Homology spheres and the ~-invariant appear to play a crucial role in many problems in low dimensional topology. For exsmple~ the existence of a 3-dimensional homology sphere ~ of ~-invariant 8 for which ~ # Z bounds an acyclic 4manifold would imply triangulability of all manifolds of dimension > 6. As Casson has remarked~ if Z has ~-invariant 8 and admits an orientation reversing homeomorphism~ then Z would satisfy this criterion.

As another example~ the intersection form of a simply connected almost parallelizable 4-manifold is even. No such closed 4-manifold with non-trivial definite form is known. A reasonable and popular procedure to collect empirical data towards the existence or non-existence of such manifolds is to study 4-manifolds with (Z/2)homology sphere boundaries~ since pasting along such boundaries preserves evenness of forms.

In this paper we have compiled some results and computations in these areas for the special case of Seifert 3-manifolds and other plumbed manifolds. For example~ we classify in section 8 those Seifert manifolds which admit orientation reversing homeomorphisms. No rational homology spheres other than lens spaces are among them.

Section i reviews the fundamentals of Seifert manifolds in a more convenient version than the usual one.

We show in sections 2~ 3~ and 4 that the class of Seifert manifolds which are homology spheres coincides precisely with a natural subclass of the class of Brieskorn complete intersections (studied in section 2) and also with a natural subclass of the class of homogeneous spaces discussed in section 3. Thus Seifert homology spheres arise as links of isolated complex surface singularities with C*action. We show~ in faet~ in section 5 that almost every Seifert manifold arises *Research supported in part by the National Science Foundation. this way, namely precisely those which do not fiber equivariantly over S 1 . We do this by describing a "canonical plumbing diagram" for such Seifert manifolds. We give a table of Seifert homology spheres with small invariants which can be made to bound even definite 4-manifolds by these methods.

In sections 6 and 7 we give various useful algorithms for computing ~-invariants for Seifert (Z/2)-homology spheres and other plumbed manifolds. A table is included.

Theorem i.I. Let M and M ~ be two Seifert manifolds with associated Seifert invariants (g ; (~i~i) ..... (~s~s)) and (gl ; ( ~ ) ..... ( ~ ) ) respectively. Then M and M I are orientation preservingly homeomorphic by a fiber preservin~ homeomorphism if and only if~ after reindexing the Seifert pairs if necessary~ there exists a k such that (i) ~i = ~i -f-or i = I~ ...~ k and ~i = ~j = I for i ~ j > k. (ii) ~i ~ ~iI (mod ~i) fo_~r i = i~ ..., k.

(iii) ~ si=l (~i/~i) = ~it=l ( ~ / ~ ) sl-bundles~ the theorem would be an easy cohomology calculation~ since e(M) = <c(M)~[F]) where c(M) E H2(F;Z) = [F,K(Z~2)] = [F,BS I] is given by the classifying map F - BS 1 of the bundle. Although M - F is not a genuine bundle~ we can make it into a genuine bundle by replacing the fibers by the " ratl"onalized circle" S(I0 ) . This replaces BS 1 by BS~0 ) = K(Q~2) ~ so the argument sketched above then goes through. To make this argument precise~ it is easiest to observel) that we do not have to "localize all the way to @". Let a be a positive integer divisible by all the ~i's occurring in M or M ~ Factor by the (Z/a)-action inside the S laction to get a diagram of maps

M M/(Z/a) f a >

M l > M'/(Z/a) We need that f can be made (Z/a)-equivariant~ but this is easily done. Now M/(Z/a) and M~/(Z/a) are both genuine sl-bundles~ so the theorem is true for fa " If we show it is true for ~ and ~ ~ then it follows for f . But for and ~ it follows from the following lemma (due to Seifert [S]).

Lemma 1.3. invariants

If M ha___~sSeifert invariants (g ; (~i,~i)) then M/(Z/a) has Seifert (g ; (~i~i)) where ~i/~i is a~i/~ i expressed in lowest terms. manifolds~ coming maybe from factoring by a group action~ then e(M) and e(M') determine each other by Theorem 1.2 while the slice types of M can generally be computed from those for M' and vice versa by elementary local computations.

In the following~ we shall often write M = M(g ; ( ~ i ~ i ) ~ i = l~...~m) as an abbreviation for '~ has Seifert invariant (g ; (~i~i)~i=l~...~m)".

Let a1~i ...~ an be integers~ a.l ~ I . Then if A = (~.j) general ( n - 2 ) X n - m a t r i x o f c o m p l e x n u m b e r s ~ t h e v a r i e t y is a sufficiently

VA(al,...,a n) = {z E cnl~ilZla I +... + ~inZna n = 0~i= l,...,n-2] is a complex surface which is non-singular except perhaps at the origin and

E 3 (al~...~an) = VA(al~...~an) ~ S 2n-I is a smooth 3-manifold which does not depend on A up to diffeomorphism. A is in fact sufficiently general if (and if all a.I ~ 2 also only if) every (n-2) × (n-2) subdeterminant of A is nonzero~ by Hamm [Ha]. We assume A satisfies this from now on.

VA(al~...,an) has a C* -action by t(Zl~°..,Zn) = (t ql Zl~... ~t qnZn) for t E C* ~ where qj = (icm ai)/a j . S I c C* acts fixed point freely on ~(al~.. ?~an) so ~(al~...~a n) is a Seifert manifold.

Theorem 2.1. E = Z (al~...~a n) where sj(tj~j) means (tj,Sj) has Seifert invariants repeated sj? times and

(g ; Sl(tl~l)~...~Sn(tn~Sn))~ t. = icmi(ai)/icmi~.J(~a.) J sj =~i#j(ai)/Icmi~j(ai) g = l(2+(n-2)~i(ai)/Icm(a i) - l j s j ) and the ~i and the Euler number e(5") are determined (up to equivalence of Seifert

Note that the latter equation can be rewritten (by dividing through by its right hand side)

I 6jqj = i where qj = icm(ai)/a."j But clearly t.i divides q~j if i ~ j and is prime to q.j if i = j . Thus modulo tj the equation becomes ~jqj N 1 (mod t j ) and hence determines ~j(mod tj)~ as claimed in the theorem.

Proof of theorem. First note that the only points of Z = ~aRl~...,a n) with nontrivial isotropy are points with some coordinate zero. The condition on the coefficient matrix A implies that VA n [z i = zi = 0] = {0} for i ~ j , so we need only consider z 6 ~ 3 with one coordinate zero, say z.J = 0. At such a point the isotropy subgroup has order gcdi~j(qi) = t..j An easy counting argument (see [NI]) can be used to see that Z 3 n {zj = 0] consists of exactly s.J orbits, but this follows also from the later discussion.

Observe that we can write

Z 3 (a I ..... an) = (VA(a I ..... an) - {0])IR+ , where R÷c C* is in the C -action.

Consider the diagram

(VA

{0})/C* = ~/S I by P(al~...~an) .

VA(al, .~a n) - [0] ~

) VA(I ..... i) - {0] Z (al,...,an)

I/sI ~ > Z (1 ...,I)

I/s1 P(al,...,a n ) ) P(I,.%.,I) a I a n with horizontal arrows induced by ~(Zl,...,Zn) = (Zl,...,z n ) ? We intend to apply

Note that @ and ~ are equivariant if we let C* and S I act non-effectively on VA(I,...,I) and ~(i,...,I) by t(zl,...,z n) = (tazl,...,taz n) , a = Iom(ai) . Thus ~ has degree a on a typical fiber.

To determine the degree of ~ , note that the group H = (Z/al) X... ×(Z/a n ) acts on each space on the left of the diagram by letting Z/aj act by multiplication by e2~i/aj in the j-th coordinate. The map ~ can be identified with the orbit map VA(al~...,an)-{0} ~ (VA(al,...~an)-{0})/H , and similarly for ~ and ~. Considering S I and H both as subgroups of Diff(Z(al,...,an)) by these actions, denote H 0 = S I n H. Now on the one hand, H 0 is isomorphic to the non-effectivity kernel of S I acting on ~(al,...,an)/H = ~(I,...,I) , so H 0 ~ Z/a , while on the other hand H 0 is the non-effectivity kernel of H acting on ~(al~...,an)/Sl = P(al~...,a n) , so the orbit map ~ of this action has degree !H/H01 = ~(ai)/a.

Now VA(I,...,I ) c Cn is a linear subspace and hence E(I,...,I) ~ P(I,...,I) is the usual Hopf map S 3 - CP I . Thus e(~(l,...~l)) = -i , so by Theorem 1.2~ e(Z(al,...,an)) = -~(ai)/a 2 .

Finally to compute g , note that the subspace z.3 = 0 of P(I~...,I) is a single point and that these points are precisely the points where branching of occurs. The argument we used to show ~ itself has degree ~a.i/Icm a.l applies with one coordinate less to show that ~ restricted the subspaces z.J = 0 of P(al~...~a n) and P(I,...,I) has degree ~i¢.ai/icmi¢.aJij = sj ~ so P(al~.°.,an) contains exactly sj points with z.J = 0 (proving, by the way, that Z(al~...~a n) n [zj =0} consists of s.J orbits, as promised earlier). The standard formula for euler characteristic of a branched cover thus gives

X(P(al . . . . . an)) = ( ~ ( a i ) / a ) x ( P ( l ' ' ' ' ' l ) ) + l j ( s j - ~--~(ai)/a)

= (2"n)~i(ai)/a + l j Sj yielding the value for g claimed in the theorem.

Remark. One can also give a very elementary computation of the Seifert pairs (tj~Sj) satisfy and the Euler number ~ j q j = i ~ then e(Z)

by observing that if the ~j are chosen to

R = [z E E l zj =r.ej i~.J~rj > 0 ~ L~8 jej ~ 0 (mod 2~)] is a section to the sl-action in the complement of the exceptional orbits which yields~ via the definition of the Seifert invariant~ the values (tj~j) for the Seifert pairs. However~ one still needs a computation like the above proof to determine g.

A completely analogous proof to the above shows more generally

Let alJ ...~ an ~ dl~ ...~ dn_ 2 be positive integers and gcd(di) = i.

V = [z Ecnl~ilZldia 1 +... + ~inZnd .ia n = 0,i = l~...~n-2} with sufficiently general coefficients G.l.j ~ and let E = V n S 2n'l .

Seifert manifold with invariant (g ; dsj(tj~j)~j =l~...~n) where are as in T h e o r e m 2 . 1 ~ d = ~d.1 ~

tJ° and s °

J and e (~) and th___~e ~j ar___~egiven by g = 1 _ "d~E~/, sj-(!di)l'T'Faj/lcm a j ] , -e(~) = I dsj ht. = d ~ a i / ( I c m a i ) 2 "

J Wagreich [OW]~[O]. The method used here is, however~ based on the case n = 3 of Theorem 2.1 done by Neumann [NI]. Brieskorn complete intersections of the type in Theorem 2.2 were first introduced by Randell [Ran].

Example 3.$. It is easy to apply this to Theorem 2.1 to see that E(al~...~a n) has the form ~\G. In this case ~ is the commutator subgroup of T = - I ( Q ) ~ where Q c PSL(2~R) is a Fuchsian group of signature (0 ;al~...~an) . This can be shown by explicit computation of Seifert invariants~ which was how we originally did it. Using automorphic forms~ one can prove a stronger version of the same result (IN2]). This has been done independently by I. Dolgacvev. The case n = 3 was done by J. Milnor [M]~and F. Klein [K] for G = SU(2) . More generally~ the manifold of Theorem 2.2 has the form *~\G if and only if is a positive integer and is prime to 5j = Icm ai/ Icm i6j a.1 holds for~example~ if d divides Z d .].

for each j . This m But det A = ~i=l~iSl "'" ~i "'" 5m = +51- "'" ~mZ (~i/si) . The condition that M is a homology sphere is thus: g = 0 and 51 ... ~m~ (~i/si) = ~I . By reversing orientation if necessary~ we can assume

~i "'" ~ m l (~i/Si) = +l .

T h i s e q u a t i o n i m p l i e s t h a t t h e ~j modulo ~j for each j .

e(M) = "I ~i/~i = -i/~i "'" ~m" It thus determines M completely for given ~I ~ "''~ ~m"

Comparing with Theorem 2.1 proves the second statement. Alternatively~a simple proof of 2.1 for this case is given by observing that by Hamm [Ha]~ ~(~l~...~m ) is a homology sphere for ~.l pairwise coprime and its S 1 action clearly has isotropy Z/~I~ ...~ Z/~ m.

One can get a simple proof of Example 3.3 for this case also by applying part (iii) of the following lemma.

Proof. Up to and including part (i)~ this is the same proof as the previous theorem. Parts (ii) and (iii) then follow by observing that Z/c c S 1 acts freely on M 1 so M c must be MI/(Z/c ) by Lemma 1.3.

Similar statements hold for g ~ 0. We leave their formulation and proof to the reader.

be the result of plumbing according to the following weighted tree -b " ~~" mbl

The [g] above means that the corresponding bundle being plumbed is the bundle of Euler number -b over a surface of genus g ; all the other bundles are bundles of Euler number b.l.j over the sphere S 2 . We omit the [g] if g = 0. We are using the notation P(F) for the four-manifold obtained by plumbing disc bundles according to F and ~P(F) for its boundary obtained by plumbing circle bundles.

By yon Randow [vR]~ in any plumbing graph F we can "blow down" vertices corresponding to a bundle of Euler number e = @i over S 2 having at most two adjacent vertices by removing that vertex and replacing the weights b i of the neighboring vertices of F by b.i - ¢ . This does not change ~P(F) . For example~ if we start with and then by iteratively blowing down -l's we can finally get to with

cij = [bij~bi~j+l~...~bi~si] ?

Thus F can be positive definite if and only if e(M) > 0 ~ and similarly for negative definite. Assume now e(M) > 0 ~ by reversing the orientation of M if necessary. Normalize the Seifert invariant of M to satisfy 0 < ~i < ~i for i = 13 ...~ m . Then ~i/~i can be uniquely expanded as acontinued fraction ~i/~i = [bil~...~bis ] with bij ~ 2.

Proof. One can do plumbing holomorphically to obtain P(~) as a complex manifold with holomorphic C -action and then apply Grauert [G~ to blow down the central configuration of curves in P(r) . That one can blow down equivariantly follows by functoriality of Grauert's theorem. Since the complex structure one puts on P(r) is in general far from unique~ one~ of course~ gets a whole family of possible singularities. One can also prove this corollary directly via the injective holomorphic C* -actions of Conner and Raymond [CRI]~ by showing that they can be compactified by a single singular point to give a complex affine variety~ if the Euler number is negative. as the dual graph of the minimal good resolution. Blowing down (in the sense of complex manifolds; we cannot do it in the sense of plumbing graphs~ since the (-l)vertex has three neighbors) can be done twice~ giving the result -?2 -4

-2 2 -2? -J2 -?2 -?2 -2J where the heavy line means a tangency of intersection number 2 between the corresponding curves of the resolution. Since Z(2~II~19) is the boundary of a regular neighborhood of the corresponding configuration of curves~ this shows that Z(2~II,19) bounds a simply connected four manifold Y with negative definite intersection form of signature -8 (which must hence be equivalent to the standard E 8 form~ by the classification of such forms~ but this can easily be seen directly).

It is of interest to know which homology spheres bound simply connected manifolds with even definite intersection forms, For Seifert homology spheres~ the minimal resolution of the corresponding singularity will sometimes provide a positive answer. For example~ the minimal good resolution for ~(2~4k-l~8k-3)~ or what is the same~ the canonical plumbing diagram~ has even form of signature -8k. It is~ in fact~ the bilinear form commonly denoted r8k. Z(2~8k-5~12k-7) is another example giving even forms of signature -8k.

The following table gives all E(p~q,r) with p < q < r pairwise coprime~ p = 2 and q < 20 ~ or p ~ 5 and q N i0 , for which the minimal resolution gives what we want. Omitted weights are -2. Double lines represent curves of the resolution intersecting tangentially with intersection number 2. Triangles represent three curves intersecting transversally in one point. One can simplify the algorithm provided by Theorems 2.1 and 5.1 for finding a -dm~sm -dm2 -dml Algorithm 5.4. -I-f al~ ...~ an fine Cl~ ... ~ cn by Expand a.J/]c.

as a continued fraction cj ~ -a I ... ~j.... an(rood a j) ~ 0 < c.j < a..j aj/cj = [djl~dj2 .... ~dj ~sj ] ~ dji ~ 2. plumbing diagram for Z(al,... ~an) by observing that if then

a.i ° is the canonical i ~ the sense of 5.2 and 5.3) plumbing graph for Z(al~...~a n) where b is best determined in practice by estimating it via the equation b = l j i/[dj~sj~'''~dj2'djl] + i/al ''" an

We leave the proof to the reader as well as the generalization to non-coprime

We first give a slightly generalized version of the usual definition of ~-in

for all d E H2(X;Z) will be called (Z/2)-homology sphere if and only where c(p~q) is the function introduced in IN1] described below, e(M) = -75i/~i and in case (ii) we have chosen the Seifert invariant so that (~i-6i) is odd for all i (possible, by replacing Bi by 6i-+ 5i if necessary for each i > i ~ and then adjusting B1 s_oo e (M) i__ssunchanged).

Here c(p~q) is defined for coprime integer pairs uniquely determined by the recursions (p~q) with p odd.

It is = q- i - 4Np~q ~ q odd where

N P~q = #[l<i-~<pi<q (mod q)} .

Proof. (i) is the special case of Theorem 6.2 with M = M(0 ; (p~q)) = L(q~p) for q odd. Equation (ii) is a way c(p~q) originally came up in [NI]. This function was renamed t(q;p) and generalized by Hirzebruch and Zagier ([HI]~ [HZ]~ especially pp. 245-246) and (iii) is a selection of the many formulae for t(q;p) given there. The last formula is especially interesting~ since by Gauss~ (-i) Np~q = (~q) is the quadratic residue symbol~ so the last formula implies

c(p~q) ~ i - 2(~) + q (mod 8) , q odd.

We have appended, at the end of this section 6, a table from IN-l] for c(p,q) . All values for p ~ 27 and q ~ 26 are given.

If p a__nnd q are coprime integers, then p/q has a continued fraction p/q = [bl,b2,...,bs] (see Theorem 5.1) with each b. even if and only if exactly one of p and q is even. There is then a unique such expansion satisfying in addition: Ibil ~ 2 fo___Kr i >i.

The proof of this lermma is an easy induction which we omit.

Applying this lemma and Theorem 5.1, we can express M 3 = ~P(F) , where F is a weighted tree

F = bll

b12 bml bm2 ~l~s 1 bm,s m with all weights even, Ibij I ~ 2 for j > i , and ~i/~i = [bil,...,bi,s. ] for each i . If we take X = P(F) , so M = ~X , then the definition of ~(M)i reduces to ~(M) = sign (X) (mod 16) . Using the diagonalization of the intersection matrix A(F) of X = P(F) described in the proof of Theorem 5.2, we see sign X = I ~(~i'~i ) + sign e(M) where ~(~i,gi ) =~[jll<j-<si,bij>0 } - #[jll<j<si~bij<0 } + sign(~i/gi ) . following recursion formulae follow directly from this definition of ~(~9) : The If we define c~(p~q) = ~(q~q-p) ~ then it is easy to deduce that c I satisfies the recursion formulae defining c ~ so c1(p~q) = c(p~q) . Thus ~(q~q-p) = c(p~q) so ~(~) = c(~- 9~) ~ completing the proof of the formula for case (ii).

The proof in case (i) can be done similarly~ although in this case M cannot be written as ~PF where F has only even weights~ which complicates this approach slightly. A more interesting proof uses the following theorem.

a(M,T) ~ ~(M) (mod 16) where ~(M,T)

is the Browder Livesay invariant.

By Hirzebruch [H2] (see also [HJ] and [AS])

~(M~T) = sign(X~T') - IF] ? [F] where IF] E H2(X;Z) on H2(X;~ ) ~ we have H2(X;Z/2) satisfies is the represented homology class. Since T ~ acts trivially sign(X,T I) = sign X . We must thus only show that IF] 2 E [F]2 ? x = x ? x for all x E H2(X;Z/2) . But x = T~x by

~(H,T) = Z (c(~i~$i) + sign ~i ) + sign e(M)

Theorem 6.5. M = ~ 3 (~i~" .. ~ n ) with ~i pairwise coprime embeds in S 5 a_~s~ fibered knot~ the signature of whose fiber V i_~s = t ( ~ l ~ 2 ~ 3) of [HZ] if n = 3.

Proof. If E 5 ( ~ i ~ . . . ~ n) is defined just like ~ 3 (~l~...~n) but using (n-3) instead of (n-2) equations~ then by Hamm [Ha]~ there is a '~ilnor fibration" of the complement of Z 3 in ~ 5 ~ whose fiber V has the above signature (see also Nirzebruch [H3]). Furthermore~ if ~i' "''~ ~n are pairwise coprime~ then ~ 5 is a homotopy sphere~ so E 5 ~ S 5 . V is stably parallelizable~ so its intersection form is even~ so ~(M) = sign(V) (mod 16) if ~(M) is defined (e.g.~ ~i pairwise coprime).

For n = 3 the function in the above theorem was denoted t ( ~ l ~ 2 ~ d 3) and studied and tabulated by Hirzebruch and Zagier ([HZ]~ table on page 118).

TABLE OF c(p~q) 3 5 7 9 ii 13 15 21 25 0 0 0 0 0 0 0 0 0 -i i -i i -i i -i i i

-2 2 -2 2 2 i -i -3 3 i -i -3 -i 3 0 0 -4 4 0 4

i -i -5 5 5 2 2 -2 -2 -6 6 -2 I -i -i i -i -7 -I -I

0 0 0 0 3 I -i -3 9 2 2 -2 2 -i0

-I 2 3 0 i 3 4 I 2 -5 0 -2 -i

I 2

Theorem 6.1 enables us to give an algorithm to compute ~(M) for an arbitrary (Z/2)-homology sphere obtained by plumbing. Note that a necessary condition that M = 5P(F) be a (Z/2)-sphere is that F be a tree and all the genus weights vanish.

bjf _ _ © where b.l and b.j are the weights of vertices i and j .

Move 1. I f b. i s even r e p l a c e F by the d i s i o i n t u n i o n r' of r I, ... ,r s. Move 2. I f b.1 i s odd~ r e p l a c e F b y

F' = of (ii) If M = 5P(F) is a (Z/2)-sphere, define a subset S(F)

inductively as follows: a) If F 0 is a set of isolated points with odd weights~ put S(F0) of the vertices equal to S(F) = S(F') U [i} ,

= s(F') ,

S(F) = S(I") U [i} if j ~ S(F') , = S(F')

if j 6 S(F') . ~(M) = sign A(F) bi

(mod 16)~

I i 6s(F) where A(F)

is the matrix of th__~egraph.

b.. i 6S(F) 1

Problem 7.2. If M is a (Z/2)-homology 3-sphere with a free orientation preserving involution, is it true that ~(M) = ~0~,T) (mod 16) ?

The answer is "yes" for Seifert Z-homology spheres, and more generally for Seifert (Z/2)-spheres M(0;~i,Si)) with pairwise distinct Seifert pairs (~i~imod ~i ). In these cases we shall show in a later paper that any free involution must be in the sl-action, putting us into the situation of the proof of part (i) of Theorem 6.2.

Theorem 8.1. If e(M) ~ 0 , then ~i "'" ~m ]e(M)] = order of torsion of HI(M;Z) ? If e(M) = 0 ~ then M fibers equivariantly over the circle. If M is not a principal circle bundle over a torus~ then M fibers over the circle if and only if the fibering is equivariant.

This is due to Orlik~ Vogt and Zieschang [OVZ] and Orlik and Raymond in certain exceptional cases. The fibering~ if it exists~ is far from unique. These fiberings have been constructed explicitly from the viewpoint of homologically injective actions by Conner and Raymond [CR2]. That these fiberings are S i-equivariant follows most easily from this viewpoint. The principal circle bundles over the torus are the only Seifert fiberings which fiber over the circle but fail to fiber equivariantly. (All but the 3-torus has e(M) # 0. ) The S I-equivariant fiberings are also constructed explicitly from a plumbing viewpoint by Neumann in [N3].

The present investigation arose from the next

Proof. Clearly (v)__--->(i)--_~_~(ii)___-->(iii)i)(iv). We show (iv)--~v). We suppose first that M has infinite fundamental group. We assume M is not the 3-torus. Then by [W] if M is sufficiently large or by [OVZ] or [CR3]~ in general, any homotopy equivalence M - -M is homotopic to a fiber preserving homeomorphism, so e(M) = e(-M) = -e(M) ? Hence, e(M) = 0 . Therefore, M fibers over S I equivariantly. In fact, since M has Seifert invariants (g ; (~i,~i)) then -M has Seifert invariants (g;(~i,-~i)) . This easily yields that the Seifert invariants of M must be expressible as

M = (g ; (2,b I) ..... (2,bs) ,(~i,~i )'(~i'-~i )) for some s , k ~ 0, ~i > 2 , i = i, ..., k. Now and since the b.i are odd, this implies s is even? for M are equivalent t__~o e(M) = 0 implies ½ E b i = 0

Thus~ the Seifert invariants

Therefore, M is the orientation double covering fixed-point free sl-manifold by Seifert [S;p. 198]. This completes the proof if ~I(M) is infinite.

Our attack for the finite fundamental groups must be different. Each Seifert manifold with finite non-abelian fundamental group appears as S3/G where G is a finite subgroup of S0(4) which acts freely on S3 , that is~ a spherical space form. The 2-Sylow subgroups of these manifolds are either cyclic of order at least 4 or a generalizedqusternionicgroup. The 2-Sylow subgroups are all conjugate and in the cyclic case, there is a unique subgroup of order 4. The generalizedquaternionicgroups have a characteristic subgroup of order 4 , namely the second term of the upper central series. The quaternion group itself has a unique conjugacy class of elements of order 4. In any case, we may pass to the unique covering space corresponding to the subgroup of order 4 since this is determined up to conjugacy.

This must be a lens space L(4,1) or L(4,3) . Whether it is L(4,1) or L(4~3) will be determined by the orientation of M . Now any self homotopy equivalence f of M must preserve the unique conjugacy class of our subgroup of order 4 . Hence, f may be lifted to a self homotopy equivalence f:L(4,a)~L(4~a). If f reverses orientation, then ~ must do the same. But, L(4~a) , a = 1 j or 3 admits no orientation reversing self-homotopy equivalence since -i is not a square modulo 4. Hence~ M could not possess an orientation reversing selfhomotopy equivalence. This completes the proof of Theorem 8.2.

We assume that M is a closed oriented Seifert 3-manifold with a non-orientable decomposition space. The fibering mapping is not the orbit mapping of an S laction and its type is distinct from the Seifert manifolds considered elsewhere in this paper. With a few exceptions~ none of these manifolds support an sl-action. They do support "local S0(2)-- actions". The Seifert invariants are written (On k ; (~i,~i)) where the On refers to orientable total space and non-orientable base. They are exactly similar to the invariants for oriented Seifert fiberings with orientable decomposition space except that k represents the non-orientable genus of the decomposition space, and so, k ~ 1. Just as before the unnormalized representation is not unique.

We first observe that there is a double covering M ~ of M which is an oriented Seifert manifold with orientable decomposition space and whose unnormalized invariants are (g = k-1 ; (~l,~l),(~l,~l) ..... (~s,~s),(%,~s)) .

One can easily deduce from this that the naturality properties of e(M) = -Z~i/~i extend to the case of non-orientable decomposition space as long as the total space M is kept orientahle. But it is also easy to reduce considerations to the case of orientable decomposition spaee~ which is the method we shall follow.

Assume~ now~ that if k = i ~ then there are at least 2 singular fibers~ if k = 2 ~ then there is at least one singular fiber~ and if k = i ~ s = 2 ~ then ((~i,~I),(~2,~2)} ~ ((2~i)~(2~-i)} . We shall treat these presently avoided cases separately.

Under the hypothesis on the invariants~ the element of the fundamental group represented by an ordinary fiber generates an infinite cyclic characteristic subgroup of ~I(M) and ~I(M ~) is the centralizer of this cyclic subgroup. It is easy to check that every automorphism of ~I(M) induces an automorphism of the subgroup ~I(M ~) . Consequently~ we may lift any self-homotopy equivalence h on M to a self-homotopy equivalence h ~ on M ~ h will be orientation reversing if and only if h ~ is orientation reversing. Therefore~ we know that M ~ must be of the type exhibited in (vi) of Theorem 8.2. Consequently~ the Seifert invariants of M must be

(Onk ; (~i,~i),(~i~-Si)) .

We now wish to show that each such M actually admits an orientation reversing involution. From p. 198 of [S]~ observe that as long as the non-orientable genus k of M is evenj M is an orientable double covering of Seifert manifolds of type (N~n~ll) and (N~n~lll) using Seifert's terminology. For all k > i ~ M is also an orientable double covering of certain non-orientable 3-manifolds closely related to the classical Seifert 3-manifolds. These manifolds described by Orlik and Raymond admit "local SO(2)-actions". Although they are not classical Seifert 3-manifolds~ they would be considered as injective Seifert 3-manifolds from the point of view of Conner and Raymond. We see from the table on page 155 of [OR] that each M (type n 2 = On) is a double covering of a non-orientable local SO(2)manifold~ provided that k > I. For k = I ~ none of the tabulated double coverings will work. However~ there exists an involution on RP 2 so that an isolated point and orientation reversing circle appear as the fixed point set. This involution is embedded in the effective SO(2)-action on RP 2 . With this involution~ one may define an involution on M where k = i and the Seifert invariants satisfy the necessary conditions for an orientation reversing homeomorphism. First~ one removes the tubular neighborhoods of the singular fibers. The resulting circle bundle with structure group 0(2) is the restriction to the deleted RP 2 of the associated sphere bundle S(001) where e is the line bundle det (TRP 2) and i denotes the trivial line bundle. It makes sense~ since the bundle has a section, to flip in the 1-direction. This carries the bundle over the region away from a M~bius band into itself by a rotation in D 2 and a flip in S I . This can be extended to the deleted tubular neighborhoods. The involution has 2 isolated fixed points and so the orbit space is not a manifold. (No free involution presumably exists in case k = I . The argument to check that no free involution exists is rather complicated and the details have not been checked.)

We turn now to the omitted cases. M = (Onl ; (2~i)~(2~-i)) has an involution as described above. M = (On2 ; (i~)) can also be regarded as a torus bundle over the circle with monodromy (--~0'-i ] ~ ~ E Z . If ~ ~ 0 ~ then the fundamental group of the torus fiber is a characteristic subgroup. The outer automorphism group of ~I(M) is calculated in Conner and Raymond [CR4;6.14]. It is readily seen from this calculation that if ~ ~ 0 ~ M admits no orientation reversing self-homotopy equivalence. For ~ = 0 ~ M can be identified with [g= 0~(2~i)(2~-I)~(2~i)~(2~-i)} which does admit an orientation reversing free involution.

The remaining cases to treat are M = M(Onl ; (~i~i)) . If (~i~$i) = (i,0) then M is RP 3 ~ RP 3 which certainly has an orientation reversing homeomorphism. If B ~ 0 ~ then M = M(0nl;(~8)) also has a Seifert fibering with orientable base as M = (0;(2,1)(2,-I)~(~,8)) . If ~ = ±i this is the lens space~L(4~,2~+l) and if 181 > i it is a nonabelian spherical space form. (This corrects a statement in [OR].) In either case it admits no orientation reversing equivalence.

We may now summarize our result for the closed case as follows: (Onk ; (~j,Sj),(~j,-Sj)) not a lens

Moreover~ if k > i ~ the orientation reversing involution can be chosen to be free.

For this case we assume ~M ~ 0. Let h denote the number of boundary components. Then the Seifert invariants for M are given by ((g,h) ; (~i~Si)) (On(k~h) ; (~i,Si)) [(~i,Si)} = {(~1,81 ) ~(~i~i- 81 ) ~-.. ~(~t~St ),(~t,~t- 8t)] As before~ involutions can be constructed on each of these manifolds. However~ when the Euler characteristic of the decomposition space is odd~ we cannot expect to find free involutions in general. Added in Proof. The reason given in 8.2 for the quaternion group is incorrect. The result is still valid since the quaternion group can be embedded in SU(2) . Consequently~ the covering space associated to each subgroup of order 4 is the lens space L(4~3) ~ or equivalently~ the lens space L(4~I) if the opposite orientation of SU(2) is used. The rest of the argument proceeds as before. [G] [H2] [H3] [K] [M] [N2] ~ Injective actions of toral groups~ Topology i0 (1971)~ 283-296.

~ Deforming homotopy equivalences to homeomorphisms in aspherical manifolds~ Bull. Amer. Math. Soc. 83 (1977)~ 36-85.

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Grauert~ H.~ Uber Modifikationen und exceptionelle analytische Raumformen~ Math. Ann. 146 (1962)~ 331-368.

Hirzebruch~ F.~ Free involutions on manifolds and some elementary number theory~ Symposia Mathematica~ Instituto Nazionale de Alta Matematica~ Roma~ Academic Press V (1971)~ 411-419. _ _ ~ Involutionen auf Mmnnigfaltigkeiten~ Proceedings of the Conference on Transformation Groups~ Tulane (1967)~ Springer (1968)~ 148-166.

~ Pontrjagin classes of rational homology manifolds and the signature of some affine hypersurfaces~ Proceedings of Liverpool Singularities Symposium II (ed. C.T.C. Wall) ~ Lecture Notes in Math. 209~ Springer-Verlag (1971)~ 207-212.

Hamm~ H.~ Exotische Spharen als Umgebungsrander in speziellen komplexen Raumen~ Math. Ann. 197 (1972)~ 44-56.

Hirzebruch~ F.;J~nich~ K.~ Involutions and singularities~ Algebraic Geometry~
Papers presented at the Bombay Coll. (1968)~ Oxford Univer

Hirzebruch~ F.; Zagier~ D.~ The Atiyah Singer theorem and elementary number theory~ Math. Lecture Series 3~ Publishor Perish Inc.~ Boston~ Berkeley~ 1974.

Klein~ F.~ Lectures on the Icosahedron and the solution of equations of the fifth degree~ Dover~ New York~ 1956.

Kervaire~M.;Milnor~ J.~ On 2-spheres in a 4-manifold~ Proc. Nat. Acad. Sci. U.S.A. 49 (1961)~ 1651-1657.

Milnor~ J.~ On the 3-dimensional Brieskorn manifold M(p~q~r) ~ Papers Dedicated to the Memory of R.H. Fox~ Ann. of Math. Studies~ Princeton University Press~ 1975~ No. 48~ 175-225.

Neumann~ W.D.~ sl-actions and the ~-invariant of their involutions~ Bonner Math. Schriften 44~ Bonn~ 1970.

~ Brieskorn complete intersections and automorphic forms~ Invent.Math. 42 (1977)~ 285-293.

S I ~ to appear.

Orlik~ P.~ Seifert manifolds~ Springer Lecture Notes~ Vol. 291 (1972). gefaserter dreidimensionaler

Thomas~ C.B.~ Homotopy classification of free actions by finite groups on S 3 to appear. von Randow~ R.~ Zur Topologie von dreidimensionalen Baummanigfaltigkeiten~ Bonner Math. S chriften 14 (1962).

Waldhausen~ F.~ Eine Klasse von 3-dimensionalen Mannigfaltigkeiten~ I and II~ Invent. Math. 3 (1967)~ 308-333~ 4 (1967)~ 87-117.