,

305

Tori in the diffeomorphism groups of simply-connected 4-nianifolds BY PAUL MELVIN University of California, Santa Barbara {Received U March 1981) Let Jf be a closed simply-connected 4-manifold. All manifolds will be assumed smooth and oriented. The purpose of this paper is to classify up to conjugacy the topological subgroups of Diff (if) isomorphic to the 2-dimensional torus T^ (Theorem 1), and to give an explicit formula for the number of such conjugacy classes (Theorem 2). Such a conjugacy class corresponds uniquely to a weak equivalence class of effective y^-actions on M. Thus the classification problem is trivial unless M supports an effective I^^-action. Orlik and Raymond showed that this happens if and only if i f is a connected sum of copies of ± CP^ and S^ x S^ (2), and so this paper is really a study of the different T^-actions on these manifolds. 1. Statement of results An unoriented k-cycle {fij... e^) is the equivalence class of an element (e^,..., e^) in Z^ under the equivalence relation generated by cyclic permutations and the relation (ei, . . . , e ^ ) - ( e ^ , . . . , e i ) .

From any unoriented cycle {e^... e^ one may construct an oriented 4-manifold P by plumbing S^-bundles over S^ according to the weighted circle shown in Fig. 1. Such a 4-manifold will be called a circular plumbing. The co^-e of the plumbing is the union S of the zero-sections of the constituent bundles. The diffeomorphism type of the pair. {P,S) determines {ef-ej^, and so there is a one-to-one correspondence between circular plumbings with specified cores and unoriented cycles. Set

Now let i f be a closed simply-connected 4-manifold, and T be a topological subgroup of Diff {if) isomorphic to T^. (The existence of such a subgroup implies that M is the connected sum of copies of ± CP^ and 8^ x S^.) We will associate with T an unoriented cycle e(T) as follows. Let S{T) be the set of all points a; in i^ for which there is a diffeomorphism t ^ tin T with tx = X. Denote a regular neighbourhood of *S'(T') by P{T). An educated glance at the orbit space of any T^-action associated to T establishes the following result (see § 3). PROPOSITION. P{T)

is a circular plumbing with core 8{T).

Now set e{T) = e{P{T),S{T)). A formula for e{T) will be given in § 3.

•

.

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PAUL MELVIN

Fisc. i .

Definition. A reducible cycle is an unoriented cycle that can be reduced to <0 0> ifsing the moves B±i:(-.c

±i d...}-^{...c+i

d^i...}.

Remarh. Using the point of view of § 2 it can be shown that an imoriented cycle is reducible if and only if its associated circular plumbing has boundary T^. 1. Let M he a connected sum of copies of ±CP^ and 8^xS^. Then the assignment T -> e{T) sets up a one-to-one correspondence between conjugacy classes of topohgical subgroups of Diff (if) isomorphic to T^ and reducible cycles (e^... e&> satisfying THEOREM

(1) k=^xm, (2) e^ + ...-\-e^ = ^

c(l) = i c{n) = S c{i)c{n~i) and c{a)) = 0 otherwise. The first few values of t{h) are as follows: t{h) = 1 for ^ < 6, ((6) = 3, ((7) = 4, ((8) = 12, t{Q) - 27, t{10) = 82, i(U) = 228, t{i2) = 733. Remark. It was brought to my attention by Craig Squler that the numbers c{n) are the Catalan numbers . . 1 (2n-i\

Tori in the dijfeomorphism groups of 4:-manifolds

307

in •3/2

2/3

.2/1

112 1/3

1/3

-1/2 -3/2

-2/3

-I/I Fig. 2. Diagram of ^^£(2, Z).

This formula is obtained from the recursive definition in Theorem 2 as follows. The generating function CO

for c{n) evidently satisfies /^ = /—x, and so

/(^) _ Wiizi^ by the quadratic formula (with the minus sign chosen to satisfy c(l) = 1). The Taylor series for tMs function yields the formula. 2. The diagram of P8L{2,I) The well known diagram in Fig. 2 will be useful as a book-keeping device. It is obtained as follows. First define a i-complex K. The vertices of K are the points of QP^ = O u {1/0}. Two vertices {p, q) and (p', q') are joined by an edge if and only if

det(^ K]-+i. Thus the oriented edges of K can be identified with the elements of PSL{2,1)]

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PAUL MELVIN

Now embed K in the Poincare disc D as the 1-skeleton of a tesselation by ideal triangles, obtained by successive reflections in the sides of one such triangle. The labelling of vertices is unique once it is given on one triangle. This labelled subcoihplex X c D is called the Diagram of PSL{2,1) (see Eig. 2). Observe that the set QP^, viewed as lines in U^ through 0 of rational slope, is invariant under the action of GL{2,1) on U^. This gives an action of GL{2, T) on the vertices of K, which readily extends to an action on Z> by hyperbolic isometries leaving K invariant. The subgroup 8L{2, Z) acts by orientation preserving isometries. Definition. A loop in K is a. closed, oriented edge path inK'^ D, and will be identified by its (oriented) cycle {pjqi... pj^k) of "v-ertices. If p^/q^ ^ p^jq^ for i ^ j , then the loop is simple. Two loops in K are conjugate if they lie in the same orbit of the action of GL{2, Z) on the set of loops given by

In order to study the conjugacy class of a loop r = (pi/qi.. - Ph/^k) ^^ ^> associate with r an unoriented cycle e(r) = < % . . . eft>,

where (*)

e, = det((^^-^ ^')(^^

Pi.i\(Pi-i

Pi.i\)

ii=.i,...,k),

with indices taken mod h. In geometric terms, the loop turns across e^ triangles at the vertex Pi/qi, with e^ < 0 if and only if the turn is clockwise. LEMMA. (1) Two loops in K are conjugate if and only if their associated unoriented cycles are equal. (2) An unoriented cycle is associated unth a loop in K if and only if it is reducible. Proof. (1) The direct implication is geometrically evident as GL{2, Z) acts on {D, K) by isometries. Alternatively, it can be verified from (*). The converse follows from the transitivity of the action of SL{2,I) on oriented edges in K. (2) Any loop r in ^ can be collapsed to a vertex using elementary collapses GQ and G±x defined in Fig. 3. For r always contains a simple subloop r' (possibly == r). If r' is just a retraced edge, then GQ can be applied. Otherwise r' encloses a triangulated polygon P in D. Viewing P as a regular Euclidean polygon, the shortest interior edge in P cuts off a triangle to which C±j can be applied. Now elementary collapses GQ and G±^ on r correspond to moves BQ and B±i on e{r) (see § 1). Thus e{r) is reducible. Conversely if e is a reducible cycle then the sequence of reducing operations corresponds to a sequence of elementary collapses, which in reverse gives instructions for building up a loop r in X with e{r) = e. \ ^

Tori in the diffeomorphism groups of 4:-mamfolds

309

A

P'/Q'

(p + p'Viq + q')

P'k'.

F i g . 3. (A) Go: E l e m e n t a r y collapse of a r e t r a c e d edge (... j?/g p'fq' pfq ...)~^i--p/q •••)• (B) 0±i_: E l e m e n t a r y collapse across a t r i a n g l e {...p/q {p±p')/{q±q') p'lq'-••)->

i-p/q

P'fq' -) (G+i shown).

3. Proofs Let JJf be a connected sum of copies of ± CP^ and 8^ x 8\ and a: T^ ^ Diff {M) be an effective y^.action on M. Following Orlik and Raymond (2), the weighted orbit space of a is an oriented 2-disc with boundary consisting of singular orbits and interior consisting of principal orbits (Fig. 4). The h distinguished points on the boundary are fixed points, and the arcs Joining them correspond to 2-spheres in M. (The Proposition in § 1 follows readily.) The weights in Q U {1/0} specify the isotropy subgroups of these 2-spheres. In particular ^/gr specifies the circle mT^ = R^/I? which is covered by the line in IR® through 0 of slope p/q. '• As the weights satisfy

det(^^ ^'A==±\

ii=^i,...,k)

(see p. 534 in (2)), with subscripts taken mod k, the cycle

T{a)^{pJq^.,.Pulqu) represents a loop in K (see § 2). In fact, the equivariant classification theorem of Orlik and Raymond (2) shows that the assignment Qc -> r{a)

sets up a one-to-one correspondence between equivalence classes of effective T^-actions on closed simply-connected 4-manifolds and loops in K. (Recall that two actions a^. T^^Jyi^iMi) {i = 0,1) are eg'ttim^ewHf there is a diffeomorphism A:lfQ->ifi with ha^ig) = ai(gf) h for all g in T^.) Remarks. (1) Let 8^ be the 2-sphere in M with weight pi/qi, and E^ be its normal

310

PAUL MELVIK

K Pilgi / Pk/?k Fig. 4.

bundle in M. The T^-action on dE^ identifies 8E^ as the lens space Lie^, i) where e^ is given by the formula {*) in § 2. (See p. 550 in (2).) Thus E^ has Euler class e^. Setting we have e{T{a)) = eir{cc)).

(2) Recall that two actions a^iT^-^ X)W{M) (i ~ 0,1) are weakly equivalent if aj and a^J. are equivalent for some automorphism A oiT^. Since AutjP^ = OL{2,1), it is straightforward to show that CCQ and cc^ are weakly equivalent if and only if either of the following conditions is satisfied: {a) r{ao) and y(ai) are conjugate in DilF(Jif), (&) r{aQ) and r{a-^) are conjugate loops in K. Proof of Theorem 1. By the Remarks above and the Lemma in § 2, the assignment T -> e{T) sets up a one-to-one correspondence between conjugacy classes of toral subgroups of the diffeomorphism groups of closed simply-connected 4-manifoIds and reducible cycles. Since connected sums of copies of ± CP^ and ;S^ x 8^ are classified by their Euler characteristics and the signatures and parities of their intersection forms, it remains to show {X)X{M)==k, (2) a{M) = (ej-!-... -|-e/,)/3, and (3) M is spin if and only if e^, ...,e^ are all even for any effective y^-action cconM with associated reduced cycle e(a) = {e^... e^). A well known theorem of P. Conner states that xi^) ^^ equal to the number of fixed points of a, giving (1). To prove (2), observe that the moves B^ and B±i reducing e{a) (see § 1) have the following effect on the underlying manifold: BQ

surger a 2-sphere in .M", JB±J blow down a 2-sphere in i f of self intersection ± 1. Thus ^0 does not affect the signature of the underlying manifold, and 5±i alters it by + 1. An inductive argument, starting with (r{8*) = 0 (corresponding to the reduced cycle {00}), establishes (2). , (3) follows from Theorem 6-3 in (l). | ^

Tori in the diffeomorpMsm groups of 4:'manifolds

311

Fig. 5.

Proof of Theorem 2. First observe that S^ x 8^ and 8^x8^^ CP^ # - CP^ each has an infinite number of conjugacy classes of toral subgroups in its diffeomorphism group, corresponding to the reducible cycles (nO-nO) with n even and odd, respectively. It follows that if i f has a connected sum factor diffeomorpMc to 8^ x 8^ or 8^ x 8^, then Diff (if) has infinitely many inconjugate toral subgroups. Now assume that M has no 8^ x 8^ or 8^ x 8^ factors. Then i f or - i f is a connected sum of copies of CP^. Without loss of generality we assume henceforth that M=

# CP2 A-2

for some & S 2. Then for any toral subgroup T of Diff {if), e{T) is reducible to <0 0> using only moves B^^. For, as observed in the proof of Theorem 1, move BQ splits off 9.8^x8^ or 8^ X 8^ factor from M, and ^±1 splits off a + CP^ factor. I t follows from the definition of move B^ that Diff (if) has only a finite number t{k) of conjugacy classes of toral subgroups. These correspond exactly to actions associated with simple loops in K of length k. Any such loop *• encloses an oriented, triangulated jfc-gon in D, whose regular Euclidean model we denote by P{r). Evidently the conjugacy class of r in JS^ (or equivalently of the associated torus in Diff (if)) is uniquely determined by P{r) up to rigid Euclidean motions (including orientation reversing motions). In particular t{k) = the number of triangulations of the &-gon (with no added vertices) up to rotations and reflections. For example, the 6-gon has three triangulations (Fig. 5) and so Diff(#CP2) has three conjugacy classes of toral subgroups. To compute t{k), we first ignore reflectional symmetry and compute the number s(k} of triangulations of the ^-gon up to rotations. Observe that every triangulation of a regular polygon P as above has at most 3-fold rotational symmetry. (This was pointed out to me by Benjamin Halpern.) For, the centre of P lies either on an edge or inside a triangle. There can be at best 2-fold rotational symmetry in the former case, and 3-fold in the latter. See Fig. 6 (the circle represents P).

312

PAtTL

MELVIN

Fig. 6.

a 6 C(A/2)

b e C(ft/3)

Fig. 7.

Let s^{h) denote the number of triangulations of the ^-gon (up to rotations) which have i-fold rotational symmetry (for i ~ t, 2, or 3). Note that

Let C{n) denote the set of all simple edge paths in K of length n joining 1/0 to 0/1 and lying in the tipper half of the Poincare disc D (i.e. with second vertex p/1 for some p > 0). Set c{n) = Gaj:dC{n).

For n> i, every path in Oin) must pass through 1/1, and so c(n) = S c{i)c{n-~i) in agreement with the definition in the statement of the theorem. For any loop r in ^ of length h, let S{r) be the set of all edge paths in C{h - 1) with vertices r^ = 1/0,r^,...,ri^ = 0/1 such that rand {ri-.-Tf^) He in the same orbit of the action of SL{2,1) on K. Evidently the sets S{r) partition C{h-i) into disjoint subsets. Observe that if Pir) has i-fold symmetry {i = 1,2, or 3), then S{r) contains h/i paths. Thus c{k~~ 1) = ks^{k) 4- Iks^m + Ihssih).

The triangulations with 2- and 3-foId rotational symmetry are evidently of the form

Tori in the diffeomorphism groups of 4:-mamfolds

a, e

CUl

313

b e C{kl2~i)

Fig.

a e C{kl4)

mg. 9. indicated in Fig. 7. One easily deduces that s^{h) = G{hji) for i s^{h) = s{k) ™ s^ik) - S3{/c), the equation displayed above yields

2 and 3. Since

Now we wish to compute t{k). The computation of s{k) ignored reflectional symmetry, and so we effectively counted triangulations without reflectional symmetry twice. Thus, setting r{k) equal to the number of triangulations with reflectional symmetry, we have But r{k) is easy to compute by hand. For the triangulations with reflectional symmetry are of one of the three forms indicated in Fig. 8 (the dotted line is the line of reflection; x represents the reflection of the edge path x through the edge joining 1/0 and 0/1). The number of triangulations (up to rotations and reflections) in each of the first two cases is

-!(•©"©)• TT

pspgi

^

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PAULMELVIN

These two cases overlap in triangulations of the form shown in Tig. 9. There are c{k[4.) such triangulations and so there are a total of c(fe/2) triangulations in the first two cases of Fig. 8. In the third case there are c((^-l)/2) triangulations. Thus r{k)

y.-fc^-^-j.

Comhining this with the formula for s{h) above, we obtain the desired formula for

m. I REFERENCES (1) MELVIH, P . On 4-manifolds with singular toms actions. Math. Ann. 256 (1981) 255-276. (2) OBME, P . and RAYMOITO, F . Actions of the torus on 4-manifolds: I. Trans. Amer. Math. Soc, 152 (1970) 531-559.

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