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(1)Branched Coverings of PL Involuations on 3-Manifolds By. Seiya NEGAMI. '. 1. Introduction. In three-manifold topology, one can find several papers where useful methods to study 3-manifolds are generalized and applied to PL involutions (self-homeomorphisms of period 2) on 3-manifolds. For example, Nagase [9] introduced characteristic Heegaard sptitting and gave a generalization of Reide-. meister-Singer theorem which connects the equivalence of PL involutions to the stable equivalence of Heegaard splittings. Kim and Tollefson [7] showed the existence of what should be called "prime decompositions" of PL involutions. In this paper, we shall deal with branched coverings of PL involutions and prove an analogy of the fact that any orientable closed 3-manifold covers a 3-sphere branched over a link [1, 4, 5, 8].. Let M'i and N'Z be two n-manifolds and let T and o be involutions on them respectively. The involution T is called a branched covering of o if there is a. branched covering map q:M'i -> IVn such that q･T = a･q. We shall use naturally several terms for branched coverings of manifolds as those for PL involutions. The following is our main result:. Theorem 1. Let T be a PL involution on an orientable closed 3-mandeld. if T admits no 1-sided sudece as a component of its fixed point set, then T is a 'i'. branched covering of a PL involution of the 3-sphere. lf the fixed point set of T contains a 1-sided sui:f}2ce, then T is a branched covering of a PL involution on the real projective 3-space.. .. Information about the downstairs involutions and the degree of the branched coverings will be obtained in detail in the proof of the theorem. The upstairs fixed point set is necessarily contained in the lift of the downstairs one.. It will be noticed that any branched covering of a PL involution on a 3-sphere has no 1-sided surface in its fixed point set.. We work in the PL category. Our manifolds are all compact and connected. In particular, we call a 2-manifold "a surface". The interior, closure and boundary of (*) will be denoted by Int(*), cl(*) and 0(*), respectively. The symbols S'i, P'i B3, D2 and I stand for the n-sphere, the real projective nspace, the 3-ball, the 2-disk and the unit interval, respectively. Department of Mathematics, Faculty of Education, Yokohama National University. L,, -.. K･ S. 'f '.

(2) S. NEGAMI. 2. 2. Lemmas In this section, we shall prepare three lemmas to reduce the problem of 3-manifolds to that of surfaces. The first two describe the structure of an involution on an orientable closed 3-manifold in terms of surfaces embedded in. the 3-manifold. The third one is a useful extension theorem･ for branched coverlngs.. Let T be a (PL) involution on an orientable 3-manifold M3 and let fixT denote the fixed point set of T. Then fixT is a disjoint union of proper. "t. L. submanifolds and finitely many isolated points in M3 and if T is orientation-. ,". preserving (or reversing) then each component of fixT (if non-empty) has odd (or even) dimension. We define fix(i)T as the union of i-dimensional components of fix T. An involution T is said to interchange two subsets A and B in M3 if T(A) =: B. We call an involution T a refZexion of a 3-manifold M3 if M3 splits. into two submanifolds Mi and M2 (M3 = Mi U M2 and Mi fi M2 = 0Mi = OM2) and if Tinterchanges Mi and M2 leaving each point on 0Mi fixed (fixT. = 0Mi)･ Lemma 1. Let T be an involution on an orientable closed 3-manifold M3. if T admits no 1-sided sudece as a component of fix(2)T, then either T is a rdiexion of M3 or there is a 2-sided sudece 4 in M3 such that fix(2)T U F. splits M3 into two. components which T interchanges. Proof Nagase [9] has already proved that for an orientation-preserving involution T on an orientable closed 3-manifold M3, there is a Heegaard splitting. (M3; U, V) with handlebodies U and V such that T interchanges U and V. (Such a Heegaard splitting is said to be characteristic for T.) In the case fix(2)T =. ip, we can show the same conclusion, following his proof without essential. modification, and take the vinvariant Heegaard surface 0U (= aV) as the desired surface F,,. Assume that fix(2)T consists of 2-sided surfaces Fi,･･･, F), (n ) 1). It is clear. ;･ ?. that if M3 - Fi U ･･･ U E, is disconnected then T is a reflexion of M3. Suppose. that the 3-manifold M' obtained from M3 by cutting open along Fi U ･･･ U E, is connected. Then M' admits an involution T' which interchanges each pair of copies ny and ny' of L (i = 1,･･･, n). Cap off each boundary component IZX (or ny') of M' with a handledbody Ui' (or q') so that the resulting closed 3manifold M. has an involution T. with T. IM' = T'. Then fix(2)T. = ￠ and hence there is a splitting surface F. in M. for T... The surface F. separates M. into two submanifolds M+ and M-. If F. meets Ui and hence Ui･' also, then replace F. n Ui' and F. n Ul' with aUi n M. and aor' n M- respectively, and push them slightly into Int M'. If the surface F. is transformed into a disconnected one, then join its components with pairs of equivariant pipes. This equivariant modification does not change the Z2homology class of F.. So F. can be chosen not to meet all of-Ui' and q' (i = 1,. /･. ).

(3) Branched coverings of PL involutions. ･･･. , n) and naturally embedded into M3. The embedding of F. is the desired. one for T. []. Lemma 2. Let T be an involution on an orientabte closed 3-manijbld M3 and Ei, ･･･. , Ee(e }i 1) the 1-sided components of fix(2)T. 71hen there is a 1-sided sudece. E. in M3 disy'oint from fix(2)T - Ei U ･･･ U Ee such that M3 - fix(2)T U E. consists oftwo components interchangable by T. 71hesudece E* can be chosen to intersect each non-orientable sui:face Ei (i = 1,･･･, V) in one circle.. Proof We shall deform M3 in a way similar to that of the previous proof, in order to obtain a closed 3-manifold M. admitting an involution c, whose fixed point set contains no 1-sided surface. First cut open M3 along Ei U ･･･ U Ee and let M' denote the resulting 3-manifold. Then Tinduces naturally an involu-. tion T' on M' which leaves each boundary component of M' invariant. The component Ei of eM' corresponding to Ei i's the orientable double covering space of Ei (i = 1,･･･,e) and the restriction t = T'IEi' is its covering transformation (Ei/t = Ei). Next cap off each Ei with a handlebody U} so that T' extends to an involution Tk on the closed 3-manifold M. = M' U Ui U ･･･ U Ue. It will not be however a trivial way, since 0Ui is T'-invariant. As' a model for an orientation-reversing involution on a closed surface of. even genus which extends to that of handledbody, we have the antipodal map on a 3-ball with several pairs of equivariant handles attached. This involution. has only one fixed point in the center of the 3-ball. As a model for a closed surface of odd genus, there is a free involution o on D2 × Si with equivariant handles for which D2 × Silois a solid Klein bottle. By the classification of free. involutions on surfaces [2], any two free orientation-reversing involutions on a. closed surface are equivalent (or conjugate). Therefore we can meet our requirement of attaching of Ui, using those models. Clearly fix(2)T. contains no 1-sided surface, so there is the surface F. in M. 1'.;. for Tl described in Lemma 1. We may assume that F. n 0Ui consists of pairwise disjoint circles whose union is q-invariant. We shall decrease the number of components of F. A aUi as well as possible.. p. Suppose that there is a component C of F. n 0U} such that q(C) n C =: ￠. If there is another circle C' of F. n 0U}, we can assume that there is an arc or in. 0U} such that or n F. n 0U} = or n Cn C' = 0or and q(or) n or = ip. Attach a pair of equivariant pipes along or and q (or) to F. , then the intersection of the. new surface with aUl consists of fewer circles than that of F.. Now suppose that each component of F. n 0Ul is t-invariant. If there are two circles in F. n 0Ui,. then a new surface whose intersection with 0U} contains two disjoint circles C and t(C) will be obtained by joining those circles with equivariant pipes. So this case can be reduced to the above. These modifications transform F into a * surface which meets 0U} in either one Q-invariant circle or two disjoint circles. interchangable by q.. 3.

(4) S. NEGAMI. 4. Let E. be the surface (F. n M')1(T'IOM') in M3 (= M'1(T'lOM')). Then E. is the desired one and it can be assumed to meet each Ei in a single circle by. the above observation. D. An m-fold branched covering q:M'i -> N'i is said to be simple if q-i(x) consists of at least m-1 points for each x E N't. .fl.. Lemma 3. Let W3 be an orientable 3-manijbld with bounda7:y components Fi,･･･, E, (n }i 1) and qi:I}. aB3 mi-fold simple branched coverings (i = 1,･･･, n) with pairwise disy'oint branched sets in OB3. if m == mi + ･･･ + m. }i: 3, then. there is an mvCbld simple branched covering q:W3 -> B3 branched over a disy'oint union ofproper arcs in B3 such that qll7> == qi (i -- 1,･･･, n).. Proof Berstein and Edmonds [3] have proved that if an orientable 3-manifold. W3 has a connected boundary then any branched covering qi:0W3 . 0B3 of degree at least 3 extends to a branched covering q:W3 -) B3 (Corollary 6.3) and it can be observed that if qi is simple then q is also a simple one branched. over proper arcs in B3. We shall prove the lemma in the case that OW3 is disconnected, assuming their result. Let D2 be a small 2-disk in 0B3 disjoint from the branched sets of qi (i = 1, , n) and DZ･･(1' = 1,･･･, mi) the components of qi i (D2). Let A3 be a 3-ball ･･･. in B3 which meets 0B3 in D2 and AZ･ ((i, J') : (1,1), (2,1)) pairwise disjoint 3-balls in W3 such that A7)･ n 0W3 == bj. Join Fi to F2 with a handle D2 × Iin. W3 so that D2 × 0I = Dai u D22i and D2 × In AZ･ = ip. We extend qi:Dft･j .. D2 to a homeomorphism q:AZ･･ -> A3 and qi U q2:D2 × al -) D2 to a standard 2-fold branched covering map,q:D2 × I -> A3 branched over a single. arc. Denote cl(W3 - D2 × I U E, A?,) and cl(B3 - A3) by Wi and B3i respectively. Then Wi has n - 1 boundary components, on one of which an (mi + m2)-fold simple branced covering has been defined and the others admit mi-fold ones (i = 3,･･･, n).. After this process repeated n-1 times, the submanifold W),-i in VV3 with. -t. :t.. connected boundary, the 3-ball B?,-i in B3 and an (mi + ･･･ + m.)-fold simple branched covering q:cl(W3 - W.-i) -> cl(B3 -B?,-i) will be obtained.. By Berstein and Edmonds' extension theorem,'the desired branched covering q:w3 -> B3 can be defined.O. 3. Proofofthetheorem It has been known that there are precisely four equivalence classes of involutions on the 3-sphere S3 [10]. First of all, we shall describe the standard. '. involutions pi (i = -1,O,1,2) to state our result in more detail. Let S3 be the. unit 3-sphere {(x,y,z,t) E R4:x2 + y2 + z2 + t2 = 1}, then they are defined as follows:. i i. I l. L. l. I. I. f. I.

(5) Branched coverings of PL involutions 5 P-i: (X,Y,Z, t) . (-X, -Y, -Z, 't). po :(x,y,z,t) -> (-x,-y,z,-t) Pi :(X,Y,Z,t) . (-X,Y,Z,'t) p, :(x,y,z,t) --> (x,y,z,-D. Each pi interchanges the upper an.d lower halves B3. of S3, where. B3. = {(x,y,z,t) E S3:t }i! O}. and B3- = {(x,y,z,t) E S3:t si 0}. The orientation-preserving involution p-i acts freely on S3 and the orbit space. is P3. The involution po is orientation-reversing and leaves the two points (O,O,±1,O) fixed. The circle {(O,y,z,O) E S3:y2 + z2 = 1} is fixed by the orientation-preserving involution pi. The involution p2 is a reflextion of S3 with fixed point set 0B3. (= oB3-). The three involutions pi (i = O,1,2) induces involutions Q, respectively, on the projective 3-space P3 identified with S31p-i, since p-i･pi = pi･p-i. The. two involutions Tb and T2 are equivalent and fixT2 consists of the 1-sided projective plane 0B3+lp-i and the single point [(O,O,O,1)] in P3. The involution Ti leaves a two-component link fixed. The projective 3-space P3 admits precisely. three involutions including another free one, up to equivalence [6]. Now we shall prove Theorem 1. Hereafter T: M3 -> M3 is a given involution on an orientable closed 3-manifold M3. 0ur proof is based on the uniqueness of involutions on surfaces [2]. We shall first define a branched covering map. from the union of fixT and the surface F. or E. in Lemma 1 or 2, and next extend it to a branched covering from the whole of M3 by Lemma 3. We have the following seven considerable cases. Theorem 1-1. if fixT consists of 2n isolated points (n }ir 1), then T is an m-fold r". branched covering of po branced over a po-invariant link L in S3 with L n fix po = ip, for any integer m ;) max {n,3}. Theorem 1-2. lf fixT consists of n circles (n ;}ir 1), then T is an m-fbld branched. -.t. covering of pi branched over a pi-invariant link L in S3 with L A fixpi = ip, for any integer m ;}ir max {n,3}. Proof of 71heorem 1-1 and 2. In these cases, there is a vinvariant surface F. in. M3 which splits M3 into two interchangable components M+ and M-. By the uniqueness of involutions on surfaces, we have a model in R3 for the restriction. T. = TIF. iHustrated in Fig. 1 (n > 1) and Fig. 2. (n = 1). The rotation around the z-axis realizes a in the case of Theorem 1-1, and the reflexion in the yzplane does in the case of Theorem 1-2. Fig. 1 and Fig. 2 show how to cover a 2sphere S2 with F. so that T. can be obtained as a lift of the rotation or the. reflexion on S2 by the branched covering q.:F. -) S2. Identify S2 with aB3..

(6) i 1. h. S. NEGAMI. 6. 1. F. z. m.. z. :. a!L. n+1. : t. i. ,. m. :. it. :. {. ::. n. ,. .. :. .. I. N. .. ,. :. J. .. d. :. l l. ,. .. 3. d ' t 1. l. :. 2. 2. : i:. i 1. 1. : :. ::. F.. :,,. F* x. x. y. 1,,. y. iq*. z. z T'.. S2. :;. S2. l. ' , 1. 1 1. 1 ,. x. x. y. Fig. 1. (n > 1). y Fig. 2. (n = 1). (= 0B3-) in the above-mentioned 3-sphere S3, then the rotation and the refiexion on S2 can be regarded as the restrictions of po and pi to aB3+, respectively. The branched covering q. can be chosen to be simple and to have an arbitrary degree more than 2. So q. extends to a branched covering q+:M+ -> B3+ branched over a disjoint union A+ of proper arcs in B3+. We define the desired branched covering q:M3 -> S3 by qlM+ = q+ and qlM- = pi･q+･T (i = O or 1). Then q will be branched over the pi-invariant link A+ U pi (A+) in S3 which is disjoint from fixpi.O. Remark. It is impossible to decrease the degree of the branched covering map q less than n, because each fixed point of pi must be covered with at least n fixed points of T. Theorem 1-3. 1[f T is orientation-reversing and f>'ee, then T is a 2m-fold branched. covering of po branched over a po-invariant link L in S3 with L n fixpo = ip, for any integer m ;ir 2.. i･'E. L.. i.

(7) Branched coverings of PL involutions 7 Theorem 1-4. if T is orientation-preserving and free, then T is a 2m-fold branched covering of pi branched over a pi-invariant link L in S3 with L n fixpi = ￠, for any integer m }ir 2.. Proof of 7-7zeorem 1-3 and 4. Since fix(2)T = ip, there is a decomposition. M3 = M. u M- with a splitting surface F. (= M+ n M-) such that T(M+) = M-. Notice that Tpreserves the orientation of M3 if and only if its restriction T.. to F. reverses that of F., and that a closed surface admitting an orientation-. preserving free involution has odd genus. In Fig.3, T. is realized as the rotation around the y-axis or the antipodal map with central point (O,O,O) in R3, according to whether T. is orientation-preserving or reversing. (Neglect the. middle hole of the surface in Fig. 3 when the genus of F. is even.) Define a simple branched covering q.:F. --> 0B3+ as indicated in Fig. 3, then Tlt induces. an involution on 0B3+ which is equivalent to polOB3+ or pilOB3+. Extend q. equivariantly, then the desired branched covering of po or pi will be obtained.o. z. 2m-1. :.1. x. 2Y 1.2m. Y. F.. , iq". T-g. T` S. aB3.. 4. . ,. 1. x y. Fig. 3. Theerem 1-5. if T is a r(zf7exion of M3 with n 2-sided sui:faces as ils fixed point set. (n }}i 1), then T is an m-fold branched covering of p2 branched over a p2-invariant link in S3, for any integer m }i max{2n,3}..

(8) ' :. S. NEGAMI. 8. i. Theorem 1-6. 1[f T is not a rEfiexion of M3 and if fix(2)T consists of n 2-sided. sudeces (n }i: 1), then T is an m-fold branched covering of p2 branched over a p2-invariant tink in S3, for any integer m >- 2n + 4.. Proof of Theorem 7-5 and 6. Let Fi,･･･, E, be the components of fix(2)T, then. Fi U ･･･ U E, (U F.) splits M3 into two submanifolds M+ and M- for which T(M+) = M-. The lower involution on S3 must be p2 since fix(2)T t ip. Choose any mi-fold simple branched coverings qi:4 . aB3+ (i = 1, ･･･, n) (so that mi + ･･･ + m. ;)r 3 if T is a reflexion) and define a 4-fold simple branched covering q.:F. -> aB 3+ as the composition F. . F.IT -> aB 3+, where the first. 1,. Kt. map is the canonical projection and the second one is a standard 2-fold branched covering. Then qi U ･･･ U q. (U q.) extends to the desired branched. covermg.D Finally we shall deal with an involution Twhose fixed point set contains 1sided surfaces. Let Ei,･･･, Ee (V ;}i 1) be the 1-sided surfaces in fix(2)T. We. call Ei an odd (or even) surface if the Euler characteristic of Ei is odd (or even), and denote the number of the odd surfaces by d. Let e be the number counting each odd surface Ei by 1 and an even one by 2, that is, e = 2e - d. Assume that Tadmits p isolated fixed points (p }}ir O) and n 2-sided surfaces Fi, '''. , Ei (n ) O) in fix(2)T. We define and integer mo by. f5+4n (if(p,e)-(1,1)) Mo ={4 (if (p,e, d, n) = (O,2,O,O,)) Lmax{p,e}+2d+4n (otherwise). Theorem 1-7. if fix(2) T contains a 1-sided sumbce, then T is an (mo + 2k)-fold branched covering of T2 branched over a T2-invariant link in P3, for any nonnegative integer k.. Proof Let E. be the surface for T in Lemma 2. The lemma states that the. ,;A. intersection of each 1-sided surface Ei with E. can be assumed to be a single circle. We however modify E. by a surgery with pairs of equivariant pipes so that afterwards E. fi Ei is a union of two circles for each even surface Ei. Then. the number of components of E. n (Ei U ･･･ U Eg) is equal to e.. Let M' be the 3-manifold obtained from M3 by cutting open along Ei U ･･･ U Ee. Each boundary compent Ei' of M' admits an orientation-reversing free involution oi such that Ei'lui = Ei and M'loi U ･･･ U oe = M3. Let T' be. the involution on M' naturally induced by T, then o) = T'IEi. The 2dimensional fixed point set fix(2)T' consists of the 2-sided surfaces Fi,･･･, E,. naturally embedded in M', and p isolated fixed points of T' lie on the 2-sided. proper surface Ek in M' such that Ek/ui U ･･･ U qe = E.. Each component of E:, n Ei' (= aE:, n Ei') is a q-invariant circle. By the property of E., the union. Fi U ･･･ U E, U Eg, (C M') splits M' into two submanifolds M+ and M- which. ttl.

(9) Branched coverings of PL involutions 9 T' interchanges, and 0Ek splits each El into two subsurfaces E,'･ (= E, n M+). and E,- (= E, n M-) which q interchanges. Let S3 = A3+ u A3- be another decomposition of the unit 3-sphere S3 into two 3-balls. A3. = {(x,y,z,t) E S3:O E{ x si 1} P. and A3- --- {(x,y,z,t) E S3:-1 s{ x s{ O}, ,. and define two involutions g, 6:S3 . S3 by. g:(x,y,z,t) . (x,-y,-z,t) g: (x,y,z, t) . (-x, -y, -z, t). Then g interchanges A3+ and A3-, and induces the same involution T2 on P3 that p2 does. Notice that 410A3. = gloA3... First we shall construct a simple branched covering bk:M+ -> B3. A A3+ for any non-negative integer k. To do so, we need to define bklOM+. Fig. 4 ((p,e) t (1,1)) and Fig. 5 ((p,e) = (1,1)) show models in R3 for the involution. z. z. ". t-V-""Lij----iV-----S-Ii--P. Ei. Ek x. bk i (O.0.0.1). oA3.nB3.. l ,,. x. Y. y. (O,O,O,1). 3A3.nB3.. .. .. a(B3. fi A3.). aB3. n zNl (1,0,O,O). Fig. 4. ((p,e) -E (1,1)). aB3.nA3.. (1,O,0,O) Fig. 5. ((p,e) - (1,1)). s(B3.nA3.).

(10) 1. l l. 10 S. NEGAMI. F. TI = T'IEI,, where Tl is realized as the rotation around the z-axis, and suggest. how to map Eg, to a(B3. A A3.) by bk so that bk･TL -- g･bk. (The way in Fig. 4 is not suitable for killing equivariant handles of E'. i'n the case (p,e) = (1,1).). Notice that p + e is always even. The map bk is required to carry each component of 0E;, onto 0B3. A aA3., the shaded part of EZ, onto OB3. A A3. and the remainder onto aA3+ n B3+. Then fix(o)T' will be sent to {(O,O,O,1), s. (1,O,O,O)}. Define bklEi' for each odd surface Ei so that bk sends a samll 2-disk. ,. D?･ in Int Ei onto aAi n B? homeomorphically and cl(Ei - D?･)covers OBi A N 2-fold and properly via bk. Let bklEi :Ei -> 0B3. n A3. be a standard 2-fold branched covering from a surface with two-component boundary to a 2-disk for each even surface Ei. Especially, modify bklEi' so that Ei" covers 0(B3. n A3.) k-fold in addition. Choose a standard 2-fold branched covering with its branched set disjoing from OB3+ n 0A3. as bki4:Ili . a(B3. n A3.). Now we shall evaluate the degree of bk, counting the number nk of circles in 0M+ (== (Ek U Ei' U ･･･ U Ee') U Fi U ･･･ U E,) which cover the circle C = 0B3. n aA3.. The e boundary components of E'., d circles aD?･ corresponding to odd surfaces Ei, 2n circles in Fi U ... U E, and k circles in Ei' always cover. C homeomorphically. Ifp < e, then no other circle is mapped onto C, so nk =. e + d + 2n + k. In the case p ) e and (p,e) ± (1,1), there are (p - e)12 additional circles in the lower part of Eg,, and hence nk = (p + e)l2 + d + 2n. + k. If (p,e) : (1,1), then d = 1 and nk =3+ 2n + k. Since e+ d }i 2 andp + e is even, no is not less than 3 except for the case (p,e,d,n) = (O,2,O,O). In. the exceptional case, replace bk with bk+i. Then nk ;}r 3 in any case, and bklOM+ extends to a simple branched covering bk:M+ -> B3. n A3. of degree. nk by Lemma 3. Pull out bk (Et U ･･･ U Ee" U Fi U ･･･ U E,) fi aA3. A B3. into B3. fi zesi and bk (Ek) n 0B3+ n A3+ into B3- n A3+,leaving aB3+ n 0A3. fixed, untill. a map rk:M+ --> S3 such that. rk(fiX(2)T' U Ei' U ''' U Ee') c 0B3. rk(fix(o)T') c fix g = {(O,o,o,±1)}. ft(Ek) c aA3. and ft･Tl - g'ft is obtained. Let g:S3 -> P3 (= S31p-i) be the canonical projection, and define. q'k:M' -> P3 by qZIM+ = g･ft and qZIM' = g･g･ft･T'. Then qh induces the desired branched covering qk:M3 (== M'!oi U ･･･ U op -> P3. Remark that for each point x E jF}･, ft(x) l g･ft･T'(x) but that they are sent to the same point in P3 by g. Each point x E Ei A M3 first splits to two pints x± in Ett. respectively, but g･ft(x+) = g･g･ft･T'(xN). The well-definedness of qklE. follows from the fourth property of ft. The degree of qk is equal to 2nk - e, because C covers g(C) 2-fold while each component of E. n Ei contributes to deg qk only by 1, Check the equation 2nk - e = mo + 2k for the integer mo. T.

(11) Branched coverings of PL involutions. 11. defined above in each case.o Remark. It should be noticed that these seven cases really happen. Lemma 1 and 2 suggest a way to construct examples of each type. In Theorem 1-5, 6 and 7, the degree of each branched covering map may be cut down if fix(2)T contains 2-sphere components. Corollary 1. Any involution on an orientable closed 3-manifold is a branched covering of an involution on P3. Proqfl In the case that fix(2)T contains no 1-sided surface, the composition of. the branched covering map q:M3 -> S3 obtained in each theorem with the canonical projection g:S3 -> P3 gives a branched covering of the involution q on P3 by T, because g･pi = Tl･･g (i = O,1,2).O. 4. Corollaries A branched covering of an involution on a 3-manifold naturally induces a branched covering of its orbit space. In particular, the results for orientation-. reversing involutions yield several interesting consequences.. Corollary 1-1. Let N3 be a non-orientable closed 3-manijbld and P2i,･･･, Pft n pairwise disy'oint2-sidedprojectiveplanes in N3 (n i}r 3). lfPi U ･･･ U P?, does not separate N3, then N3 can be expressed as an n-fold branched covering of P2 × Si branched over a link so that each Pi covers a common projective plane P2. in P2 × Si homeomorphically. Proof Let N' be the 3-manifold obtained from N3 by cutting open along P2i U U P?, and let M' be the orientable double covering space of N' with the ･･･ covering transformation T'. Cap off each 2-sphere boundary component with a 3-ball, then the resultjng closed 3-manifold M3 admits an involutjon Tsuch that l,. TIM' = T' and each 3-ball added to M' contains an isolated fixed point of T. By. Theorem 1-1, there js an n-fold branched covering map from M31Tto S31po Its restriction to M'/T(== N') can be regarded as a branched covering map from N' b. to P2 × I, since S31plo is homeomorphic to a suspension of P2. Glue the boundaries of N' and P2 × I respectively, then the desired branched covering will be obtained.O Corollary 1-3. Any non-orientable closed 3-manijbld is a 4-fold branched covering of a suspension of P2 branched over a link and the two vertices.. Proof Apply Theorem 1-3 to the covering transformation on the orientable double covering space of a given non-orientable closed 3-manifold.o Corollary 1-5. Any orientable 3-mainfold with non-empty boundary is a branched covering of B3 branched over a link..

(12) i !. i. S. NEGAMI. 12. Proof Consider the double of a given 3-manifold and its reflexion. However, This corollary should be called a direct consequence of Lemma 3 rather than that of Theorem 1-5.D Corollary 1-6 & 7. Any non-orientable 3-maiofold with non-empty boundaiy is branched covering of a cone over P2 branched over a link and the vertex. 1. Proof Let N3 be a non-orientable 3-manifold with non-empty boundary and M' its orientable double covering space with the covering transformation T'. Then the closed 3-mainfold M3 = M'/(T'laM') admits the involution Tnaturally induced by T', for which fix(2)T = 0M'1(T'lOM') C M3. Each 1-sided surface in fix(2)Tcorresponds to a non-orientable component of 6VV3 and a 2-sided one to an orientable one. By Theorem 1-6 and 7, T covers the involution T2 on P3. The branched covering by Tinduces the desired one, since M31T and P3!T2 are. homeomorphic to N3 and a cone over P2 respectively.o References [1] ALExANDER, J.W.: Note on Riemann spaces. Bull. Amer. Math. Soc. 20, 370-372 (1920) [2] AsoH, T.: Classification of free involutions on surfaces. Hiroshima Math. J. 6, 171-181. (1976) [3] BERsTEiN,I.,EDMoNDs,A.L.:Ontheconstructionofbranchedcoveringsoflow-dimensional manifolds. Trans. Amer. Math. Soc. 247, 87-124 (1979) [4] HiLDEN, H.M.: Three-fbld branched coverings of S3. Amer. J. Math. 98, 989-997 (1976) [5] HiRscH, U.: Uber offene Abbildungen auf die 3-Sphare. Math. Z. 140, 203-230 (1974) [6] KiM, P.K.: PL involutions on lens spaces and other 3-manifolds. Proc. Amer. Math. Soc. 44,. 467-473 (1974) [7] KiM, P.K., ToLLEFsoN,J.L.: Splitting the PL involutions of nonprime 3-manifolds. Michigan. Math. J. 27, 259-274 (1980) [8] MoNTEsioNs, J.M.: Three-manifolds as 3-fold branched covers of S3. Quart. J. Math. Oxford. Ser. (2) 27, 85-94 (1976) [9] NAGAsE, T.: A generalization of Reidemeister-Singer theorem on Heegaard splittings. Yoko-. hama Math. J. 27, 23-47 (1979) [10] WALDHAusEN, F.: Uber Involutionen der 3-Sphare. Topology 8, 81-91 (1969). ･'r,. ,.

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