANR’s The questions of this section were posed by K. Borsuk in his books on shape theory [5] and on retract theory [4] (see also [8]): Problem 3 [8, Problem (2.6)]. Do there exist two FANR’s X and Y such that Sh(X) Sh(Y ), Sh(Y ) Sh(X) and Sh(X) = Sh(Y )? Problem 4 [4, Chapter IX, (12.7)]. Do there exist two ANR’s X and Y such that X Y , Y X and X Y ? Applying Corollaries 1 and 2, we obtain that such examples could not exist among FANR’s or ANR’s whose first shape group or fundamental group, respectively, is polycyclic-by-finite.
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Theorem 2. Let X, Y ∈ FANR and πˇ 1 (X, x) be polycyclic-by-finite for some x ∈ X. Then Sh(X) Sh(Y ) and Sh(Y ) Sh(X) implies that Sh(X) = Sh(Y ). Proof. We apply Corollary 2.
2
Theorem 3. Let X, Y ∈ ANR and π1 (X) be polycyclic-by-finite. Then X Y and Y X implies that X Y . Proof. This is a consequence of Corollary 1 or Theorem 2.
2
Theorem 4. Let X ∈ FANR and πˇ 1 (X, x) be polycyclic-by-finite, for some x ∈ X. Then every shape domination h : X → X is a shape equivalence. Proof. We obtain it by Corollary 2.
2
Theorem 5. Let X ∈ ANR and π1 (X) be polycyclic-by-finite. Then every homotopy domination h : X → X is a homotopy equivalence. Proof. This follows from Corollary 1 or Theorem 4.
2
Recall the following definitions: Definition 1 (S. Mardeši´c, J. Segal, see [24]). Let X and Y be two compacta. X is said to be Y -like, if for every ε > 0 there exists a continuous map f of X onto Y such that, for y ∈ Y , the diameters of all sets f −1 (y) are less than ε. Definition 2 (K. Kuratowski, S. Ulam, see [21]). Two compacta X and Y are quasihomeomorphic if X is Y -like and Y is X-like. K. Borsuk proved that there exist two quasi-homeomorphic compacta of the different shapes (see [5, p. 231], [1]) and posed: Problem 5 (See [5, Problem (12.7), p. 233]; see also [8]). Is it true that the shapes of two quasi-homeomorphic ANR’s are necessarily equal? By the previous results of this section, we obtain: Theorem 6. Let X, Y ∈ ANR and π1 (X) be polycyclic-by-finite. If X and Y are quasihomeomorphic, then X Y . Proof. We apply the result of Borsuk [1, Theorem (2.1)] that if X and Y are two compacta, Y ∈ ANR and Y is X-like, then Sh(Y ) Sh(X) (see also [5, (12.6) p. 233]). Thus X Y , and Y X, and our claim follows from Theorem 3. 2 Let us finish with the conjecture that each polyhedron P whose fundamental group π1 (P ) is polycyclic-by-finite has finite depth D(P ):
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Conjecture. Let P be a polyhedron with polycyclic-by-finite fundamental group π1 (P ). Then there exists an integer k such that each sequence · · · Xi+1 Xi · · · X2 X1 P contains at most k different homotopy types. References [1] K. Borsuk, A note on the shape of quasi-homeomorphic compacta, Comment. Math. 14 (1970) 25–34. [2] K. Borsuk, On fundamental deformation retracts and on some related notions, Fund. Math. 86 (1975) 261– 270. [3] K. Borsuk, Some problems in the theory of shape of compacta, Russian Math. Surveys 34 (6) (1979) 24–26. [4] K. Borsuk, Theory of Retracts, Monografie Matematyczne, vol. 44, PWN, Warsaw, 1967. [5] K. Borsuk, Theory of Shape, Monografie Matematyczne, vol. 59, PWN, Warsaw, 1975. [6] M. Cohen, A Course in Simple Homotopy Theory, Graduate Texts in Mathematics, vol. 10, Springer, Berlin, 1970. [7] J. Dydak, Pointed and unpointed shape and pro-homotopy, Fund. Math. 107 (1980) 58–69. [8] J. Dydak, A. Kadlof, S. Nowak, Open Problems in Shape Theory, Mimeographed Notes, University of Warsaw, 1981. [9] J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. [10] D.A. Edwards, R. Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. (2) 101 (1975) 521–535. [11] H.M. Hastings, A. Heller, Homotopy idempotents on finite-dimensional complexes split, Proc. Amer. Math. Soc. 85 (4) (1982) 619–622. [12] H.M. Hastings, A. Heller, Splitting homotopy idempotents, in: Shape Theory and Geom. Top. Proc., Dubrovnik, 1981, in: Lecture Notes in Mathematics, vol. 870, Springer, Berlin, 1981, pp. 25–36. [13] P. Hilton, An Introduction to Homotopy Theory, Cambridge University Press, Cambridge, 1961. [14] V.A. Hiremath, Hopfian rings and Hopfian modules, Indian J. Pure Appl. Math. 17 (7) (1986) 895–900. [15] A. Karras, D. Solitar, On free products, Proc. Amer. Math. Soc. 9 (1958) 217–221. [16] D. Kołodziejczyk, Homotopy dominations within polyhedra, Fund. Math. 178 (2003) 189–201. [17] D. Kołodziejczyk, Polyhedra with finite fundamental group dominate only finitely many different homotopy types, Fund. Math. 180 (2003) 1–9. [18] D. Kołodziejczyk, Simply-connected polyhedra dominate only finitely many diff erent shapes, Topology Appl. 112 (3) (2001) 289–295. [19] D. Kołodziejczyk, There exists a polyhedron dominating infinitely many polyhedra of different homotopy types, Fund. Math. 151 (1996) 39–46. [20] D. Kołodziejczyk, There exists a polyhedron with infinitely many left neighbors, Proc. Amer. Math. Soc. 129 (2001) 303–309. [21] K. Kuratowski, S. Ulam, Sur un coefficient lie aux transformations continues d’ensembles, Fund. Math. 20 (1933) 244–253. [22] S. Lang, Algebra, Addison-Wesley, Reading, MA, 1965. [23] A.I. Malcev, On isomorphic representations of infinite groups by matrices, Mat. Sb. 8 (1940) 405–422. [24] S. Mardeši´c, J. Segal, ε mappings onto polyhedra, Trans. Amer. Math. Soc. 109 (1963) 146–164. [25] S. Mardeši´c, J. Segal, Shape Theory. The Inverse System Approach, North-Holland Mathematical Library, vol. 26, North-Holland, Amsterdam, 1982. [26] D. Passman, The Algebraic Structure of Group Rings, New York, 1977. [27] D.J.S. Robinson, A Course in the Theory of Groups, Springer, Berlin, 1982. [28] J. Rutter, Spaces of Homotopy Self-Equivalences, Lecture Notes in Mathematics, vol. 1662, Springer, Berlin, 1997. [29] C.T.C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965) 56–69. [30] J.E. West, Compact ANR’s have finite type, Bull. Amer. Math. Soc. 81 (1975) 163–165. [31] J.H.C. Whitehead, Combinatorial homotopy I (and II), Bull. Amer. Math. Soc. 55 (1949) 213–245, 453–496.
Homotopy dominations by polyhedra with polycyclic-by-finite fundamental groups Danuta Kołodziejczyk 1 Faculty of Mathematics and Informational Sciences, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warsaw, Poland Received 1 February 2003; received in revised form 22 September 2003; accepted 22 September 2003
Abstract In the previous papers, in connection with a question of K. Borsuk, we proved that there exist polyhedra with polycyclic fundamental groups homotopy dominating infinitely many different homotopy types. Here we consider a few problems of K. Borsuk concerning infinite chains of polyhedra or FANR’s ordered by the relation of domination (in homotopy or shape category) and obtain that for polyhedra with polycyclic-by-finite fundamental groups, there are no pathology similar to the above. 2004 Elsevier B.V. All rights reserved. MSC: 55P55; 55P15; 55P10 Keywords: Polyhedron; ANR; CW-complex; FANR; Compactum; Homotopy domination; Homotopy type; Shape domination; Shape
1. Introduction In this paper every polyhedron, ANR, FANR and CW-complex is assumed to be connected (only for convenience). We expect that the reader is familiar with the basic notions and facts of shape theory (see [5,9,25]) and of retract theory (see [4]).
E-mail address: [email protected] (D. Kołodziejczyk). 1 Address for correspondence: ul. Jasna 8/18, 00-013 Warsaw, Poland.
0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2003.09.016
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Define the capacity C(A) of a compactum A as the cardinality of the class of the shapes of all compacta X such that Sh(X) Sh(A). Let us say that a system consisting of k compacta X1 , X2 , . . . , Xk is a chain of length k for a compactum A if Sh(X1 ) < Sh(X2 ) < · · · < Sh(Xk ) Sh(A) (Sh(X) < Sh(Y ) if and only if Sh(X) Sh(Y ) holds but Sh(Y ) Sh(X) fails). The depth D(A) of a compactum 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 . The above notions were introduced by K. Borsuk in 1979, at the Topological Conference in Moscow, together with some relevant problems (see [3]). In [19], answering the most interesting question: Is the capacity of each polyhedron finite? (see also [8]), we showed an example of a polyhedron dominating infinitely many different homotopy types of polyhedra (see also [20]). Such phenomena are even frequent: in [16] we proved that, for every non-Abelian poly-Z-group G, there exists a polyhedron P with π1 (P ) ∼ = G dominating infinitely many polyhedra of different homotopy types. Thus, there exist polyhedra with polycyclic or nilpotent fundamental groups with this property. Using those examples, we gave also a positive answer to the other question from [3] (Question (8), see also [8, Problem (6.1)]) stating that there exist polyhedra (with polycyclic fundamental groups) with infinite capacity and finite depth. We conjecture that while there exist polyhedra with polycyclic fundamental groups and infinite capacity, every polyhedron with such a fundamental group has finite depth. The problem, if there exists a polyhedron with infinite depth, remains unsolved. It is an obvious observation that finite capacity of a given polyhedron implies finite depth. Let us mention that we showed that all the polyhedra with finite fundamental groups, nilpotent polyhedra, and some others, dominate only finitely many different shapes (or homotopy types), see [18] and [17]. It is also known, by the classical results of shape theory, that the capacity of each 1-dimensional polyhedron is finite. Observe that a polyhedron P has finite depth if and only if: there exists an integer k such that each sequence · · · Sh(Xi ) · · · Sh(X1 ) Sh(P ) contains at most k different shapes (compare [8, Problem (6.1)]). Here we show that for polyhedra with polycyclic-by-finite fundamental groups there are no infinite sequences of this kind. This implies the solutions of some questions of K. Borsuk included in his books [4,5] concerning infinite sequences of shape or homotopy dominations between FANR’s or ANR’s, for FANR’s with polycyclic-by-finite first shape groups and ANR’s with such fundamental groups (recall that by the well-known result of J. West [30], every ANR has the homotopy type of a polyhedron). In the sequel X Y will denote that X is homotopy dominated by Y . Let us remark that by the classical results in shape theory (see [11] or [12]; [9, Theorem 2.2.6]; [10]) 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 (not necessarily finite) homotopy dominated by this polyhedron (in both, pointed and unpointed cases). Hence, if X and Y are two compacta shape dominated by a polyhedron, and Sh(X) Sh(Y ), then X Y , for corresponding CW-complexes X and Y (respectively), and conversely. Thus considering questions on shape dominations between FANR’s we may work in the homotopy category of CW-complexes and homotopy classes of cellular maps between them. Also in the definitions of the capacity and the depth of a polyhedron
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one can investigate dominations in the homotopy category of CW-complexes. Moreover, there is no difference which case, pointed or unpointed, is studied (see [7]).
2. Algebraic preliminaries Let us recall some definitions (see [27]): A group G is polycyclic if it has a finite normal series G = G0 ⊇ G1 ⊇ · · · ⊇ Gl = 1, (Gi G, for i = 1, . . . , l) for which each factor Gi−1 /Gi is infinite cyclic or finite cyclic (this is true for all the finitely generated nilpotent groups, see, for instance, [28, 15.4, p. 92]). G is a poly-Z-group if it has a finite normal series with factors Z. A group G is called polycyclic-by-finite if it is an extension of a polycyclic group by a finite group, i.e., there exists a polycyclic group H G such that G/H is finite. It is said that a group G satisfies ascending chain condition (or max-condition) if and only if every ascending sequence of subgroups of G, G1 ⊆ G2 ⊆ · · · , where Gi = Gi+1 , is finite. A group G is called residually finite if and only if their subgroups of finite index have trivial intersection. Recall that a group G is Hopfian if every epimorphism f : G → G is an automorphism (equivalently, N = 1 is the only normal subgroup for which G/N ∼ = G). Let R be a ring. A right R-module M is said to be noetherian if every R-submodule of M is finitely generated, equivalently, M satisfies ascending chain condition, i.e., every ascending sequence of submodules of M: M1 ⊆ M2 ⊆ · · · such that Mi = Mi+1 is finite (compare [26, p. 419], or [22, Chapter VI,1]). We call a ring R noetherian if it is a noetherian module as a left module over itself. A module M is called Hopfian (see [14]) if every epimorphism f : M → M is an isomorphism (equivalently M is not isomorphic to a proper quotient of itself).
3. Main results Let us begin with the following: Lemma 1. Assume that G is a polycyclic-by-finite group. Let G = G0 → G1 → · · · → Gi → Gi+1 → · · · be a sequence of groups with epimorphisms fi : Gi−1 → Gi , for i = 1, 2, . . . , then there exists an integer i0 such that for i > i0 , fi is an isomorphism. Proof. Assign to each group Gi a normal divisor Ni G such that Ni = ker si , where si = fi · · · f2 f1 : G → Gi (i = 1, 2, . . .). Consider the following sequence of subgroups of G G ⊇ · · · ⊇ Ni+1 ⊇ Ni ⊇ · · · ⊇ N1 .
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As a polycyclic-by-finite group, G satisfies the max-condition (see [27, Chapter 3, p. 69 or (5.4.12), p. 152], hence there exists an integer i0 such that Ni+1 = Ni , for i i0 . Then for i i0 , we have Gi+1 ∼ = Gi . Every subgroup of a polycyclic-by-finite group is a group of this kind. Moreover, it is known that every polycyclic-by-finite group is Hopfian. Indeed, by the result of Hirsch, every polycyclic-by-finite group is residually finite (see [27, (5.4.17)]), and by the result of Malcev, if a finitely generated group G is residually finite, then it is Hopfian (see [23,15]). Obviously, an epimorphism between isomorphic Hopfian groups is an isomorphism. Therefore, for every i > i0 , fi : Gi−1 → Gi is an isomorphism. This ends the proof of the lemma. 2 In what follows we will also use the following: Lemma 2 (See [29, the proof of Theorem A, implication (ii) ⇒ (i), p. 60]). Let X P , where P is a finite CW-complex, then there exists a finite CW-complex P such that X P and π1 (P ) ∼ = π1 (X). Proof. Assume that X P and P is a finite CW-complex. Then, by attaching finitely many 2-cells, we may change P into a finite CW-complex P such that X P and π1 (P ) ∼ = π1 (X). Indeed, without loss of generality one can assume P to have just one 0-cell. Let P(P ) = gi | rj be a presentation of π1 (P ) in which the generators correspond to the 1-cells of P and the relations are given by the attaching maps of the 2-cells of P . Let P = gi | rj , gi−1 ωi be a presentation which is obtained from P(P ) by adding one new relation gi−1 ωi for each generator gi of P(P ), where ωi is the following word on generators of P: If j : π1 (X) → π1 (P ) and r : π1 (P ) → π1 (X) are homomorphisms of fundamental groups induced by a map converse to the domination P X and this domination (respectively) and p is a natural epimorphism of a free group on generators {gi } on the group π1 (P ), then for each i, there exists a word ωi on the generators {gi } such that j rp(gi ) = p(ωi ). One can see that P is a presentation of π1 (X) (see [29, the proof of Lemma 1.3, p. 60]). Let P be a CW-complex naturally corresponding to the presentation P . Since all of the relations gi−1 ωi hold in π1 (X), a given domination of P over X can be extended to P . Thus X P . 2 will denote, as usually, the universal covering space of X. In what follows, X Let f : X → Y be a cellular map of CW-complexes such that f (x) = y, for vertices x ∈ X, y ∈ Y . Choose x˜ ∈ p −1 (x), y˜ ∈ p −1 (y) (where p denote the projections). Then →Y such that p f˜ = fp and f˜(x) there exists a unique map f˜ : X ˜ = y. ˜ The map f˜ induces → Hk (Y ), for k = 1, 2, . . . , and we say that f˜k is induced by homomorphisms f˜k : Hk (X) f (see [13, p. 107]). Theorem 1. Assume that P is a polyhedron with polycyclic-by-finite fundamental group π1 (P ). Let P = X0 , X1 , X2 , . . . be a sequence of CW-complexes with maps
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fi : Xi−1 → Xi , for i = 1, 2, . . . , inducing epimorphisms π1 (Xi−1 ) → π1 (Xi ) and epii−1 ) → Hk (X i ), for k = 2, 3, . . . . Then there exists an integer i0 such morphisms Hk (X that for i > i0 , fi : Xi−1 → Xi are all homotopy equivalences. Proof. Let π1 (P ) ∼ = G. By Lemma 1, without loss of generality, we may consider a sequence in which homomorphisms fi : Xi−1 → Xi , for i = 2, 3, . . . , induce isomorphisms of fundamental groups π1 (Xi−1 ) → π1 (Xi ). Let π1 (Xi ) ∼ = H , for i = 1, 2, . . . , where H is a subgroup of G. Since H is a polycyclic-by-finite group, ZH is a noetherian ring (P. Hall proved that if H is a polycyclic-by-finite group and R is right noetherian ring with identity, then the group ring RG is right noetherian (see [27, p. 446, (15.3.3)]). Hence every finitely generated ZH module is noetherian (see [22, Chapter VI]). Then any submodule of a finitely generated ZH -module is also finitely generated. Recall that for every finite CW-complex L, the kth chains in the cellular complex of have a structure of a finitely generated module over Zπ1 (L) Ck (L), the universal cover L, with the basis corresponding to the k-cells of L (Whitehead [31], or see, for example, [6, Chapter 2, p. 28]). is also a finitely generated Zπ1 (L)-module. Indeed, since Zk (L) is a Then Hk (L) submodule of Ck (L), then Zk (L) is a finitely generated Zπ1 (L)-module. Thus Hk (L), a quotient module of Zk (L), is also a finitely generated Zπ1 (L)-module. i ), for i = 1, 2, . . . , are all finitely generated as ZH -modules. Therefore the Hk (X Indeed, by Lemma 2, we may replace P by a polyhedron P with π1 (P ) ∼ = H. This is also a corollary to Theorems A and B in [29] which imply that if ZH is a noetherian ring, then for every CW-complex X with π1 (X) ∼ = H which is dominated by a (for k 2) are finitely generated. polyhedron, all the ZH -modules Hk (X) i−1 → X i be the lifting to the universal covers of fi : Xi−1 → Xi . Let f˜i : X i ) and consider a sequence of ZH -modules For i = 1, 2, . . . , take Ai = Hk (X A1 → · · · → Ai → Ai+1 → · · · i−1 → X i , for i = 2, 3, . . . . We will with epimorphisms f˜i∗ : Ai−1 → Ai induced by f˜i : X ˜ show that there exists an integer i0 such that, for i > i0 , fi∗ : Ai−1 → Ai are all isomorphisms. To do this, assign to each ZH -module Ai , for i = 2, 3, . . . , a ZH -module Mi = ker ki∗ , where ki∗ = f˜i∗ · · · f˜3∗ f˜2∗ : A1 → Ai . Then · · · ⊇ Mi+1 ⊇ Mi ⊇ · · · ⊇ M2 constitute an ascending sequence of ZH -submodules of A1 . The ZH -module A1 is noetherian, hence there exists an integer i0 , such that Mi+1 = Mi , for i i0 . Therefore also Ai+1 ∼ = Ai , for i i0 . Since every finitely generated noetherian module is Hopfian (see, for example, [14, Proposition 6]), f˜(i+1)∗ : Ai → Ai+1 are all isomorphisms, for i i0 . Thus, for each sequence of CW-complexes P = X0 , X1 , X2 . . . , where P is a polyhedron and π1 (P ) is polycyclic-by-finite, with homomorphisms fi : Xi−1 → Xi , for i = 1, 2, . . . , satisfying the assumptions of the theorem, there exists an integer i0 such that
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i−1 ) → Hk (X i ), for for i > i0 , fi induce isomorphisms π1 (Xi−1 ) → π1 (Xi ) and Hk (X k = 2, 3, . . . . By the Whitehead Theorem (see, for example, [13, Theorem 3.1, p. 107]), if X and Y are CW-complexes and there exists a map f : X → Y inducing isomorphisms π1 (X) → π1 (Y ) → Hk (Y ), for all k = 2, 3, . . . , then f is a homotopy equivalence. Thus the and Hk (X) proof is complete. 2 Therefore we obtain: Corollary 1. Assume that P is a polyhedron and P = X0 , X1 , X2 , . . . is a sequence of spaces with dominations di : Xi−1 → Xi and converse maps ui : Xi → Xi−1 (i.e., di ui idXi ), for i = 1, 2, . . . . If π1 (Xi ) of some Xi in this sequence is polycyclic-by-finite, then there exists an integer i0 such that for i > i0 , di and ui are all homotopy equivalences. Proof. Without loss of generality, each Xi may be assumed to be a CW-complex, not necessarily finite (by the known results of J.H.C. Whitehead, each space homotopy dominated by a polyhedron has the homotopy type of some CW-complex, see also [29]). Applying Theorem 1 and Lemma 2 we obtain that there exists an integer i0 such that for i > i0 , di : Xi−1 → Xi are all homotopy equivalences. Then for i > i0 , ui : Xi → Xi−1 induce also isomorphisms π1 (Xi ) → π1 (Xi−1 ) and i ) → Hk (X i−1 ), for k = 2, 3, . . . . Hence by the Whitehead Theorem, for i > i0 , ui Hk (X are also homotopy equivalences and the proof is finished. 2 Remark. Let X and Y be FANR’s, x ∈ X, y ∈ Y , and f : X → Y be shape equivalence (domination). Then there exists a pointed shape equivalence (domination) f : (X, x) → (Y, y) equal to f if the base points are omitted (by [7, Theorem 5.1]). Corollary 2. Assume that (X1 , x1 ), (X2 , x2 ), . . . is a sequence of FANR’s with shape dominations d i : Xi−1 → Xi and converse shape morphisms u i : Xi → Xi−1 (i.e., d i u i = id Xi ), for i = 2, 3, . . . . If πˇ 1 (Xi , xi ) of some (Xi , xi ) in this sequence is polycyclic-byfinite, then there exists an integer i0 such that for i > i0 , d i and u i are all shape equivalences. Proof. According to the remark in the Introduction, we may consider CW-complexes (not necessarily finite) corresponding to the FANR’s Xi . Then we apply Corollary 1. 2
4. On some problems of K. Borsuk on infinite sequences of retracts In his monograph [5] K. Borsuk stated the following questions (see also [8]): Problem 1 [5, Problem (8.2), p. 285]. Is it true that if X1 , X2 , . . . ∈ ANR’s and Xi+1 is a retract of Xi , for i = 1, 2, . . . , then Xi ∈ FANR?
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Problem 2 [5, Problem (8.3), p. 285]. Is it true that for every sequence X1 , X2 , . . . of ANR’s such that Xi+1 is a retract of Xi , there exists an index i0 such that Xi+1 is a deformation retract of Xi , for i i0 ? With the assumption that π1 (Xi ) is polycyclic-by-finite for some Xi , the answer to any of these two questions is affirmative. In fact, applying Corollary 2, we obtain more: Corollary 3. Assume that (X1 , ∗), (X2 , ∗), . . . is a sequence of FANR’s such that Xi+1 is a fundamental retract of Xi and πˇ 1 (Xi , ∗) of some (Xi , ∗) in this sequence is polycyclicby-finite. Then there exists an index i0 such that Xi+1 is a fundamental deformation retract of Xi , for i i0 . Proof. Let (X1 , ∗), (X2 , ∗), . . . be a sequence of FANR’s such that Xi+1 is a fundamental retract of Xi and πˇ 1 (Xi , ∗) of some (Xi , ∗) in this sequence is polycyclic-by-finite. By Corollary 2, there exists i0 such that for i i0 , the fundamental retraction between Xi and Xi+1 and the converse shape morphism induced by the inclusion Xi+1 ⊂ Xi are both shape equivalences. Thus, for i i0 , Xi+1 is a fundamental deformation retract of Xi . 2 Note that a fundamental deformation retraction between two ANR’s is a deformation retraction. K. Borsuk [5, Chapter 8, Theorem (8.1), p. 285] stated that if X1 , X2 , . . . is a sequence of ANR’s such that Xi+1 is a deformation retract of Xi , for every i = 1, 2, . . . , then Xi ∈ FANR. As a consequence of Corollary 3 we also obtain: Corollary 4. If (X1 , ∗), (X2 , ∗), . . . ∈ FANR’s such that Xi+1is a fundamental retract of Xi and πˇ 1 (Xi , ∗) of some (Xi , ∗) is polycyclic-by-finite, then Xi ∈ FANR. Proof. By Corollary 3, there exists an index i0 such that, for i i0 , Xi+1 is a fundamental deformation retract of Xi . K. Borsuk showed thatif X1 , X2 , . . . ∈ FANR’s such that Xi+1 is a fundamental deformation retract of Xi , then Xi ∈ FANR [2, Corollary (4.26)]. This ends the proof. 2 5. On some problems of K. Borsuk concerning two ANR’s and FANR’s The questions of this section were posed by K. Borsuk in his books on shape theory [5] and on retract theory [4] (see also [8]): Problem 3 [8, Problem (2.6)]. Do there exist two FANR’s X and Y such that Sh(X) Sh(Y ), Sh(Y ) Sh(X) and Sh(X) = Sh(Y )? Problem 4 [4, Chapter IX, (12.7)]. Do there exist two ANR’s X and Y such that X Y , Y X and X Y ? Applying Corollaries 1 and 2, we obtain that such examples could not exist among FANR’s or ANR’s whose first shape group or fundamental group, respectively, is polycyclic-by-finite.
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Theorem 2. Let X, Y ∈ FANR and πˇ 1 (X, x) be polycyclic-by-finite for some x ∈ X. Then Sh(X) Sh(Y ) and Sh(Y ) Sh(X) implies that Sh(X) = Sh(Y ). Proof. We apply Corollary 2.
2
Theorem 3. Let X, Y ∈ ANR and π1 (X) be polycyclic-by-finite. Then X Y and Y X implies that X Y . Proof. This is a consequence of Corollary 1 or Theorem 2.
2
Theorem 4. Let X ∈ FANR and πˇ 1 (X, x) be polycyclic-by-finite, for some x ∈ X. Then every shape domination h : X → X is a shape equivalence. Proof. We obtain it by Corollary 2.
2
Theorem 5. Let X ∈ ANR and π1 (X) be polycyclic-by-finite. Then every homotopy domination h : X → X is a homotopy equivalence. Proof. This follows from Corollary 1 or Theorem 4.
2
Recall the following definitions: Definition 1 (S. Mardeši´c, J. Segal, see [24]). Let X and Y be two compacta. X is said to be Y -like, if for every ε > 0 there exists a continuous map f of X onto Y such that, for y ∈ Y , the diameters of all sets f −1 (y) are less than ε. Definition 2 (K. Kuratowski, S. Ulam, see [21]). Two compacta X and Y are quasihomeomorphic if X is Y -like and Y is X-like. K. Borsuk proved that there exist two quasi-homeomorphic compacta of the different shapes (see [5, p. 231], [1]) and posed: Problem 5 (See [5, Problem (12.7), p. 233]; see also [8]). Is it true that the shapes of two quasi-homeomorphic ANR’s are necessarily equal? By the previous results of this section, we obtain: Theorem 6. Let X, Y ∈ ANR and π1 (X) be polycyclic-by-finite. If X and Y are quasihomeomorphic, then X Y . Proof. We apply the result of Borsuk [1, Theorem (2.1)] that if X and Y are two compacta, Y ∈ ANR and Y is X-like, then Sh(Y ) Sh(X) (see also [5, (12.6) p. 233]). Thus X Y , and Y X, and our claim follows from Theorem 3. 2 Let us finish with the conjecture that each polyhedron P whose fundamental group π1 (P ) is polycyclic-by-finite has finite depth D(P ):
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Conjecture. Let P be a polyhedron with polycyclic-by-finite fundamental group π1 (P ). Then there exists an integer k such that each sequence · · · Xi+1 Xi · · · X2 X1 P contains at most k different homotopy types. References [1] K. Borsuk, A note on the shape of quasi-homeomorphic compacta, Comment. Math. 14 (1970) 25–34. [2] K. Borsuk, On fundamental deformation retracts and on some related notions, Fund. Math. 86 (1975) 261– 270. [3] K. Borsuk, Some problems in the theory of shape of compacta, Russian Math. Surveys 34 (6) (1979) 24–26. [4] K. Borsuk, Theory of Retracts, Monografie Matematyczne, vol. 44, PWN, Warsaw, 1967. [5] K. Borsuk, Theory of Shape, Monografie Matematyczne, vol. 59, PWN, Warsaw, 1975. [6] M. Cohen, A Course in Simple Homotopy Theory, Graduate Texts in Mathematics, vol. 10, Springer, Berlin, 1970. [7] J. Dydak, Pointed and unpointed shape and pro-homotopy, Fund. Math. 107 (1980) 58–69. [8] J. Dydak, A. Kadlof, S. Nowak, Open Problems in Shape Theory, Mimeographed Notes, University of Warsaw, 1981. [9] J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Mathematics, vol. 688, Springer, Berlin, 1978. [10] D.A. Edwards, R. Geoghegan, Shapes of complexes, ends of manifolds, homotopy limits and the Wall obstruction, Ann. of Math. (2) 101 (1975) 521–535. [11] H.M. Hastings, A. Heller, Homotopy idempotents on finite-dimensional complexes split, Proc. Amer. Math. Soc. 85 (4) (1982) 619–622. [12] H.M. Hastings, A. Heller, Splitting homotopy idempotents, in: Shape Theory and Geom. Top. Proc., Dubrovnik, 1981, in: Lecture Notes in Mathematics, vol. 870, Springer, Berlin, 1981, pp. 25–36. [13] P. Hilton, An Introduction to Homotopy Theory, Cambridge University Press, Cambridge, 1961. [14] V.A. Hiremath, Hopfian rings and Hopfian modules, Indian J. Pure Appl. Math. 17 (7) (1986) 895–900. [15] A. Karras, D. Solitar, On free products, Proc. Amer. Math. Soc. 9 (1958) 217–221. [16] D. Kołodziejczyk, Homotopy dominations within polyhedra, Fund. Math. 178 (2003) 189–201. [17] D. Kołodziejczyk, Polyhedra with finite fundamental group dominate only finitely many different homotopy types, Fund. Math. 180 (2003) 1–9. [18] D. Kołodziejczyk, Simply-connected polyhedra dominate only finitely many diff erent shapes, Topology Appl. 112 (3) (2001) 289–295. [19] D. Kołodziejczyk, There exists a polyhedron dominating infinitely many polyhedra of different homotopy types, Fund. Math. 151 (1996) 39–46. [20] D. Kołodziejczyk, There exists a polyhedron with infinitely many left neighbors, Proc. Amer. Math. Soc. 129 (2001) 303–309. [21] K. Kuratowski, S. Ulam, Sur un coefficient lie aux transformations continues d’ensembles, Fund. Math. 20 (1933) 244–253. [22] S. Lang, Algebra, Addison-Wesley, Reading, MA, 1965. [23] A.I. Malcev, On isomorphic representations of infinite groups by matrices, Mat. Sb. 8 (1940) 405–422. [24] S. Mardeši´c, J. Segal, ε mappings onto polyhedra, Trans. Amer. Math. Soc. 109 (1963) 146–164. [25] S. Mardeši´c, J. Segal, Shape Theory. The Inverse System Approach, North-Holland Mathematical Library, vol. 26, North-Holland, Amsterdam, 1982. [26] D. Passman, The Algebraic Structure of Group Rings, New York, 1977. [27] D.J.S. Robinson, A Course in the Theory of Groups, Springer, Berlin, 1982. [28] J. Rutter, Spaces of Homotopy Self-Equivalences, Lecture Notes in Mathematics, vol. 1662, Springer, Berlin, 1997. [29] C.T.C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965) 56–69. [30] J.E. West, Compact ANR’s have finite type, Bull. Amer. Math. Soc. 81 (1975) 163–165. [31] J.H.C. Whitehead, Combinatorial homotopy I (and II), Bull. Amer. Math. Soc. 55 (1949) 213–245, 453–496.