SIMPLIFYING STABLE MAPPINGS INTO THE PLANE FROM A GLOBAL VIEWPOINT

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1 TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 348, Number 7, July 1996 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE FROM A GLOBAL VIEWPOINT MAHITO KOBAYASHI AND OSAMU SAEKI Abstract. Let : M R 2 be a C stable map o an n-dimensional maniold into the plane. The main purpose o this paper is to deine a global surgery operation on which simpliies the coniguration o the critical value set and which does not change the dieomorphism type o the source maniold M. For this purpose, we also study the quotient space W o, which is the space o the connected components o the ibers o, and we completely determine its local structure or arbitrary dimension n o the source maniold M. This is a completion o the result o Kushner, Levine and Porto or dimension 3 and that o Furuya or orientable maniolds o dimension 4. We also pay special attention to dimension 4 and obtain a simpliication theorem or stable maps whose regular iber is a torus or a 2-sphere, which is a reinement o a result o Kobayashi. 1. Introduction Let : M R 2 be a proper stable map o an n-dimensional maniold into the plane (n 3). It is oten an important problem to change by some process to obtain another stable map g : M R 2 which is simpler than the original stable map. For example, Levine [L1] hasshownthat,imis orientable, we can change by a homotopy so that the new map g has at most one cusp point. In this paper, we will deine a new surgery operation, called an R-operation, which converts a given stable map to another one which is simpler than the original one in the sense that the coniguration o the critical value set is simpler. This is motivated by the work o the irst author [Kob1], who deined two surgery operations on certain stable maps o simply connected 4-maniolds. Unortunately, one o the operations he deined, called an S-operation, does not preserve the dieomorphism type o the source maniold in general, although it preserves the homeomorphism type. Our R-operation is deined in a totally dierent manner, but has the same eect as an S-operation without changing the dieomorphism type o the source maniold. Furthermore, our R-operation is also deined or certain higher dimensions. In order to deine an R-operation in a general context, we will need to deine a pseudo quotient space W and a pseudo quotient map q : M W,whichare generalizations o the quotient space W and the quotient map q : M W o a stable map : M R 2 respectively (see, or example, [BdR, L2, KLP, Sa2]). Received by the editors October 24, Mathematics Subject Classiication. Primary 57R45; Secondary 57R35, 57M99. The second author has been partially supported by CNPq, Brazil, and by Grant-in-Aid or Encouragement o Young Scientists (No ), Ministry o Education, Science and Culture, Japan c 1996 American Mathematical Society License or copyright restrictions may apply to redistribution; see

2 2608 MAHITO KOBAYASHI AND OSAMU SAEKI For this reason, we need to completely determine the local structure o the quotient space W o a stable map into the plane. This will be done in 2 (Theorem2.2 and Corollary 2.12). Note that the local structure o a quotient space has already been determined by Kushner, Levine and Porto [KLP, L2] or dimension 3 and by Furuya [Fu, PF] or the orientable case o dimension 4. Our result is a completion o their results or arbitrary dimensions. In particular, we will see that the quotient space o a stable map into the plane is a locally inite polyhedron such that each point has a standard conic neighborhood. This is a undamental result in itsel, and it is important in studying the global topology o stable maps into the plane. The paper is organized as ollows. In 2, we study the topology o the quotient space o a stable map and we determine its local structure. In 3, we deine pseudo quotient spaces and pseudo quotient maps, which are generalizations o quotient spaces and quotient maps o stable maps respectively. One o the advantages o this generalization is that, in this generalized category, we can deine a surgery operation, called an R-operation, which is deined in 4. We also obtain our simpliication theorem o mappings(theorem 4.13) in 4. In 5, we consider the existence problem o an immersion into the plane o a given pseudo quotient space so that we can deine the R-operation in the category o stable maps, not in the category o pseudo quotient maps. Finally in 6, we study stable maps o 4-dimensional maniolds and we obtain a reinement o a result o [Kob1]. We also obtain some important corollaries o this reinement on 4-dimensional topology. In 7, as an example which well illustrates our simpliication theorem, we consider pseudo quotient maps with only spherical regular ibers and we completely determine the dieomorphism type o the source maniold o such a map. Throughout the paper, all maniolds are in C class and the symbol = denotes a dieomorphism between maniolds or an appropriate isomorphism between algebraic objects. 2. Local structure o a quotient space In this section, we study the topology o the quotient space o a stable map into the plane and we completely determine its local structure. Note that the case where the dimension o the source maniold is equal to 3 has already been studied by Kushner, Levine and Porto [KLP, L2], and the case where the source maniold is orientable o dimension 4, by Furuya [Fu, PF]. Let : M R 2 be a proper smooth map o an n-maniold into the plane. We denote the set o such maps by C (M,R 2 ), which is equipped with the Whitney C -topology. We say that is stable i the orbit o by the natural action o A = Di(M) Di(R 2 )isanopensetoc (M,R 2 ). Note that the set o stable maps is open and dense in C (M,R 2 )([Ma]) and that the ollowing characterization is known (see, or example, [L1, L2, GG]). In the ollowing, S() denotes the singular set o. Proposition 2.1. Let : M R 2 be a proper smooth map. Then is stable i and only i the ollowing local and global conditions are satisied. For every p S(), there exist local coordinates (u, z 1,,z n 1 ) centered at p and (U, Y ) centered at (p) such that n 1 (L 1 ) U = u, Y = ±zi 2 (p : a old point) i=1 License or copyright restrictions may apply to redistribution; see

3 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2609 or there exist local coordinates (u, x, z 1,,z n 2 ) centered at p and (U, Y ) centered at (p) such that and n 2 (L 2 ) U = u, Y = ±zi 2 + xu + x 3 (p : acusppoint), i=1 (G 1 ) or every cusp point p S(), 1 ((p)) S() ={p} and, (G 2 ) (S()\{cusp points}) is an immersion with normal crossings. We call a old point p a deinite old point i all the signs appearing in the summation in (L 1 ) are the same. Otherwise, we call p an indeinite old point. We denote by S 0 () the set o the deinite old points o. Note that i is stable, S() is a closed regular 1-dimensional submaniold o M. Now we recall the notion o the Stein actorization o a smooth map (see [BdR, KLP, L2]). Let g : M N be a smooth map o an n-maniold into a p-maniold (n >p). For x, y M, deine x y i g(x) =g(y)andx, y are in the same connected component o g 1 (g(x)) = g 1 (g(y)). We denote by W g the quotient space o M by the equivalence relation and by q g : M W g the quotient map. Note that there exists a unique map ḡ : W g N such that g =ḡ q g.thespace W g or the commutative diagram g M N q g ḡ W g is called the Stein actorization o g. The ollowing theorem has been proved by Kushner, Levine and Porto [KLP, L2] orn= 3 and by Furuya [Fu, PF]orn=4. Theorem 2.2. Let : M R 2 be a proper stable map o an n-maniold into the plane (n 3). Then the quotient space W is a Hausdor space and every point x W has a conic neighborhood as in Figure 1, where the thick lines indicate q (S()). Remark 2.3. We can replace the target maniold R 2 by arbitrary 2-dimensional maniolds and Theorem 2.2 still holds or this larger class o mappings. Note that stable mappings into 2-dimensional maniolds are deined in the same way as in the case o R 2 and that Proposition 2.1 is also valid. In this paper, or simplicity, we almost always consider stable mappings into the plane. Lemma 2.4. Let be as in Theorem 2.2. Then the quotient space W o is a Hausdor space. Remark 2.5. The above lemma does not hold or non-proper maps in general. For example, consider the map u = π (R 2 \{0}) :R 2 \{0} R,whereπ:R 2 Ris the standard projection. Then the quotient space o u is not a Hausdor space. License or copyright restrictions may apply to redistribution; see

4 2610 MAHITO KOBAYASHI AND OSAMU SAEKI (a) x (b) x (b 1) x : boundary point x (b 2) x : branch point x (b 3) (c) x x (c 1) cuspidal branch (c 2) (d) x x (d 1) double cone (d 2) x x (d 3) (d 4) x (d 5) trident Figure 1 License or copyright restrictions may apply to redistribution; see

5 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2611 Proo o Lemma 2.4. Take two distinct points x, y W. I (x) (y), then there exist disjoint open neighborhoods U x and U y o (x) and (y) respectively in R 2.SetV x = 1 (U x )andv y = 1 (U y ). Then we see that V x and V y are disjoint open sets o W which separate x and y. Thus we suppose (x) = (y)=a R 2. The ollowing lemma is proved in [L2, p.7]orn= 3 and the same argument works also or general n. Hence we omit the proo here. Lemma 2.6. Set I =( 1,1) and J =[ 1,1]. For each point a (M),there exists an embedding ψ : I J R 2 such that a = ψ(0, 0) and that the composition h : 1 (ψ(i J)) ψ(i J ) ψ 1 I J p1 I is a trivial bundle, where p 1 is the projection to the irst actor. Let Z be the interior o ψ(i J) andletu x and U y be the connected components o 1 (Z) containing q 1 (x) andq 1 (y) respectively. Note that q (U x )andq (U y ) are open sets o W,sinceq 1 (q (U x )) = U x and q 1 (q (U y )) = U y are open sets o M. Recall that q 1 (x) q 1 (y) =,sincex y. For the proo o Lemma 2.4, it suices to show that U x U y =. Case 1. The case where q 1 (x) contains a cusp point. By the global condition (G 1 ) o Proposition 2.1, q 1 (y) contains no singular point o, and hence U y : U y Z is a submersion by the proo o Lemma 2.6. In particular, U y contains no cusp points, and hence U x U y =. Case 2. The cases other than Case 1. Set T x =(h 1 (Z)) 1 (0) U x and T y =(h 1 (Z)) 1 (0) U y. Note that T x (resp. T y ) is connected since U x (resp. U y ) is dieomorphic to T x ( 1, 1) (resp. T y ( 1, 1)) by Lemma 2.6. Now suppose that a point x 1 in q 1 (x) is connected by a path in 1 (Z) toapointy 1 in q 1 (y). Since h 1 (Z) is a trivial ibration, we see that x 1 and y 1 are connected by a path in T 0 =(h 1 (Z)) 1 (0). Since Z does not contain the image o a cusp point, we see that the composition g 1 : T 0 ψ(i IntJ) ψ 1 I IntJ p2 IntJ is a Morse unction with at most two critical points and at most one critical value c, wherep 2 is the projection to the second actor and c = g 1 (x 1 )=g 1 (y 1 ). Then, by an argument using a gradient like vector ield ([Mi]), we see that there exists a retraction r : T 0 g1 1 (c). Thus we see that x 1 and y 1 are connected by a path in g1 1 (c). Since g 1 1 (c) 1 ( (x)), this is a contradiction. This completes the proo. Proo o Theorem 2.2. Take a point x W.Letψ:I J R 2 be an embedding or a = (x) as in Lemma 2.6 and let h be the map deined in the lemma. Let T denote the connected component o h 1 (0) containing q 1 (x). Note that T is a compact (n 1)-maniold with boundary. Since h : 1 (ψ(i J)) I is a trivial ibration, there exist an embedding φ : I T M and a smooth map g : I T J such that φ = ψ (id g) :I T R 2,whereφ(I T)is the connected component o 1 (ψ(i J)) containing q 1 (x) (see[l2, Deinition License or copyright restrictions may apply to redistribution; see

6 2612 MAHITO KOBAYASHI AND OSAMU SAEKI (p. 9)]). Then by [L2, Lemma 2 (p.11)], there exists an embedding θ : W id g W such that the ollowing diagram is commutative: I T q id g φ q M θ id g W id g W id g I J ψ R 2. Note that we can prove the above lemma or every dimension n, byusingthesame argument as in [L2] orn=3. NotealsothatU=θ q id g (I Int T )isanopen set o W,sinceq 1 (U) is an open set Int φ(i T )om. The ollowing lemma is not diicult to prove. Lemma 2.7. There exists an embedding η : D 2 ψ(i IntJ) such that η(0) = (x) and the composition ξ : 1 (η(d 2 \{0})) η(d 2 \{0}) η 1 D 2 \{0} α S 1 (0, 1] p 2 (0, 1] is a trivial bundle, where α is an appropriate dieomorphism and p 2 is the projection to the second actor. In particular, there exists a dieomorphism τ : 1 (η(d 2 \{0})) 1 (η( D 2 )) (0, 1] such that the ollowing diagram commutes: 1 (η(d 2 \{0})) η(d 2 \{0}) η 1 D 2 \{0} τ 1 (η ( D 2 )) (0, 1] id (η id) α η 1 η ( D 2 ) (0, 1] η 1 id α D 2 (0, 1]. Set S = η( D 2 )andx= 1 (S) φ(i T). Furthermore, set k = X : X S and let S be the connected component o 1 (η(d 2 )) containing q 1 (x). Note that S = 1 (η(d 2 1 )) φ(i T )andthatτ maps S\q (x) dieomorphically onto X (0, 1]. Now deine γ : X [0, 1] W by γ(p, t) =q (τ 1 (p, t)) i t 0and γ(p, 0) = x (p X). Lemma 2.8. γ is continuous. Proo. Take an open set U o W contained in q (φ(i T )). I x U, then it is easy to see that γ 1 (U) isopeninx [0, 1]. I x U, thenγ 1 (U)= (X {0}) τ(q 1 (U\{x})). Since q 1 (U) is an open neighborhood o q 1 (x) in S,τ 1 (X (0,ε)) q 1 (x) q 1 (U) or a small positive real number ε. Thus X (0,ε) τ(q 1 (U\{x})), and hence γ 1 (U) isopeninx [0, 1]. License or copyright restrictions may apply to redistribution; see

7 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2613 It is easy to see that γ induces an injective continuous map γ : W k [0, 1]/W k {0} W such that the ollowing diagram commutes: X [0, 1] q k id W k [0, 1] γ π γ W k [0, 1]/W k {0}, where π is the natural projection. Since W k [0, 1]/W k {0}is compact and W is a Hausdor space by Lemma 2.4, γ is an embedding. Note that the image o γ is a neighborhood o x in W. Thus we have proved the ollowing. Lemma 2.9. There exists a neighborhood o x in W which is homeomorphic to theconeoverw k,wherek:x Sis the proper stable map o the (n 1)-maniold X into S( = S 1 ). Note that Theorem 2.2 has been proved in [L2] orn= 3. Thus in the ollowing, we assume that n 4. Note that we do not assume that M is orientable. Lemma There exists a point a η(d 2 )\(S()) such that 1 (a ) S is connected. Proo. I q 1 (x) S() =, then the result is obvious. I q 1 (x) contains a cusp, set a = ψ(0,ε) or a suiciently small non-zero real number ε. Near the cusp point, the composition has the normal orm 1 (ψ(0 J)) W ψ(0 J) ψ 1 0 J n 2 (x, z 1,,z n 2 ) ±zi 2 + x3 (see (L 2 ) o Proposition 2.1). Then it is not diicult to see that 1 (a ) S is connected. Now suppose that q 1 (x) contains a singular point which is not a cusp point. Then q 1 (x) contains at most two singular points p 1 and p 2, which are old singular points. For a point b R 2 close to (x), 1 (b) S and q 1 (x) dier topologically only near p 1 and p 2. For p 1, there exists a hal disk region 1 in η(d 2 )such that, or b 1 \(S()), 1 (b) S is connected near p 1, since n 4. We also have a similar hal disk region 2 or p 2.Thenapointa 1 2 \(S()) is a required point. i=1 Now we continue the proo o Theorem 2.2. Case 1. x q (S()). In this case k is just a smooth iber bundle over S with connected ibers. Thus W k is homeomorphic to S 1 and x has a neighborhood as in Figure 1 (a). License or copyright restrictions may apply to redistribution; see

8 2614 MAHITO KOBAYASHI AND OSAMU SAEKI x Figure 2 Figure 3 Case 2. q 1 (x) contains exactly one singular point which is not a cusp point. In this case k has exactly two critical points. Then it is not diicult to see that W k is as in Figure 1 (b). Case 3. q 1 (x) contains a cusp point. In this case k has two critical points. By using the same arguments as in the proo o Lemma 2.10, we see easily that W k is as in Figure 1 (c) or as in Figure 2. Suppose that W k is as in Figure 2. Then or some δ I\{0}, the unction l : 1 (ψ(δ J)) φ(i T ) ψ(δ J ) ψ 1 δ J has exactly two critical points. Note that l is a Morse unction and its quotient space W l is homeomorphic to the space as in Figure 3. In particular π 1 (W l ) = Z whose generator does not come rom the regular iber l 1 (δ 1) o l. On the other hand, by Lemma 2.6, 1 (ψ(δ J)) φ(i T ) is dieomorphic to 1 (ψ( δ J)) φ(i T ), which is dieomorphic to l 1 (δ 1) J, since the map corresponding to l or δ in place o δ has no critical point. Let π + : 1 (ψ(δ J)) φ(i T ) l 1 (δ 1) be the natural projection. Then we see that the induced homomorphism (π + ) on the undamental groups has kernel containing Z by the above observation. This is a contradiction. Thus W k is not as in Figure 2. Case 4. q 1 (x) contains two singular points o. Firstly note that the two singular points are indeinite old points and that a small neighborhood o (x) is divided into 4 regions by C() =(S()). Note also that a regular iber 1 (b) (b R 2 \C()) o is a closed (n 2)-dimensional maniold and that, every time b passes through (S()\S 0 ()), the regular iber changes by a surgery along an embedded i-sphere (0 i n 3) in general. When 1 i n 4, the number o components o the regular iber is obviously unchanged. Denoting by F the connected component 1 (a ) S o a regular iber License or copyright restrictions may apply to redistribution; see

9 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2615 as in Lemma 2.10, we denote by S 1 and S 2 the two surgery spheres embedded in F corresponding to the two singular points. I the dimension o S 1 and/or S 2 is equal to 0, then we can choose another connected component o a regular iber so that both o the surgery spheres have dimension greater than 0. Thus we may assume that dim S 1 dim S 2 1. Firstly we consider the case where dim S 1 =dims 2 =n 3. In this case, since the indices o the corresponding handles are equal, we may assume that S 1 S 2 =. Furthermore S 1 and S 2 have trivial normal bundles N(S 1 )andn(s 2 )inf respectively. We denote by A j and B j (j =1,2) the two connected components o N(S j ). The our connected sets A 1,B 1,A 2 and B 2 are divided into some classes by the equivalence relation deined by the existence o an arc in F \(IntN(S 1 ) IntN(S 2 )) which connects given two sets. Since F is connected, we may assume that A 1 and A 2 belong to the same class. Case 4-1. When A 1,A 2,B 1 and B 2 all belong to the same class. A surgery along S 1 and/or S 2 does not aect the connectivity o F. Hence, by Lemma 2.9, we see that a neighborhood o x in W is homeomorphic to the space as in Figure 1 (d 2). Case 4-2. When the our sets are divided into the two classes {A 1,A 2 }and {B 1,B 2 }. We see that a single surgery along S 1 or S 2 does not aect the connectivity o F. However, the successive surgeries along S 1 and S 2 make F disconnected. Hence, we see that a neighborhood o x in W is homeomorphic to the space as in Figure 1 (d 4). Case 4-3. When the our sets are divided into the two classes {A 1,A 2,B 1 } and {B 2 }. By an argument similar to that in Case 4-2, we see that a neighborhood o x in W is homeomorphic to the space as in Figure 1 (d 3). Case 4-4. When the our sets are divided into the three classes {A 1,A 2 },{B 1 } and {B 2 }. Similarly we see that a neighborhood o x in W is homeomorphic to the space as in Figure 1 (d 5). When dim S 1 = n 3and1 dim S 2 n 4, the situation is the same as in Case 4-1 or 4-3 above. I both o the surgery spheres S 1 and S 2 have dimensions between 1 and n 4, the situation is as in Case 4-1. Thus we have shown that a neighborhood o x in W must be homeomorphic to one o the spaces as in Figure 1 (d). This completes the proo o Theorem 2.2. For a uture reerence, we call the conic neighborhood as in Figure 1 (c 1) a cuspidal branch, (d 1) a double cone and (d 5) a trident. Note that the double cone does not occur or n 4. Remark We do not know i a similar result holds also or stable maps into R p with n>p 3. I a result like Lemma 2.9 holds also in this case, we would be able to prove such a result by the induction on the target dimension p. Theorem 2.2 has an immediate corollary as ollows. License or copyright restrictions may apply to redistribution; see

10 2616 MAHITO KOBAYASHI AND OSAMU SAEKI Corollary Let : M R 2 be a proper stable map o an n-maniold into the plane (n 3). Then W is a locally inite polyhedron. Remark Note that all conic neighborhoods as in Figure 1 actually arise or some stable map. For example, see [ML, MPS]. Remark Note that every conic neighborhood as in Figure 1 is equipped with a natural map into R 2, which corresponds to the map : W R 2 or a stable map : M R 2. We also note that every component o W \q (S()) is canonically oriented by the local homeomorphism, once an orientation o R 2 is ixed, and that their orientations are consistent along q (S()). In the next section, we deine generalizations o quotient spaces and quotient maps so that we have more reedom in changing a given map. 3. Pseudo quotient maps and coniguration triviality In this section, we irst deine a generalization o quotient maps, namely pseudo quotient maps, and then we introduce a notion o coniguration triviality o the critical value set o a pseudo quotient map. The advantage o this generalization is that we can deine a surgery operation, which simpliies a given map in the generalized category, which will be seen in 4. Deinition 3.1. Let W be a 2-dimensional locally inite polyhedron. We say that W is a pseudo quotient space i every point x o W has a conic neighborhood as in Figure 1 in 2. Furthermore, we denote by Σ(W ) the set o the non-maniold points o W, including the boundary points. A pseudo quotient space W is said to be oriented i each connected component o W \Σ(W ) is oriented and the orientations are consistent along Σ(W ) in the sense o Remark The quotient space W o a proper stable map : M R 2 o an n-maniold M is always a pseudo quotient space in the above sense by Theorem 2.2 and Corollary Furthermore, it is oriented, once an orientation o R 2 is ixed. Furthermore, a branched surace as deined in [Sa2] is a pseudo quotient space in the above sense. Deinition 3.2. Let q : M W be a proper continuous surjective map o an n-maniold into a pseudo quotient space W (n 3). We say that q is a pseudo quotient map i or every point x W, there exist a conic neighborhood U as in Figure 1 and a continuous map ϕ : U R 2 as in Remark 2.14 such that ψ = ϕ q q 1 (U) :q 1 (U) R 2 is a proper stable map and that its quotient map is C 0 -let equivalent to q q 1 (U); i.e., there exists a homeomorphism α : U W ψ such that α 1 q ψ = q q 1 (U). In the above deinition, the ollowing diagram is commutative. U q 1 (U) q q ψ α W ψ ϕ ψ R 2. License or copyright restrictions may apply to redistribution; see

11 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2617 Note that the map ϕ is inite-to-one, since ψ is proper. Note also that the quotient map q : M W o a proper stable map : M R 2 is always a pseudo quotient map. For example, a old map as deined in [Sa2] is a pseudo quotient map in the sense o Deinition 3.2. For a pseudo quotient map q : M W, we deine its singularities as the union o the corresponding singularities o the stable maps ϕ q q 1 (U) as in Deinition 3.2. We denote by S(q) the set o the singularities o q and by S 0 (q) the set o the deinite old points. Furthermore, we set C(q) = q(s(q)), which is the set o the critical values o q. It is not diicult to see, with the help o the above commutative diagram, that W \C(q) admitsaunique smooth structure so that ϕ (U\C(q)) is an immersion or every pair (U, ϕ) as in Deinition 3.2. For a pseudo quotient map q : M W,iS(q)=, then W is a smooth nonsingular surace and q is a smooth bundle projection. Thus, in this paper, we always assume that a pseudo quotient map has nonempty singular set. In the next section we will introduce a geometric operation on pseudo quotient maps which simpliies the coniguration o the critical value set in the ollowing sense. Deinition 3.3. Let q : M W be a pseudo quotient map o an n-maniold. A component R o W \C(q) is said to be a spherical region i, or every point x R, q 1 (x) is dieomorphic to the standard sphere S n 2.NotethatiR q(s 0 (q)), i.e., i R is adjacent to q(s 0 (q)), then R is a spherical region. Deinition 3.4. We say that a pseudo quotient map q : M W is coniguration trivial with respect to the spherical regions i or every spherical region R such that R is compact and that R q(s 0 (q)) =, R is homeomorphic to the open 2-disk. Remark 3.5. The above deinition may seem a little strange. However, or stable maps o simply connected 4-maniolds studied in [Kob1], it coincides with the deinition o the coniguration triviality deined in [Kob1]. Lemma 3.6. Suppose that q : M W is a pseudo quotient map o an n-maniold. I q is coniguration trivial with respect to the spherical regions and W is connected, then W \( i R i) is connected, where R i runs over the spherical regions R o W \C(q) such that R is compact and that R q(s 0 (q)) =. Furthermore, i H 1 (M; Z 2 )= 0and W \( i R i) is connected, then q is coniguration trivial with respect to the spherical regions. Note that when M is compact, the condition that R is compact is not necessary in Deinition 3.4 and Lemma 3.6. ProooLemma3.6.Suppose that q : M W is coniguration trivial. Then each R i is homeomorphic to the open 2-disk. Now take two arbitrary points z,w W\( i R i). Then, since W is connected, we can ind a path α in W which connects z and w. By transversality on each R i, we may assume that there exists a point r i in each R i which does not intersect α. Then we can deorm α on each R i \{r i } ixing the end points o α so that R i α R i \R i. This implies that z and w are connected by a path in W \( i R i). This completes the irst part o Lemma 3.6. Suppose that H 1 (M; Z 2 )=0andthatW\( i R i) is connected. Take a spherical region R such that R is compact and that R q(s 0 (q)) =. There exists a compact License or copyright restrictions may apply to redistribution; see

12 2618 MAHITO KOBAYASHI AND OSAMU SAEKI surace R with boundary whose interior is homeomorphic to R. (For example, R\IntN is such a compact surace, where N is a small regular neighborhood o C(q) inw. Note that R may not be homeomorphic to R.) We will show that R is homeomorphic to the 2-disk. Suppose that R is not connected. Then we have a properly embedded arc α in R and a simple closed curve β in Int R which intersect each other transversely in one point. Let α and β be the corresponding curves in R. Since W\( i R i) is connected, we can connect the end points o α in W \( i R i) by a curve α 1. Let α be the closed curve in W which is the union o α and α 1. Set V = q 1 (β ), which is a compact codimension-1 submaniold o M. Then it is not diicult to see that we have a smooth closed curve α in M such that q( α)=α and that α and V intersect each other transversely in one point, by modiying α 1 i necessary. This implies that [V ] [ α] 0, where [V ] H n 1 (M; Z 2 )and[ α] H 1 (M;Z 2 ) are the homology classes represented by V and α respectively and [V ] [ α] Z 2 denotes the intersection number. This contradicts the assumption that H 1 (M; Z 2 )=0. Hence Rshould be connected. Next suppose that R is not homeomorphic to the 2-disk. Then we have two simple closed curves γ and δ in R which intersect each other transversely in one point. Then an argument similar to the above one leads to a contradiction. Hence R is homeomorphic to the 2-disk. This completes the proo. In the ollowing sections, we will be mainly interested in the case where M is simply connected. In this case, by the above lemma, the coniguration triviality is equivalent to the connectedness o the complement o certain spherical regions. For two pseudo quotient maps q i : M i W i (i =1,2) such that M i and W i are possibly oriented, that n =dimm 1 =dimm 2 and that S 0 (q 1 ) S 0 (q 2 ), we deine a new pseudo quotient map q 1 q 2 : M 1 M 2 W 1 W 2, called the connected sum o q 1 and q 2, as ollows, where means the connected sum and the boundary connected sum. Let a i (i =1,2) be a point in W i = q i (S 0 (q i )) and U i its small regular neighborhood in W i.letλ i (i=1,2) be the closure o U i \ W i,whichis an arc joining two points in W i. Note that, when W i is oriented, λ i is naturally oriented as a part o the boundary o W i \U i. We take U i so that λ i is normal to W i. It is not diicult to see that q 1 i (U i ) is dieomorphic to the n-disk and that q 1 i (λ i )= (q 1 i (U i )). Let ϕ : (M 2 \q2 1 (U 2)) (M 1 \q1 1 (U 1)) and ψ : λ 2 λ 1 be (orientation reversing) dieomorphisms satisying the equation (q 1 q1 1 (λ 1)) ϕ = ψ (q 2 q2 1 (λ 2)) on q2 1 (λ 2). Reer to [Kob1, p. 336] or the existence o such dieomorphisms ϕ and ψ. Set M 1 M 2 = (M 1 \q1 1 (U 1)) ϕ (M 2 \q2 1 (U 2)) and W 1 a1,a 2 W 2 = (W 1 \U 1 ) ψ (W 2 \U 2 ). Now we deine q 1 a1,a 2 q 2 : M 1 M 2 W 1 a1,a 2 W 2 by (q 1 a1,a 2 q 2 ) (M i \q 1 i (U i )) = q i (M i \q 1 i (U i )) (i =1,2). The map q 1 a1,a 2 q 2 is obviously a pseudo quotient map. It is also obvious rom the construction that i b i (i =1,2) is a point in the same connected component o License or copyright restrictions may apply to redistribution; see

13 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2619 W i as a i,thenq 1 b1,b 2 q 2 is right-let equivalent to q 1 a1,a 2 q 2. Following [Kob1], we use the notation (M 1 M 2,q 1 a1,a 2 q 2 )=(M 1,q 1 ) a1,a 2 (M 2,q 2 ). We oten omit the subscripts a 1 and a 2 i there is no conusion, or instance, in the case that W 1 and W 2 have connected boundaries. Note that the above operation is commutative and associative up to right-let equivalence. In the next section, we will deine a global surgery operation, called an R- operation, on a pseudo quotient map, which simpliies the coniguration o the critical value set. Unortunately this operation can be deined only or certain dimensions n o the source maniold. In order to describe this restriction, we prepare some notations. For the standard (n 2)-sphere S n 2,Di + (S n 2 ) denotes the group o the orientation preserving dieomorphisms o S n 2,andπ 0 (Di + (S n 2 )) denotes its identity component. It is known that π 0 (Di + (S n 2 )) is isomorphic to the group o homotopy (n 1)-spheres Θ n 1 ([KM, C2]) or n 7andthat π 0 (Di + (S n 2 )) = 0 or n =2,3,4,5,7,13, ([C1, Sm, H]). The R-operation mentioned above will be deined or n 4withπ 0 (Di + (S n 2 )) = 0 in the next section. The ollowing lemma will be helpul or getting a geometric interpretation o the coniguration triviality. Lemma 3.7. Suppose that π 0 (Di + (S n 2 )) = 0 and n 3. A pseudo quotient map q : M W o a closed connected n-maniold M with H 1 (M; Z 2 )=0is coniguration trivial with respect to the spherical regions i and only i there exists a decomposition o the orm (M,q) =(M 1,q 1 ) (M 2,q 2 ) (M k,q k ), where each q i : M i W i is a pseudo quotient map such that each spherical region not adjacent to q i (S 0 (q i )) is homeomorphic to the open 2-disk and that each spherical region adjacent to q i (S 0 (q i )) is homeomorphic to the open annulus or the open 2-disk. Proo. Firstly suppose that q is coniguration trivial with respect to the spherical regions. Take a spherical region R o W \C(q). Since W is compact in our case, R is always compact. I R q(s 0 (q)) =, thenris homeomorphic to the open 2-disk. I R q(s 0 (q)), thenris an open planar surace by the proo o Lemma 3.6. Let R be the compact surace with boundary whose interior is dieomorphic to R. I Ris connected, then R = W \q(s 0 (q)), and we have easily the conclusion (k =1and(M,q) =(M 1,q 1 )). I R has exactly two components, then R is homeomorphic to the open annulus. I R has three or more components, let C be a component o R which has a 1-dimensional intersection with R q(s 0 (q)). Then there exist properly embedded disjoint arcs γ 1,,γ l in R such that the end points o γ i are in C q(s 0 (q)) and that the closures o each connected component o R\( l i=1 γ i) is homeomorphic to the annulus. We may assume that γ i are normal to R. Furthermore, we may assume that γ i are embedded in R W. Then X i = q 1 (γ i ) is a smooth closed (n 1)-dimensional maniold which admits a Morse unction with exactly two critical points. Then, by our dimension assumption, we License or copyright restrictions may apply to redistribution; see

14 2620 MAHITO KOBAYASHI AND OSAMU SAEKI see that X i is dieomorphic to the (n 1)-sphere. Furthermore, by an argument similar to the proo o Lemma 3.6, we see that each γ i separates W and that each X i separates M. In other words, M\ l i=1 X i has (l + 1) connected components, whoseclosureswedenotebym 1,,M l+1. Note that each boundary component o M j is dieomorphic to the (n 1)-sphere by our assumption on n. Then by a standard argument, we can construct pseudo quotient maps q j : M j W j such that M j is the closed maniold obtained by attaching n-disks to the boundaries o M j and that (M,q) =( M 1,q 1) ( M l+1,q l+1). (For example, see [Sa1].) I we apply the same operation to all the spherical regions adjacent to q(s 0 (q)), then we obtain a desired decomposition. Conversely, i q hasthedecompositionasabove,itisclearthatqis coniguration trivial with respect to the spherical regions. This completes the proo. Remark 3.8. We note that the above decomposition lemma is valid or all n 3, without the assumption on π 0 (Di + (S n 2 )), in the C 0 -category; i.e., (M,q) isc 0 - right-let equivalent to (M 1,q 1 ) (M 2,q 2 ) (M k,q k ), where each q i : M i W i is a C 0 pseudo quotient map. In this case, M i is just a topological maniold and may not have a smooth structure. 4. R-operation In this section we introduce a surgery operation, called an R-operation, o pseudo quotient maps o compact maniolds, an operation which does not change the dieomorphism type o the source maniold and which simpliies the coniguration o the critical value set C(q) o a given pseudo quotient map q : M W in a global sense. Let q : M W be a pseudo quotient map o a closed connected n-maniold (n 3) into a pseudo quotient space W. We assume that H 1 (M; Z 2 )=0. Takea simple closed curve c in a spherical region R. Lemma 4.1. W \c has exactly two connected components. Proo. Suppose that W \c is connected. Then we can ind a simple closed curve in M which intersects q 1 (c) transversely in one point, as in the proo o Lemma 3.6. This contradicts the assumption on H 1 (M; Z 2 ). We denote by A and B the closures o the two connected components o W \c. Lemma 4.2. Suppose that H 1 (M; Z) = 0 and H 2 (W ; Z) = 0. Then either H 1 (A; Z) or H 1 (B; Z) vanishes, and the other is isomorphic to the ininite cyclic group Z. For the proo o Lemma 4.2, we need the ollowing. Lemma 4.3. Suppose that q : M W is a pseudo quotient map. Then the induced homomorphism q : π 1 (M) π 1 (W ) is surjective. Proo. Take a loop l in W. We may assume that l does not pass through the image o the cusps and that l q(s 0 (q)) =, bymovinglby a homotopy. Then it is easy to see that l can be lited to a loop in M. This completes the proo. License or copyright restrictions may apply to redistribution; see

15 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2621 Proo o Lemma 4.2. By Lemma 4.3 and our assumption, we see that H 1 (W ; Z) =0. Consider the ollowing Mayer-Vietoris exact sequence with coeicients in Z: H 2 (W ) H 1 (A B) H 1 (A) H 1 (B) H 1 (W ). Since H 1 (W )=0=H 2 (W)andA B=c,weseethatH 1 (A) H 1 (B) =Z.This completes the proo. In what ollows, we adopt the convention that H 1 (A; Z) = 0 and hence that H 1 (B; Z) = Z. We also assume that π 0 (Di + (S n 2 )) = 0 and n 3. Set à = q 1 (A) and B=q 1 (B), which are codimension 0 submaniolds o M with boundary. It is obvious that M = à B. Furthermore à B = à = B and q à B : à B c is an orientable smooth iber bundle with iber S n 2,sincec lies in a spherical region. Then, by our assumption on n, we see that it is a trivial bundle and that à B is dieomorphic to S 1 S n 2. Now we deine two maniolds M 1,M 2 and two pseudo quotient maps q 1, q 2 o M 1,M 2 respectively as ollows. Since q à and hence q B are the trivial S n 2 - bundles, there exist dieomorphisms ϕ i and ψ i (i =1,2) which make the ollowing diagrams commutative: (4.4) (S 1 D n 1 ) p 1 ϕ 1 à q à (D 2 S n 2 ) p 2 ϕ 2 B q B S 1 ψ 1 c, S 1 ψ 2 c, where p 1 and p 2 are the obvious projections. Let g 1 : S 1 D n 1 S 1 [0, 1] be the mapdeinedbyg 1 (t, x) =(t, x 2 ), where D n 1 is the unit disk in R n 1 and x denotes the usual norm o x R n 1. Note that g 1 has deinite old points along S 1 {0} S 1 D n 1 and that g 1 (S 1 D n 1 )=p 1 : (S 1 D n 1 ) S 1 {1}. Furthermore, let g 2 : D 2 S n 2 D 2 be the projection to the irst actor. Set M 1 = à ϕ 1 S 1 D n 1 and W 1 = A ψ1 S 1 [0, 1], where ψ 1 is regarded as a dieomorphism rom S 1 1toc A. We deine the pseudo quotient map q 1 : M 1 W 1 by Furthermore, set q 1 à = q à and q 1 (S 1 D n 1 )=g 1. M 2 = B ϕ2 D 2 S n 2 and W 2 = B ψ2 D 2. We deine the other pseudo quotient map q 2 : M 2 W 2 by q 2 B = q B and q 2 (D 2 S n 2 )=g 2. License or copyright restrictions may apply to redistribution; see

16 2622 MAHITO KOBAYASHI AND OSAMU SAEKI Deinition 4.5. Let q : M W be a pseudo quotient map o a closed connected n-maniold M. We assume that π 0 (Di + (S n 2 )) = 0, n 3, H 1 (M; Z) =0and H 2 (W;Z)=0. Letq 1 :M 1 W 1 and q 2 : M 2 W 2 be the pseudo quotient maps which were constructed in the previous paragraph. We also assume that S 0 (q 1 )ands 0 (q 2 ) are both non-empty. Then, an R-operation is the procedure which converts q to the pseudo quotient map q 1 a,b q 2 : M 1 M 2 W 1 a,b W 2 or some points a W 1 = q 1 (S 0 (q 1 )) and b W 2 = q 2 (S 0 (q 2 )). For the change o the target pseudo quotient space and the coniguration o the critical value set, see Figure 4 as an example. We note that the reverse operation o an R-operation can also be deined. Remark 4.6. When π 0 (Di + (S n 2 )) 0, we cannot deine an R-operation in the C -category. However, it is known that π 0 (Homeo + (S n 2 )) = 0 or all n 2, where Homeo + (S n 2 ) is the group o the orientation preserving homeomorphisms o S n 2. By using this, we can still deine an R-operation in the C 0 -category; i.e., in a situation as in Deinition 4.5, without the assumption on n, we can ind topological maniolds M i and C 0 pseudo quotient maps q i : M i W i (i =1,2) and we can construct a new C 0 pseudo quotient map q 1 a,b q 2 : M 1 M 2 W 1 a,b W 2. Note that an R-operation preserves the dieomorphism type o the source maniold provided that it is simply connected, which will be veriied by the lemmas below. Lemma 4.7. Assume that π 1 (M) =1and that n =dimm 4. Then an R- operation perormed on a pseudo quotient map q : M W as in Deinition 4.5 preserves the dieomorphism type o M; namelym =M 1 M 2. To show the above lemma we need the ollowing. Lemma 4.8. I π 1 (M) =1and n =dimm 4,thenπ 1 (Ã)=1. Proo. In the ollowing, we consider homology groups with Z-coeicients. Firstly consider the Meyer-Vietoris exact sequence: (j1) (j2) (4.9) H 1 (à B) H 1 (Ã) H 1( B ) H 1 (M ), where j 1 : à B à and j 2 : à B B are the inclusion maps and H 1 (M) =0 by our assumption. Furthermore we have the commutative diagram H 1 (à B) (j 2) H 1 ( B ) (q à B ) (q B ) (i 2 ) H 1 (c) H 1 (B ), where i 1 : c A and i 2 : c B are the inclusion maps, (q à B) is an isomorphism and (i 2 ) is also an isomorphism, by the proo o Lemma 4.2. Together with the exact sequence (4.9), this implies H 1 (Ã) = 0,H 1( B) = Z and (j 2 ) : H 1 (à B) H 1 ( B) is an isomorphism. Consider the commutative diagram π 1 (à B) τ (j 2) π 1 ( B ) τ H 1 (à B ) (j 2 ) H 1 ( B ), License or copyright restrictions may apply to redistribution; see

17 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2623 c W W 1 W 2 W 1 W 2 Figure 4 where τ and τ are the abelianizations. Since τ and the lower (j 2 ) are isomorphisms, we have ω (j 2 ) =id π1(ã B),whereω=τ 1 (j 2 ) τ :π 1 ( B) π 1 (Ã B). Then, by using van Kampen theorem, we see that there exists a homomorphism = ξ : π 1 (M) π 1 (Ã) π 1(Ã B) π 1( B ) π 1 (Ã) such that ξ (k 1) =id:π 1 (Ã) π 1 (Ã), where k 1 : Ã M is the inclusion map. Since π 1(M) is trivial, this implies that π 1 (Ã) =1. License or copyright restrictions may apply to redistribution; see

18 2624 MAHITO KOBAYASHI AND OSAMU SAEKI Remark The above proo shows that π 1 ( B) is normally generated by (j 2 ) (α), where α π 1 (à B) = Z is a generator. This implies that π 1 (M 2 )=π 1 ( B ϕ2 D 2 S n 2 )=1. Note also that 1 = π 1 (Ã) = π 1 (à ϕ 1 S 1 D n 1 )=π 1 (M 1 ). Proo o Lemma 4.7. Firstly set ϕ = ϕ 1 1 ϕ 2 : (D 2 S n 2 ) (S 1 D n 1 ), which is a dieomorphism, where ϕ 1 and ϕ 2 are the dieomorphisms as in diagram (4.4). Then it is easy to see that (S 1 D n 1 ) ϕ (D 2 S n 2 ) is dieomorphic to S n. By Lemma 4.8 and van Kampen theorem, we see that M 1 = à ϕ 1 S 1 D n 1 is simply connected. Set K = S 1 D n 1 M 1. Then, since n =dimm 1 4, we see that there exists an embedded n-disk D in M 1 such that K Int D. Thenwe see that à = M 1 \IntK =(M 1 \IntD) (D\IntK) = M 1 ((D n D)\IntK) = M 1 (S n \S 1 D n 1 ) = M 1 D 2 S n 2. Here, à and (M 1 D 2 S n 2 )= (D 2 S n 2 ) are identiied by the dieomorphism ϕ 1 ϕ 1 1. Hence we see that M = à id B = (M 1 D 2 S n 2 ) ϕ 1 ϕ 1 1 = M 1 (D 2 S n 2 ϕ 1 B) 2 = M 1 M 2. id B This completes the proo o Lemma 4.7. Recall that we have assumed that H 2 (W ; Z) = 0 in order to deine an R-operation or a pseudo quotient map q : M W. Furthermore, i π 1 (M) =1,thenπ 1 (W)=1 by Lemma 4.3. Since W is a 2-dimensional polyhedron, this implies that W is contractible. Lemma Let W 1 and W 2 be the pseudo quotient spaces as in Deinition 4.5. I W is contractible, then so are W 1 and W 2. Proo. In the ollowing, we consider homology groups with Z-coeicients. Consider the Mayer-Vietoris exact sequence: H 2 (c) H 2 (A) H 2 (B) H 2 (W ) H 1 (c) H 1 (A) H 1 (B) H 1 (W ). Since H 2 (c),h 2 (W)andH 1 (W) vanish, we see that H 2 (A),H 2 (B)andH 1 (A)also vanish and that H 1 (B) = Z. Then, by the same argument as in the proo o Lemma 4.8, we see that π 1 (A) =1andthatπ 1 (B) is normally generated by the class [c] π 1 (B). Then we see that W 1 = A ψ1 S 1 [0, 1] and W 2 = B ψ2 D 2 License or copyright restrictions may apply to redistribution; see

19 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2625 are simply connected. Furthermore, since W 1 is homeomorphic to A, we have H 2 (W 1 ) = H 2 (A) = 0, and by the Mayer-Vietoris exact sequence H 2 (B) H 2 (D 2 ) H 2 (W 2 ) H 1 (c) H 1 (B) H 1 (D 2 ), we also see that H 2 (W 2 )=0. HenceW 1 and W 2 are contractible. This completes the proo. The ollowing lemma will be proved in the next section. Lemma Let q : M W be a pseudo quotient map o a closed connected n-maniold into a compact pseudo quotient space such that H (W ; Z 2 ) = H ({point}; Z 2 ) and that W contains no trident (Figure 1 (d 5)). Then W contains a boundary point (Figure 1 (b 1)). Now we state the main theorem o this section as ollows. Theorem Let q : M W be a pseudo quotient map o a closed connected n- maniold such that π 0 (Di + (S n 2 )) = 0,n 4,π 1 (M)=1,H 2 (W;Z)=0and that W contains no trident (Figure 1 (d 5)). Then, by a inite iteration o R-operations perormed on q, we can convert q to a pseudo quotient map q : M W which is coniguration trivial with respect to the spherical regions. Proo. Take a spherical region R o W \q(s 0 (q)) such that R q(s 0 (q)) =. Firstly note that R is an open planar surace by the proo o Lemma 3.6. Let R be the compact surace with boundary whose interior is homeomorphic to R. Inthe ollowing, we identiy R with Int R. Furthermore, we deine the number o boundary components o R to be that o R. Now,iRis not homeomorphic to the open 2-disk, take a simple closed curve c in R which is parallel to a component o R. Then perorm the R-operation along c to obtain a new pseudo quotient map q : M W. Here we note that, by Lemma 4.11, the two target pseudo quotient spaces o the two pseudo quotient maps constructed just beore Deinition 4.5 or q are contractible and contain no trident. Thus, by Lemma 4.12, we can perorm the R-operation as desired. Note that, by this procedure, the number o spherical regions not adjacent to q(s 0 (q)) is unchanged and that the new spherical region not adjacent to q (S 0 (q )) corresponding to R is homeomorphic to the open 2-disk or it has ewer boundary components than R. Furthermore, the new pseudo quotient map satisies the conditions as in the theorem. Since M is compact, W has a inite number o spherical regions. Thus, by perorming the above operations initely many times, we inally obtain a pseudo quotient map q : M W such that each spherical region not adjacent to q(s 0 ( q)) is homeomorphic to the open 2-disk, i.e., q is coniguration trivial with respect to the spherical regions. This completes the proo. Remark By the proo o Theorem 4.13, q and q have the same number o spherical regions. Furthermore, the numbers o spherical regions not adjacent to the images o the deinite old points are also the same. License or copyright restrictions may apply to redistribution; see

20 2626 MAHITO KOBAYASHI AND OSAMU SAEKI 5. Immersing Z 2 -acyclic pseudo quotient spaces into R 2 Deinition 5.1. Let W be a pseudo quotient space. A proper continuous map η : W R 2 is a topological immersion i, or each point x W, there exists a conic neighborhood as in Figure 1 on which η is C 0 right-let equivalent to the canonical map into R 2 as mentioned in Remark Let q : M W be a pseudo quotient map o an n-maniold into a pseudo quotient space. We say that a proper topological immersion η : W R 2 is a generic smooth immersion with respect to q i the map η q : M R 2 is a C stable map. For a pseudo quotient space W, we denote by BR(W) thesetothepoints in W which have a conic neighborhood as in Figure 1 (b-2). We call a point in BR(W)abranch point. It is easy to see that BR(W) consists o some open arcs and circles. Note that or every branch point x BR(W), its regular neighborhood V x is homeomorphic to the trivial Y -bundle over a closed interval, where Y = {re 2πiθ C;0 r 1,θ = 0,1/3,2/3}. Suppose that there exists a pseudo quotient map q : M W o an n-maniold into W. Then, by the deinition o a pseudo quotient map, we have a continuous map ϕ : V x R 2 as in Deinition 3.2. I V x is suiciently small, then the three components A 1,A 2 and A 3 o V x \BR(W) aremappedbyϕ homeomorphically into the plane and two o them, say A 1 and A 2,aremapped on the same side o ϕ(v x BR(W)). We call a point in A 1 A 2 an arm point and a point in A 3 a stem point. Note that this deinition does not depend on the choice o the map ϕ, provided that the regular neighborhood V x is suiciently small. Furthermore, we call the set o the arm points the arm o W with respect to q and the set o the stem points the stem. Note that the arm and the stem satisy a certain consistency condition in the neighborhoods o the non-simple points o W (i.e., the points o W which have conic neighborhoods as in Figure 1 (d)). Let W be a pseudo quotient map. Suppose that a structure o arm and stem around BR(W) which satisy the consistency condition at non-simple points is already given. Furthermore, we suppose that W is oriented (see Deinition 3.1). Then it is not diicult to see that the regular neighborhood o Σ(W ) can be topologically immersed into R 2 so that the structure o arm and stem is respected. Let q : M W be a pseudo quotient map. I there exists a topological immersion η : W R 2 which respects the structure o arm and stem o W with respect to q, then we can ind another topological immersion η : W R 2 which is an approximation o η and which is a generic smooth immersion with respect to q : M W. The purpose o this section is to prove the ollowing. Proposition 5.2. Let W be a compact pseudo quotient space such that H (W ; Z 2 ) = H ({point}; Z 2 ). We suppose that a structure o arm and stem which satisies the consistency conditions at non-simple points is given to W. Furthermore, suppose that W contains no cuspidal branch (Figure 1 (c 1)) and no trident (Figure 1 (d 5)). Then there exists a topological immersion η : W R 2 which respects the structure o arm and stem. By using Proposition 5.2, we can prove Lemma 4.12 as ollows. Proo o Lemma Firstly note that a structure o arm and stem it is already given to W, since we have a pseudo quotient map q : M W. Suppose that W License or copyright restrictions may apply to redistribution; see

21 SIMPLIFYING STABLE MAPPINGS INTO THE PLANE 2627 (d 1 ) (d 4 ) Figure 5 contains no boundary point. Then W contains no cuspidal branch. By Proposition 5.2, there exists a topological immersion η : W R 2. Since W contains no boundary point, we easily see that η is an open map. Thus η(w ) is a nonempty open compact set, which is a contradiction. Proo o Proposition 5.2. First, or each point in W which has a conic neighborhood as in Figure 1 (d 1) or (d 4), replace the conic neighborhood with the space as in Figure 5 (d 1 )or(d 4 ) respectively. Denote by W the resulting polyhedron. Then W is a pseudo quotient space and inherits a structure o arm and stem induced rom that o the original space W, since the original structure satisies the consistency conditions at non-simple points. Note that W is homotopy equivalent to W and that, i W is immersed into R 2 respecting the structure o arm and stem, then so is W. Thus we may assume that each point o W has a conic neighborhood as in Figure 1 (a) or (b). Since it has a structure o arm and stem, W is a branched surace in the terminology o [Sa2]. Then we can represent W by its associated graph G W (or details, see [Sa2, 3]). In the ollowing, we use the terminology and notations introduced in [Sa2]. The ollowing lemma is easy to prove by using the Mayer-Vietoris exact sequence o homology groups in Z 2 -coeicients. Lemma 5.3. I H 1 (W ; Z 2 )=0, then the associated graph G W is a tree. Now we go back to the proo o Proposition 5.2. By the above lemma together with the assumption H 1 (W ; Z 2 )=0,weseethatW is orientable. Then we see that, i G W has no double edge, W can be immersed into R 2 by [Sa2, Corollary 3.9]. (In [Sa2], H 2 (W ; Z) = 0 is assumed. However, the same argument works also under the assumption H 2 (W ; Z 2 ) = 0.) We will prove the proposition by using the same argument as in the proo o Corollary 3.9 o [Sa2]. When the number n o vertices o G W is greater than or equal to 3, we have to consider the orms as in Figure 6 (d), (e), () and (g) o G W other than the three orms (a), (b) and (c) o [Sa2, p.90]. Suppose that G W is o the orm (d). Let R be the component o W\Σ(W ) corresponding to the white vertex, where Σ(W ) is as deined in Deinition 3.1 or in [Sa2]. Then we see that there exists a non-zero element γ H 1 (R; Z 2 )which cannot vanish in H 1 (W ; Z 2 ). This is a contradiction. Thus the case (d) does not occur in our case. License or copyright restrictions may apply to redistribution; see