Annales de l’institut Fourier (AIF)

Xiao, Gang

Galois covers between $K3$ surfaces

Sottile, Frank

Pieri's formula for flag manifolds and Schubert polynomials

Kohler, Franz

ANNALES DE L’INSTITUT FOURIER
FRANZ KOHLER
Feuilletages holomorphes singuliers sur les surfaces
contenant une coquille sphérique globale
Annales de l’institut Fourier, tome 46, no 2 (1996), p. 589
<http://www.numdam.org/item?id=AIF_1996__46_2_589_0>
© Annales de l’institut Fourier, 1996, tous droits réservés.
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(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
46, 2 (1996), 589
ERRATUM
FEUILLETAGES HOLOMORPHES SINGULIERS
SUR LES SURFACES CONTENANT
UNE COQUILLE SPHÉRIQUE GLOBALE
par Franz KOHLER
(Article paru dans le tome 45 (1995), fascicule 1, pp. 161-182)
Je remercie le Professeur Marco Brunella de m'avoir signalé l'exis-
tence de feuilletages sur les surfaces non génériques. Cette conclusion était
due à une erreur page 178, ligne 5 '.^^{y') = y " . Il faut alors reformuler la
partie 3) du théorème 2.3.1, page 171 :
DÉFINITION. — Soit S(F) une surface de Kato où F = TUT. S'oit
w une 1-forme holomorphe invariante par F et F le feuilletage associé. Si
chaque point 0^ est singulier pour le feuilletage défini au voisinage de Ci par
l'image réciproque de w on dit que le feuilletage est totalement dégénéré.
THÉORÈME. — Soit S une surface de Kato. Si il existe sur S un
feuilletage totalement dégénéré alors S est une surface d'Inoue-Hirzebruch.
Preuve. — cf. preuve du théorème 2.3.1, partie A.
COROLLAIRE. — Les surfaces d'Inoue possèdent un unique feuille-
tage holomorphe.
Angers, le 15 janvier 1996.
Franz KOHLER,
Université d'Angers
Département de Mathématiques
2 boulevard Lavoisier
49045 ANGERS CEDEX 01 (France).


Feuilletages holomorphes singuliers sur les surfaces contenant une coquille sphérique globale

Fourrier, Laurence

Topologie d'un polynôme de deux variables complexes au voisinage de l'infini

Dal'bo, Françoise

Peigné, Marc

Groupes du ping-pong et géodésiques fermées en courbure -1

Kervaire, Michel

Corps quadratiques, $GL_2({\bf Z})$ et polynômes de Dickson

Baker, Mark D.

ANNALES DE L’INSTITUT FOURIER
MARK D. BAKER
On boundary slopes of immersed
incompressible surfaces
Annales de l’institut Fourier, tome 46, no 5 (1996), p. 1443-1449
<http://www.numdam.org/item?id=AIF_1996__46_5_1443_0>
© Annales de l’institut Fourier, 1996, tous droits réservés.
L’accès aux archives de la revue « Annales de l’institut Fourier »
(http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé-
nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa-
tion commerciale ou impression systématique est constitutive d’une in-
fraction pénale. Toute copie ou impression de ce fichier doit conte-
nir la présente mention de copyright.
Article numérisé dans le cadre du programme
Numérisation de documents anciens mathématiques
http://www.numdam.org/
Ann. Inst. Fourier, Grenoble
46, 5 (1996), 1443-1449
ON BOUNDARY SLOPES OF IMMERSED
INCOMPRESSIBLE SURFACES
by Mark D. BAKER
1. Introduction.
Let M be a compact, orientable, irreducible 3-manifold with 9M a
torus. Hatcher [H] showed that there are only finitely many slopes on 9M
realized by boundary curves of embedded incompressible, ^-incompressible
surfaces in M.
In this paper we show that there can be infinitely many slopes on 9M
realized by the boundary curves of immersed, incompressible, cMncom-
pressible surfaces in M which are embedded in a neighborhood of 9M.
2. Notation and statement of results.
Let To be the torus with an open disk removed, pictured in Figure 1.
Let Dx (resp. Dy) denote the right handed Dehn twist about the loop x
(resp. the loop y).
Consider the manifold M, a once punctured torus bundle over 51,
given by
M^Tox[0 , l ] / ( / i (5 ) ,0 )~(5 , l )
Key words: 3-Manifold - Injective surface - Boundary slope.
Math. classification: 57N10 - 57M50.
1444 MARK D. BAKER
where h = Dy^ o D^. Fix a basepoint b € 9To. The loops a = b x [0,1]/ ~,
f3 = 9To give a coordinate system for 9M, the loop ap'f3x in <9M is
represented by the pair (a, A) € Z2 and is said to have slope fi/X.
Now, if S C M is an immersed surface, properly embedded in a
neighborhood of 9M, then 9S consists of parallel simple closed curves
in 9M parametrized by a coprime pair (/^, A).
THEOREM. — For M as above, the coprime pair (//, A) is realized
by the boundary curves of an immersed, incompressible, 9-incompressible
surface provided p, > 1 and [A| > p,.
Remarks.
1) It suffices to prove the theorem for (/A, A) with p, > 1 and A > ^
since M admits an involution sending the curves a, f3 to a, —f3. Indeed, let
k: To -—> To be a reflection in the diagonal (see Figure 1) followed by Dy^.
Then k2 is isotopic to Dy^ o D^ and the map (x, t) ^—> {k(x),t + |) induces
the desired involution on M.
2) The immersed surfaces of the theorem are virtually embedded
in M: they lift to embedded surfaces in finite covers of M (see §3) and one
obtains virtually Haken manifolds by Dehn filling on M with respect to
these boundary slopes.
3) There exist manifolds with torus boundary (N Seifert fibered for
example) for which only finitely many slopes on 9N are realized as the
boundary curves of essential immersed surfaces.
3. Proof of theorem.
3.1. — We prove our result by constructing, for each (fi, A), a finite
covering space M —> M such that:
(i) The loop 0^/3^ lifts to loops in each of the four components of 9M.
(ii) In ^fi(M;Z) there exists a relation of the form 71 — 72 4- 73 — 74 = 0
where 7^ is a lift of ap'|3x to the i-tb. component of 9M.
Property (ii) implies that M contains an incompressible, 9-
incompressible surface 5" whose boundary consists of the loops 71,..., 74.
Indeed, if we consider a triangulation for M and simplicial homology, the
fact that 71 — 72 + 73 — 74 is a primitive element in H\(QM\V^ that is
ON BOUNDARY SLOPES OF IMMERSED INCOMPRESSIBLE SURFACES 1445
zero in H^M'^) implies that these loops bound an oriented 2-complex, AT,
in M with property that an even number of triangles meet at each interior
edge, with oriented sum equal to zero. Thus by cutting and pasting along
the edges of K, and then pulling apart at vertices, we obtain an embedded
surface which can then be compressed. Now by property (i) this surface
projects to an immersed incompressible surface S in M with boundary
consisting of four parallel copies of a^/3\
The cover M is obtained by constructing a cover F —> To to which
h = Dy^oD^ lifts to a homeomorphism h: F —» F. Then M is the mapping
torus of the pair (-F, h).
3.2. — We construct F —^ To by cutting and pasting together copies
of the following two covers of To:
a) The four-fold cover X' —> To corresponding to the kernel of the
map 6: TTiCZo) -^ Z/2 C Z/2 defined by e{[x}) = (1,0) and 0([y]) = (0,1).
b) The two-fold cover Y ' —> TQ corresponding to the kernel of the map
0: Ti-i (To) -> Z/2 defined by 0([x}) = 0 and 0([y}) = 1.
Note that X' (resp. Y ' ) is a torus with four (resp. two) boundary
circles.
Now alter X' (resp. V) by making two (resp. one) vertical cuts TI, r^
(resp. r) between boundary circles as shown in Figure 2a (resp. Figure 2b).
Denote the cut surfaces by X and Y.
xi
X2
Xl xi
c
—>—
^ c
—>—
yi
(
.)
T
3 ,
y
b)
c
, c
—>—
/i V2 y
(a)
F
,)
TI
^ ,
C
, c
' )
T2
^ ,
11
igure 2
3.3. — Since A > ^, we have that A — p. = 2r + s for 0 <: s < 1. Then
F —> To is obtained by gluing together ^ + 2r copies of X and 2s copies
1446 MARK D. BAKER
of V as indicated in steps i) - v) below. The cover F for (/^,A) = (2,7),
with edges glued as numbered, is illustrated in Figure 3.
row 1
row 2
row 3
Fig-ure 3
i) First arrange IJL copies of X, indexed Xi, . . . , X^, in a row, followed
by the remaining 2r copies of X, indexed X^,. . . ,X^ in a second row.
The 25 copies of V, indexed Vi , . . . , Y^s form the third row.
Note that row 1 is never empty whereas either row 2 or row 3 (but not
both) can be.
ii) In the first row, glue the right edge of 7-1 (resp. left edge
of T2) in Xi to the left edge of 7-1 (resp. right edge of 7-2) in X^+i for
z = l , . . . , / z — l . I n the second row, perform the same gluing as in the first
row, first on the Xf , . . . , X^ for i = 1,.. . , r — 1 and then on X^.^,..., X^
for i = r + 1,... ,2r — 1.
The gluing pattern for the remaining edges depends on the values of
r and s. The three distinct cases are treated in iii) - v) below.
iii) The case r > 0 and s = 1. There are sixteen remaining edges to
glue in this case. Glue the left edge of 7-1 in X\ to the right edge of 7-2 in Xf;
the right edge of 7-2 in Xi to the left edge of 7-1 in X^; the right edge of 7-1
ON BOUNDARY SLOPES OF IMMERSED INCOMPRESSIBLE SURFACES 1447
in X^ to the left edge of TI in Xf; the left edge of r^ in Xy, to the right edge
ofT2inX^i.
Finally, attach Y\ to Xr and Yz to Xsr by, in each case, gluing the left
edge (resp. right edge) of r to the right edge of TI (resp. left edge of r^).
iv) The case r > 0 and s = 0. Glue the copies of X as in iii). Finish
by gluing the right edge of r\ to the left edge of T2, first in X^ and then
inXi.
v) The case r = 0 and s = 1. There are eight edges to glue. Glue the
left edge of TI (resp. right edge of r^) in Xi to the right edge of T in Vi
(resp. left edge of T in Y^).
Finally, pair the right edge of TI (resp. left edge of r^) in X^ with the
left edge of T in Yi (resp. right edge of T in Y^).
Note that the surface F is indeed a cover of To; some of its properties
are given in:
LEMMA 1. — The surface F is a 4A-foid cover of To. It is of genus
(2A — 1) with four boundary components, each of which projects A to 1
onto f3 = QTQ .
3.4. — Now the loop x (resp. y) in To is covered by loops in F
which project d to 1 onto x for d C 1,2,4 (resp. 2 to 1 onto y). Thus D4,
and D^ lift to appropriate powers of Dehn twist homeomorphisms in F,
whence h = Dy4 o D^ lifts to a homeomorphism h: F —^ F, which fixes 9F
pointwise. Denote by M the mapping torus of (F, h):
M=Fx[0 , l ] / ( f t (5 ) ,0 )~(5 , l ) .
From the construction of M, it is clear that we can choose on each
boundary torus, T^, of M a pair of loops %,/3z which cover a,/3 in 9M.
The loop di (resp. /^) projects 1 to 1 onto a (resp. A to 1 onto /3). We
will index the four tori in all cases so that the labelling matches that of X\
in Figure 3.
3.5. — It is clear from the preceding paragraph that the loops a^ft
in 9M project homeomorphically to 0^/3^ in 9M. Thus property (i) of §3.1
is verified and all that remains to show is that property (ii) holds.
1448 MARK D. BAKER
3.6. LEMMA 2. — Let 7i denote the loop 5 .̂ Then, in ffi(M; Z),
7i - 72 + 73 - 74 = 0.
Hence there is a properly embedded incompressible surface S/ C M
with 9S/ the collection of boundary curves 5^ which projects to an
immersed incompressible surface S in M with boundary consisting of four
parallel copies of Q^/?^.
Proof. — It suffices to show that in -Hi(M;Z) the following relation
holds:
(*) (/25i + A) - (/^52 + /?2) + (/^ + A) - (^4 + /?4) = 0.
Note that
(a) /?i + /?2 + /?3 4- /?4 = 0 the relation being given by the fiber surface F,
and
(b) /^(5i - 02 + 03 - 04) + 2(/3i + /?3) = 0.
Relations (a) and (b) imply (*).
One sees (b) as follows: Consider the loops ^/i, $2 on X, pictured in
Figure 2, and let y\^ and ^2,1 denote the corresponding loops in Xi C -F,
i = 1,..., fji (see §3.3). Now on each of these Xi, let 7^ C Xi (resp. <$i C Xi)
denote a horizontal, properly embedded arc between f3\ and /?2 crossing ^2,1
(resp. between /3^ and /?4 crossing ^1,1). Then the disks 7^ x I c F x I and
6i x I C F x I provide the relations
5i - £2 4- (/^(7i) * 7i~1) = 0»
53-S4+(/l(^^)*^- l)=0,
where ft-(7i) * 7^~1 (resp. /i(^) * <^~1) denotes path composition and is equal
to 2^2,z (resp. -2^,z).
Combining the 2/^ relations thus obtained gives:
/2(5i - £2 + £3 - 04) + 2(^2,1 + • • • + y2, P') - 2(^1,1 + • • • + yi^) = 0
and hence
^(5i - 02 + 03 - 54) + 2(/3i + A?) = 0
since the sum of loops about the boundary circles contributing to /?i and /?3
in the Xf and Yz are all homologous to zero. D
ON BOUNDARY SLOPES OF IMMERSED INCOMPRESSIBLE SURFACES 1449
4. Concluding remarks.
1) A once-punctured torus bundle will have infinitely many slopes
realized by immersed incompressible surfaces if its characteristic homeo-
morphism is of the form D^ o Dy o • • • o D^ o D^ where 2 | Si and 4 | r^,
i = 1,..., fc, provided that «i + • • • + Sk 7^ 0. Hence the same will be true
for any once-punctured torus bundle whose monodromy (in SL2(Z)) has a
power that is conjugate to the monodromy of one of the above bundles.
2) The cut and paste techniques in §3 can also be used to produce
families of once-punctured surface bundles over S1 of any given genus g > 1
having infinitely many slopes realized by immersed incompressible surfaces.
BIBLIOGRAPHY
[H] A. HATCHER, On the boundary curves of incompressible surfaces, Pacific J.
Math., 99 (1982), 373-377.
Manuscrit recu Ie 26 juin 1995,
accepte Ie 11 mars 1996.
Mark D. BAKER,
IRMAR
Universite de Rennes 1
Campus de Beaulieu
35042 Rennes Cedex (France).
baker@univ-rennesl. fr


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