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AN INVARIANT OF PLUMBED HOMOLOGY SPHERES

Walter D. Neumann

If A is a ring, a 3-dimensional A-sphere will mean a closed oriented 3-manifold M 3 with the same A-homology as S 3.

3 and 3 is

An A-homology bordism between two A-spheres M! M 2 a compact oriented 4-manifold W 4 with ~W = M I + (-M 2) , such that the inclusions M. c___~ W induce isomorphisms in A-homology. i Here, and in the following, + means disjoint union, - means reversed orientation, and all manifolds will be assumed smooth, compact, and oriented. Diffeomorphlsms should preserve orient- ation. The set of A-homology bordism classes of 3-dimensional A-spheres forms a group under connected sum, which we denote ~A 3" We are most interested in A = ~/2 or • . Every ~-sphere is a ~/2-sphere and every ~/2-sphere is a rational sphere. If M 3 is a ~/2-sphere, then its ~L-invariant (see for instance [ 3 ] , where the opposite sign convention is used) can be defined as ~I(M 3) = sign(W 4) (mod 16) , where W 4 is any simply connected parallelizable compact manifold with }W 4 = M 3 . This invariant is a ~/2-homology bordism invariant and its range of values is summed up in the commutative diagram ~3 .~ 8~/] 6~.. f ~3 ' .~) 2=/1 6~- A The groups ~3 are of interest in their own right, but also because of applications. The most important application is the result of Galewski and Stern [~J and Matumoto [ ~ ] that for any Research partially supported by the NSF. The hospitality of the Sonderforschungsbereich 40 in Bonn during the preparation of this paper is also gratefully acknowledged. 126
n ~ 5 , all closed TOP n-manifolds are simplicially triangulable contains a subgroup ~/2 on which /~t is an if and only if ~3 isomorphism. Another application is to the classical knot concordance group C 3 . Given a knot (S3,K) one can do a p/q-Dehn surgery

3 3 M 3 with on it to get a homology lens space Mp/q(S ,K) = p/q

HI(M3, ;~) = Z/p Thus if p is invertible in A then M 3 P/q p/q A is an A-sphere. The map C 3 • ~3 given by (S3,K) ~

[M3, (S3,K)] - [e(p,p-q)] is well defined though possibly not p/q A could yield new a homomorphism. Thus information about ~3 knowledge of C 3 Other similar constructions can be used to the same purpose. Our main results are the following time-dependent theorem and the non time-dependent consequences, theorems 6.! to 6.3, of section 6 below. THEOREM. There exists an invariant ~(M 3) ~ ~ with the followin$ ~roperties: (i). ~(M 3) is defined for ~/2-sDheres M 3 which are plumbed manifolds ( = graph manifolds in Waldhausen's sense [IO]) and is an oriented diffeomorphism type invariant. (ii). ~(M 3) (mod |6) = ~(M 3) (iii) For all known (Ma~ |979) ~/2-homolo y bordi of t ................ ~ sms such ~/2-spheres, ~ is invariant. If ~ is in fact a YL/2-homology bordism invariant and if it can be defined for any ~/2-sphere M 3 , then of course we have a negative answer to the triangulation problem. If on the other hand ~ turns out not to be a Z~/2-homology bordism in- variant, it will hopefully be due to some new idea in construct- ing such bordisms which then might lead to an example which solves the triangulation problem positively. Moreover, /~ might be invariant under some stronger bordism relation of geo- metric significance; we discuss this briefly in the last section. Some consequences of our results are: plumbed ~/2-spheres with non-zero ~-invariant admit no orientation reversing homeo- morphisms; the Kummer surface cannot be pasted together from two simply connected plumbed 4-manifolds. The latter (see 6.2) depends on a generalization of ~ described in section 4. 127

2. The invariant.

For the purpose of this paper we define a connected plumbing graph r to be a connected graph with no cycles, each of whose vertices, which we label i = 1,2, ... ,s , carries an integer weight e. The oriented 4-manifold P(F) obtained by plumbing i according t_~o ~ is then defined in the usual way, namely, for each vertex i we let E. be the D2-bundle over S 2 of euler I number e. and then plumb these together according to the edges l of ~ (see for instance [3 ]). We call M(~) = ~P(~) the oriented

3-manifold obtained by ~lumbing at.cording t~o

More generally, we allow disconnected plumbing graphs whose components are as above by making the convention that if = ~l + ~2 is the disjoint union of ~| and ~2 ' then P(~) = P(PI ) ~ P(~2 ) (boundary connected sum), so M(~) = M(F|)~M(~ 2) (connected sum). We allow also the empty graph by defining P(~) = D 4 and M(@) = S 3 It is more customary, when plumbing, to allow graphs with cycles and bundles over surfaces of higher genus, but, since such plumbing could never yield homology spheres, our more restricted plumbing graphs include all we want. They are precisely the plumbing graphs for which P(~) is simply connected. If matrix is a plumbing graph in our sense, recall that the A(~) (aij)i,j=l'" ..... s a.. = e. if i = j 13 I = l if i is connected to = 0 otherwise , j by an edge is the intersection matrix for the natural basis of H2(P(r);~.) namely the basis represented by the zero-sections of the plumbed bundles. THEOREM 2.1. If r i_~s ~ plumbing graph as above then M(P) is a ~./2-sphere if and only if detA(p) is odd. In this case there exists a unique subse t SCVert(~) = If, ... ,s~ such that the following condition holds for any j = |, ... ,s : 128
(~) ~. a.. ~ a.. (mod 2) S lj 3J

The inteser

~(M(T)) = signA(P) - Z a.. (recall a.. = e.) j~S jj JJ J then only depends o__n_n M(| TM) and not on tion is the usual ~-invariant ~(M(P)) . Its modulo 16 reduc- Proof. Everything except the oriented diffeomorphism type invarlance of ~(M(P)) was proved in L~ ]. We describe the main ingredient however, since we need it later. Suppose M 3 is a 7Z/2-sphere and M 3 = @W 4 where W 4 is oriented. Recall that an integral Wu class for W 4 is a class d 6 H2(W;~) such that (dot represents intersection number): d.x ~ x.x (mod 2) for all x ~ H2(W;~) We assume that such a class d exists and moreover that it can be chosen to be spherical , that is, it is representable by a smoothly embedded sphere in W . Then the ~-invariant can be computed as

2~t(M 3) = sign(W) - d.d (mod 16) .

It is easily seen that d does exist if HI(W;~) has no even torsion and is unique up to even multiples of elements of H2(W;~) if HI(W;~) = 0 . In particular, if ~ is as in the theorem, then there is a unique subset SC Vert(P) such that d = ~'j~S ~" is a Wu class for P(P) , where ~], ... ,~ is j s the natural basis of H2(P(C);~) The defining property of d translates to condition (~) of the theorem, so this set is the set S of the theorem. Condition (~) implies that no two adjacent vertices of ~ can both be in S . It follows that d is spherical and d.d = ~'j~S ajj , so ~(M(p)) = signP(~) - d.d ~2R(M(p)) modulo 16. To see ~(M(p)) only depends on M(~) we need the follow- ing proposition. 129
PROPOSITION 2.2. If ~ ] and F 2 are two plumbing graphs as defined above, then M(PI) ~ M(F2) if and only if Pl can be obtained from ~2 by ~ sequence of th____~e following moves and their inverses• la. Delete a comFonent of ~ consisting of an isolated vertex with weight ~] e+l +I e lb. "./ • ) .'.~ Ic. e ±I ±I e2+l .-.--.~..~1 ~_...~ ~..~l e2_...~. e 0 e 2 e.+e 2 2 .---..___I ---~. ~'---'--2---/J~ • ~ • _ ~ :

3 o )r I + .,. +r t (disjoint

union). Move I is called blowing down, and its inverse blowing up. This proposition is proved in ~ , theorem 3.2~ and I under- stand a forthcoming paper of Bonahon and Siebenmann will also contain a proof. To complete the proof of theorem 2.1 we need only verify that ~(M(~)) is invariant under the above moves. This is a trivial computation, so we leave it to the reader. COROLLARY 2.3. The formulae of [~ , theorem 6.2] for the ~-invariants of Seifert manifolds which are 7£/2-spheres are actually formulae for ~ , if one does not reduce modulo 16. 130

3. Examples.

A quite general method of generating homology spheres which are homology null-bordant is given by the following simple lemma. LEMMA3.1. Le___~t A be a principal ideal domain and let V 4 be a connected oriented 4-ma.nifold with connected boundary ~V 4

N 3 such that

H|(V;A) = A s , H2(V;A) = H3(V;A) = O

L.et M 3 be an A-sphere obtained b__~y performing A series o__n_n N . Then M represents zero in --~3 s index 2 sur- Proof. Let ~.: SIX D 2 ) N , i = 1, ... ,s , be the em- i beddings on which the surgeries are performed. Then ~ S 1 1 i = I, ... ,s , clearly represent an A-basis of H|(N;A) = H| (V;A) = A s , the first equality here being by the exact homol- ogy sequence for the pair (V,N) . Let W 4 be the result of adding s 2-handles to V along the ~, , so ~W = M 3 . Then i

W is clearly A-acyclic, proving the lemma.

Applied to V 4 SIx D 3 this lemma shows that any

A-sphere M 3 which results by a single index 2 surgery on SI~S 2 represents zero in ~ . Such manifolds are called

Mazur manifolds.

PROPOSITION 3.2. Let ~I and ~2 be the following two plumbing graphs: k c c I c ..._~r - - 6 2 ___ = O~ a £| | ~2 ....... ~s ~bl b 2 b t b b 2 b I 0 al+c 1 a 2 a C 8r Then M(~l) results by doing a single index 2 surgery o nn M(~ 2) and vice versa. Thus if one of them is sl~ S 2 then the other, s if an A-sphere, is a Mazur manifold and regresents zero in ~ 131
PROPOSITION 3.3. If M is a plumbed ~/2-sphere which is shown to be a Mazur manifold by proposition 3.2 then ~(M) = O . Before giving proofs we make some remarks. Firstly it is not hard to write down many propositions of the form of 3.2. The above was the most productive one I found. Secondly it is a very easy matter to check if a plumbing graph P represents S ] K S 2 : it follows from [ ~ ~ that this is so if and only if ~ can be reduced by the moves of proposition 2.2 (without using the inverse moves) to an isolated vertex with weight O . For Seifert mani- fold plumbing graphs it is even easier.

Recall from [ ~ ] that the plumbing graph

all alr ...... :2rl b • ........ "~" ", sr ...... S yields the Seifert manifold

M(O; (l,-b), (~l'~l) ..... (° (unnormalized Seifert invariants, as in [ 6 ], [ ~ ] ) with ~i/~i = Jail ..... air.] , i = I ..... S 1

We are using the continued fraction notation

I I

Ix I ..... x r] = x I - 32 - _

I X r and we assume ~. # 0 for all i . i This Seifert manifold is S l x S 2 if and only if: ~. = +I i for at least s-2 of the indices i , and b - Z~i/~i = 0 0 It is thus a simple but tedious matter to list all Seifert manifolds which are shown to be Mazur manifolds by proposition 132

3.2. One obtains all the examples of Casson and Harer [ 2 3 plus

the following three parameter family of Seifert manifolds:quotesdbs_dbs47.pdfusesText_47

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