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Heegaard Splittings and Splitting Homomorphisms HEEGAARD SPLITTINGS AND SPLITTING HOMOMORPHISMS BY WILLIAM JACOO 1. Introduction. Let M be a closed, orientable three-manifold and suppose (U, V) is a Heegaard splitting of M. Denote the common boundary of U and V by S. Let u: (S, s0) ->({/, s0) and v: (S, s0) -> (V, s0) be the inclusion maps. In [9] J. Stallings introduced a natural homomorphism to investigate; namely, the homomorphism («*, K*) : ""I(ft S0) -> TTl(U, S0) X TTy(V, s0) where (w*, u*)(/) = («*(/), f *(/)). The homomorphism (w*, v*) is called the splitting homomorphism of ^(ft s0) induced by (U, V). In studying the results and questions presented by Stallings in [9], a rather obvious question comes to mind. With the three-manifold M and the Heegaard splitting (U, V) of M is related the algebraic configuration called the splitting homomorphism induced by (U, V), is it possible that all such algebraic configura- tions occur in precisely this way? That is, suppose S is a closed, orientable surface of genus «>0. Let ft and ft denote free groups of rank n and suppose r¡1, r¡2 are homomorphisms of tt(S, s0) onto ft and ft, respectively. If (vi,V2)-^i(S,s0)^F1xF2 is the homomorphism defined as (-n^ r¡2)(l) = (-n1(l), r¡2(l)), then is there a closed, orientable three-manifold M and a Heegaard splitting (U, V) of M so that the splitting homomorphism induced by (U, V) is (in some sense) (-qlt 7)2)1In Theorem 5.2 this question is answered in the affirmative. The homomorphism (r¡u r¡2), as described above, is called a splitting homo- morphism. In Theorem 5.3, a one-one correspondence is established between equivalence classes of Heegaard splittings and equivalence classes of splitting homomorphisms. In §6, it is shown that two conjectures made by J. Stallings in [9] are equivalent to Poincaré's Conjecture that a simply-connected, closed three-manifold is topo- logically equivalent to the three-sphere. These conjectures appear in §6 as Con- jecture B and Conjecture D (using the notation of [9]). It is interesting that Conjecture D can be stated using only the language of algebra. Received by the editors October 14, 1968. O This paper is part of the author's doctoral dissertation written at the University of Wisconsin under the supervision of Professor D. R. McMillan, Jr. The author was supported by the Wisconsin Graduate Research Committee-WARF Funds. 365 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 366 WILLIAM JACO [October The proofs of Theorem 5.2 and Theorem 5.3 are dependent on the study of mappings of closed surfaces into wedges given in §3 and §4. Using techniques of surgery on PL maps, it is shown that certain mappings of closed surfaces into wedges have mapping cylinders which are handlebodies. Perhaps of independent interest is Corollary 3.3 of §3. It states that there is a homomorphism of the fundamental group of a closed surface with Euler charac- teristic 2—p onto a free group of rank k if and only if k ^ [p/2] where [p/2] is the greatest integer less than or equal to p/2. The basic definitions appear in §2. Some other definitions appear in the particular sections in which they are used. 2. Basic definitions and notation. The term map is used to mean continuous function. Let /denote the closed unit interval [0, 1]. Suppose/: S -+ X is a map. Consider the disjoint union oí Sx I and X and introduce the equivalence relation jx 1 ~/(s) for each se S. The identification space [2] of this relation is called the mapping cylinder of fand is denoted C(f). The notation of simplex, complex, simplicial map, etc. may be found in [2, Chapter 1]. If K is a complex, then the Euler characteristic of K is denoted x(K)- A map of the underlying point set of a complex K into the underlying point set of a complex L is called apiecewise linear (PL) map if and only if there is a subdivision K' of K and a subdivision L' of L so that the map is simplicial from K' to L'. If X is a space and x0 is a point of X, then for k ^ 1, irk(X, x0) denotes the kth homotopy group of X based at x0 [1]. The group ttx(X, x0) is also called the funda- mental group of X based at x0. If X is arcwise connected, then the kth homotopy group is sometimes simply written Trk(X). Suppose/is a map from (S, s0) to (X, x0), then the homomorphism of ^(S, sQ) into ttx(X, x0) induced by/is denoted /*. An «-manifold is a connected metric space in which each point has a neighbor- hood homeomorphic to the closed unit ball in En. If M is an «-manifold, then the interior of M (Int M) is the set of points of M which have an open neighborhood homeomorphic to the interior of the unit ball in En. The boundary of M (Bd M) is defined to be M—Int M. If M is a compact «-manifold and Bd M= 0, then M is said to be a closed n-manifold. The term surface is synonomous with 2-manifold. A three-manifold is called a handlebody of genus n, n ¡g 0, if it is a regular neighbor- hood of a finite, connected graph with Euler characteristic 1 —«. In the case that the handlebody is orientable, it is often referred to as a cube-with-handles of genus n. The term regular neighborhood is used in the sense of [14, Chapter 3]. Sometimes it is convenient to denote the point set interior of a subset Y of a space X by Y°, where there can be no confusion that this refers to the topology of X. An n-manifold M is said to be simply connected if n±(M) is the trivial group. For definitions of free group, direct product of groups, and free product of groups, the reader is referred to [4]. 3. Mapping surfaces into wedges. Let T denote a wedge at t0 of k simple closed License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1969] HEEGAARD SPLITTINGS AND SPLITTING HOMOMORPHISMS 367 curves Tly..., Tk. In this section a study of mappings of closed surfaces into Tis considered. Let x denote a point of T distinct from t0. Then there is an embedding <f>of xx [-1, 1] into T so that </>(x,0) = x. Define U(x)=<p(xx [-1, 1]). Then U+(x)=</>(xx [0, 1]) is called the positive side of x in U(x) while t/-(x) = #cx[-l,0]) is called the negative side of x in ¡7(x). A PL map/of a closed surface »Sinto T is said to be transverse with respect to x if it is true that (1) Each component of/_1(x) is a polyhedral simple closed curve in S and (2) there is a neighborhood U(x) of x in T, as above, so that for each component J of/_1(x), the component U(J) of/" 1(C/'(x))containing / is a product neighbor- hood of J in S where the parametrization of U(J) as a product is that induced by <£; i.e. the parametrization of U(J) as a product may be realized by an embedding ip of / x [ -1, 1] into S with i/i(s, 0)=s for every s e J and with / mapping each arc >p(sx [-1,1 ]) homeomorphically onto <f>(f(s)x [— 1,1]) where/° >p(s,r)=<f>(f(s),r), Suppose {Xi,..., x,,} is a disjoint collection of points of T with x4 e T—{r0}, 1 Si^p. The map/is said to be transverse with respect to {xx,..., xp} if/is trans- verse with respect to each x¡. In what follows the collection of points {x1(..., xp} will always be chosen so that no two members belong to the same simple closed curve Tt of the wedge T. Notice that if/ is transverse with respect to {xl5..., xp}, then the product structure for each component of/_1(i/(x()) is defined so that/ maps the positive side, U+(J) (negative side, U~(J)), oí J to the positive side (negative side) of x¡ in t/(x¡). Suppose / is a PL map of (S, s0) into (T, t0) and for some collection of points {xx,..., Xp} in T—{t0} the map / is transverse with respect to {xj,..., x„}. Each component of/" J(x¡) is a two-sided polyhedral simple closed curve in S. Suppose J0 and Jx are distinct components of/_1(xJ) for a fixed j. Suppose that A is a polyhedral arc in S-ÍSo} from /„ to Jx such that A n (Jf=1f~1(xt) = Bd A. Let Qo^Jo and Qx^Jx be small arcs each containing an endpoint of A in its interior. Suppose QcS—{s0} is a regular neighborhood of A so that A is a spanning arc of Q, and Q meets (J?=i /-1(*0 only in ôo u ôicBd g. Then the closure of Bd Q —(Qo u {?!) consists of two disjoint arcs /10 and Ax from /0 to Jx. Lemma 3.1. Iff maps the arc A into the trivial loop of T based at /(Bd A), then there is a PL map g of S into T so that (i) g is homotopic /o/(Rel {s0}), (iï)g-1(xi)=f-1(xi),iïj, (iii) g'Kx^f'^x,) u (A0u ^)-(ôo u Oí), a«¿ (iv) g is transverse with respect to {xx, ■■ ■, xp}.
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