- Email: [email protected]
Strong surjections from two-complexes with trivial top-cohomology onto the torus
Topology and its Applications 210 (2016) 63–69
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Strong surjections from two-complexes with trivial top-cohomology onto the torus Marcio Colombo Fenille Universidade Federal de Uberlândia – Faculdade de Matemática, Av. João Naves de Ávila 2121, Sta Mônica, 38408-100, Uberlândia-MG, Brazil
a r t i c l e
i n f o
Article history: Received 13 April 2016 Received in revised form 4 July 2016 Accepted 14 July 2016 Available online 20 July 2016 MSC: primary 55M20 secondary 57M20
a b s t r a c t We build a countable collection of two-dimensional CW complexes with trivial second integer cohomology group and, from each of them, a strong surjection onto the torus. Furthermore, we prove that such two-complexes are the simplest with these properties. This answers, for dimension two, a problem originally proposed in the 2000’s for dimension three. © 2016 Elsevier B.V. All rights reserved.
Keywords: Strong surjection Two-dimensional complexes Zero top-cohomology Equations in free groups Root theory
1. Introduction Let K be a finite and connected two-dimensional CW complex (a two-complex, to shorten). By the Hopf–Whitney Classification Theorem [12, Corollary 6.19 on p. 244], if the top-cohomology group H 2 (K) is trivial (integer coefficients is subtended), then all maps from K into the two-sphere are homotopic to a nonsurjective map, but we do not know about maps into the others closed surfaces. This problem becomes more interesting when we note that a two-complex K with H 2 (K) = 0 is (co)homologically like a one-complex, since by the Universal Coefficient Theorem we have, in such case, H2 (K) = 0 and H 1 (K) ≈ H1 (K) torsion free. This fact aggravates the problem: since H 2 is not able to detect the existence of (non-collapsible) two-cells in K, is it able to detect the existence of strong surjections from K into a closed surfaces? Here, by a strong surjection we mean a map whose homotopy class contains only surjective maps. Of course, the same question makes sense also in greater dimensions. In the 2000’s, C. Aniz and D.L. Gonçalves approached the problem in dimension three. In [1], C. Aniz proved that every map from E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.07.010 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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a 3-complex K with H 3 (K) = 0 into S 1 × S 2 is homotopic to a non-surjective map, but for Y the nonorientable S 1 -bundle over S 2 , there exists a strong surjection f : K → Y from such a 3-complex. In [2], C. Aniz proved that there is no strong surjection from a 3-complex K with H 3 (K) = 0 into the orbit space of the 3-sphere S 3 with respect to the action of the quaternion group Q8 determined by the inclusion Q8 ⊂ S 3 . In 2008, D.L. Gonçalves introduced us to the problem, in dimension two, and conjectured that, as in dimension three, there should be a strong surjection from a two-complex with trivial top-cohomology onto some closed surface. It is noteworthy that the dimension two is often left out, in several contributions to topological root theory, since it does not permit the use of special techniques as obstruction theory. In order to attack our problem itself, also the Nielsen root theory is not feasible, despite the Wecken property for roots established by D.L. Gonçalves and P. Wong in [9], since there is no practical mechanisms for calculating the Nielsen root number. Our approach is based in combinatorial group theory, in special equation in free groups, what has been used successfully in coincidence and root theory for maps between closed surfaces, as we see, for instance, in [3,7] and [8]. In [6], also via combinatorial group theory, we study indirectly the problem considering only maps into the real projective plane RP2 . We prove that for a finite and connected two-dimensional complex K, the condition [K; RP2 ]∗ = 0 implies H 2 (K) = 0, and the opposite implication is true if the number of two-cells of K is equal to the first Betti number of its one-skeleton K 1 . In this article, we establish the conjecture proposed by D.L. Gonçalves. Specifically, we build a collection of two-complexes Kp, q with trivial top-cohomology, for all coprime integers p, q ≥ 2, and a strong surjection from each Kp, q onto the torus T = S 1 × S 1 . To ensure the success of our approach, two things were particularly important: first, the necessary and sufficient condition, expressed in terms of equation in free groups, to a map from a two-complex into a closed surface to be strongly surjective, presented in [4]; second, the classical result of R.C. Lyndon and M.P. Schutzenberger, published in [10], which states that if a, b and c are elements of a free group and am = bn cp , for integers m, n, p ≥ 2, then a, b and c are contained in a cyclic subgroup. After we build the aforementioned collection of two-complexes with trivial top-cohomology and strong surjections from them onto the torus, we prove, in the final section of the article, that such collection is as simple as possible. We explain: each two-complex Kp, q has five cells and the first Betti number of its one-skeleton is three; we prove that if K is a finite and connected two-complex with H 2 (K) = 0 and either K has less than five cells or β1 (K 1 ) ≤ 2, then there is no strong surjection from K into the torus. 2. On the equation (Xw1 )p (Y w2 )q (Zw3 )r = 1 In this section, we prove a simple result in combinatorial group theory, which is directly used, in the next section, in the proof of our main theorem. Let p, q and r be integers greater than or equal to two and let w1 , w2 and w3 be words in the free group F n = F (a1 , . . . , an ) of rank n ≥ 2. Put 1 to be the empty word in F n . In what follows we consider the equation (Xw1 )p (Y w2 )q (Zw3 )r = 1
(1)
in the free group F n . A solution for Equation (1) is a group homomorphism φ : F (x1 , x2 , x3 ) → F n for which the triad (X, Y, Z) = (φ(x1 ), φ(x2 ), φ(x3 )) satisfies the equation. In such a case, we say also that such triad is a solution for Equation (1). Of course, Equation (1) has an obvious solution, namely, the triad (w1−1 , w2−1 , w3−1 ). If φ is a solution for Equation (1) whose image is contained in a subgroup H of F n , we say that φ is a solution for Equation (1) over H. We are particularly interested in the case in which H is the commutator subgroup of F n . Therefore, let ξ : F n → Zn be the abelianization homomorphism and, for each index
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1 ≤ λ ≤ n, put ξλ = πλ ◦ ξ : F n → Z, in which πλ : Zn → Z is the projection onto the λth coordinate. Thus, ξλ (ai ) = δλi , the Kronecker delta, and ξ(w) = (ξ1 (w), . . . , ξn (w)), for each w ∈ F n . Moreover, ξλ (w) is the sum of all powers of the letter aλ in the word w. If pξ(w1 ) + qξ(w2 ) + rξ(w3 ) = 0, then it makes sense to ask if Equation (1) has a solution over ker ξ. The following result presents a condition under which Equation (1) has not one. Proposition 2.1. If for some indexes 1 ≤ i, j ≤ 3 and 1 ≤ k, l ≤ n one has ξk (wi ) = 0, ξl (wi ) = 0 and ξk (wj ) = 0, then Equation (1) has no solution over ker ξ. Proof. Let φ : F (x1 , x2 , x3 ) → F n be a solution for Equation (1), which certainly exists. Then the triad (φ(x1 )w1 , φ(x2 )w2 , φ(x3 )w3 ) is a solution for the equation ap bq cr = 1 in F n . By the main result of [10] there exists a word w ∈ F n such that, for each index 1 ≤ s ≤ 3, one has φ(xs )ws = wts for some ts ∈ Z. Hence, for each index 1 ≤ λ ≤ n, we have ts ξλ (w) = ξλ (wts ) = ξλ (φ(xs )ws ) = ξλ (φ(xs )) + ξλ (ws ). Suppose that φ is actually a solution for Equation (1) over ker ξ. It follows from the assumption on the word wi that ti ξk (w) = ξk (φ(xi )) + ξk (wi ) = 0
and
ti ξl (w) = ξl (φ(xi )) + ξl (wi ) = ξl (wi ) = 0, which implies ξk (w) = 0. On the other hand, it follows from the assumption on the word wj that tj ξk (w) = ξk (φ(xj )) + ξk (wj ) = ξk (wj ) = 0, which implies ξk (w) = 0. Thus, we have a contradiction and, therefore, there is not a solution for Equation (1) over ker ξ. 2 As a special example in which Proposition 2.1 works, take integers p, q ≥ 2 and consider the equation (Xaq bp )p (Y a−p b−q )q (Zbq−p )p+q = 1
(2)
in the free group F (a, b) of rank two. For the abelianization homomorphism ξ : F (a, b) → Z ⊕ Z given as before, we have pξ(aq bp ) + qξ(a−p b−q ) + (p + q)ξ(bq−p ) = 0. Therefore, it makes sense to investigate the existence of solutions for Equation (2) over ker ξ. However, as a simple consequence of Proposition 2.1, we have: Corollary 2.2. For all integers q > p ≥ 2, Equation (2) has no solution over ker ξ. Proof. For the words w1 = aq bp and w3 = bq−p we have ξ1 (w3 ) = 0, ξ2 (w3 ) = q − p = 0 and ξ1 (w1 ) = q = 0. Therefore, Proposition 2.1 applies. 2 Corollary 2.2 is the (algebraic) key for our main theorem (Theorem 3.2), which we prove in the next section.
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3. Strong surjections onto the torus Let p and q be positive integers and let Kp, q be the model two-complex induced by the group presentation P = x, y, z | xp y q z p+q , that is, the two-complex whose one-skeleton is the bouquet Sx1 ∨ Sy1 ∨ Sz1 of three circles and whose single two-cell is attached according to the relator xp y q z p+q . See [11] for details. The next proposition is a simple exercise of elementary Algebraic Topology, for which we provide a proof using results on root theory and homotopy theory, in order to contextualize. This simple result is important to the conclusion of the paper. Proposition 3.1. H 2 (Kp, q) = 0 if and only if p and q are coprime. Proof. By the Hopf–Whitney Classification Theorem [12, Corollary 6.19 on p. 244], the integer cohomology group H 2 (Kp, q) is in one-to-one correspondence with the set [Kp, q; S 2 ] of the homotopy classes of maps from Kp, q into the two-sphere. On the other hand, it follows from [5, Theorem 1.1] that [Kp, q; S 2 ] = 0 if and only if the diophantine linear system px + qy + (p + q)z = d has an integer solution for all d ∈ Z. Obviously, this last condition is equivalent to gcd(p, q) = 1. Therefore, H 2 (Kp, q) = 0 if and only if gcd(p, q) = 1. 2 Of course, a direct proof for Proposition 3.1, independently on results on root and homotopy theory, may be given by analyzing the cellular chain complex of Kp, q. Theorem 3.2. For all integers q > p ≥ 2, there exists a strong surjection from the two-complex Kp, q onto the torus. Proof. Put K = Kp, q and consider its one-skeleton K 1 = Sx1 ∨ Sy1 ∨ Sz1 = e0 ∪ e1x ∪ e1y ∪ e1z . Let τ : F (x, y, z) → F (a, b) be the homomorphism given by τ (x) = aq bp , τ (y) = a−p b−q and τ (z) = bq−p . Let T be the torus and consider its 1-skeleton T1 = Sa1 ∨ Sb1 = c0 ∪ c1a ∪ c1b . Let f 1 : K 1 → T1 be the cellular map which carries e1s into T1 exactly as τ carries s into F (a, b), for s = x, y, z. Then, up to the obvious identifications, we have 1 τ = f# : π1 (K 1 ) ≡ F (x, y, z) −→ F (a, b) ≡ π1 (T1 ).
Let l : T1 → T be the natural inclusion and let l# : π1 (T1 ) → π1 (T) be the homomorphism induced by l on fundamental groups. Then, again up to identifications, l# = ξ : F (a, b) → Z ⊕ Z, the abelianization homomorphism. And we have: (l# ◦ τ )(xp y q z p+q ) = l# ((aq bp )p (a−p b−q )q (bq−p )p+q ) = 0. Hence, l ◦ f 1 : K 1 → T extends to a cellular map f : K → T. Since π2 (T) = 0, such map is convenient, in the sense of [4], and by [4, Theorem 7.2], the map f may be deformed by homotopy to a non-surjective map if and only if the equation (Xaq bp )p (Y a−p b−q )q (Zbq−p )p+q = 1 has a solution over ker ξ. But such a solution does not exist, by Corollary 2.2. Therefore, the map f is strongly surjective. 2 As a consequence of Proposition 3.1 and Theorem 3.2, for each pair of coprime integers p, q ≥ 2, the corresponding model two-complex Kp, q has trivial top-cohomology and there exists a strong surjection from it onto the torus.
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4. The minimality of Kp, q In this section we prove that the previously built two-complexes Kp, q, for coprime integers p, q ≥ 2, are the simplest two-complexes with trivial top-cohomology from which there exist strong surjections onto the torus. The simplicity here is understood in two related senses: with regards to the first Betti number β1 of the one-skeleton and with regard to the total number of cells of the complex. Of course, the first Betti number of the one-skeleton of Kp, q is three and the total number of cells of Kp, q is five. And we have: Theorem 4.1. Let K be a finite and connected two-complex with trivial second cohomology group. If either β1 (K 1 ) ≤ 2 or K has less than five cells, then there is no strong surjection from K into the torus. In order to shorten the proof of Theorem 4.1 presented below, we anticipate some notions and notations. Firstly, we remember that the fundamental group of a model two-complex KP induced by a (finite) group presentation P = x | r is isomorphic to the group F (x)/N (r) presented by P, in such a way that the homomorphism induced on fundamental groups by the skeleton inclusion KP1 → KP may be naturally identified with the epimorphism Ω corresponding to the presentation P, which we summarize writing simply Ω : F (x) ≈ π1 (KP1 ) → π1 (KP ) ≈ F (x)/N (r). Secondly, as we have seen in the proof of Theorem 3.2, if we consider the torus T with its minimal cellular decomposition, then the homomorphism l# : π1 (T1 ) → π1 (T) induced on fundamental groups by the skeleton inclusion l : T1 = Sa1 ∨ Sb1 → T corresponds to the abelianization homomorphism ξ : F (a, b) → Z ⊕ Z. We use all theses identifications in the proof of Theorem 4.1 without previous notification. We also use the following lemma: Lemma 4.2. Let X, Y and Z be topological spaces. If X and Y are homotopy equivalent, then there exists a strong surjection from X onto Z if and only if there exists a strong surjection from Y onto Z. Proof. Let ϕ : X → Y and ψ : Y → X be inverse homotopy equivalences. Suppose that there is no strong surjection from Y into Z and let f : X → Z be a map. By the assumption, the composed map f ◦ ψ : Y → Z is homotopic to a non-surjective map f : Y → Z. It follows that f ◦ ϕ : X → Z is non-surjective and, moreover, f is homotopic to f ◦ ϕ. This proves that there is no strong surjection from X into Z. The converse is analogous. 2 Now we present the proof of Theorem 4.1. Proof of Theorem 4.1. Firstly, we remark that it is sufficient to prove the result under the condition β1 (K 1 ) ≤ 2, since the opposite condition β1 (K 1 ) ≥ 3 implies that K has at least four cells, namely, one 0-cell and three 1-cells. Furthermore, the bouquet L = S 1 ∨ S 1 ∨ S 1 is the unique complex with four cells for which one has β1 (L1 ) = 3, and obviously there is no strong surjection from L into the torus T. Let K be a finite and connected two-complex with H 2 (K) = 0 and β1 (K 1 ) ≤ 2. Let us prove that there is no strong surjection from K into T. By [11, Theorem 1.9 on p. 61] the skeleton pair (K, K 1 ) is homotopy equivalent to that of a model two-complex KP induced by a finite group presentation P = x | r. In this case, #x = β1 (KP1 ) = β1 (K 1 ) and #r is the number of two-cells of KP , where the symbol # indicates cardinality. Moreover, H 2 (KP ) = 0 and, by the Universal Coefficient Theorem, this forces #r = #x − β1 (KP ); see [6, Proposition 2.1]. In particular, #r ≤ #x.
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By Lemma 4.2, it is sufficient to prove that (under the assumption #x ≤ 2) there is no strong surjection from KP into the torus T. Now, since #r = 0 implies that KP has no two-cell, we just need consider the cases #r = #x = 1
and
1 ≤ #r ≤ #x = 2.
In the first case, KP is the closed two-disc D2 and the result is obvious. For the second case, we take x = {x, y} and we consider the two possibilities: #r = 1 or #r = 2. We handle each one separately below. For now, let f : KP → T be a map. We want to prove that f is homotopic to a non-surjective map. By [4, Theorem 2.6], this happens if and only if f# lifts through ξ in the diagram below: F (x, y)
F (a, b) ξ
Ω
π1 (KP )
f#
Z⊕Z
We prove the existence of such a lifting: Certainly, there exists a (unique) homomorphism f : H1 (KP ) → Z ⊕ Z such that f# = f ◦ ρ, where ρ : π1 (KP ) → H1 (KP ) is the Hurewicz epimorphism. Thus, if #r = 2, then the identity #r = #x − β1 (KP ) implies β1 (KP ) = 0 and so the homomorphisms f and f# are both trivial. In this case, f# lifts trivially through ξ and, therefore, the result follows. If #r = 1, we say r = {r}, then β1 (KP ) = 1 and so the image subgroup (in common) of the homomorphisms f and f# is cyclic. If it is trivial, then the results follows as in the previous case. Else, the image subgroup of the composed homomorphism f# ◦ Ω is cyclic and nontrivial, so that f# ◦ Ω(x) = s(c, d) and f# ◦ Ω(y) = t(c, d), for some (0, 0) = (c, d) ∈ Z ⊕ Z and some s, t ∈ Z. Furthermore, 0 = f# ◦ Ω(r) = (ξ1 (r)s + ξ2 (r)t)(c, d)
and so
ξ1 (r)s + ξ2 (r)t = 0,
where ξλ = πλ ◦ ξ : F (x, y) → Z, for λ = 1, 2, as in Section 2. We remark that we are using ξ to denote indistinctly the both abelianization homomorphism F (x, y) → Z ⊕ Z and F (a, b) → Z ⊕ Z. Take w ∈ F (a, b) such that ξ(w) = (c, d). Define φ : F (x, y) → F (a, b) to be the homomorphism given by φ(x) = ws and φ(y) = wt . Then ξ ◦ φ = f# ◦ Ω and we have φ(r) = wξ1 (r)s+ξ2 (r)t = w0 = 1. Thus, there exists a (unique) homomorphism Φ : π1 (KP ) → F (a, b) such that φ = Φ ◦ Ω. We claim that Φ is a lifting of f# through l# . In fact: for each z¯ = Ω(z) ∈ π1 (KP ), we have f# (¯ z ) = f# ◦ Ω(z) = ξ ◦ φ(z) = ξ ◦ Φ ◦ Ω(z) = ξ ◦ Φ(¯ z ). This concludes the proof of the theorem. 2 Acknowledgements I thank D.L. Gonçalves for proposing the problem and O.M. Neto for some helpful conversations on the subject of the problem. I thank also the partial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq – Grant 448829/2014-2.
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References [1] C. Aniz, Strong surjectivity of mappings of some 3-complexes into 3-manifolds, Fundam. Math. 192 (2006) 195–214. [2] C. Aniz, Strong surjectivity of mappings of some 3-complexes into MQ8 , Cent. Eur. J. Math. 6 (4) (2008) 497–503. [3] S. Bogatyi, D.L. Gonçalves, H. Zieschang, The minimal number of roots of surface mappings and quadratic equations in free groups, Math. Z. 236 (3) (2001) 419–452. [4] M.C. Fenille, O.M. Neto, Root problem for convenient maps, Topol. Methods Nonlinear Anal. 36 (2) (2010) 327–352. [5] M.C. Fenille, O.M. Neto, Strong surjectivity of maps from 2-complexes into the 2-sphere, Cent. Eur. J. Math. 8 (3) (2010) 421–429. [6] M.C. Fenille, The trivial homotopy class of maps from two-complexes into the real projective plane, Topol. Methods Nonlinear Anal. 46 (2015) 603–615. [7] D.L. Gonçalves, H. Zieschang, Equations in free groups and coincidence of mappings on surfaces, Math. Z. 237 (1) (2001) 1–29. [8] D.L. Gonçalves, E. Kudryavtseva, H. Zieschang, Roots of mappings on nonorientable surfaces and equations in free groups, Manuscr. Math. 107 (3) (2002) 311–341. [9] D.L. Gonçalves, P. Wong, Wecken property for roots, Proc. Am. Math. Soc. 133 (9) (2005) 2779–2782. [10] R.C. Lyndon, M.P. Schützenberger, The equation am = bn cp in a free group, Mich. Math. J. 9 (1962) 289–298. [11] A.J. Sieradski, Algebraic topology for two-dimensional complexes, in: C. Hog-Angeloni, W. Metzler, A.J. Sieradski (Eds.), Two-Dimensional Homotopy and Combinatorial Group Theory, Cambridge University Press, 1993, pp. 51–96. [12] G.W. Whitehead, Elements of Homotopy Theory, Springer-Verlag, Inc., New York, 1978.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
Strong surjections from two-complexes with trivial top-cohomology onto the torus Marcio Colombo Fenille Universidade Federal de Uberlândia – Faculdade de Matemática, Av. João Naves de Ávila 2121, Sta Mônica, 38408-100, Uberlândia-MG, Brazil
a r t i c l e
i n f o
Article history: Received 13 April 2016 Received in revised form 4 July 2016 Accepted 14 July 2016 Available online 20 July 2016 MSC: primary 55M20 secondary 57M20
a b s t r a c t We build a countable collection of two-dimensional CW complexes with trivial second integer cohomology group and, from each of them, a strong surjection onto the torus. Furthermore, we prove that such two-complexes are the simplest with these properties. This answers, for dimension two, a problem originally proposed in the 2000’s for dimension three. © 2016 Elsevier B.V. All rights reserved.
Keywords: Strong surjection Two-dimensional complexes Zero top-cohomology Equations in free groups Root theory
1. Introduction Let K be a finite and connected two-dimensional CW complex (a two-complex, to shorten). By the Hopf–Whitney Classification Theorem [12, Corollary 6.19 on p. 244], if the top-cohomology group H 2 (K) is trivial (integer coefficients is subtended), then all maps from K into the two-sphere are homotopic to a nonsurjective map, but we do not know about maps into the others closed surfaces. This problem becomes more interesting when we note that a two-complex K with H 2 (K) = 0 is (co)homologically like a one-complex, since by the Universal Coefficient Theorem we have, in such case, H2 (K) = 0 and H 1 (K) ≈ H1 (K) torsion free. This fact aggravates the problem: since H 2 is not able to detect the existence of (non-collapsible) two-cells in K, is it able to detect the existence of strong surjections from K into a closed surfaces? Here, by a strong surjection we mean a map whose homotopy class contains only surjective maps. Of course, the same question makes sense also in greater dimensions. In the 2000’s, C. Aniz and D.L. Gonçalves approached the problem in dimension three. In [1], C. Aniz proved that every map from E-mail address: [email protected]. http://dx.doi.org/10.1016/j.topol.2016.07.010 0166-8641/© 2016 Elsevier B.V. All rights reserved.
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a 3-complex K with H 3 (K) = 0 into S 1 × S 2 is homotopic to a non-surjective map, but for Y the nonorientable S 1 -bundle over S 2 , there exists a strong surjection f : K → Y from such a 3-complex. In [2], C. Aniz proved that there is no strong surjection from a 3-complex K with H 3 (K) = 0 into the orbit space of the 3-sphere S 3 with respect to the action of the quaternion group Q8 determined by the inclusion Q8 ⊂ S 3 . In 2008, D.L. Gonçalves introduced us to the problem, in dimension two, and conjectured that, as in dimension three, there should be a strong surjection from a two-complex with trivial top-cohomology onto some closed surface. It is noteworthy that the dimension two is often left out, in several contributions to topological root theory, since it does not permit the use of special techniques as obstruction theory. In order to attack our problem itself, also the Nielsen root theory is not feasible, despite the Wecken property for roots established by D.L. Gonçalves and P. Wong in [9], since there is no practical mechanisms for calculating the Nielsen root number. Our approach is based in combinatorial group theory, in special equation in free groups, what has been used successfully in coincidence and root theory for maps between closed surfaces, as we see, for instance, in [3,7] and [8]. In [6], also via combinatorial group theory, we study indirectly the problem considering only maps into the real projective plane RP2 . We prove that for a finite and connected two-dimensional complex K, the condition [K; RP2 ]∗ = 0 implies H 2 (K) = 0, and the opposite implication is true if the number of two-cells of K is equal to the first Betti number of its one-skeleton K 1 . In this article, we establish the conjecture proposed by D.L. Gonçalves. Specifically, we build a collection of two-complexes Kp, q with trivial top-cohomology, for all coprime integers p, q ≥ 2, and a strong surjection from each Kp, q onto the torus T = S 1 × S 1 . To ensure the success of our approach, two things were particularly important: first, the necessary and sufficient condition, expressed in terms of equation in free groups, to a map from a two-complex into a closed surface to be strongly surjective, presented in [4]; second, the classical result of R.C. Lyndon and M.P. Schutzenberger, published in [10], which states that if a, b and c are elements of a free group and am = bn cp , for integers m, n, p ≥ 2, then a, b and c are contained in a cyclic subgroup. After we build the aforementioned collection of two-complexes with trivial top-cohomology and strong surjections from them onto the torus, we prove, in the final section of the article, that such collection is as simple as possible. We explain: each two-complex Kp, q has five cells and the first Betti number of its one-skeleton is three; we prove that if K is a finite and connected two-complex with H 2 (K) = 0 and either K has less than five cells or β1 (K 1 ) ≤ 2, then there is no strong surjection from K into the torus. 2. On the equation (Xw1 )p (Y w2 )q (Zw3 )r = 1 In this section, we prove a simple result in combinatorial group theory, which is directly used, in the next section, in the proof of our main theorem. Let p, q and r be integers greater than or equal to two and let w1 , w2 and w3 be words in the free group F n = F (a1 , . . . , an ) of rank n ≥ 2. Put 1 to be the empty word in F n . In what follows we consider the equation (Xw1 )p (Y w2 )q (Zw3 )r = 1
(1)
in the free group F n . A solution for Equation (1) is a group homomorphism φ : F (x1 , x2 , x3 ) → F n for which the triad (X, Y, Z) = (φ(x1 ), φ(x2 ), φ(x3 )) satisfies the equation. In such a case, we say also that such triad is a solution for Equation (1). Of course, Equation (1) has an obvious solution, namely, the triad (w1−1 , w2−1 , w3−1 ). If φ is a solution for Equation (1) whose image is contained in a subgroup H of F n , we say that φ is a solution for Equation (1) over H. We are particularly interested in the case in which H is the commutator subgroup of F n . Therefore, let ξ : F n → Zn be the abelianization homomorphism and, for each index
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1 ≤ λ ≤ n, put ξλ = πλ ◦ ξ : F n → Z, in which πλ : Zn → Z is the projection onto the λth coordinate. Thus, ξλ (ai ) = δλi , the Kronecker delta, and ξ(w) = (ξ1 (w), . . . , ξn (w)), for each w ∈ F n . Moreover, ξλ (w) is the sum of all powers of the letter aλ in the word w. If pξ(w1 ) + qξ(w2 ) + rξ(w3 ) = 0, then it makes sense to ask if Equation (1) has a solution over ker ξ. The following result presents a condition under which Equation (1) has not one. Proposition 2.1. If for some indexes 1 ≤ i, j ≤ 3 and 1 ≤ k, l ≤ n one has ξk (wi ) = 0, ξl (wi ) = 0 and ξk (wj ) = 0, then Equation (1) has no solution over ker ξ. Proof. Let φ : F (x1 , x2 , x3 ) → F n be a solution for Equation (1), which certainly exists. Then the triad (φ(x1 )w1 , φ(x2 )w2 , φ(x3 )w3 ) is a solution for the equation ap bq cr = 1 in F n . By the main result of [10] there exists a word w ∈ F n such that, for each index 1 ≤ s ≤ 3, one has φ(xs )ws = wts for some ts ∈ Z. Hence, for each index 1 ≤ λ ≤ n, we have ts ξλ (w) = ξλ (wts ) = ξλ (φ(xs )ws ) = ξλ (φ(xs )) + ξλ (ws ). Suppose that φ is actually a solution for Equation (1) over ker ξ. It follows from the assumption on the word wi that ti ξk (w) = ξk (φ(xi )) + ξk (wi ) = 0
and
ti ξl (w) = ξl (φ(xi )) + ξl (wi ) = ξl (wi ) = 0, which implies ξk (w) = 0. On the other hand, it follows from the assumption on the word wj that tj ξk (w) = ξk (φ(xj )) + ξk (wj ) = ξk (wj ) = 0, which implies ξk (w) = 0. Thus, we have a contradiction and, therefore, there is not a solution for Equation (1) over ker ξ. 2 As a special example in which Proposition 2.1 works, take integers p, q ≥ 2 and consider the equation (Xaq bp )p (Y a−p b−q )q (Zbq−p )p+q = 1
(2)
in the free group F (a, b) of rank two. For the abelianization homomorphism ξ : F (a, b) → Z ⊕ Z given as before, we have pξ(aq bp ) + qξ(a−p b−q ) + (p + q)ξ(bq−p ) = 0. Therefore, it makes sense to investigate the existence of solutions for Equation (2) over ker ξ. However, as a simple consequence of Proposition 2.1, we have: Corollary 2.2. For all integers q > p ≥ 2, Equation (2) has no solution over ker ξ. Proof. For the words w1 = aq bp and w3 = bq−p we have ξ1 (w3 ) = 0, ξ2 (w3 ) = q − p = 0 and ξ1 (w1 ) = q = 0. Therefore, Proposition 2.1 applies. 2 Corollary 2.2 is the (algebraic) key for our main theorem (Theorem 3.2), which we prove in the next section.
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3. Strong surjections onto the torus Let p and q be positive integers and let Kp, q be the model two-complex induced by the group presentation P = x, y, z | xp y q z p+q , that is, the two-complex whose one-skeleton is the bouquet Sx1 ∨ Sy1 ∨ Sz1 of three circles and whose single two-cell is attached according to the relator xp y q z p+q . See [11] for details. The next proposition is a simple exercise of elementary Algebraic Topology, for which we provide a proof using results on root theory and homotopy theory, in order to contextualize. This simple result is important to the conclusion of the paper. Proposition 3.1. H 2 (Kp, q) = 0 if and only if p and q are coprime. Proof. By the Hopf–Whitney Classification Theorem [12, Corollary 6.19 on p. 244], the integer cohomology group H 2 (Kp, q) is in one-to-one correspondence with the set [Kp, q; S 2 ] of the homotopy classes of maps from Kp, q into the two-sphere. On the other hand, it follows from [5, Theorem 1.1] that [Kp, q; S 2 ] = 0 if and only if the diophantine linear system px + qy + (p + q)z = d has an integer solution for all d ∈ Z. Obviously, this last condition is equivalent to gcd(p, q) = 1. Therefore, H 2 (Kp, q) = 0 if and only if gcd(p, q) = 1. 2 Of course, a direct proof for Proposition 3.1, independently on results on root and homotopy theory, may be given by analyzing the cellular chain complex of Kp, q. Theorem 3.2. For all integers q > p ≥ 2, there exists a strong surjection from the two-complex Kp, q onto the torus. Proof. Put K = Kp, q and consider its one-skeleton K 1 = Sx1 ∨ Sy1 ∨ Sz1 = e0 ∪ e1x ∪ e1y ∪ e1z . Let τ : F (x, y, z) → F (a, b) be the homomorphism given by τ (x) = aq bp , τ (y) = a−p b−q and τ (z) = bq−p . Let T be the torus and consider its 1-skeleton T1 = Sa1 ∨ Sb1 = c0 ∪ c1a ∪ c1b . Let f 1 : K 1 → T1 be the cellular map which carries e1s into T1 exactly as τ carries s into F (a, b), for s = x, y, z. Then, up to the obvious identifications, we have 1 τ = f# : π1 (K 1 ) ≡ F (x, y, z) −→ F (a, b) ≡ π1 (T1 ).
Let l : T1 → T be the natural inclusion and let l# : π1 (T1 ) → π1 (T) be the homomorphism induced by l on fundamental groups. Then, again up to identifications, l# = ξ : F (a, b) → Z ⊕ Z, the abelianization homomorphism. And we have: (l# ◦ τ )(xp y q z p+q ) = l# ((aq bp )p (a−p b−q )q (bq−p )p+q ) = 0. Hence, l ◦ f 1 : K 1 → T extends to a cellular map f : K → T. Since π2 (T) = 0, such map is convenient, in the sense of [4], and by [4, Theorem 7.2], the map f may be deformed by homotopy to a non-surjective map if and only if the equation (Xaq bp )p (Y a−p b−q )q (Zbq−p )p+q = 1 has a solution over ker ξ. But such a solution does not exist, by Corollary 2.2. Therefore, the map f is strongly surjective. 2 As a consequence of Proposition 3.1 and Theorem 3.2, for each pair of coprime integers p, q ≥ 2, the corresponding model two-complex Kp, q has trivial top-cohomology and there exists a strong surjection from it onto the torus.
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4. The minimality of Kp, q In this section we prove that the previously built two-complexes Kp, q, for coprime integers p, q ≥ 2, are the simplest two-complexes with trivial top-cohomology from which there exist strong surjections onto the torus. The simplicity here is understood in two related senses: with regards to the first Betti number β1 of the one-skeleton and with regard to the total number of cells of the complex. Of course, the first Betti number of the one-skeleton of Kp, q is three and the total number of cells of Kp, q is five. And we have: Theorem 4.1. Let K be a finite and connected two-complex with trivial second cohomology group. If either β1 (K 1 ) ≤ 2 or K has less than five cells, then there is no strong surjection from K into the torus. In order to shorten the proof of Theorem 4.1 presented below, we anticipate some notions and notations. Firstly, we remember that the fundamental group of a model two-complex KP induced by a (finite) group presentation P = x | r is isomorphic to the group F (x)/N (r) presented by P, in such a way that the homomorphism induced on fundamental groups by the skeleton inclusion KP1 → KP may be naturally identified with the epimorphism Ω corresponding to the presentation P, which we summarize writing simply Ω : F (x) ≈ π1 (KP1 ) → π1 (KP ) ≈ F (x)/N (r). Secondly, as we have seen in the proof of Theorem 3.2, if we consider the torus T with its minimal cellular decomposition, then the homomorphism l# : π1 (T1 ) → π1 (T) induced on fundamental groups by the skeleton inclusion l : T1 = Sa1 ∨ Sb1 → T corresponds to the abelianization homomorphism ξ : F (a, b) → Z ⊕ Z. We use all theses identifications in the proof of Theorem 4.1 without previous notification. We also use the following lemma: Lemma 4.2. Let X, Y and Z be topological spaces. If X and Y are homotopy equivalent, then there exists a strong surjection from X onto Z if and only if there exists a strong surjection from Y onto Z. Proof. Let ϕ : X → Y and ψ : Y → X be inverse homotopy equivalences. Suppose that there is no strong surjection from Y into Z and let f : X → Z be a map. By the assumption, the composed map f ◦ ψ : Y → Z is homotopic to a non-surjective map f : Y → Z. It follows that f ◦ ϕ : X → Z is non-surjective and, moreover, f is homotopic to f ◦ ϕ. This proves that there is no strong surjection from X into Z. The converse is analogous. 2 Now we present the proof of Theorem 4.1. Proof of Theorem 4.1. Firstly, we remark that it is sufficient to prove the result under the condition β1 (K 1 ) ≤ 2, since the opposite condition β1 (K 1 ) ≥ 3 implies that K has at least four cells, namely, one 0-cell and three 1-cells. Furthermore, the bouquet L = S 1 ∨ S 1 ∨ S 1 is the unique complex with four cells for which one has β1 (L1 ) = 3, and obviously there is no strong surjection from L into the torus T. Let K be a finite and connected two-complex with H 2 (K) = 0 and β1 (K 1 ) ≤ 2. Let us prove that there is no strong surjection from K into T. By [11, Theorem 1.9 on p. 61] the skeleton pair (K, K 1 ) is homotopy equivalent to that of a model two-complex KP induced by a finite group presentation P = x | r. In this case, #x = β1 (KP1 ) = β1 (K 1 ) and #r is the number of two-cells of KP , where the symbol # indicates cardinality. Moreover, H 2 (KP ) = 0 and, by the Universal Coefficient Theorem, this forces #r = #x − β1 (KP ); see [6, Proposition 2.1]. In particular, #r ≤ #x.
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By Lemma 4.2, it is sufficient to prove that (under the assumption #x ≤ 2) there is no strong surjection from KP into the torus T. Now, since #r = 0 implies that KP has no two-cell, we just need consider the cases #r = #x = 1
and
1 ≤ #r ≤ #x = 2.
In the first case, KP is the closed two-disc D2 and the result is obvious. For the second case, we take x = {x, y} and we consider the two possibilities: #r = 1 or #r = 2. We handle each one separately below. For now, let f : KP → T be a map. We want to prove that f is homotopic to a non-surjective map. By [4, Theorem 2.6], this happens if and only if f# lifts through ξ in the diagram below: F (x, y)
F (a, b) ξ
Ω
π1 (KP )
f#
Z⊕Z
We prove the existence of such a lifting: Certainly, there exists a (unique) homomorphism f : H1 (KP ) → Z ⊕ Z such that f# = f ◦ ρ, where ρ : π1 (KP ) → H1 (KP ) is the Hurewicz epimorphism. Thus, if #r = 2, then the identity #r = #x − β1 (KP ) implies β1 (KP ) = 0 and so the homomorphisms f and f# are both trivial. In this case, f# lifts trivially through ξ and, therefore, the result follows. If #r = 1, we say r = {r}, then β1 (KP ) = 1 and so the image subgroup (in common) of the homomorphisms f and f# is cyclic. If it is trivial, then the results follows as in the previous case. Else, the image subgroup of the composed homomorphism f# ◦ Ω is cyclic and nontrivial, so that f# ◦ Ω(x) = s(c, d) and f# ◦ Ω(y) = t(c, d), for some (0, 0) = (c, d) ∈ Z ⊕ Z and some s, t ∈ Z. Furthermore, 0 = f# ◦ Ω(r) = (ξ1 (r)s + ξ2 (r)t)(c, d)
and so
ξ1 (r)s + ξ2 (r)t = 0,
where ξλ = πλ ◦ ξ : F (x, y) → Z, for λ = 1, 2, as in Section 2. We remark that we are using ξ to denote indistinctly the both abelianization homomorphism F (x, y) → Z ⊕ Z and F (a, b) → Z ⊕ Z. Take w ∈ F (a, b) such that ξ(w) = (c, d). Define φ : F (x, y) → F (a, b) to be the homomorphism given by φ(x) = ws and φ(y) = wt . Then ξ ◦ φ = f# ◦ Ω and we have φ(r) = wξ1 (r)s+ξ2 (r)t = w0 = 1. Thus, there exists a (unique) homomorphism Φ : π1 (KP ) → F (a, b) such that φ = Φ ◦ Ω. We claim that Φ is a lifting of f# through l# . In fact: for each z¯ = Ω(z) ∈ π1 (KP ), we have f# (¯ z ) = f# ◦ Ω(z) = ξ ◦ φ(z) = ξ ◦ Φ ◦ Ω(z) = ξ ◦ Φ(¯ z ). This concludes the proof of the theorem. 2 Acknowledgements I thank D.L. Gonçalves for proposing the problem and O.M. Neto for some helpful conversations on the subject of the problem. I thank also the partial support of Conselho Nacional de Desenvolvimento Científico e Tecnológico – CNPq – Grant 448829/2014-2.
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