<<

. 3
( 10)



>>

Vn(Pl, ... ,Pk)). The large piece, with X = 12n - 3, is independent of PI,.·. ,Pk,
and will be denoted ˜n. It can be shown that tP n is actually diffeomorphic to a
well-known manifold, namely the Milnor fiber of ˜(2, 3, 6n -1). This is the locus
of x 2 +y3 +z6n-l = e in the closed unit ball in C 3 • The boundary, ˜(2, 3, 6n-l),
has at most two self-diffeomorphisms (up to isotopy) [4], and the nontrivial one
(when it exists) extends over the Milnor fiber as complex conjugation. Thus,
there is a canonical procedure for reconstructing Vn(Pl, ... ,Pk) from its nucleus
49
Gompf: On the topology of algebraic surfaces

N n (p1, ... ,Pk). (Simply glue on the Milnor fiber by any diffeomorphism of the
houndaries.) In particular, Vn(Pt, ... ,Pk) and Vn (Q1, ... ,ql) will be diffeomor-
I)hic if their nuclei are diffeomorphic.
We consider the nuclei in detail. It is not hard to verify from the construc-
t.ion we will give that the inclusion Nn(Pl' ... ,PI.) '--+ Vn(P1' ... ,Pk) induces an
i˜()morphism of fundamental groups. We will restrict attention to the simply
("onnected case: N n (p, q), p, q relatively prime. In this case, the nucleus has
"I = 0 and ˜ = 2. It follows from Freedman theory that two of the manifolds
Vn(p, q) will be homeomorphic if and only if their nuclei are. We have already
H(-en that nondiffeomorphic Vn(p, q)'s have nondiffeomorphic nuclei. (The con-
VPfse also holds in the cases where two Vn(p, q)'s are known to be the same. That
iN, the nuclei N 1 (p) are all diffeomorphic.) It follows that our families of home-
( H Ilorphic but nondiffeomorphic elliptic surfaces yield families of homeomorphic

IHlt nondiffeomorphic nuclei. Thus, we may trim away much useless complexity
fn)in the elliptic surfaces, and see their most important topological properties
c'apt,ured in the much simpler nuclei. As a measure of the simplicity of these
"tI(˜lei, consider the ones Nn(p) with a single multiple fiber. (Note that these are
n.11 nondiffeomorphic if n > 1.) Each Nn(p) has a handle decomposition with only
three handles: One O-handle and two 2-handles. (Equivalently, it has a perfect
Morse function with only three critical points.) These handle decompositions
C"Ul be drawn explicitly.


The art of drawing pictures of 4-manifolds as handle:bodies is called Kirby
ftl.l(:1Llus. (See, for example, [18].) Suppose, for example, that H is a 4-dimensional
Illt.lldlebody built with one O-handle and k 2-handles. The O-handle is just a 4-ball
//,1, and each 2-handle is a copy of D2 X D2, glued onto the boundary 8B4 = S3
,,1()1l˜ S1 X D 2 , by some embedding Ii : Sl x D 2 ˜ S3 (1 ::; i ˜ k). Figure 1 is
" Hchematic picture of this, which is literally a pair of 2-dimensional l-handles
fI."H˜d to a O-handle (D2) to yield a punctured torus. To specify fi up to isotopy,
1 x {OJ up to isotopy,
\\ff' need two pieces of information. First, we specify filS

wl.iell is just a knot in S3. Then, we specify the twisting of the normal vectors
'''I follows: If p i= 0 is a point in D2, then li(Sl x {OJ) and li(Sl x {p}) are a
1)l1i .. of disjoint knots in S3. Their linking number, an integer called the framing
•• f f,ll˜ 2-handle, specifies the normal twisting as required. To completely deter-
IIlill(' H, we must simultaneously specify all gluing maps of 2-handles. This is
u,·I,i(-vcd by a framed link, or a link f : Ilk S1 '--+ S3 with an integer attached to
Gompf: On the topology of algebraic surfaces
50

each component. Identifying 8 3 with R3 U {oo}, we may draw the framed link to
obtain a complete picture of H. (For comparison, the 2-dimensional handlebody
drawn literally in Figure 1 is specified by the nontrivial link of two O-spheres in
8 1 , together with a framing in Z2 for each I-handle. If we change one framing,
i.e., put a half-twist in one 1-handle, the manifold is changed to a punctured
Klein bottle.)
Now we can see the manifolds Nn(P). The simplest pictures, Nn,(O) and
Nn, = Nn,(l) are shown in Figure 2. The boxes indicate twists in the link. In the
picture for Nn,(O), for example, the box indicates n 3600 left-handed twists. Our
previous discussion implies

Theorem.
a) For any fixed odd integer n ˜ 3, the two bandlebodies shown in Figure 2 are
homeomorphic but not diffeomorphic.
˜
b) For each homeomorphism type shown in Figure 2 (n 1 arbitrary) there are .
infinitely many diffeomorphism types.
':;
˜
Figure 2 is undoubtably the most visuaJizable example known of homeomorphic
but nondiffeomorphic manifolds, with the possible exception of Akbulut's ex-
ample [1] of two such manifolds, each built with a single 2-handle. (His proof
depends on a calculation in Donaldson-Floor theory by Fintushel and Stem and,
of course, Freedman theory for the topological part.)
Figure 3 shows a general Nn,(p). The ribbon at the top represents a spiral
with p loops.

Theorem.
Fix n ˜ 2. If n is even, fix the mod 2 residue of p. Then the manifolds in tbe
infinite family given by Figure 3 are all homeomorphic, but no two are diffeo-
morphic.

It remains to sketch the construction of the nuclei. Details (and further
properties of the nuclei) can be found in [13]. First, consider Vn,- By perturbing
the elliptic fibration if necessary, we may assume that Vn, has a singular fiber of
. a type called a cusp fiber. This is a 2-sphere in Vn, which is smoothly embedded
except at one non-locally flat point where it is locally a cone on a trefoil knot
(i.e., the zero locus of x 2 + y3 in C2 ). A regular neighborhood of this will be a
handlebody made by attaching a 2-handle to a O-framed trefoil knot. We may
51
Gompf: On the topology of algebraic surfaces

also find a section for V n , i. e., a smoothly embedded 2-sphere intersecting each
fiber transversely in a single point. (In V1 , any of the nine cpt's created by the
blow-ups will work. Vn is made by fiber sum from n copies of V1 , and a section is
obtained by splicing together sections of Vi.) The section will have normal Euler
number -no Let N fa C Vn denote a regular neighborhood of the section union
the cusp fiber. The reader can check that this is obtained from. a handlebody
on a O-framed trefoil by adding a 2-handle along a -n-framed meridian. This is
Reen explicitly in Figure 2. An easier task is to verify that the intersection form
of N n is [˜ '!nl which is unimodular, proving that fJNn is indeed a homology
sphere. A routine computation with Kirby calculus shows that fJNn , as seen in
Figure 2, is -˜(2, 3, 6n -1). More work shows that the closed complement ˜n is
the Milnor fiber. (Question: Can this be seen directly by algebraic geometry?)
To construct nuclei in general, observe that by construction ·Nn contains a
neighborhood of a singular fiber. Such a neighborhood contains a continuous
fn.mily of regular fibers. Change Vn to Vn (P1,'" ,pic) by performing logarithmic
t.ransforms on k regular fibers in the interior of N n . This will not disturb the
homology sphere or ˜n, but it will change N n to a new manifold Nn(Pt, .•• ,pic),
which is the nucleus of Vn(Pl"" ,pic). Figure 3 may now be derived by Kirby
('I"lculus. Details appear in [13].


Ileferences


(1] Akbulut, S. An exotic 4-manifold. Preprint.

(2] Atiyah, M. (1988)..New invariants of 3 and 4 dimensional manifolds. In The
Mathematical H.eritage of Herman Weyl. Proc. Symp. Pure Math., 48. Amer.
Math. Soc.

[a] Barth, W., Peters, C. ˜ Van de Yen, A. (1984). Compact complex surfaces.
In Ergebnisse de·˜ Mathematik, Series 3, vol. 4, Springer.

III] Boileau, M. & Otal, J. (1986). Groupe des diffeotopies de certaines varietes
de Seifert. C.R. Acad. Sc., (Series I) 303, 19-22.

Ir.1 Donaldson, S. (1987). Irrationality and the h-cobordism conjecture. J. Diif.
Geom., 26, 141-168.

IU) Donaldson, S. Polynomial invariants for smooth four-manifolds. Preprint.
52 Gompf: On the topology of algebraic surfaces

[7] Fintushel, R. & Stern, R. Instanton homology of Seifert fibered homology ,
three spheres. Preprint.

[8] Freedman, M. (1982). The topology of four-dimensional manifolds. J. Diff.
Geom., 17, 357-453.

[9] Friedman, R. & Morgan, J. (1988). On the diffeomorphism types of certain
algebraic surfaces, I and II. J. DifJ. Geom., 27, 297-398.

[10] Friedman, R. & Morgan, J. To appear.

[11] Gieseker, D. (1977). Global moduli for surfaces of general type. Invent.,:
Math., 43, 233-282.

[12] Gompf, R. (1988). On sums of algebraic surfaces. Invent. Math., 94, 171-174. :.'
[13] Gompf, R. Nuclei of elliptic surfaces. To appear.

[14] Gompf, R. Sums of elliptic surfaces. To appear.

[15] Griffiths, P. & Harris, J. (1978). Principles of algebraic geometry. New York: ';
John Wiley & Sons.

[16] Harer, J., Kas, A. & Kirby, R. (1986). Handlebody decompositions of com-··;'
plex surfaces. Memoirs, Amer. Math. Soc., 62, (350).

[17] Kas, A. (1977). On the deformation types of regular elliptic surfaces. In Com-'
plez Analysi8 and Algebraic Geometry, pp. 107-111. Cambridge: Cambridg˜'
University Press.

1374·˜:.
[18] Kirby, R. (1989). The topology of 4-manifolds. Lecture Notes in Math.,
Springer.

[19] Kodaira, K. (1969). On the structure of compact complex analytic surfac "
IV. Am. J. Math., 90, 1048-1066.

[20] Mandelbaum, R. (1979a). Decomposing analytic surfaces. In Geometric Top'
09Y, Proc. 1977 Georgia Topology Conference, 147-218.

[21] Mandelbaum, R. (1979b). Irrational connected sums. 7'rans. Amer. Matl(
Soc., 247, 137-156.
.!
[22] Mandelbaum, R. (1980). Four-dimensional topology: an introduction. BullJ
Amer. Math. Soc., 2, 1-159. ,I
53
Gompf: On the topology of algebraic smfaces

[23] Moishezon, B. (1977). Complex surfaces and connected sums of complex
projective planes. Lecture NotelJ in Math., 603, Springer.

okonek, C. & V˜ de Ven, A. (1986). Stable bundles and differentiable
[24]
structures on certain elliptic surfaces. Inventiones Math., 86, 357-370.

[25] Wall, C.T.C. (1964&). Diffeomorphisms of 4-manifolds. J. London Math.
Soc., 39, 131-140.
(26] Wall, C.T.C. (1964b). On simply-connected 4-manifolds. J. London Math.
Soc., 39, 141-149.
54 Gompf: On the topology of algebraic surfaces




Figure 1




Figure 2



2
-np +p-l




Figure 3
The topology of algebraic surfaces with
irregularity and geometric genus zero
I)IETER KOTSCHICKf)
Queens' College, Cambridge, CB3 9ET, England, and
'I'he Institute for Advanced Study, Princeton, NJ 08540 USA




Interest in algebraic surfaces with pg = hO(O(K» = 0 goes back to the work
c.r Enriques and Castelnuovo in the 19th Century. After Clebsch had proved that
.'urves with pg = 0 are rational, these authors considered the analogous question
rOl'surfaces. It was clear to them that in this case the irregularity q = h1 (O(K»
III\H to be controlled as well.
III 1894 Enriques constructed his now famous surface, which is irrational with
II :˜ pg = 0, disproving the most obvious rationality criterion. Two years later
(!Mt,clnuovo proved that the modified conditions q = P2 = 0 do imply rationality.
'1'11\18 he substituted the second plurigenus from PA: = hO(O(K k » for the first. (For
th˜ Enriques surface K is a 2-torsion bundle, so the bigenus is one.)
()ver the next forty years more examples of irrational surfaces with q = pq = 0
W"I'(˜ constructed. Like the Enriques surface they were all elliptic. Only in 1931 did
(Jud.˜RUX [G] find a surface of general type with these invariants. His construction
3
WAM disarmingly simple: divide the Fermat quintic in CP by the standard free Zs-
'u,t.ioll on the coordinates. Campedelli also gave an example of a surface of general
'Y1H', introducing his "double plane" construction. This has Hl (X, Z) = Z˜.
()u the basis of these examples Severi conjectured in 1949 that the two conditions
II. (<It, Z) = 0 and Pg = 0 should imply rationality. (Recall that q = 0 implies
II,()(,Q) = 0.) This was disproved by Dolgachev, who in 1966 gave examples
,at -.ililply connected irrational elliptic surfaces with vanishing geometric genus, d .
. Illvl. Then, in her 1982 Warwick thesis, R. Barlow constructed a simply connected
Itil fh,(˜(˜ of general type with Pg = 0, cf. [B1]. To this day it is the only such example.
III this paper we want to summarize the progress made in understanding alge-
tunic surfaces with pg = 0 since the survey of Dolgachev [Dv]. In view of the
Ilu c·t,,,,eular advances in 4-manifold topology initiated by Freedman and Donaldson
1


Wf' .˜x t,(lud the classification programme from the algebraic or analytic to the smooth
....1 topological categories. Although the results are not yet complete, a coherent
,,,·t,tln. emerges. Namely, if the topology of a surfa.ce is sufficiently complicated (e.g.
r"'111'· f, ludamental group), then it determines the˜ smooth structure. On the other
hn... I, if the topology is simple (e.g. rational surfaces) then there are homeomorphic
lUI f",·.'s which are not diffeomorphic to the given one.

t1t˜If,.purl.cd by NSF Grant Number DMS-8610730
Kotsehick: The topology of algebraic surfaces with q = Pg = 0
56

The reader should be warned that nothing is proved in this article. For back-
ground for the papers referred to in the third section consult the forthcoming book '
by S. K. Donaldson and P. B. Kronheimer (Oxford UP). The philosophy is well .:
explained in [FM2].
I would like to thank R. Gompf for pointing out an erroneous claim about (non- '
algebraic) 4-manifolds with 2-torsion in HI (X, Z) made in the oral version of this'
lecture. (

1. ALGEBRAIC CLASSIFICATION
The following theorem is part of the classification of surfaces [BPV], Chapter:
VI.
1 (Enriques-Kodaira). Let X be a smooth minimal compact complex
THEOREM
algebraic surface with q =pg = o. Then X is one of the fonowing: "
A) a minimal rational surface
B) an Enriques surface
C) a minimal properly elliptic surface
D) a minimal surface of general type.

We can describe these surfaces in more detail:
A) The minimal rational surfaces are p2 and the Hirzebruch surfaces En, n >



0,2,3,4, .... Here En is the pI-bundle P(O $ O(n» over pl. Thus EO = pI X p˜,
The surface El is not minimal, it is p2 blown up once. Note that K 2 = 9 for .
and K 2 =8 for En.
B), C) Let Xg be the rational elliptic surface obtained from p2 by blowing up t ,/
nine base points of a generic cubic pencil. Then Dolgachev [Dv] proved that t :
surfaces in B), C) are precisely those obtained from X g by performing logant .
transformations on at least two different smooth fibers. We denote the surf ;:'
obtained in this way by X (PI , ... ,Pi), where the Pi are the multiplicities of t
logarithmic transformations, and call it a Doigachev surface. (Some authors r
this name for the case when there are only two multiple fibers and their multiplicitii
are relatively prime.) Up to the deformation X(pI, ... ,Pi) does not depend on t',
choices involved in the construction. Now X(2,2) is the Enriques surface. It:':;,
distinguished from the other Doigachev surfaces by its Kodaira dimension (= ,:
rather than 1). Note second K 2 = 0 for all of these surfaces, just like for X9. l
D) For a minimal surface of general type we have c˜ > 0 and C2 > o. If in addit "
q = Pg = 0, then c˜ + C2 = 12 by Noether's formula, and K 2 = c˜ E {I, 2, ... , ˜
Examples of surfaces realizing all these values of K2 are contained in the ta
on page 237 of [BPV]. The reader can also find there an overview of difF
methods used in those constructions. Among them are the classical Godeaux .
Campedelli constructions mentioned in the introduction. For a modern discussi
of these see [R3J. The author knows of only three new constructions which he;
appeared since [BPV]. One is J. H. Keum's method using branched double covers
57
Kotschick: The topology of algebraic surfaces with q = Pg = 0

Enriques surfaces [Ke]. He uses this to give examples of surfaces with'K2 = 2,3,4.
Some of them are known to coincide with surfaces constructed earlier by different
methods and contained in the table of [BPV], whereas others are new. Another
recent method of construction is due to M. Oka [02], who uses singularity theory
and toroidal embeddings. Unfortunately this method has not yet produced new
examples. Finally, a new surface with K 2 = 2 is contained in Xiao Gang's book on
˜enus 2 fibrations [X] , Example 4.11, cf. also [Be].
Having long lists of examples is interesting, but it cannot be the ultimate aim of
the theory. Rather the aim is to give complete constructions of all such surfaces.
To this end one introduces another invariant, besides q, pg and K 2 • This is the
p;roup TorsX of torsion divisors. H q = 0 then TorsX ˜ H 1 (X, Z). For given q,
l'g and K 2 one tries to pin down the possible groups for Tors X, and then gives an
('xhaustive construction for the whole moduli scheme of surfaces with a fixed torsion
p.;roup.
Let us look at the case q = pg = 0, K 2 = 1 in detail. These surfaces are
('ltIled numerical Godeaux surfaces, in honor of Lucien Godeaux who constructed
the first such surface [G]. In this case a result of Deligne and Bombieri [Bo] implies
\'l'orsXI ˜ 6. This has been refined by Reid [RI], who has also determined the
Ilu)duli spaces in some cases, by using explicit presentations of the canonical ring.
Hili result is:
For a numerical Godeaux surface TorsX is one 0[0, Z2, Z3,
'I'IIEOREM 2 (Reid).
Z.t or Zs. In the last three cases the moduli schemes are irreducible 8-dimensional
\'I1.,.h˜ties.

It. is natural to conjecture that the irreducibility result will extend to the cases
ur t.orsion 0 or Z2. However, the problem becomes harder for smaller groups, and
twC) recent attempts on the case of Z2 by Reid [R2] and by Catanese-Debarre [CD]
Af'C'·11l to have failed. At least the existence of such surfaces is known. Examples
httv(' been constructed by Barlow [B1] for the case of no torsion and by Oort-Peters
I().») and by Barlow [B2] for the case TorsX = Z2. (The Oort-Peters surface is a
.˜I. Ui(' cousin of Xiao's surface mentioned above. Both of them are genus 2 fibrations
IX),)
course one would like to have an analogue of Theorem 2 for K 2 > 1 as well.
()f
1I.,w('ver, except for some unpublished work of Reid on the case K 2 = 2, not much
1,11'ln:-l t,o be known.

2. HOMEOMORPHISM TYPES

II. c.rder to identify the homeomorphism types of some of the surfaces described
1.1 I, W(˜ have to determine their fundamental groups. The rational surfaces are, of
•·.."HU·, simply connected. For the elliptic surfaces we have the result of Dolgachev
II tvl: 11"1 (X(p, q» = Zk, with k = g.c.d.(p, q). In particular X(p, q) is simply
H'IIIIC·Ct.(˜d if p and q are relatively prime. The fundamental group is non-abelian if
t1"'1 t' IU'(˜ more than two multiple fibers.
=0
58 Kotsehick: The topology of algebraic smfaces with q = Pg


For surfaces of general type the answers are less coherent. In general it is very ,:'
hard to control 11"1 in complicated constructions. Only for Oka's method is there.:
a general theorem [01], which says that 11"1 is always finite cyclic. This should˜'
encourage attempts to use his methods to find new surfaces (say simply connected -
ones). Here are some other results. If K 2 = 9, then Yau's uniformization result ':
implies that the fundamental group is infinite. H K 2 = 8, then the only known;'
examples have infinite 11"1, and for K 2 = 7, the fundamental group is unknown. For'
K 2 = 6 the example cited in [BPV] has infinite 1rl. For 1 :5 K 2 < 6 surfaces with
various finite fundamental groups are known. For K 2 = 4 there is also an example'!
of J. H. Keum [Ke] with fundamental group Z" )4 (Z2)2.
As in the previous section, more is known in the case of numerical Godeaux:
=1.Here HI (X, Z) =0, Z2, Z3, Z4 or Zs by Theorem 2. Moreover,:
sUrfaces, i.e. K 2
1rl =HI for all surfaces with HI = Z3, Z4 or Zs, and for the other known surfaces';
except possibly the Oort-Peters example. This is because the Barlow surface [Bl},
is simply connected by construction, and the construction in [B2] gives 1rl = Z2.'
In the cases of torsion Z3, Z4 or Zs all surfaces have the same topological type
because of the irreducibility of moduli (Theorem 2). Thus it is enough to exhibit
one example with 1r1 = HI for each case. This is done in [Be]l for Z3 and in [M}
for Z4. For Zs it is obvious in view of the classical Godeaux construction. ':
We now give the homeomorphism classification in simple cases, which are th ˜
only ones where it is known. For the simply connected case we call on Freedman'˜
work [Fl], who proved that smooth 4-manifolds are classified by their intersectio '
forms.
3 (Freedman). The simply connected Dolgachev surfaces are homeomor+
THEOREM
pbic to Xg, and the Barlow surface is homeomorphic to X s . Moreover any simp]'
connected minimal surface of general type with pg = 0 is homeomorphic to X 9 - K .,
ortoS2 xs'l. '
In the non-simply connected case we can only deal with the case 11"1 = Zk, du
Y




to the work of Hambleton-Kreck. Building on the fact that surgery works [F2
they extend Freedman's classification to the case 1r1 = Zk with k odd [HKlt
They have also dealt with the case of even k [HK2]. Here the classification ·
more complicated, because it involves the so-called w2-type of the manifold. Som˜
consequences of their results are:
4 (Hambleton & Kreck). A numerical Godeaux surface with 11"1 = Zk
THEOREM
Z"
homeomorphic to Xs#'Ek, where Ek is any rational homology sphere with 7r1 =
Similarly the Dolgachev surfaces X(p, q) and X(q', q') are homeomozphic ifand onJ
t
if g.c.d.(p, q) = g.c.d.(p', q') = k and, when k is even, f + i == + of mod 2.
1 Beware, the construction of a simply connected surface suggested in [Be] does not work. On th
other hand, the constructions using double covers of Enriques surfaces are essentially contained i
[Ke].
=Pg = 0
Kotschick: The topology of algebraic surfaces with q 59

Here the complication in the case of even k comes precisely from the w2-type.
If k is odd and in the case i + t == 1 mod 2 for even k the intersection form of
.\((p,q) is diagonal and X(p,q) is homeomorphic to Xg#'Ek. In the remaining case
t1w intersection form is H (D E a , where H = (˜ ˜) is the standard hyperbolic, and
Jt,'H is taken to be negative definite. This case includes the Enriques surface.

3. DIFFERENTIABLE STRUCTURES
In the previous section we did not discuss the homeomorphism classification for
t.hose manifolds for which it coincides with the diffeomorphism classification. This
'N t.he case for the Hirzebruch surfaces ˜n, which by construction are diffeomorphic
t.o S2-bundles over SJ. It is a trivial consequence of Wall's work [W] that ˜n is
....t'J X S2 for n even and p2#p2 for n odd. A much deeper result is the following
C'uuHcquence of a theorem of Ue [U]:
(Ue). The diffeomorphism type of a Dolgachev surface with three or
'I'IIEOREM 5
'''Ul'(˜ multiple fibers is determined by its fundamental group.

Although this r˜sult is far from trivial, its proof is in a sense elementary. On
t.lu' other hand, Theorems 3 and 4 given in the previous section are proved us-
'II˜ Freedman's surgery [Fl), [F2]. This means that they leave room for non-
.lilr(˜(>lnorphic surfaces in the same homeomorphism type. Indeed, in many cases
",u·ll non-diffeomorphic pairs can be found using gauge theory. (This of course
,UfI4lH'oves the 4-dimensional h-cobordism conjecture [Dl], [D2].)
'I'h(˜ first result of this type was obtained by Donaldson [Dl], [D2], who proved
tIIU.t, X(2,3) is not diffeomorphic to X g • His argument was extended by Friedman
Itutl Morgan [FMl] to show, among other results, the following:
6 (Friedman & Morgan). No Dolgachev surface is diffeomorphic to Xg_
'I'III':()REM
A'f.,'(˜()ver the map from deformation types to diffeomorphism types of simply con-
tut."/.t·d Dolgachev surfaces is nnite-to-one.

It'I'i(-<!man and Morgan [FMl] also showed that this is still true after an arbitrary
IIUlllb(˜r of blowups. As a corollary one finds that XI: has infinitely many smooth
"t., IIt't,ures for k ˜ 9. We have given a simpler proof of Theorem 6 in [K3]. This
U˜"N n new invariant <PI defined using gauge theory on 80(3)-bundles with non-
h ivilLl Stiefel-Whitney class, instead of Donaldson's r-invariant coming from 8U(2)-
t'"lIcll(˜H [Dl], [D2]. Using this type of argument we have also proved [K2]:

7 (Kotschick). The Barlow surface is not diffeomorphic to X s .
II'III':()HEM

'I'll(' generalization of this to arbitrary blowups is very complicated, and has not
'tf˜f'U (·oInpleted yet. Thus, to prove the optimal Theorem 8 below, we have reverted
i I. 11\ 4], see also [K1], Chapter V, to the case of the Donaldson invariant instead
ttl. HII' own invariants from [K3].
Kotschick: The topology of algebraic surfaces with q =Pg = 0
60

8 (Kotschick). For all k the k-fold blowup of the Barlow surface is not
THEOREM
diffeomorpbic to a rational or (blown up) Dolgacbev surface.
The proof of Theorem 8 is rather interesting, because it uses naturally the ge- \
ometry of surfaces of general type, and suggests a way of extending the proof to :
arbitrary such surfaces. H this is successful it will prove the following folk conjecture :'
in the case pg = 0 and 1r} = 0: '
Surfaces of different Kodaira dimension are not diffeomozphic.
CONJECTURE.

Now let us look at the case of non-simply connected surfaces. In view of Theorems '
4 and 5 the interesting cases are those of finite (cyclic) fundamental groups. For:;
these one can obtain results by going over to the universal cover and applying the<
theory of Donaldson polynomials [D3]. This is done for elliptic surfaces in [FM3]';
and for the Godeaux surface in [HKl]. However, direct arguments are possible 88,:'1
:
˜.
Thus Maier [Mal extended the work of Donaldson [DIl, [D2] and of Friedman and?
Morgan [FMl] to non-simply connected Dolgachev surfaces, proving that they give;;
infinitely many smooth structures on every X9#'Ek. Similarly Okonek [Ok] treated:˜,
the case of elliptic surfaces homeomorphic to the Enriques surface. Technically this˜j
is the simplest possible case, because the r-invariant [Dl], [D2] takes a simple)
form, allowing arguments of the type used for SO(3)-invariants in [K3l. Finally,:˜
our own method of proof using SO(3)-invariants [K2], [K3] works uniformly for:,;
any fundamental group, as long as the intersection form is odd. Thus we can deal.˜
with non-simply connected numerical Godeaux surfaces and with those Dolgachev'
surfaces which have diagonal intersection forms. Our method does not apply to the;,
Enriques surface (at least not in the naive form given in [K3] d. the discussion in'
§4 of [K3]).

Constructions of algebraic surfaces: ,;.,
[Bl] R. N. Barlow, A simply connected surface of general type with pg = 01
Invent. math. 79, 293-301 (1985).
[B2] R. N. Barlow, Some new surfaces with Pg = 0, Duke Math. Joumal51, No..
4 (1984), 889-904.
[BPV] W. Barth, C. Peters and A. Van de Ven, Compact Complex Surfaces.';
Springer-Verlag, Berlin 1984. "
[Be] A. Beauville, A few more surfaces with pg = 0, Letter dated Sept. 3, 1984.,{
[Bo] E. Bombieri, Canonical models for surfaces of general type, Pub!. Math˜
IHES 42 (1973), 171-219. >




[CD] F. Catanese and O. Debarre, Surfaces with K 2 = 1, Pg = 1, q = 0, J. rein˜
angew. Math. 395 (1989), 1-55.
, 11


[Dv] I. Dolgachev, Algebraic surfaces with q = Pg = 0, in "Algebraic Surfaces";"!
˜
CIME 1977, Liguore Editore Napoli (1981), 97-215.
Kotschick: The topology of algebraic surfaces with q = Pg = 0 61

[G} L. Godeaux, Sur une surface algebrique de genre zero et de bigenre deux, Atti
Acad. Naz. Lincei 14 (1931), 479-481.
[Ke] J. H. Keum, Some new surfaces ofgeneral type with pg =0, (Univ. of Utah
preprint).
[M] Y. Miyaoka, 'lHcanonical Maps of Numerical Godeaux Surfaces, Invent.
luath. 34, 99-111 (1976).
[01] M. Oka, On the resolution of Hypersurface Singularities, Advanced Study
ill Pure Ma.th. 8 (1986), 405-436.
[02] M. Oka, Examples of algebraic surfaces with q = 0 and pq :5 1 which are
locally hypersurfaces, in "A Fete of Topology", ed. Y. Matsumoto et. al., Academic
Press, Boston, 1987.
[OP] F. Oort and C. Peters, A Campedelli surface with torsion group Z/2, Inda-
I(lttiones Math. 43, Fasc. 4 (1981), 399-407.
[Rl] M. Reid, Surfaces with pg = 0, K 2 = 1, Journal of Faculty of Science, Univ.
c.f Tokyo, Sec. lA, Vol. 25, No.1 (1978), 75-92.
[R2] M. Reid, Infinitesimal view of extending a hyperplane section - deforma-
t.ion theory and computer algebra, in "Hyperplane sections and Related Topics",
(L'Aquila, May 1988), 00. L. Livomi, Springer LNM (to appear).
[R3] M. Reid, Carnpede11i versus Godeaux, Proc. of the 1988 Cortona com. on
AI˜ebraic Surfaces, INdAM and Academic Press (to appear).
[Xl G. Xiao, Surfaces fibres en courbes de genre deux, Springer LNM 1137, 1985.

Ilotneomorphism types:
[Fl] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential
(:('0. 17 (1982), 357-453.

(F2] M. H. Freedman, The disk theorem for four-manifolds, Proc. ICM Warsaw
(IHS4),647-663.
IIII{l] I. Hambleton and M. Kreck, On the Classmcation ofTopologica14-Manifolds
\\'it.Jl Finite Fundamental Group, Math. Ann. 280, 85-105, (1988).
II-II{2] I. Hambleton and M. Kreck, Smooth structures on algebraic surfaces with
'J'('lie fundamental group, Invent. math. 91, 53-59 (1988).

I )in'(˜rentiable structures:
(1)1] S. K. Donaldson, La topologie differentielle des surfaces complexes, C. R.
Ae...<1. Sci. Paris Sere I Math. 301, (1985), 371-320.
(I )2] S. K. Donaldson, Irrationality and the h-cobordism conjecture, J. Differential
(:.'0. 26 (1987), 141-168.
)3] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology
11
appear).
( to
Kotschick: The topology of algebraic surfaces with q = Pg = 0
62

[FM1] R. Friedman and J. W. Morgan, On the diffeomorphism types of certain
algebraic surfaces I, J. Differential Goo. 27 (1988), 297-369.
[FM2] R. Friedman and J. W. Morgan, Algebraic surfaces and 4-manifolds: some
conjectures and speculations, Bull. AMS 18 (1) (1988), 1-19.
[FM3] R. Friedman and J. W. Morgan, Complex versus differentiable classification
of algebraic surfaces, (Columbia Univ. preprint).
[K1] D. Kotschick, On the geometry of certain 4-manifolds, Oxford thesis, 1989.
[K2] D. Kotschick, On manifolds homeomorphic to CP2#8Cp2, Invent. math. '
95, 591-600 (1989).
[K3] D. Kotschick, SO(3)-invariants for 4-manifolds with bt = 1, (Cambridge and
lAS preprint).
[K4] D. Kotschick, Positivity versus rationality of algebraic surfaces, (to appear).
[Ma] F. Maier, On the diffeomorphism type of elliptic surfaces with cyclic funda- ,
mental group, Tulane thesis, 1987.
[Ok] C. Okonek, Fake Enriques surfaces, Topology Vol. 27, No.4 (1988), 415-427.
[U] M. Ue, On the diffeomorphism types of elliptic surfaces with multiple fibres,
Invent. math. 84, 633-643 (1986).
[W] C. T. C. Wall, Diffeomorphisms of 4-manifolds, J. London Math. Soc. 39
(1964), 131-140.
ON THE HOMEOMORPmSM CLASSIFICATION OF SMOOTH KNOTTED
SURFACES IN THE 4-SPHERE

Matthias KRECK
Max-Planck-Institut fiir Mathematik, Bonn




1. In [FKV] an infinite family of smooth (real) surfaces F(k) embedded in S4
was constructed which has the following properties:

+l
The knottings (S4,F(k» and (8 4,F(l» are not diffeomorphic for k .
i)
F(k) = #10(IRP 2)
ii)
2"1(S4 - F(k» = 712
iii)
The normal Euler number (with local coefficients) of F(k) in 84 is 16.
iv)


The knottings (S4,F(k» are constructed from the Dolgachev surfaces
D(2,2k+l). There are antiholomorphic involutions c on D(2,2k+l) with fixed
point set F(k) = #10(lRp 2) and orbit space D(2,2k+1)/c diffeomorphic to 84.
Thus the diffeomorphism type of D(2,2k+1), the ramified covering along the
knotting, is an invariant and one can distinguish these Dolgachev surfaces by
Donaldson's r-type invariants [D), [FM], lOY]. It was also proved in [FKV] that
the number of homeomorphism types of these knottings is finite and it was con-
jectured that they are all homeomorphic to the standard embedding (S4,F) with
normal Euler number 16. The main result of this note is an affirmative answer to '
this conjecture.


More precisely consider the standard embedding of IRp2 into S4 with nor-
mal Euler class -2. This can be considered as the fixed point set of the standard
aotiholomorphic involution c on (p2 embedded into (p2Ic ˜ S4. Then the
standard embedding (S4,F) with normal Euler class 16 is obtained by taking the
(:onnected sum (84,iP 2)#9(-84,IRP 2).

rrbeorem: Let 8 = #10(lRp 2) be embedded into 84 with normal Euler number
64 Kreck: Homeomorphism classification of smooth knotted surfaces

16 and r (S4-8) = 71 . Th˜ (S4,S) is homeomorphic to (S4,F), the standard
1 2
embedding with normal Euler number 16. The homeomorphism can be chosen as a
diffeomorphism on a neighborhood of S and F.

Corollary: The knottings (S4,F(k» are all homeomorphic to (S4 JF ) implying
that the standard knotting (S4,F) has infinitely many smooth structures.

Remark: Recently R. Gompf [G) constructed non-diffeomorphic embeddings of a
4
punctured Klein bottle K (= Klein bottle minus open 2-ball) into n with
4
r 1(D - K) = 712 and intersection form of the 2-fold ramified covering along K
equal to <1> • <-1> . The same methods as used for the proof of our Theorem
show that they are pairwise homeomorphic if they have same relative normal Euler
number and the knots 8K in S3 are equal. We will comment the necessary modi-
fications of the proof in section 5. I was informed by o. Viro that he has similar
knottings of K in D4 which 'are related to the construction in [V).

2. Proof: Since F and S have isomorphic normal bundles we can choose a linear
identification of open tubular neighborhoods and denote the complements by C
and 0'. We identify the boundaries, so that DC = DC' =: M. We want to
extend the identity on M to a homeomorphism from 0 to C'. Since C and
C' are Spin-manifolds a necessary condition for this is that we can choose
Spin-structures on C and C' which agree on the common boundary. Another
necessary condition is that the diagram




1 1
(1)

1"1(0') - - -..... 1/2


commutes. One can show that by choosing the linear i.dentification of the tubular
neighborhoods appropriately one can achieve these two necessarr conditions. I am
indepted to O. Viro for this information. To obtain condition (1), choose sections
s and 8' from FO resp. SO (delete an open 2-disk) to M such that the
composition with the inclusion to C and C' reap. are trivial on . Since the
1('1


normal Euler numbers of the knottings are equal one can choose the linear
Kreck: Homeomorphism classification of smooth knotted surfaces 65

identification of the tubular neighborhoods such that they commute with s and
8' resp_ yielding (1). To obtain the compatibility of Spin structures on M it is
enough to control them on the image of s and 8' . Note that for each embedded
circle a in FO , sea) bounds an immersed disk D in C. The normal bundle of
IIJD .The
a determines a l-dimensional subbundle of lJ(D) Spin structure on the
image of s is characterized by the obstruction mod 4 to extending this
subbundle to JJ(D) and gives a quadratic form q: H1(Fo) - + 1/41 [GM]. Thus
we have to control that the identification of F and S respects this fonn or
equivalently that the Brown invariants in lL/81L agree. But this follows from the
generalized Rochlin formula [GM].

In the following we will assume that the Spin-stmctures on IJC = DC' =M
agree and the diagram (1) commutes. There is another obvious invariant to be
controlled, the intersection form on the universal covering. For this we assign to
our knotted surface the 2-fold ramified covering along F denoted by X. A simple
calculation shows that X is l-eonnected, e(X) = 12 and sign(X) =-8.. Thus the
intersection form on X is indefinite and odd (since otherwise the signature were
divisible by 16 by Rochlin's Theorem). By the classification of indefinite fonns, the
intersection form on X is <1>.9 <-1>. The long exact homology sequence
combined with excision and Poincare duality leads to an exact sequence

0--+ H (F) .... ˜(˜) --+ ˜(X) --+ H2 (F) --+ 0
1

1˜2
and the map H2(X) --+ H2(F) =1 2 is a ..---. a 0 [F], the mod 2 intersection
number of a with F. Since tj is Spin and X is not Spin (see above) the map
a .-+ a 0 [F] is given by w2(X): a 0 [F] = <w2(X),a>. Since the image of
U (F) is contained in the radical of the intersection form on H2(˜) and the form
1

on H2(X) restricted to the kernel of w2 is non-singular, the image of H1(F) is

the radical of the form on H2(˜). The form on H2(˜)/rad is the restriction of

<1> • 9<-1> ˜ ES • <1> • <-1> to the kernel of x t-+ x x which is
0


J<˜8lB [˜˜] ˜ ES + 2(˜ ˜l
66 Kreck: Homeomorphism classification of smooth knotted surfaces

We know that the covering transformation 'T acts trivially on H1(F) and
by -Ion H2(X) (since XI r = S4). Thus, if we take the A = 1l(1l2J module
on H2(˜) into account we have an exact sequence
structure given by 'T




where + or - indicates the trivial or non-trivial A-action. Moreover one can
show that ˜(˜) =1l_ $ A9 ([FKV), Lemma S.2A). We can summarize these
considerations as follows:


H2(˜) - 1- EBA 9 ,
˜ .
the radical of the intersection fonn is H2(˜)+' the +1 eigenspace;
(2)

H2(˜)/rad is E S $ 2 [˜ ˜).
the fonn on


The proof is finished by the following proposition which is the main step.


Proposition: Let C and C' be 4--dimensional Spin manifolds with fundamental
group "'8.2' DC = DC' =M and inducing same Spin-structure on M such that
the conditions (1) and (2) are fulfilled. Then there is a homeomorphism from C to
0' inducing on M the identity.


3. Proof of the Proposition. We use the method of [K]. The normall-type of C is
the fibration p: B =IRp lD )( B Spin ˜ BO and a nonnal smoothing of X in
(B,p) is given by the non-trivial map C -+ IRpm and a Spin-structure on C
(given by a lift of the normal GauD map to B Spin). Thus it is uniquely deter-
mined by a Spin-structure. By assumption there exist normal smoothings of C
and C' in (B,p) which agree on the common boundary. Thus we can form
C U(-e / ), a closed manifold with (B,p)--structure. An easy computation with the
Atiyah-Hirzebruch spectral sequence shows that 04(B,p) ˜ I, detected by the sig-
nature. Since sign C = sing C', C u-e' is zero bordant in (B,p).
Kreck: Homeomorphism classification of smooth knotted surfaces 67

e lS(7L/2)
Let W be a zero bordism. Then there exists an obstruction B(W,C)

such that C is h-eobordant to C' reI. boundary if and only if B(W,C) is zero
bordant [K]. This implies our statement using the topological h-eobordism
Theorem [Fl.

We will not repeat the definition of 8(W,C). Instead we formulate some
elementary properties which are enough to show that in our situation B(W,C) is
zero bordant. Elements in lS(7L/2) are represented by pairs (H(Ar),U), where
H(Ar ) is the hyperbolic form on Ar )( Ar and U C Ar )( Ar is a half rank free
direct summand. Note that the difference to the oldinary Wall groups is, that
there U is an addition self annihilating (a hamiltonian). Note also that we can
forget here the quadratic refinement of the form since it is determined by it. Since
the ordinary Wall group LS(7L ) vanishes one can characterize zero bordant ele-
2
ments in lS(7L2 ) as follows:


(3) [H(AI),u] E i 5(1L/2) is zero bordant if U has a hamiltonian complement V.


By construction of B(W,C) and some elementary considerations it has the
following properties:

(4) If (H(A1),U) represents B(W,C) then (H(AI ),U.1.) represents B(W,C').

(5) There exists a surjective homomorphism d: U --+ H (e) inducing an isome-
2
try of the fOlm on U with the intersection form on H (e).
2

f
(6) If V = AS - - I - t H2(e) is a free A-resolution, 8(W,C) has a representative
(H(As),V) such that d occurring in (5) is equal to f.

=1_ ED A9 =A10
H2(e)
Since we can take V with the obvious map

V -+-t H2(˜).
f:
68 Kreck: Homeomorphism classification of smooth knotted surfaces

The natural thing for showing that B(W,C) is zero bordant is to prove
that in the restriction of (H(As),V) to the :J:l-eigenspaces, V:I: have hamiltonian
complements and then to construct from them a hamiltonian complement for V.

The restriction of the hyperbolic form b on H(A8) to the ='=1 eigenspaces is
twice the hyperbolic form on H(7z8). In particular the restriction to VoJ: is divi-I
˜
:l
sible by two. After dividing by 2 we c&1l this form b:l: and V:I: sits isometri-

j
·."1
cally in H(7z8).

By assumption (2) the form b+ vanishes identically on V+ and thus
={OJ.
(H(A)+,V+) represents an element in the ordinary L-group L5


A9 --+ H2(e)jrad = 71.˜O
V ˜ A10 ˜ H2(e) =71._ •
We have and
flV_ maps onto 21l˜O. Thus the form b_ on V_is




Since by (4») (H(A10),Vl.) represents 8(W,C') and the form on ˜(e') is
v.: is -b_. Thus
minus the form on ˜(e), we know from (5) that the form on
10
=b_fa (-b-->
we have an isometric embedding V_ fB V: into H(1l ) and we are
searching for a hamiltonian complement of V_in H(1l 10 ).


The different isometry classes of embeddings of a pair of direct summands V_
10
and V: (they are direct summands since V and VJ. are so) into H(1 ) = H .
are equivalently classified by analyzing in how many different ways the hyperbolic
form can be reconstructed from the sublattice V_ • V:. To do this we consider
Adb_ : V_ --+ V˜. Denote the cokemel of Adb_
the adjoint by L, a finite
o.
abelian group since Det b_ '/: On L we have an induced quadratic form '
q:L--+ Q/71. given by q([x])= ˜ b_((Adb-->-1( ILI·x), (Ad b-->-1( 1LI • x».
Kreck: Homeomorphism classification of smooth knotted surfaces 69

Similarly starting with V: we get a quadratic form denoted by (LJ..,qJ..).
Of course (L,q) and (LJ..,-q.l) are isometric and by means of this isometry iden-
tify them with (L,q). We can reconstruct H and the embeddings of V_and V:
H =Ker (V˜ )( (V:>*--+ L), V_ =Ker P2 : V˜)( (V:>*--+ (V:>*,
as follows.
V:= Ker PI : V˜x(V:>* --+ V˜. Here the map V˜ lC (V:>* --+ L is the diffe-
rence of the projections onto L. This reconstruction follows from a standard argu-
ment similar to ([W], p. 285 ff).

Thus we have to analyze the isometries between (L,q) and (LJ..,-q.1.) = (L,q)
modulo those which can be lifted to isometries of V˜. Indeed, (H,V-> is zero bor-
dant if and only if the corresponding isometry of (L,q) can be lifted to V˜. This
follows since if V_ has a hamiltonian complement, (H,V.J is isomorphic to an
(˜Iement which corresponds to Id on L. On the other hand the element correspon-
ding to a liftable isometry of (L,q) has an obvious hamiltonian complement.

Unfortunately there exist isometries of (L,q) which cannot be lifted to
v˜. We have to show that the corresponding elements of lSC1l2) don't occur in
our geometric situation. The key for this is that we know that since C and C'
they are stably diffeomorphic [K], i.e.
'Lre bordant reI. boundary in 04(B,p)
(˜#r(S2xS2) is diffeomorphic to C' #r(S2xS2) fOI some r and in particular
there exists a bordism W between C#r(S2)(5 2) and C' #r(5 2 )(5 2) with
U(W,C#r(S2xS2» zero bordant. Obviously W is bordant to W#r(S2 xD 3)
#r(S2 xD3) where the boundary connected sum takes place along C and C'
wspectively and W is appropriately chosen. If (H(A2),V) represents B(W,C)
(H(AS+2r ), V4BH(A r )( {O}» represents B(w,C#r(S2)( S2». Denote
t.heR
V_ ED H(Alx {OJ)_. Then L = L • H(7Lr )/2. We know that the isometry of
V_ :=
r
(L,q) corresponding to B(W,C)_ can after adding Id on H(1L )/2 be lifted to
t*. We call an isometry (L,q) with this property a restricted
'Ln isometry of
iHometry.

Lemma: The group of restricted isometries of (L,q) modulo those induced by
iHometries of V* is trivial.
70 Kreck: Homeomorphism classification of smooth knotted surfaces

Before we prove this Lemma we finish our argument that B(W,C) is zero
bordant, i.e. V in H(A10) has a hamiltonian complement T. We know that V:I:
have hamiltonian complements T:-J:. We also know that V is a direct summand
1
(over A) in H(A 0) =H. Choose ll-bases ai of V+' bi of V-' ci of T+ and
+ b.)/2 is a A-base of V and a.1 0 c· = b.1 0 d. = 26...
d. of T ,such that (a.
J J
1 - 1 1 IJ
0i E V+' Pi E V_
Then we know that for each di there are elements and ,
=0 mod 2
7·E T+ such that a. + IJ.. 7· + d.
+ in H and p.:= (a·+!J,·+1·+d.)/2
1 1 1 1 1 1 1111

form a A-basis of H/V. We want to choose these elements so that they generate a
hamiltonian, i.e. the form is trivial between those base elements.

Since a.+ b. = 0 mod 2 we can assume IJ..1 = o. Write a·1 = Ea.IJ· a.1 and
1 1

7i= E7ifj with aij E {O,:l:1} and 7ij E {O,l}. A simple computation with eva-
luation of the form implies 7ij = 6ij and thus 7i =ci . Similarly one can show
=O. Since we are free to change the sign of
a.. = a.. mod 2 and a·· a·. we can
J1
IJ IJ
11

=-aji for i f j . With these assumptions it is easy to check that
assume aij
Pi 0 Pj = 0 for all i,j and we are finished.


4. Proof of the Lemma. In an equivalent formulation we have to study the
following situation. Consider in H(ll) e E S the lattice 4· H(ll) ED 2· E S and
consider L = H(7I)/4H(1l) • ES/2ES =LIED L2 with the induced "quadratic form
=1 =i
b(x,x) and ,
q which is on L1 given by q[x] b(x,x) and on L2 by q[x]
L1 ..L L2. A simple calculation shows that the only isometries of (L 1,q IL1) are,
:1:1 and :I: [˜ ˜] , which obviously can be lifted to L1 = H(ll). The nontrivial :
analogous lifting statement holds for L2 ([BS], p. 416). Thus we are finished if
modulo isometries of H(l). ES each restricted isometry of L preserves L1 and

L 2·

We denote the standard symplectic basis of H(l) by e and f. Let
g:(Ltq) --+ (L,q) be a restricted isometry. Write g[e] = are] + b[f] + [x] with
Kreck: Homeomorphism classification of smooth knotted surfaces 71

eE S. Since g[e] has order 4, a or b must be odd. Since g is restricted,
x
r
g CD Id on L CD H(7l)/2 can be lifted to an isometry of H(71) CD E CD H(71 ) under
S
amod
which e is mapped to ae + t)f +x + 2y + 2z where a = 4, b = t) mod 4,
r ). Computing the quadratic form of this element yields
y eE and z eH(ll
S
+ (x + 2y) 0 (x + 2y) = 0 mod 8.
2ab

Since a or b is odd we can after acting with an appropriate liftable
= [e] + b[˜ + [x].
a= 1
isometry assume or g[e] Now consider
A
2b + (x + 2y) 0 (x + 2y) =Sc. Then
g(e) := e + (b - 4e)f + x + 2y, where
˜(e) • ˜(e) = O. We can extend ˜ to an isometry of H(71) CD E S by setting
˜(f) = f. Then i(e) and i(f) span a hyperbolic plane in H(71) CD ES whose
orthogonal complement is isometric to E S and we use this isometry to extend ˜.


a-I h[˜
= [e]. Since h[e] 0 =˜
After composing with we obtain h with h[e]

=ale] + [ij + [y]. By the same argument as above we obtain an
we must have h[ij
H(71) CD ES with A(e) =e and A[fj =ale] + [fj + [y] and after
Isometry A of
A-1 we obtain an isometry which preserves H(1l)/4H(1l)
<:omposing again with
finishing our proof.

4
5. Some knottings in n . Let K be the punctured compact Klein bottle with
houndary 81. We consider smooth embeddings of (K,DK) into (D4,S3) with
4
fixed relative normal number, r l (D - K) = 712 ' intersection form of the 2-fold
ramified covering equal to < 1 > ED < -1 > and (S3,1JK) a fixed knot. We
(˜Iaim that two such knottings (D4,K) and (D4,K') are homeomorphic rel.
boundary. The proof is similar as for our Theorem and we indicate the necessary
changes.


As in section 2 we choose linear identifications of open tubular neighbor-
hoods of K and K I and denote their complements by C and C I. We identify
IJ(˜ = IJC I = M and choose our identification such that the Spin structures on M
"'Kree and the diagram (1) commutes. A similar consideration as in section 2 shows
tha.t H2(˜) =7l_ CD A and the radical of the intersection form is 7l+ = H2(˜)+

H2(˜)/rad is 2 (˜ ˜].
ILlld the form on
72 Kreck: Homeomorphism classification of smooth knotted surfaces

Then we proceed as in section 3. Most of the arguments there don't make
any special assumptions which are not fulfilled in our situation. The only difference
2
is in the analysis of (H(A )_,V-l. Again this is determined by an isometry of

= coker 4 [˜ ˜] ,q). The situation is easier than in section 4, since the lifting
(L
problem is simpler. The problem is here whether any isometry on (L,q) is induced
from an isometry of H(lI). But as mentioned in section 4 this holds, finishing the
argument.


I would like to thank R. Gompf, O. Viro and C.T.C. Wall for useful conver-
sation and M. Kneser for the information about a reference.




References

[BS] F. van der Blij and T. Springer: The arithmetic of octaves and of the
group G ) Indag. Math. 21 (1959), 406-418
2

[D) S. Donaldson: Irrationality and the h-eobordism conjecture, J. Diff.
Geom. 26 (1987), 141-168.

S.M. Finashin, M. Kreck, O.Ya. Viro: Non-diffeomorphic but
[FKV]
homeomorphic knottings in the 4-sphere, in SLN 1346 (1988), 157-198

[F] M.R. Freedman: The disk theorem for 4-manifolds, Proc. Int. Congress
Math., Warsaw 1983, 647-663

R. Friedman and J. Morgan: On the diffeomorphism type of certain
[FM]
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[GM]
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Topology 2 (1964), 281-298
Flat Algebraic Manifolds

F.E.A. JOHNSON
DEPARTMENT OF MATHEMATICS, UNIVERSITY COLLEGE LONDON. LONDON WClE, 6BT




The relationship between the class S, of "fundamental
groups of compact Klthler manifolds, and the class 9,
of fundamental groups of smooth complex projective
˜
varieties, is not well understood; one clearly has C
S, but, although some compact KKhler manifolds are
non-algebraic, there is, at present, no known example
˜ ˜.
of a group in which is definitely not in It is
known that membership oT S is severely restricted [5].


In this paper, we consider the subclasses Sflat' 9 f1at
consisting of fundamental groups of compact KKhler
(resp. complex projective) manifolds whose underlying
Riemannian manifold is flat ; we show


˜flat
Theorem I: The classes and 9 flat are identical.


This Tollows easily from


Theorem II: A smooth compact flat Riemannian manifold
K˜hler manifold
X admits the structure of a flat if
and only if it also admits the structure of a smooth
flat complex projective variety.
Johnson: Flat algebraic manifolds

In a previous paper with E.G. Rees ([6]), we showed
tttat STlat may be characterised as the class oT
extension groups G oT the form


o -. Z2n --. G 1
---+ • -.


in which is finite, G is torsion free, and the
4>
operator homomorphism p • ---+ GL 2n (Z) admits a complex
structure; that is, the image of p may be conjugated by
a real matrix so as to be contained within the
subgroup GLn(C). By an amalgamation of Bertini's
Theorem and the Lefschetz Hyperplane Theorem, we see
that every group in is the Tundamental group oT a
c:P
smooth complex algebraic surface (see, for example,
(1.4) oT [7]). Thus we obtain;


Theorem III Let G be a torsion free group occurring
0 -. Z2n ˜ G ---+ 'I -.
in an extension of the Torm 1
where • is Tinite and the operator homomorphism admits
a complex structure; then there exists a smooth
G = Wt(X).
complex projective surface X such that


Our starting point is the Tormal similarity between
the rational group ring of a finite group and the ring
oT rational endomorphisms of an abelian variety ; each
is semisimple and admits a positive involution. The
proof proceeds by an analysis OT the rational holonomy
˜ GL 2n (Q), using Albert's
nprsenutioo p •
classification of positively involuted semisimple
algebras [1]. For restricted classes of holonomy group
., for example, the symmetric groups, the Tull
complication of the proof may not emerge when. is
nilpotent, a short prooT using only classical
representation theory has been given by the author's
student N.C. Carr ( [3J).
Johnson: Flat algebraic manifolds

positively
The paper is organised as Tollows
involuted algebras are dealt with in §1, abel ian
varieties in §2, and rational representation theory in
Theorems I and are proved in as
§3. II §4, (4.3)
and (4.2) respectively.
The problem investigated here arose out OT joint work
with Elmer Rees, to whom the author would I ike to
express his gratitude Tor many interesting
conversations and much help and good advice, not least
his unsparing (and oTten unsuccess-ful) efforts to
instill the virtues of brevity into the author's
prose.


§ Positive division algebras:
1
Let A be a finite dimensional semisimple algebra over a
field K. A is an isomorphism of A
An involution r on
with its opposite algebra such that r 2 = l A . When k is
be˜mw
a real field, the involution r is said to when
>0
Trl(xr(x» Tor all nonzero x where 'TrK'
A,
E
=
denotes When the class oT such
'reduced trace'. Q,
I(

positively involuted algebras has been determined by
[1], [12]. We recall his results.
Albert
= Ale.
express A in the Torm A Am, where

We may
each Ai is a simple two-sided ideal. If r is a positive
=
involution on A, it Tallows easily that r(A i ) Ai Tor
each i, so it suffices to consider the case where A is
=
A Mn(D) , D a
simple; that is, where is finite
dimensional division algebra over K. An involution tT

on D extends to an involution u on A thus
=
U«Xij) (u(xji»
with transposed indices as indicated. By the Skolem-
Noether Theorem, each involution on Mn(D) has this
form. Moreover, U is positive if and only if is
U

positive : that is
Johnson: Flat algebraic manifolds

Proposition Let be a positively involuted
1.1: (A,r)
Tinite dimensional semi simple algebra over a real
Tield K. Then there is an isomorphism oT involuted K-
(A,r) -
algebras (Mnl(Dl),Tt)x ... x(Mnm(Dm),rm) where
(Di,ri) is a positively involuted division algebra .


Let F be a field, s a field automorphism oT F oT order
n , and let be a nonzero element OT such that
F
a

= (F,s,a) is constructed as
sea) The
a. cyc/icalgebra
Tollows (F, s ,a) is two sided F-vector space oT
a
([Xr])O˜r˜n_l' subject
dimension n, with basis to the
relations
= (l E F)


={ =
and is an algebra with centre E x E F: sex) x}
and multiplication

< n-l
o ˜ r
[X] [x r ]
= n-l
r

=
In the case =
n we may take Tor some
2, F E.Jb
bEE. (F,s,a)
nonzero Then is isomorphic to the
basis {l, i,j ,k} over E,
quatemion algebra (a b) with subject
i
= a.l;
= k; i 2
ij = -J1 j2 = b.l
to the relations
oT a simple algebra . A is said to be
An involution r
first kind when it restricts to the identity on the
of the
OT A;
˜ ofMeA˜ndkm˜
centre otherwise, r is said to be
A quaternion algebra (aib) admits two essentially
distinct involutions OT the first kind, namely conjugation,
c, and reversion, r, clef ined thus
+ Xli + Xo - Xii - x2 j
=
x2 j + x3 k )
c(xO xgk
-




Albert [1] ,[12], showed that a positively involuted
(D,r),
division algebra finite dimension over
OT Q,
Johnson: Flat algebraic manifolds

K are
falls into one of Tour classes here E and
algebraic number fields.


= E is totally real and r= 1 E ;
I: D


= (aib),
II: D where E is totally real, a is totally
positive, b is totally negative, and is reversion;
T



= (aib),
III : D where E is a totally real, a and b
are both totally negative, and r is conjugation;


D= (K,s,a), where s is an automorphism of Ie
I-Y.-J
is
whose fixed point field an imaginary qua.dratic
E
extension, E = EO(˜b), of a totally real field.E O' and
a E E; moreover, if L is a maximal totally real
subfield oT K, there exists a totally positive element
dEL such that =


Albert's results [12] may be summarised thus


Theorem 1.2 : The finite dimensional rational division
algebras which admit a positive involution oT the first
or
Icind are precisely those of type III
II
I
those which admit a positive involution OT the eoondkmd
are precisely those of type IV.


§2 : Riemann matrices and abelian varieties:
Let K be a subring oT the real number field R; a Riemann
ma˜xovu is a pair (A,t) where A is a free K-module aT
k
finite rank, and t is a complex structure on the real
vector space : A®K R ˜ A@K R is a
that is, t
A®E R
t2
real rkK(A) must
linear map such that -1
=
oT Riemann
then be even. There category ˜Ak
is a
matrices over Ie whose morphisms t/J : (A 1 ,t 1 ) ---+ (A2 ,t 2 )
t2(˜@1)
are that
t/J : A 1 ---+ A2 such =
K-linear maps
Johnson: Flat algebraic manifolds

(˜˜l)tl. ˜jLK mmp˜ i˜
An object (A,t) in is over K and
only i"f A contains no nontrivial K-submodule B such
=
t(B˜KR) B˜KR. ˜AK
that has "finite products given
=
by (A 1 ,t 1 ) x (A 2 ,t 2 ) (A l x A2 , t 1 x t 2 ). By a memann
form on (A,t) e ˜AK we mean a nonsingular skew K-
p: A x A ˜
bilinear £orm K such that


˜
(i) iT PR:A@I(R x A01(R R denotes the R-bilinear Torm
obtained from (J, then "for each x,y e A˜KR,

=
PR(tx,ty) PR(x,y)

P: A0 KR x A0 KR ˜ R,
(ii) the associated "form
=
P(x,y) is positive de"finite symmetric.
PR(tx,y)


A Riemann ma.trix is when it admits Riemann
algebraic a.

e ˜AK' we write (A,t)* = (A*,t*) where
Torm.IT (A,t)
A* = HomK(A,K) is the k-dual o"f A, t* is the R-clual oT
A*˜KK
t, making the identifications - Homl(A,R) -
HomR(A®I(R,R). The correspondence (A,t) (A,t)* gives
1-+


a contravariant "functor * :
K ˜ CJ,A K . A Riemann
CJ,A
P
form on (A,t) gives rise to an injective K-linear
map P A ˜ A* such that P(x)(y) = P(x,y) Tor all
is a field, p
x, yEA. When I( is bijective, and,
p*
identifying A with A**, we obtain a Riemann Torm on
(A,t)* thus ;
{J* (e, 'I) =
(A,t) is algebraic precisely when (A,t)* is algebraic
moreover, a Riemann Torm on (A,t) induces an
P ˜ ˜AK.
isomorphism (A,t) (A,t)* in In this
ca.se, if k is a subfield oT R, the K-algebra Endl(A,t)

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