A DECOMPOSITION INTO HOMEOMORPHIC HANDLEBODIES WITH NATURALLY EQUIVALENT INVOLUTIONS

Downing [6] extended the well-known result that any closed 3-manifold X contains a handlebody H such that el(X-H) is homeomorphic to H in the case where X is a compact 3-manifold with nonvoid boundary. We show that if X is a compact 3-manifold with involution h having 2-dimensional fixed point set, then X contains an h-invariant handlebody H such that the involutions i,,duced on H and el(X-If) are naturally

Let X be a closed 3-manifold with triangulation T. If S is the l-skeleton of T and H is a regular neighborhood of S, then it is a classical result, see Seifert and Threlfa]l [5,p.219], that H and H 2 el(X-H) are homeomorphic handlebodies.Downing [6] extended this result, showlug that every 3-manifold with nonvoid boundary also contains a handlebody which is homeo,orph[c to its complement.
It is the purpose of this paper to show that these res,Its can urther extended so that certain symmetries are respected.
These snmetries ale those realized as involutions of X with orientable 2-dimensional fixed po.nt sets.
Recall that an involution of X ;s a h,meomorphsm on X of period two.h i / Xi, i l, are equivalent, denoted h h 2 if there exists a Involdtlons :X ix ,Y[h(x) x}.We shall prove the following: THEOREM I. Let X be an orlentable 3-manlfold and h:X X an involution with 2- dimensional, orlentable fixed point set F. Then there exist complementary, h- invarlant handlebodies H and H 2 in X such that hlH hlH 2.
THEOREM 2. Let f:H / be the homeomorphlsm such that hlH f-I o (hiM 2) o f thereby establlshlng the equivalence hlH h H 2 of Theorem I. Then flHl may be assumed to be the identity on LEMMA I. Let X be a copact 3-manlfold and h:X X an Involutlon wlth 2- dimensional, orlentable fixed point set F.
1en there exlst complementary, h- invariant handlebodies H! and H 2 in K such that H OF H 2 F.
The 3-manlfold X may be either orlentable or nonorlentable in this lemma.X mast be assned to be orlentable in Theorem in order to invoke Nelson [4; ThE.C].
Section 2 contains the proof of thls lemma in the case when X is a closed manifold.
The proof is modified in section 3 to cover the case when X has nonvold boundary.The proof of Theorem 2 is provided In section 4. Both sections 3 and 4 are modifications of proofs which appear elsewhere.These sections refer heavily to the proofs which they modify.

)[ IS CLOSED.
Our approach is to pick an initial handlebody H that will be repeatedly modified until H Fi= H 2 F I for each component F.I of F. Any such surface admits an involution gl with a separating fixed point set + consisting of at most two simple closed curves.This separates Fiinto F I and F I + where gl (FI) Fi Let X be the orbit space of h.Since F is 2-dimenslonal, is a bordered 3- manifold with B F. Triangulate B so that the triangulation of component F.z is gl-invarlant.Since BX is collared, Rouke and Sanderson [I" Cot. 2.26], the triangulation of X can be extended to a trfangulatlon T* of X Let T be the llft + of T* to X.
Then T is an h-lnvarlant triangulation of palr (X,F) such that F i is triangulated identically to F i for each i.

Denote by T (n) the nth derived subdivision of T and by S (n) the collection of all
edges and vertices of T (n) lying on the l-skeleton S of T. Let HI= N(S (2), T(2)) and H2--(X-HI).H and H 2 are homeomorphlc orlentable handlebodles.Since T triangulates F, H 10 F i is the connected, orientable surface F i minus k disks, where k is the number of 2-simplices in the triangulation of F i H 20 F i consists of k disks.
Modification Procedures: Two procedures are used to repeatedly modify T near each F i The result of modification by either procedure is that the modified handlebody I is isomorphic to I plus a 3-handle of index one and the new complementary handlebody is isomorphic to H 2 plus a 3-handle of index one.Hence, after each modification we are again left with homeomorphl c, complementary handlebodies.
The 3-simpllces of T are in h-invarlant pairs.We say such a pair is near F i if the two 3-simplices of the pair intersect in F i The procedures below correspond to the cases where 3-simplices intersect in an edge or 2-face in F i We will not need to employ modifications exploiting s implices intersecting F i in a vertex.
Procedure =: The net result of this procedure is to remove one edge from TO F i Intuitlvely, H 0 F.I contains "less" of F i than H O F i Let e be any edge of T lying in F i Then e is the common edge of two 3- slmpllces, s and s 2 such that h(s I) s 2 and s 0 s2--{e} Let e and e 2 be the edges of T (I) connecting the vertices of e with the barycenter of s Modify S (2) to get (2) by deleting from S (2) all vertices and edges of T (2) lying on e and adding to S(2)all vertices and edges of T(2)lying on (elDe2) h(elU e2).We replace H by l--NI(2) T( 2)) I is homeomorph[c to H plus a 3-handle of index one H2--cI(X-H I) is homeomorphic to plus a 3-handle of index one, the cocore of which lies in a disk bounded by (clUe2)U h(elUe2).
Procedure B: The net result of this procedure is to add an h-lnvarlant edge to T which intersects F in a point.
As a result, HI0 F i contains a disk component not in H 0 F i Let ala2a 3 be any 2-simplex of T in F. Then ala2a3 is the common face of 3-slmplices ala2a3a4 and ala2a3a5 such that h(a 4) a 5 The union of these slmpllces is a double tetrahedron we shall retriangulate.The new triangulation consists of the slmpllces ala2a4a 5 a la3a4a 5 a2a3a4a 5 and their faces.It is easily checked that this new triangulation is h-invariant.
Let T be the triangulation of X obtained by replacing the former simplices of the double tetrahedron by the new ones.

Substitute
for Hl, I--IN((2)' ( 2))I" I is homeomorphlc to H plus a 3-handle whose core is the edge a4a5. 2 clIX-H2) is also homeomorphlc to H plus a 3-handle of index one.
Employing the Modifications: The and 8modifications are repeated independently near each F i Let f be the number of 2-simpllces and p the number of vertices in the triangulation of F (and also of F).We assume that the original T was chosen so that p < 2f for each fixed point component F i CASE I.
p > f.First apply p-f time,; where ala2a3eF In order to carry this out we ralst have that p-f -< f.But this is equivalent to the statement t1at (F i 3f e which is immediate when (F x(F i) 0 and easily checked in the remaining case when F.= S 2. Next, apply, procedure a to every edge in F Then i both II lfi F and H2 F consist of p disk components and one component homeomorphic + to F (or F minus p disks.+ CASE 2. f > p.First apply procedure B f-p times to sl.nplices in F.
Then use + procedure a to remove all edges of T from F.
The res,lt is that both H IO F and + H20 F consist of disk components and one component ho,.eomorphic to F (or F less f disks. 3. X IS NONVOID.
In this section we make the adjustments necessary in the above proof for the case X * .Notice that only tle initial choice of H was depeudent upon X having empty boundary.The modification procedures do not depend upon assumptions about X.
First, it was noted that F. is a ,nlosed, orientable surface.If K * , this is no longer necessarily true.
floweret, F will be a closed or[entable surface less a collection of disks.
Such a surface also admits an involution gi with a separating fixed point set consisting o simple closed curves and arcs.
No longer is it true that a reg,lar neighborhood of the l-skeleton of triangulation of X will have a homeo,norphic complement in X.Rather, we must show that Downing's construction of the handlebody H can be carried out equivariantly.
Let B., i I, n be the boundary components of X and let X'= X Uk(UMi) where M. is the handlebody with M B. and klB mapping B homeomorphically to 8M i i X' is a closed 3-manifold and accordingly has a classical decomposition into homeomorphic handlebodies H and q cl(X H{).cl((X' -UNi) HI). i The isotopy G of X' moving each Mi to N yields a homeomorphism f:X' X' such that f(N.) M. for all i. f(H;) and f(H)_ are homeomorphic handlebodies such that f(H 2) cl(x-f(Hl)).We do not yet know, how.ever, that f(H I) and f(H 2) are h- invariant.
H and H may, however, be assumed h-invar[ant as in section 2. But if (i) h can be extended to an involt[on ' :K' X' and (ii) if the isotopy G can be constructed to commute with h', then f(H1) and f(H 2) will be h-invariant as needed.
To extend h to h' we need only know that any involution (free or with l- dimensional fixed point set) on M. is equivalent by kl--klBi to the restriction to Bi of an involution on Mi.Then attaching M[ to X by kl allows h to be extended into M

Let M
A I where A is a 2-spher minus disks.If h is a free involution i H[ H Io H 2 H 2 By Nelson [4; ThE.C] the involutions hlH and hlH 2 are equivalent If and only If (i) their fixed point sets are homeomorphic and (If) elther both fixed point sets separate or both fall to separate.
By Downing[6;Lemma I], each Mi is isotoped by G in X' onto Ni where Nij= Ni H, j 1,2, are specially positioned in H and H 2. (M. and N are respectively, just regular neighborhoods of the l-dimensional sets denoted Y and Y in Downlng's proof.)