FUNCTIONS ON SURFACES AND INCOMPRESSIBLE SUBSURFACES 0 SERGIY MAKSYMENKO 1 0 2 Abstract. LetM be a smoothconnected compactsurface, P be n either a real line R or a circle S1. Then we have a natural right a action of the groupD(M) of diffeomorphisms of M on C∞(M,P). J For f ∈ C∞(M,P) denote respectively by S(f) and O(f) its sta- 8 bilizer and orbit with respect to this action. Recently, for a large class of smooth maps f : M → P the author calculated the ho- ] T motopy types of the connected components of S(f) and O(f). It G turnedoutthatexceptforfewcasestheidentitycomponentofS(f) . is contractible, πiO(f) = πiM for i ≥ 3, and π2O(f) = 0, while h π1O(f) it only proved to be a finite extension of π1Did(M)⊕Zl t a for some l ≥ 0. In this note it is shown that if χ(M) < 0, then m π1O(f) = G1 ×···×Gn, where each Gi is a fundamental group [ of the restriction of f to a subsurface Bi ⊂ M being either a 2- disk or a cylinder or a Mo¨bius band. For the proof of main result 2 incompressible subsurfacesandcellularautomorphismsofsurfaces v 6 are studied. 4 3 1 . 1 1. Introduction 0 0 Let M be a smooth compact connected surface and P be either the 1 real line R or the circle S1. Consider the right action of the group : v D(M) of diffeomorphisms of M on C∞(M,P) defined by i X h·f = f ◦h−1 r a for h ∈ D(M) and f ∈ C∞(M,P). For every f ∈ C∞(M,P) let O(f) = {f ◦h | h ∈ D(M)}, S(f) = {h | f = f ◦h, h ∈ D(M)} be respectively the orbit and the stabilizer of f with respect to this action. We will endow D(M), S(f), C∞(M,P), and O(f) with the Date: 08.01.2010. 2000 Mathematics Subject Classification. 37C05,57S05,57R45. Key words and phrases. Incompressiblesurface,diffeomorphisms group,cellular automorphism, homotopy type. This research is partially supported by grant of Ministry of Science and Educa- tion of Ukraine, No. M/150-2009. 1 2 SERGIY MAKSYMENKO corresponding topologies C∞. Denote by S (f) the identity path com- id ponent of S(f) and by O (f) the path component of f in O(f). In [10] f the author calculated the homotopy types of S (f) and O (f) for all id f Morse maps f : M → P. Moreover, in [12] the results of [10] were extended to a large class of maps with (even degenerate) isolated critical points satisfying certain “non-degeneracy” conditions. Infact there were introduced three types of isolated critical points (called S, P, and N) and the following three axioms for f: (Bd) f takes constant value at each connected component of ∂M and Σ ⊂ IntM. f (SPN) Every critical point of f is either an S- or a P- or an N-point. (Fibr) The natural map p : D(M) → O(f) defined by p(h) = f ◦h−1 is a Serre fibration with fiber S(f) in topologies C∞. Recall that if f : (C,0) → (R,0) is a smooth germ for which 0 ∈ C is an isolated critical point, then there exists a homeomorphism h : C → C such that h(0) = 0 and ±|z|2, if z is a local extremum, [3], f ◦h(z) = Re(zn),(n ≥ 1) otherwise, so z is a saddle, [15], (cid:26) Examples of the foliation by level sets of f near 0 are presented in Figure 1.1. Figure 1.1. Isolated critical points From this point of view S-points are saddles, while P- and N-points a local extremes. Moreover, P-points admit non-trivial f-preserving circle actions (as non-degenerate local extremes do), while N-points admit only Z -action preserving f. We will not give precise definitions n but recall a large class of examples of such points. Example 1.1. [10]. Let f : R2 → R be a homogeneous polynomial without multiple factors with degf ≥ 2, so f = L ···L ·Q ···Q , a+2b ≥ 2, 1 a 1 b FUNCTIONS ON SURFACES AND INCOMPRESSIBLE SUBSURFACES 3 where every L is a linear function and every Q is an irreducible over i j R (i.e. definite) quadratic form such that Li/Li′ 6= const for i 6= i′ and Qj/Qj′ 6= const for j 6= j′. If a ≥ 1, so f has linear factors and thus 0 is a saddle, then the origin 0 ∈ R2 is an S-point for f. If a = 0 and b = 1, so f = Q , then the origin 0 ∈ R2 is a P-point 1 for f. Otherwise, a = 0 and b ≥ 2, so f = Q ···Q . Then the origin 1 b 0 ∈ R2 is an N-point for f. Lemma 1.2. [10]. Let f : M → P be a C∞ map satisfying (Bd), and such that every of its critical points belongs to the class described in Example 1.1, in particular, f also satisfies (SPN). Then f also satisfies (Fibr). It follows from Morse lemma and Example 1.1 that non-degenerate saddles are S-points while non-degenerate local extremes are P-points. Now the main result of [12] can be formulated as follows. Theorem 1.3. [10, 12]. Suppose f : M → P satisfies (Bd) and (SPN). If f has at least one S- or N-point, or if M is non-orientable, then S (f) is contractible. id Moreover, if in addition f satisfies (Fibr), then π O (f) = π M for i f i i ≥ 3, π O (f) = 0, and for π O(f) we have the following short exact 2 f 1 sequence 1 → π D(M)⊕Zl → π O (f) → G → 1, 1 1 f for a certain finite group G and l ≥ 0 both depending on f. Thus, the information about the fundamental group π O (f) is not 1 f complete. The aim of this note is to show that the calculation of π O (f) can be reduced to the case when M is either a 2-disk, or 1 f a cylinder, or a M¨obius band, see Theorems 1.7 and 1.8 below. The obtained results hold for a more general class of maps M → P than the one considered in [12]. 1.4. Admissible critical points. We will now introduce a certain type of critical points for f. Let F be a vector field on M, V ⊂ M be anopen subset, andh : V → M be anembedding. Say thathpreserves orbits of F if for every orbit o of F we have that h(V ∩o) ⊂ o. Definition 1.5. Let f : M → P be a C∞ map and z ∈ IntM be an isolated critical point of f which is not a local extreme (so z is a saddle). Say that z is admissible if there exists a neighbourhood U of z containing no other critical points of f and a vector field F on U having the following properties: 4 SERGIY MAKSYMENKO (1) f is constant along orbits of F and z is a unique singular point of F. (2) Let (F ) be the local flow of F on U. Then for every germ t of diffeomorphisms h : (M,z) → (M,z) preserving orbits of F there exists a C∞ germ σ : (M,z) → R such that h(x) = F(x,σ(x)) near z. This definition almost coincides with the definition of an S-point, c.f. [12]. The difference is that for S-points it is also required that the correspondence h 7→ σ is continuous with respect to topologies C∞. In particular every S-point is admissible. Now put the following two axioms for f both implied by (SPN): (Isol) All critical points of f are isolated. (SA) Every saddle of f is admissible. 1.6. Main result. Let D (M) be the identity path component of the id group D(M) and ′ S (f) = S(f)∩D (M) id be the stabilizer of f with respect to the right action of D (M). Thus id S′(f) consists of diffeomorphisms h isotopic to id and preserving F, M i.e. f ◦h = f. For a closed subset X ⊂ M denote by S′(f,X) the subgroup of S′(f) consisting of diffeomorphisms fixed on some neighbourhood of X. The aim of this note is to prove the following theorem: Theorem 1.7. Suppose χ(M) < 0. Let f : M → P be a C∞ map satisfyingthe axioms(Bd), (Isol), and(SA). Thenthere existsa compact subsurface X ⊂ M with the following properties: (1) f is locally constant on ∂X and every connected component B of M \X is either a 2-disk or a 2-cylinder or a M¨obius band. Moreover, ∂B ⊂ X and B contains critical points of f. (2) Let h ∈ S′(f,X) and B be a connected component of M \X, thus h is fixed on some neighbourhood of ∂B. Then the restriction h| B is isotopic in B to id with respect to some neighbourhood of ∂B. B (3) The inclusion i : S′(f,X) ⊂ S′(f) induces a group isomorphism i : π S′(f,X) ≈ π S′(f). 0 0 0 The proof of this theorem will be given in §7. We will now show how to simplify calculations of π O(f) using Theorem 1.7. 1 Let X be the surface of Theorem 1.7 and let B ,...,B be all the 1 l connected components of M \X. For every i = 1,...,l denote by FUNCTIONS ON SURFACES AND INCOMPRESSIBLE SUBSURFACES 5 D (B ,∂B ) the group of diffeomorphisms of B fixed on some neigh- id i i i bourhood of ∂B and isotopic to id relatively to some neighbour- i Bi hood of B . Let also S′(f| ,∂B ) be the stabilizer of the restriction i Bi i f| : B → P with respect to the right action of D (B ,∂B ). Then Bi i id i i we have an evident isomorphism of groups: l ′ ′ (1.1) ψ : S (f,X) ≈ × S (f| ,∂B ), ψ(h) = (h| ,...,h| ), Bi i B1 Bl i=1 It is easy to show that ψ is in fact a homeomorphism with respect to the corresponding C∞ topologies. Theorem 1.8. Under assumptions of Theorem 1.7 suppose that f also satisfies (Fibr). Then we have an isomorphism: l ′ π O (f) ≈ × π S (f| ,∂B ). 1 f 0 Bi i i=1 Proof. It is easy to show that if f satisfies (Fibr), then O (f) is the f orbit of f with respect to the action of D (M) and the projection id p : D (M) → O (f) is a Serre fibration as well, see [11]. Hence we get id f the following part of exact sequence of homotopy groups ′ ··· → π D (M) → π O (f) → π S (f) → π D (M) → ··· 1 id 1 f 0 0 id Since χ(M) < 0, we have π D (M) = 0, [5, 4, 7]. Moreover, D (M) 1 id id is path-connected, whence together with Theorem 1.7 we obtain an isomorphism: ′ i0 ′ (1.1) l ′ π O (f) ≈ π S (f) ≈ π S (f,X) ≈ × π S (f| ,∂B ). 1 f 0 0 0 Bi i i=1 (cid:3) Theorem is proved. Thus a general problem of calculation of π O (f) for maps satisfying 1 f the above axioms completely reduces to the case when χ(M) ≥ 0. A presentation for π O (f) will be given in another paper. 1 f 1.9. Structure of the paper. In next four sections we study incom- pressible subsurfaces N ⊂ M. §2 contains their definition and some elementary properties. In §3 we show how such subsurfaces appear in studying maps M → P with isolated singularities. In §4 and §5 we ex- tend results of W. Jaco and P. Shalen [8] about deformations of incom- pressible subsurfaces and periodic automorphisms of surfaces. §6 con- tains two technical statements about deformations of diffeomorphisms preserving a map M → P. Finally in §7 we prove Theorem 1.7. 6 SERGIY MAKSYMENKO 2. Incompressible subsurfaces The following Lemma 2.1 is well-known, see e.g. [14, Pr. 2.1]. It was also implicitly formulated in [8, page 359]. Lemma 2.1. 1) Let M be a connected surface, and N ⊂ IntM be a proper compact (possibly not connected) subsurface neither of whose connected components is a 2-disk. Then the following conditions are equivalent: (a) for every connected component N of N the inclusion homomor- i phism π N → π M is injective; 1 i 1 (b) none of the connected components of M \N is a 2-disk. If these conditions hold, then N will be called incompressible, see [8, Def. 3.2]. Corollary 2.2. If N ⊂ M is incompressible, then χ(M) ≤ χ(N). Corollary 2.3. Let R ⊂ IntM be a proper compact connected subsur- face. Then the following conditions are equivalent: (R1) the homomorphism ξ : π R → π M is trivial; 1 1 (R2) R is contained in some 2-disk D ⊂ M. Proof. The implication (R2)⇒(R1) is evident. (R1)⇒(R2). Suppose R is not contained in any 2-disk. We will show that ξ is non-trivial. Let N be the union of R with all of the connected components of M \N which are 2-disks. Then by our assumption N is not a 2-disk and by Lemma 2.1 N is incompressible. Notice that ξ is a product of homomorphisms induced by the inclusions R ⊂ N ⊂ M: α β ξ = β ◦α : π R → π N → π M. 1 1 1 Also notice that α is surjective and by Lemma 2.1 β is a non-trivial monomorphism. Hence ξ is also non-trivial. (cid:3) Corollary 2.4. Let R ⊂ IntM be a proper (possibly non connected) subsurface such that neither of its connected components is contained in some 2-disk. Then every connected component B of M \R which is not a 2-disk is incompressible. Proof. Let C be a connected component of M \B. Due to Lemma 2.1 itsufficestoshowthatC isnota2-disk. NoticethatC∩R 6= ∅, whence it contains some connected component R of R. By Corollary 2.3 the i product of homomorphisms π R → π C → π M is non-trivial, and 1 i 1 1 therefore π C → π M is also non-trivial. This implies that C is not a 1 1 (cid:3) 2-disk. FUNCTIONS ON SURFACES AND INCOMPRESSIBLE SUBSURFACES 7 3. Incompressible subsurfaces associated to a map M → P 3.1. Singular foliation ∆ of f. Let f : M → P be a map satisfy- f ing axioms (Bd) and (Isol). Then f induces on M a one-dimensional foliation ∆ with singularities defined as follows: a subset ω ⊂ M is a f leaf of ∆ if and only if ω is either a critical point of f or a connected f component of the set f−1(c)\Σ for some c ∈ P. Thus the leaves of ∆ f f are 1-dimensional submanifolds of M and critical points of f. Local structure of ∆ near critical points of f is illustrated in Figure 1.1. f Denote by ∆reg the union of all leaves of ∆ homeomorphic to the f f circle and by ∆cr the union of all other leaves. The leaves in ∆reg f f (resp. ∆cr) will be called regular (resp. critical). Similarly, connected f components of ∆reg (resp. ∆cr) will be called regular (resp. critical) f f components of ∆ . It follows from (Bd) that ∂M ⊂ ∆reg. It is also f f evident, that every critical leaf of ∆cr either is homeomorphic to an f open interval or is a critical point of f. 3.2. Atoms and canonical neighbourhoods of critical compo- nents of ∆ . For every critical component K of ∆ define its regular f f neighbourhood R as follows. Let c ,...,c be all the critical values K 1 l of f and the values of f on ∂M. Since M is compact, it follows from axioms (Bd) and (Isol) that l is finite. For each i = 1,...,l let W ⊂ P i be a closed connected neighbourhood (i.e. just an arc) of c containing i no other c . We will assume that W ∩W = ∅ for i 6= j. j i j Now let K be a critical component of ∆ . Then f(K) = c for f i some i. Let R be the connected component of f−1(W ) containing K. K i Evidently, R is a union of leaves of ∆ . Following [2] we will call R K f K an atom of K, see Figure 3.1. Figure 3.1. Evidently, R is a regular neighbourhood of K with respect to some K triangulation of M. Similarly to [8] define the canonical neighbourhood N of K to be the union of R with all the connected components of K K M \R being 2-disks. If N is not a 2-disk, then by Lemma 2.1 N K K K is incompressible in M. Notice that (3.1) ∂R = f−1(∂W ) ∩ R . K i K 8 SERGIY MAKSYMENKO Let K′ be another critical component of ∆ such that f(K′) = f(K). f Since RK′ is also constructed via Wi, we obtain from (3.1) that f takes on ∂RK′ the same values as on ∂RK. This technical assumption is not essential, however it will be useful for the proof of Theorem 1.7. Lemma 3.3. Let K and K′ be two distinct critical components of ∆ . f (i) Then RK ∩RK′ = ∅, while NK and NK′ are either disjoint or one of them, say NK, is contained in NK′. In the last case NK is a 2-disk. (ii) Suppose f(K) = f(K′) and there exists h ∈ S(f) such that h(K) = K′. Then h(RK) = RK′ and h(NK) = NK′. Proof. (i) follows from the assumption that W ∩ W = ∅ for i 6= j, i j (cid:3) and (ii) follows from (3.1). We leave the details for the reader. Lemma 3.4. Let K be a critical component of ∆ such that N is a f K 2-disk. Then either (i) M is a 2-disk itself, or (ii) NK is contained in a unique canonical neighbourhood NK′ of another critical component K′ of ∆f such that NK′ is not a 2-disk. Proof. Let R be the union of atoms of all critical components of ∆ . f Then every connected component B of M \R is diffeomorphic to the cylinder S1 ×[0,1] and the restriction f| has no critical points. B Notice that M \N is connected since N is a 2-disk. Also, there K K exists a unique connected component B (being a cylinder S1 ×[0,1]) of M \R such that ∂N ⊂ B. Then N ∪B is also a 2-disk. K K Let n be the total number of critical components of ∆ in M \N . f K If n = 0, then N ∪B = M. Whence M is a 2-disk. K Suppose that n ≥ 1. Let γ be another connected component of ∂B distinct from ∂NK. Then there exists an atom RK′ of some critical component K′ of ∆f such that γ ⊂ ∂RK′. Since NK∪B is a 2-disk, we see that it is contained in NK′. If NK′ is not a 2-disk, then the lemma is proved. Otherwise, the number of critical components in M \NK′ is less than in M \N and the lemma holds by the induction on n. (cid:3) K Example 3.5. Let T2 be a 2-torus embedded in R3 as shown in Fig- ure 3.2 and f : T2 → R be the projection onto the vertical line. Fig- ure 3.2a) shows the critical components of level-sets of f, and Fig- ure 3.2b) presents blackened canonical neighbourhoods of three critical components of ∆ containing canonical neighbourhoods of all other f critical components of ∆ . f FUNCTIONS ON SURFACES AND INCOMPRESSIBLE SUBSURFACES 9 a) b) Figure 3.2. 3.6. Canonical neighbourhoods of negative Euler character- istic. Suppose M is not a 2-disk. Let K ,...,K be all the crit- 1 r ical components of ∆ whose canonical neighbourhoods are not 2- f disks. By Lemma 3.4 this collection is non-empty and by Lemma 3.3 N ∩N = ∅ for i 6= j. Moreover, again by Lemma 3.4, any other Ki Kj critical component of ∆ is contained in some N . It follows that f Ki M \∪r N contains no critical points of f, whence it is a disjoint i=1 Ki union of cylinders S1 ×I. Therefore r (3.2) χ(M) = χ(N ). Ki i=1 X The following two statements will be used for the construction of a surface X of Theorem 1.7, see §7. Lemma 3.7. The following conditions are equivalent: (1) χ(M) < 0; (2) χ(N ) < 0 for some i = 1,...,r. Ki Proof. (1)⇒(2). As χ(M) < 0, we get from (3.2) that χ(N ) < 0 for Ki some i. The implication (2)⇒(1) follows from Corollary 2.2. (cid:3) Corollary 3.8. Let K ,...,K be all the critical components of ∆ 1 k f whose canonical neighbourhoods have negative Euler characteristic and R ,...,R be their atoms. Put R := ∪k R . If R 6= ∅, K1 Kk <0 i=1 Ki <0 then every connected component B of M \R is either a 2-disk, or a <0 cylinder, or a M¨obius band. Proof. Since the homomorphism π R → π M is non-trivial for each 1 Ki 1 i, it follows from Corollary 2.4 that B is incompressible. Suppose χ(B) < 0. Notice that f takes constant values of ∂B. Then by Lemma 3.7thereexistsacriticalcomponentK ⊂ B of∆ suchthatthe f canonicalneighbourhoodN ofK withrespecttof| hasnegativeEuler B 10 SERGIY MAKSYMENKO characteristic. Itfollowsthatthehomomorphisms π N → π B → π M 1 1 1 induced by the inclusions N ⊂ B ⊂ M are monomorphisms, so N is incompressible in M. This implies that N is a canonical neighbour- hood of K with respect to f. But since χ(N) < 0, we should have that N ⊂ R , which contradicts to the assumption. (cid:3) <0 4. Deformations of incompressible subsurfaces The aim of this section is to extend some results of [8] concerning incompressible subsurfaces, see Proposition 4.5. 4.1. ±-twist. Let γ ⊂ IntM be a two-sided simple closed curve, U be its regular neighbourhood diffeomorphic to S1 × [−1,1] so that γ correspond to S1 × 0. Take a function µ : [−1,1] → [0,1] such that µ = 0 near {±1} and µ = 1 on some neighbourhood of 0. Define the following homeomorphism g : M → M by γ (ze2πiµ(t),t), x = (z,t) ∈ S1 ×[−1,1] ∼= U (4.1) g (x) = γ x, x ∈ M \U, ( see Figure 4.1. Then g is fixed on some neighbourhood of M \U and γ isotopic to id via an isotopy supported in IntU. Evidently, g is a M γ product of Dehn twists in opposite directions along the curves parallel to γ. Therefore we will call g a ±-twist near γ. γ Figure 4.1. ±-twist The following lemma is a particular case of [6, Lm. 6.1]. Lemma 4.2. [6, Lm. 6.1]. Suppose χ(M) < 0. Let γ ⊂ IntM be a simple closed curve which does not bound a 2-disk nor a M¨obius band, h : M → M be a homeomorphism homotopic to id and such that M h(γ) = γ. Let also H : M × I → M be any homotopy of id to h. M Then there exists another homotopy G : M ×I → M of id to h such t M that G (γ) = γ and G = H on M \U for all t ∈ I. t t t Moreover, there exists m ∈ Z and a homotopy G′ : M ×I → M of id to gm ◦h such that G′ = G outside U and G′ is fixed on γ for all M γ t t t ∈ I. The following statement is also well-known.