数理解析研究所講究録
第 1660 巻 2009 年 30-34

30

The mapping class group andthe Meyer function for plane curves
Yusuke Kuno
In this note, a new example of Meyer function and its application to the signature of
4-manifolds are presented.

1

The first MMM class and Meyer’s signature cocycle

be a closed oriented-surface of genus andthe group of all orientation
preserving diffeomorphisms of, endowed with
-topology. The mapping class groupof , denoted by , is defined to be the group of connected components of.
By the result of Earle-Eells[2], each connected component ofis contractible
for $g\geq 2$ . It follows that the classifying spaceis the Eilenberg-MacLane space. Let $p:Earrow B$ be an oriented
bundle. For each $i\geq 1$ , the i-th MMM class$e_{i}=e_{i}(p)$ is defined by

Let$C^{\infty}$$\Sigma_{g}$$C^{\infty}$$\Sigma_{g}$$\Sigma_{g}$$Diff^{+}\Sigma_{g}$$g$$Diff^{+}\Sigma_{g}$$\Gamma_{g}$$Diff^{+}\Sigma_{g}$$BDiff^{+}\Sigma_{g}$$K(\Gamma_{g}, 1)$$\Sigma_{g}$$e_{i}(p):=p_{!}(e^{i+1})\in H^{2i}(B)$.

Here, $e\in H^{2}(E)$ is the Euler class of the tangent bundle along the fiber of and is the
Gysin map. Since these classes are natural with respect to bundle maps between oriented
bundles, one can think of as the cohomology class in$p$$\Sigma_{g}$$p!$$e_{i}$$H^{2i}(BDiff^{+}\Sigma_{g})\cong H^{2i}(\Gamma_{g})$.

In this note, we consider the first MMM class , which is closely related to the signatureof 4-manifolds as follows.
Let $p:Earrow B$ be an oriented
bundle over a closed oriented surface $B$ . Then the total
space $E$ is a closed 4-manifold endowed with the natural orientation. By the Hirzebmch
signature formula, we have
Sign
.
(1)
$e_{1}$$\Sigma_{g}$$(E)= \frac{1}{3}\langle e_{1}(p),$$[B]\rangle$

There is also a 2-cocycle ofusing the signature of 4-manifolds. This is Meyer’s
signature cocycle. Here we briefly recall its definition.
Let $P$ denote the pair of pants, i.e.,
Int where$i=1,2$ , and 3 are
the three disjoint closed disks in the 2-sphere . Choose a base point
Int $P$ and fix a
loop
based
and such that is free homotopic to the loop traveling once the boundary
by counter clockwise manner $(i=1,2)$ . For, we can construct anoriented
bundle $E(f_{1}, f_{2})$ over $P$ such that the topological monodromy
sends
tofor $i=1,2$ . $E(f_{1}, f_{2})$ is a compact-manifold of dimension 4 endowedwith the natural orientation. Then the signature of $E(f_{1}, f_{2})$ is defined and we set$\Gamma_{g}$$\tau_{g}[9]$$P=S^{2} \backslash \bigcup_{i=1}^{3}$$D_{i}$$S^{2}$$\ell_{1}$$\ell_{2}$$D_{i},$$p_{0}\in$$\ell_{i}$$\partial D_{i}$$(f_{1}, f_{2})\in\Gamma_{g}\cross\Gamma_{g}$$\pi_{1}(P)arrow\Gamma_{g}$$\Sigma_{g}$$[\ell_{i}]$$C^{\infty}$$f_{i}$$\tau_{g}(f_{1}, f_{2})$$:=$-Sign $(E(f_{1}, f_{2}))$ .

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By the Novikov additivity of the signatureequation (1) shows that$\tau_{g}$tums out to be a 2-cocycle of$3[\tau_{g}]=e_{1}\in H^{2}(\Gamma_{g})$2Triviality of$e_{1}$$\Gamma_{g}$, and the.

over rationals

Let $p:Earrow B$ be an oriented
bundle or continuous family of compact Riemann surface
of genus .
Now we are interested in the triviality of the rational cohomology class $e_{1}(p)\in H^{2}(B;\mathbb{Q})$ .
If this is the case,pulled back tovanishes and there exists a -valued 1cochain
cobounding pulled back toby the topological monodromyof . Moreover, if $H^{1}(\pi_{1}(B);\mathbb{Z})=0$ , such a l-cochain is unique. Then we call the Meyerwith respect to $p:Earrow B$ .
function of
There are several examples:$\Sigma_{g}$$g$$H^{2}(\pi_{1}(B);\mathbb{Q})$$[\tau_{g}]$$\phi:\pi_{1}(B)arrow \mathbb{Q}$$\mathbb{Q}$$\pi_{1}(B)$$\tau_{g}$$p$$\phi$$\pi_{1}(B)$

1. W. Meyer [9] showed thatis torsion for $g=1,2$ . Thus, $e_{1}=0\in$
for $g=1,2$ . In the case $g=1,$
is isomorphic to. Meyeralso gave an explicit formula for the Meyer function
using theRademacher function.
$[\tau_{g}]\in H^{2}(\Gamma_{g};\mathbb{Z})$$H^{2}(\Gamma_{g};\mathbb{Q})$$SL(2;\mathbb{Z})$$\Gamma_{1}$$\phi_{1}:SL(2;\mathbb{Z})arrow\frac{1}{3}\mathbb{Z}$2. The hyperelliptic mapping class groupis defined as the centralizer of a
hyperelliptic involution. As was shown by F. Cohen [4] and N. Kawazumi[6] independently,
is -acyclic. In particular,. Later, H.
Endo [3] directly showed that the existence and the uniqueness of the Meyer function
using a finite presentation of
by J. Birmann-H. Hilden.
$\Gamma_{9}^{H}\subset\Gamma_{g}$$\iota\in\Gamma_{g}$$\Gamma_{g}^{H}$$e_{1}=0\in H^{2}(\Gamma_{g}^{H};\mathbb{Q})$$\mathbb{Q}$$\phi_{g}^{H}:\Gamma_{g}^{H}arrow\frac{1}{2g+1}\mathbb{Z}$$\Gamma_{g}^{H}$

3. In contrast, W. Meyer [9] showed that
has infinite order therefore
is non-trivial for $g\geq 3$ . In fact, J. Harer [5] showed thatfor $g\geq 3$ , and combining this with Meyer’s computation, it follows thatis equal to 4 times a generator.
$[\tau_{g}]\in H^{2}(\Gamma_{g};\mathbb{Z})$$e_{1}\in H^{2}(\Gamma_{g};\mathbb{Q})$$H^{2}(\Gamma_{g};\mathbb{Z})\cong$$\mathbb{Z}$$[\tau_{9}]\in$$H^{2}(\Gamma_{g};\mathbb{Z})\cong \mathbb{Z}$4. D. Mumford [10] observed that if $p:Earrow B$ is a family of non-hyperelliptic curvesof genus 3, $e_{1}(p)=0\in H^{2}(B;\mathbb{Q})$ . His proof uses the Grothendieck-Riemann-Rochformula.
In this note, we give an altemative proof of Mumford’s observation, and generalize
it to the case of family of plane curves. Our approach is to show the existence and theuniqueness of the Meyer function of the group $\Pi(4)$ defined in the next section, whichis universal for families of non-hyperelliptic curves of genus 3. Our approach is purely
topological, although some algebraic geometry is used to compute examples.

3

The mapping class group for plane curves

In this section we construct the group $\Pi(d)$ and state the main result of this note.
Henceforth is a fixed integer. Let be the complex vector space of homogeneouspolynomials of degree in the determinates, and , and let
be the projectivization of . Each point
determines the complex plane curveof degree$d$$\geq 2$

$d$

$V$

$V$

$x,$ $y$$a\in \mathbb{P}$$z$$\mathbb{P}=\mathbb{P}(V)$$C_{a}\subset \mathbb{P}^{2}$

32

. Let

be the set of all points
such that the corresponding curveis singular.
$D$
is called the discriminant locus and known to be irreducible and of codimension 1. Set

$d$

$D$

$a\in \mathbb{P}$$C_{a}$$\mathcal{F}:=\{(a,p)\in(\mathbb{P}\backslash D)\cross \mathbb{P}^{2};p\in C_{a}\}$.

The group of automorphisms of , namely $PGL(3)$ , acts naturally on
as change ofvariables. Then the first projection
is a complex analytic family of compact
Riemann surfaces of genus 3 and the projection map is $PGL(3)$ -equivariant. Here, theaction of $PGL(3)$ on is diagonal.
Taking Borel construction $(\mathbb{P}\backslash D)_{PGL(3)}=EPGL(3)\cross PGL(3)(\mathbb{P}\backslash D)$ , we obtain acontinuous family$\mathbb{P}^{2}$$\mathbb{P}\backslash D$$\mathcal{F}arrow(\mathbb{P}\backslash D)$$\mathcal{F}$$p_{u}:\mathcal{F}_{PGL(3)}arrow(\mathbb{P}\backslash D)_{PGL(3)}$of compact Riemann surfaces of genus 3.
We denote by $\Pi(d)$ the fundamental group of
mapping class group for plane curves of degree . Let$(\mathbb{P}\backslash D)_{PGL(3)}$and call this group the$d$$\rho;\Pi(d)arrow\Gamma_{9}$be the topological monodromy of . Here, $g= \frac{1}{2}(d-1)(d-2)$ .
Recall that the usual mapping class group is the fundamental group of the classifying
space BDi
. The name “mapping class group” for $\Pi(d)$ comes $kom$ the following
universal property of.
$p_{u}$$\Gamma_{g}$$ff^{+}\Sigma_{g}$$(\mathbb{P}\backslash D)_{PGL(3)}$Theorem 3.1. For any topological space$B$, there is a natural bijection$[B, ( \mathbb{P}\backslash D)_{PGL(3)}]\cong\frac{\{familyofnon-singularp1anecurvesofdegreedoverB\}}{\sim isotopy}$(2)

induced by pulling back . Here, the left hand side is the set of homotopy classes ofcontinuous maps from $B$ to.
$p_{u}$$(\mathbb{P}\backslash D)_{PGL(3)}$The following are the main results of this note:

Theorem 3.2.

$\rho^{*}([\tau_{g}])=0\in H^{2}(\Pi(d);\mathbb{Q})$.

Theorem 3.3.
$H_{1}(\Pi(d);\mathbb{Z})=\{\begin{array}{ll}\mathbb{Z}/3(d-1)^{2}\mathbb{Z} if d\equiv Omod3,\mathbb{Z}/(d-1)^{2}\mathbb{Z} if d\equiv 1 or 2 mod 3.\end{array}$

As a consequence of Theorems 3.2 and 3.3, there exists the unique l-cochain
such that. We call this cochain the Meyer function for plane curves of degree. One can easily see thatis a class function on $\Pi(d)$ .
Let be an element oftraveling once around $D$ . Such is called a lasso
around . We also denote by the image of by the natural surjection.
is well-defined up to conjugacy.
$\phi^{d}:\Pi(d)arrow$$\delta\phi^{d}=\rho^{*}\tau_{g}$$\mathbb{Q}$$d$$\phi^{d}$$\sigma$$\pi_{1}(\mathbb{P}\backslash D)$$D$$\sigma$$\sigma$$\pi_{1}(\mathbb{P}\backslash D)arrow\Pi(d)$$\sigma$$\sigma$

Proposition 3.4. For $d\geq 3$ ,$\phi^{d}(\sigma)=-\frac{d+1}{3(d-1)}$.

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By this proposition, we can see the order of the integral cohomology class$H^{2}(\Pi(d);\mathbb{Z})$$\rho^{*}[\tau_{g}]\in$.

Here we briefly explain how to prove Theorem 3.2. Letin. There is a natural map
map induces an injective homomorphism$\tilde{D}\subset V$

be the union of all lines. We can see that this$V\backslash \tilde{D}arrow \mathbb{P}\backslash Darrow(\mathbb{P}\backslash D)_{PGL(3)}$$D\subset \mathbb{P}$$H^{2}(\Pi(d);\mathbb{Q})arrow H^{2}(\pi_{1}(V\backslash \tilde{D});\mathbb{Q})$.

cobounding
pulled back to
Moreover we can construct a l-cochain. The construction is based on the signature of 4-manifolds. Thus we also have
. For details, see Section 3 of [7].
$c:\pi_{1}(V\backslash \tilde{D})arrow \mathbb{Z}$$\tau_{g}$$\pi_{1}(V\backslash \tilde{D})$$\rho^{*}([\tau_{g}])=0\in H^{2}(\Pi(d);\mathbb{Q})$4

The local signature

As an application, we define the local signature for each fiber germs of 4-dimensional fiber
spaces whose general fibers are non-hyperelliptic curves of genus 3. We first introduce aclass of 4-manifolds we consider. By a 4-dimensional non-hyperelliptic fibration of genus 3
is meant a following data:-map,
1. $E(resp. B)$ is an oriented 4(resp. 2)-manifold and $\pi:Earrow B$ is a$C^{\infty}$

2. there exist finitely many points
Int $(B)$ such that the restriction of-family of non-hyperelliptic curves of genus 3.
has a structure of$b_{1},$$\ldots,$$b_{n}\in$

$\pi$

to

$C^{\infty}$$B\backslash \{b_{i}\}_{i}$Letbe the (possibly singular) fiber germ arounda small closed 2-disk
andcentered at . Now for the case $d=4$ , the right hand side of (2) in Theorem 3.1 isnaturally isomorphic to$\mathcal{F}_{i}$$b_{\mathfrak{i}}$$D_{i}$$b_{i}$$\frac{\{continuousfamilyofnon-hypere11ipticcurvesofgenus3overB\}}{\sim isotopy}$

Thus, by restricting
to
we obtain the classifying map(determined up to homotopy). Define$\pi$$D_{i}\backslash \{b_{i}\}$.

$g^{i}:D_{i}\backslash \{b_{i}\}arrow$$(\mathbb{P}\backslash D)_{PGL(3)}$loc.sig2$(\mathcal{F}_{i})$$:=\phi^{4}(g_{*}^{i}(\partial D_{i}))_{*}(\gamma))+$Sign$(N( \pi^{-1}(b_{i})))\in\frac{1}{9}\mathbb{Z}$.

Here, $N(\pi^{-1}(b_{i}))$ denotes a fiber neighborhood of the singular fiber. It is easy to seethat this value only depends on the germ of around (note thatis a class function
on $\Pi(4))$ . We remark here that this definition originates from Y. Matsumoto [8] where
Lefschetz fibrations of genus 2 are discussed.
By using this local signature, we can now formulate the signature formula in our setting:
Theorem 4.1 (The signature formula). Let $p:Earrow B$ be a 4-dimensional non-hyperelliptic
fibration of genus 3. If $E$ is closed,$\pi^{-1}(b_{i})$$\pi$Sign $(E)= \sum_{l}$ loc.sig$\phi^{4}$$b_{i}$$Q(\mathcal{F}_{i})$.

In this case of non-hyperelliptic family of genus 3, T. Ashikaga-K. Konno [1] and K.
Yoshikawa [11] have already defined local signature independently. The definition of [1]is algebro geometric and that of [11] is complex analytic. Computing some examples ofvalues of our local signature, we observe that they coincide with those computed in [1] and[11]. For details of the computations, see [7].

34References[1] T. Ashikaga and K. Konno, Global and local properties of pencils of algebraic curves,
Algebraic Geometry 2000 Azumino, Advanced Studies in Pure Mathematics 36 (2000),

1-49.

[2] C. J. Earle and J. Eells, A fibre bundle description of Teichm\"uller space, J. Differential
Geometry, 3 (1969), 19-43.

[3] H. Endo, Meyer’s signature cocycle and hyperelliptic fibrations, Math. Ann. 316(2000), 237-257.
[4] F. R. Cohen, Homology of mapping class groups for surfaces of low genus, Contemp.
Math. 58 (1987), 21-30.
[5] J. Harer, The second homology group of the mapping class group of an orientablesurface, Invent. Math. 72 (1983), 221-239.

[6] N. Kawazumi, On the homotopy type of the moduli space of n-point sets of
J.Fac.Sci.Univ. Tokyo Ser. IA. 37 (1990), 263-287.

$\mathbb{P}^{1}$,[7] Y. Kuno, The mapping class group and the Meyer function for plane curves, to appear
in Math. Ann. (preprint arXiv:math.GT/0707.4332)[8] Y. Matsumoto, Lekchetz fibrations of genus two-a topological approach-, Proceedings of the 37th Taniguchi Symposium on “Topology and Teichm\"uller Spaces”, World
Scientific, Singapore, 1996, 123-148.
[9] W. Meyer, Die Signatur von Fl\"achenb\"undeln, Math. Ann. 201 (1973), 239-264.

[10] D. Mumford, Towards an Enumerative Geometry of the Moduli Space of Curves,
Arithmetic and Geometry, Progress in Math. vol.36, Birkh\"auser, Boston, 1983, 271-

328.

[11] K. Yoshikawa, A local signature for generic l-parameter deformation germs of a complex curve, in: Algebraic Geometry and Topology of Degenerations, Coverings andSingularities (2000), 188-200 (in Japanese).

YUSUKE KUNOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES,
THE UNIVERSITY OF TOKYO,
3-8-1 KOMABA MEGURO-KU TOKYO 153-0041, JAPANE-mail address:kunottiQms.u-tokyo.ac.jp