On the Farrell Cohomology of Mapping Class Groups

H. H. Glover, G. Mislin and Y. Xia

Introduction.

For Γ any group of finite virtual cohomological dimension and a prime p, we say that Γ is p-periodic, if there exists a positive integer k such that the Farrell cohomology groups Ĥ^{i}(Γ; M) and Ĥ^{i+k}(Γ; M) have naturally isomorphic p-primary components for all i ∈ and Γ-modules M. The p-period of Γ is defined as the least value of k (cf. [B]). For instance, if Γ is p-torsion free, then Γ is p-periodic of period one.

The mapping class group, Γ_{g}, is defined to be the group of path components of the group of orientation preserving homeomorphisms of the oriented closed surface S_{g} of genus g. For instance, Γ_{1} ≅ SL(2, ) and the cohomology is well known and easy to compute in this case. By writing SL(2, ) as an amalgamated product of /4 and /6 over /2, one finds

Ĥ^{∗}(Γ_{1}; ) ≅ ( /12)[x, x^{−1}]

with x of degree two. Thus Γ_{1} is 2- and 3-periodic, with periods equal to two.

It is well known that Γ_{g} is of finite virtual cohomological dimension and, if g > 1, vcd(Γ_{g}) = 4g − 5 (cf. [H]). In the sequel we will always assume that g > 1. Recall from [B] that a group of finite vcd is p-periodic if and only if it does not contain a subgroup isomorphic to /p × /p. Because for g > 1 the mapping class group Γ_{g} contains always a subgroup isomorphic to /2 × /2, Γ_{g} is never 2-periodic. However, for an odd prime p, Γ_{g} is p-periodic for almost all values of g. This corresponds to the intuitively obvious fact that it is hard to find two “different” homeomorphisms of order p on S_{g}, which commute with each other. The third author determined in [X1] all the genera g for which Γ_{g} is p-periodic. In particular, Γ_{g} is 3-periodic if and only if g ≢ 1 mod(3). For an odd prime p and genus g ≢ 1 mod(p), Γ_{g} is always p-periodic. Moreover, there are only finitely many “exceptional values” of g with g ≡ 1 mod(p) for which Γ_{g} is p-periodic.

Recall that for a finite p-periodic group G and p an odd prime, the Sylow p-subgroup G_{p} of G is cyclic and, if G_{p} ≠ 1, the p-period of G equals 2|N(G_{p})/C(G_{p})|, where N(G_{p}) (respectively C(G_{p})) denotes the normalizer (respectively centralizer) of G_{p} in G; in particular, the p-period of G divides 2(p − 1). Unlike the case of finite groups , the p-period of a p-periodic infinite group, p a fixed prime, may be arbitrarily large. A simple example is given by the group /p^{n} =< a, b|a^{p^{n}} = 1, bab^{−1} = a^{p+1} >. For an odd prime p, the p-period of /p^{n} equals 2p^{n−1}.

In this paper, however, we will show the surprising result that for a p-periodic mapping class group Γ_{g}, the p-period is bounded by 2(p − 1). The precise theorem reads as follows.

THEOREM 1. Let p be an odd prime and assume that Γ_{g} is p-periodic. Then the p-period of Γ_{g} is given by

lcm{2[N(π) : C(π)]|π ∈ S}

where π ranges over S, a set of representatives of conjugacy classes of subgroups of order p of Γ_{g}, and N(π) (respectively C(π)) denotes the normalizer (respectively centralizer) of π in Γ_{g}. In particular, the p-period of Γ_{g} divides 2(p − 1).

We use the convention that lcm{2[N(π) : C(π)]|π ∈ S} = 1 in case S is empty (the p-period of Γ_{g} equals one in that case).

In case of the prime 3, one can find suitable subgroups of Γ_{g} to get the following even simpler result.

THEOREM 2. Let g > 1 and assume that Γ_{g} is 3-periodic. Then the 3-period of Γ_{g} equals 4.

Indeed, as we will see, it is also possible to give a more explicit description of the p-period of the mapping class group in the general case. In particular, for g ≡ 1 mod(p) one finds the following.

THEOREM 3. Let p be an odd prime and g ≡ 1 mod(p). If Γ_{g} is p-periodic, then the p-period of Γ_{g} is 2(p − 1).

The basic idea is to study for each subgroup π of order p of Γ_{g} the action of the normalizer N(π) of π in Γ_{g} on a spherical space of the form ^{d} × ( ^{k} − {0}), which comes up as a subspace of the normal bundle of the fixed point set of the action of π on the Teichmüller space of S_{g}. The numbers d and k in ^{d} × ( ^{k} − {0}) turn out to depend only on the orbifold S_{g}/π. We use then the fact that the p-periods of the subgroups of the form N(π) of Γ_{g} determine the p-period of Γ_{g}.

The rest of the paper is organized as follows. In Section 1 we provide some background material on groups acting on spherical spaces. In Section 2 we will establish an upper bound for the p-period of Γ_{g}. In Section 3 we finish the proof of Theorem 1 in the case that g ≢ 1 mod(p), and we also establish Theorem 2. Section 4 is devoted to the case g ≡ 1 mod(p), and in the last Section 5 we discuss an explicit formula for the p-period; in particular, we will prove Theorem 3.

We would like to thank the referee for valuable suggestions.

1. Groups acting on spherical spaces

We call a (not necessarily compact) manifold X a spherical space, if X is homotopy equivalent to a sphere. A classical result on finite groups G states that if G acts freely on an odd dimensional sphere S^{k−1}, then Ĥ^{∗}(G; ) is periodic, of period dividing k. For the convenience of the reader we provide below an adaption of the classical proof to the setting of Farrell cohomology of groups of finite virtual cohomological dimension acting on spherical spaces.

(1.1) LEMMA. Let Γ denote a group of finite virtual cohomological dimension acting properly discontinuously on a spherical space X homotopy equivalent to an odd dimensional sphere S^{k−1}. Assume that the stabilizer of any point x ∈ X is a finite p-torsion free group, p a fixed prime. Suppose furthermore that Γ acts trivially on H^{k−1}(X; ). Then Γ is p-periodic of p-period dividing k.

Proof. For every finite p-subgroup P of Γ, P acts freely on X and there is an orientable spherical fibration (up to homotopy)

S^{k−1} ≃ X ⟶ X/P ⟶ BP .

The Gysin sequence of this spherical fibration shows that the Euler class e(P) ∈ H^{k}(P; ) induces, for n > dimX, an isomorphism

− ∪ e(P) : H^{n}(P; ) ⟶ H^{n+k}(P; ) .

It follows that Ĥ^{∗}(P; ) is periodic, and e(P) maps under the canonical map

H^{∗}(P; ) ⟶ Ĥ^{∗}(P; )

to a unit ê(P) ∈ Ĥ^{k}(P; ). If we write e(Γ) for the Euler class of

S^{k−1} ≃ X ⟶ EΓ ×_{Γ} X ⟶ BΓ

we conclude that ê(Γ) ∈ Ĥ^{k}(Γ; ) restricts to the unit ê(P) for every finite p-subgroup P < Γ. This implies (cf. [B]) that ê(Γ)_{(p)} ∈ Ĥ^{k}(Γ; )_{(p)} is a unit. As a result, the p-period of Γ divides k.

2. An upper bound for the p-period of Γ_{g}

Let S_{g} be a closed oriented surface of genus g > 1. It is classical that the Teichmüller space T_{g} of S_{g} is homeomorphic to ^{6g−6}, and T_{g} admits a complex structure such that Γ_{g} acts on T_{g} properly discontinuously by holomorphic automorphisms. If a subgroup G < Γ_{g} stabilizes a point of T_{g}, then G is necessarily finite and there exists a complex structure θ on S_{g} such that G lifts to a group of holomorphic automorphisms of (S_{g}, θ) (cf. [K]). When no confusion can arise, we will in our notation not distinguish between G and the lifted group (isomorphic to G) acting on S_{g}.

According to [M-H], the fixed point set of a finite group G < Γ_{g} acting on T_{g} is a submanifold homeomorphic to the Teichmüller space T ( ) where is a Fuchsian group containing π_{1}(S_{g}) with G ≅ /π_{1}(S_{g}). If the Fuchsian group admits a presentation of the form

=< x_{1}, y_{1}, …, x_{h}, y_{h}; e_{1}, …, e_{r}| e_{i}^{m_{i}} = 1 , ∏_{j=1}^{h}[x_{j}, y_{j}] ∏_{k=1}^{r} e_{k} = 1 > ,

where all m_{i} ’s are greater than one, then the Teichmüller space T ( ) is then homeomorphic to ^{6h−6+2r}.

Applied to the case of G ≅ /p, we obtain the following.

(2.1) PROPOSITION. Let p be a prime and π < Γ_{g} a subgroup of order p. Then the fixed point set (T_{g})^{π} is homeomorphic to ^{6h−6+2n}, where h is the genus of S_{g}/π and n the number of fixed points of the π-action on S_{g}.

Proof. Consider the branched covering S_{g} ⟶ (S_{g}/π) =: S_{h} and define to be the orbifold fundamental group of S_{g}/π, that is,

=< x_{1}, y_{1}, …, x_{h}, y_{h}; e_{1}, …, e_{n}|e_{1}^{p} = … = e_{n}^{p} = 1, ∏[x_{j}, y_{j}] ∏ e_{k} = 1 > .

Let p_{1}, …, p_{n} ∈ S_{g} be the fixed points of the π-action on S_{g}. Then, writing p̄_{1}, …, p̄_{n} for the images of the p_{i} ’s in S_{h}, one obtains a regular p-sheeted covering space

S_{g} − {p_{1}, …, p_{n}} ⟶ S_{h} − {p̄_{1}, …, p̄_{n}} .

The induced map of fundamental groups

φ : π_{1}(S_{g} − {p_{1}, …, p_{n}}) ⟶ π_{1}(S_{h} − {p̄_{1}, …, p̄_{n}})

gives rise to an injective map

φ̄ : π_{1}(S_{g}) ⟶

with image a normal subgroup of index p. It follows that T ( ) ≅ ^{6h−6+2n}.

Let S_{g}, g > 1, be a closed oriented surface of genus g and π < Γ_{g} a subgroup of order p, p a fixed prime. We will write n(π) for the number of fixed points of the π-action on S_{g}, and h(π) for the genus of S_{g}/π. We will establish the following upper bound for the p-period of Γ_{g}.

(2.2) THEOREM. Let g > 1 and p a prime. Assume that Γ_{g} is p-periodic. Then the p-period of Γ_{g} divides

lcm{6(g − h(π)) − 2n(π)|π ∈ S}

where S denotes a set of representatives of conjugacy classes of subgroups of order p of Γ_{g}.

Proof. It is well-known that there is a differentiable structure on T_{g} such that Γ_{g} acts smoothly. Let π < Γ_{g} be a subgroup of order p. The normalizer N(π) acts on (T_{g})^{π} ≅ ^{6h(π)−6+2n(π)} as well as the normal bundle E of (T_{g})^{π} in T_{g}; of course, E is homeomorphic to ^{6h(π)−6+2n(π)} × ^{6(g−h(π))−2n(π)} since T_{g} ≅ ^{6g−6}. We now consider the N(π)-action on the spherical space E − E_{0}, where E_{0} denotes the zero section of the bundle E ⟶ (T_{g})^{π}. First we check that the stablizer N(π)_{e} < N(π) of every point e ∈ E − E_{0} is a finite group of order prime to p. The projection E ⟶ (T_{g})^{π} is N(π)- equivariant so that N(π)_{e} is mapped injectively to N(π)ē, which is contained in the finite stabilizer (Γ_{g})ē of ē ∈ (T_{g})^{π} ⊂ T_{g}, where ē denotes the image of e in (T_{g})^{π}. Let x ∈ N(π)_{e}. Assume that x^{p} = 1. Since x ∈ N(π), it normalizes π and, as x^{p} = 1, it follows that x even centralizes π. Thus, the subgroup generated by x and π, < x, π >, is isomorphic to /p × /p, or it is equal to π. Because N(π) is a subgroup of Γ_{g}, it is p-periodic too, and it follows that < x, π >≅ /p × /p. Thus x ∈ π and if x ≠ 1, we have < x >= π. Therefore the (linear) x-action on Vē − {0} must be free; here, Vē denotes the fibre over ē of the bundle projection E → (T_{g})^{π}. Since xe = e, we conclude thus that x ≠ 1 implies that e ∈ E_{0}, the 0-section of the bundle E ⟶ (T_{g})^{π}, and it follows that the action of N(π) on (E − E_{0}) ≅ ^{d} × ( ^{k} − {0}) has finite stabilizers of order prime to p, with d = 6h(π) − 6 + 2n(π), k = 6(g − h(π)) − 2n(π) . The action is properly discontinuous on E and therefore on E −E_{0}, since it is properly discontinuous on (T_{g})^{π} ⊂ T_{g}, and the fibers of E ⟶ (T_{g})^{π} are locally compact. Also, N(π) acts trivially on H^{∗}(E −E_{0}; ), as one can see as follows. The N(π) action on T_{g} and (T_{g})^{π} is by complex automorphisms with respect to some fixed complex structure on T_{g}. The normal bundle of (T_{g})^{π} in T_{g} is the quotient bundle of τ(T_{g})|(T_{g})^{π} by τ(T_{g}^{π}), where τ(T_{g}) (respectively τ(T_{g}^{π})) denotes the tangent bundle of T_{g} (respectively (T_{g})^{π}). All these bundles have a natural orientation, induced by the complex structure of T_{g}, and these orientations are preserved by the N(π)-action. It is then obvious that N(π) acts trivially on H^{∗}(E−E_{0}; ). We are therefore in the situation of (1.1) and conclude that the p-period of N(π) divides k = 6(g−h(π))−2n(π). By a result of Brown [B], the fact that Γ_{g} is p-periodic implies that Ĥ^{∗}(Γ_{g}; )_{(p)} ≅ ∏_{π∈S} Ĥ^{∗}(N(π); )_{(p)}, where S denotes a set of representatives for the conjugacy classes of subgroups of order p of Γ_{g}. The conclusion of the theorem now follows.

In view of the application in Section 3, it is useful to rewrite our result on the pperiodicity of Γ_{g} as follows.

(2.3) COROLLARY. Let g > 1 and π < Γ_{g} a subgroup of order p, p a fixed prime. Assume that Γ_{g} is p-periodic . Then the p-period of N(π) divides

(3m + k)p − 3(m + k) − 2i

where the integers k ≥ 0, m > 0 and i with 0 ≤ i ≤ p − 1 are uniquely determined by the equations 2g − 2 = mp − i, and n(π) = kp + i.

Proof. From the proof of (2.2) we infer that the p-period of N(π) divides 6(g −h(π))− 2n(π), where h(π) denotes the genus of S_{g}/π and n(π) the number of fixed points of the π-action on S_{g}. By the Riemann-Hurwitz formula applied to the branched covering space S_{g} ⟶ S_{g}/π one has 2 − 2g = p(2 − 2h(π)) − n(π)(p − 1). Also, if one writes 2g − 2 in the form mp−i with m > 0 and 0 ≤ i ≤ p−1, then, the Riemann-Hurwitz formula shows that n(π) = kp+i for some k ≥ 0. It follows that 2g = mp−i+2 and 2h(π) = 2+m−k(p−1)−i so that 6(g − h(π)) − 2n(π) = (3m + k)p − 3(m + k) − 2i.

3. The p-period of Γ_{g} for g ≢ 1 mod(p)

We will make repeated use of the following Lemma.

(3.1) LEMMA. Let p be a prime and N a group of finite virtual cohomological dimension which is p-periodic. Suppose N contains a normal subgroup π < N of order p. Then the following holds.

(a) If x ∈ Ĥ^{∗}(N; ) restricts to a unit in Ĥ^{∗}(π; ) and degx ≠ 0, then the p-period of N divides |degx|.

(b) The p-period of N has the form 2[N : C(π)]p^{α}, where C(π) < N denotes the centralizer of π in N and α ≥ 0 an integer.

Proof. We can write x ∈ Ĥ^{∗}(N; ) uniquely as a sum of p-primary elements x_{(p)} ∈ Ĥ^{∗}(N; )_{(p)}. If the reduction x̄ ∈ Ĥ^{∗}(N; /p) is a unit, then so is x_{(p)} ∈ Ĥ^{∗}(N; )_{(p)}. This follows from the fact (cf. [B; Chapter X, 6.6]) that the reduction map

λ : Ĥ^{∗}(N; )_{(p)} ⟶ Ĥ^{∗}(N; /p)

has the property that ker(λ) is nilpotent and for every u ∈ Ĥ^{∗}(N; /p) there is an integer k such that u^{p}^{k} ∈ Im(λ). Namely, if y is an inverse for x̄ and y^{p}^{k} = λ(z), then (x^{p}^{k}z−1)_{(p)} is nilpotent and thus x_{(p)} is a unit. Thus, if x̄ is a unit and degx̄ ≠ 0 , then the period of N divides |degx|. Because N is p-periodic and π < N is normal, π is the only subgroup of order p of N. Quillen’s F-isomorphism theorem then implies (cf. [B]) that the restriction map

φ : Ĥ^{∗}(N; /p) ⟶ Ĥ^{∗}(π; /p)^{N(π)/C(π)} =: H^{∗}

has the property that ker(φ) is nilpotent and that for every v ∈ H^{∗} there is an integer s such that v^{p^{s}} ∈ Im(φ). As before, we conclude that if φ(x̄) is a unit, then so is x̄. But φ(x̄) ∈ H^{∗} is invertible if and only if it is invertible as an element in Ĥ^{∗}(π; /p), and the invertible elements of Ĥ^{∗}(π; ) ≅ /p[w, w^{−1}], degw = 2, are precisely those, which map to invertible elements in Ĥ^{∗}(π; /p), proving (a). For (b) we observe that the p-period of N is the smallest positive integer k for which Ĥ^{k}(N; )_{(p)} contains a unit. By the discussion above, this is, up to a p^{th}-power, the smallest positive degree for which Ĥ^{∗}(N; /p) contains a unit and, using the map φ, this is up to a p^{th}-power the smallest positive degree for which H^{∗} contains a unit. But H^{∗} is periodic with period 1 if p = 2, and period 2[N(π) : C(π)] if p is odd. Therefore, the p-period of N has the form 2[N(π) : C(π)]p^{α}. We will be interested in the case where π < Γ_{g} and N = N(π), the normalizer of π in Γ_{g}. To prove our main theorem stated in the introduction, it suffices to show that α = 0 in (3.1, (b)) for N = N(π) < Γ_{g}. In this section, we settle the case g ≢ 1 mod(p).

(3.2) PROPOSITION. Let g > 1 and assume that p is an odd prime. If g ≢ 1 mod(p) and π < Γ_{g} a subgroup of order p, then N(π) is p-periodic with p-period equal to 2[N(π) : C(π)].

Proof. As observed in the introduction, if p is an odd prime and g ≢ 1 mod(p), then Γ_{g} is p-periodic. Thus N(π) is p-periodic and to prove the proposition, we need, because of (3.1, (b)), only show that the p-period of N(π) is not divisible by p. There will be two cases to consider. In the first case, the upper bound (2.3) for the p-period is prime to p and we are done. For the other case, we will construct an element c_{p−1}(ρ) ∈ H^{2(p−1)}(N(π); ) with the property that it restricts to a unit in Ĥ^{2(p−1)}(π; ); this implies by (3.1, (a)) that the p -period of N(π) divides 2(p − 1), which is prime to p. To this end, consider the natural action of Γ_{g} on H_{1}(S_{g}; ) and write

ρ : Γ_{g} ⟶ GL(2g, )

for the corresponding representation over . Since π ≅ /p admits only two irreducible -representations, the trivial one, which we denote by τ, and the reduced regular representation δ : π ⟶ GL(p − 1, ), the character of μ = ρ|π satisfies

χ_{μ} = a(π)χ_{τ} + b(π)χ_{δ}  (3.3)

for some natural numbers a(π) and b(π). It is easy to check (cf. [G-M]) that the Chern class c_{(p−1)}(μ) ∈ H^{2(p−1)}(π; ) is non-zero if and only if b(π) is relatively prime to p. The number b(π) depends on the number of fixed point of the π action on S_{g} and can be determined as follows. Let x ∈ π be a generator and 1 ∈ π the neutral element. Then, for the reduced regular representation δ one has χ_{δ}(x) = −1 and therefore (3.3) yields

χ_{ρ}(x) = a(π) − b(π), χ_{ρ}(1) = 2g = a(π) + (p − 1)b(π)  (3.4)

On the other hand, the Lefschetz Trace Formula shows that

2 − χ_{ρ}(x) = n(π),  (3.5)

where n(π) denotes the number of fixed points of the x-action on S_{g}. It is convenient to write 2g − 2 in the form mp − i with 0 ≤ i ≤ p − 1. Then, from the Riemann-Hurwitz Formula, one has 2g − 2 ≡ −n(π) mod(p) and thus n(π) = kp + i for some k ≥ 0. Solving (3.4) and (3.5) for b(π), yields

b(π) = m + k  (3.6)

As 2g−2 = mp−i and g ≢ 1 mod(p), we have i ≢ 0 mod(p). By (2.2) the p-period of N(π) divides (3m+k)p−3(m+k)−2i. As observed, if p does not divide (3m+k)p−3(m+k)−2i, we are done, because then necessarily α = 0 in (3.1). On the other hand, if p divides (3m + k)p − 3(m + k) − 2i, then 3(m + k) ≡ −2i ≢ 0 mod(p) since p is odd and i ≢ 0 mod(p); of course, p ≥ 5 in that case. It then follows that m + k ≢ 0 mod(p) and by (3.6), that b(π) is not divisible by p. We conclude then that c_{p−1}(ρ) ∈ H^{2(p−1)}(π; ) is nonzero, and it is clear that then c̄_{p−1}(μ) ∈ Ĥ^{2(p−1)}(π; ) is necessarily a unit , concluding the proof of the proposition.

(3.7) COROLLARY. The 3-period of every 3-periodic mapping class group Γ_{g} is 4.

Proof. According to [X1], Γ_{g} is 3-periodic if and only if g ≢ 1 mod(3). Thus, by Proposition (3.2), the 3-period of a 3-periodic Γ_{g} is 2 or 4. To rule out the value 2, it suffices to find a subgroup of Γ_{g} whose 3-period is 4. We claim that for g ≢ 1 mod(3), Γ_{g} contains the Dihedral group D_{6} of order 6. Namely, by [X3], Γ_{2}, Γ_{3}, Γ_{5} and Γ_{6} contain D_{6}, and, as shown in [X1], if Γ_{g} contains a finite subgroup G of order |G| then so does Γ_{g+|G|}. Thus, every Γ_{g}, g ≢ 1 mod(3), contains D_{6}.

4. The p-period of Γ_{g} for g ≡ 1 mod(p)

Suppose that Γ_{g} is p-periodic and 1 < g ≡ 1 mod(p). As discussed in Section 3, to show that the p-period of Γ_{g} is given by the formula of Theorem 1 in the introduction amounts to showing that for π < Γ_{g} any subgroup of order p, the p-period of N(π) is relatively prime to p. This will be done by constructing for each such π a symplectic characterstic class

d_{k(π)}(ρ) ∈ H^{2k(π)}(Γ_{g}; )

where k(π) satisfies 1 ≤ k(π) ≤ p − 1, such that d_{k(π)}(ρ) restricts to a unit in Ĥ^{∗}(π; ). These characteristic classes arise as follows. We look at the natural representation of ρ : Γ_{g} ⟶ Sp(2g, ), by letting Γ_{g} act on H^{1}(S_{g}; ), preserving the symplectic form given by the cup-product. Recall that

H^{∗}(BSp(2g, ); ) ≅ [d_{1}, …, d_{g}]

where the d_{i}’s are such that they restrict to the universal Chern classes of a maximal compact subgroup U(g) < Sp(2g, ). The images of the d_{i}’s under the induced map

ρ^{∗} : H^{∗}(BSp(2g, ); ) ⟶ H^{∗}(Γ_{g}; )

gives rise to symplectic characteristic classes

d_{i}(ρ) := ρ^{∗}(d_{i}) ∈ H^{2i}(Γ_{g}; ), 1 ≤ i ≤ g .

For our application, the following algebraic Lemma will be useful.

(4.1) LEMMA. Let p be a prime and f(x) ∈ _{p}[x] a polynomial satisfying f(x) ≡ 1 mod(x^{p}) and which factors completely over _{p} into linear factors. Then each root of f has multiplicity divisible by p.

Proof. Let λ_{1}, …, λ_{n} be the roots of f with multiplicities. The assumption on f implies that the first (p−1) elementary symmetric functions in the λ ’s vanish. As a result

∑_{i=1}^{n} λ_{i}^{k} = 0, 1 ≤ k ≤ p − 1 .

If n_{μ} ≥ 0 denotes the multiplicity of μ ∈ _{p} as a root of f, then we infer

∑_{μ∈ _{p}−{0}} n_{μ}μ^{k} = 0, 1 ≤ k ≤ p − 1 .

Since the Van der Monde matrix [μ^{k}] with μ ∈ _{p} − {0} and 1 ≤ k ≤ p − 1 is regular, we conclude that n_{μ} ≡ 0 mod(p) for all μ ∈ _{p} − {0}.

The following useful Lemma on Chern classes of representations of /p is now an easy consequence.

(4.2) LEMMA. Let φ : /p ⟶ U(n) be a representation of /p, p a fixed prime. Assume that c_{i}(φ) ∈ H^{2i}( /p; ) is zero for 1 ≤ i ≤ p − 1. Then φ is of the form pψ ⊕ τ, with τ a trivial representation.

Proof. Decompose φ into one-dimensional representations, φ = ⊕_{k=0}^{p−1}n_{k}ω^{k}, with ω a faithful one-dimensional representation of /p. We put τ = n_{0}ω^{0} and need to show that each n_{i}, 1 ≤ i ≤ p − 1, is divisible by p. Consider the injective ring homomorphism

p[x] ⟶ H^{∗}( /p; /p)

given by mapping x to c̄_{1}(ω) ∈ H^{2}( /p; /p), the reduction mod(p) of c_{1}(ω). Note that the polynomial f(x) ∈ _{p}[x] defined by ∏_{k=0}^{p−1}(1+kx)^{n_{k}} is mapped to the total Chern class c̄(φ) = ∑ c̄_{i}(φ). Since c̄_{i}(φ) = 0 for 1 ≤ i ≤ p − 1, we conclude that f(x) ≡ 1 mod(x^{p}). By the previous Lemma, we conclude that n_{k} is divisible by p for 1 ≤ k ≤ p − 1.

We now return to the study of the representation

ρ : Γ_{g} ⟶ Sp(2g, )

and complete the proof of Theorem 1.

(4.3) PROPOSITION. Let p be an odd prime and g ≡ 1 mod(p). Assume that Γ_{g} is p-periodic. Then, for every subgroup π < Γ_{g} of order p there exists an integer k(π) with 1 ≤ k(π) ≤ p − 1 such that d_{k(π)}(ρ|π) ∈ Ĥ^{2i}(π; ) is a unit, where i = k(π).

Proof. The representation ρ : Γ_{g} ⟶ Sp(2g, ) induces a representation ρ|π : π ⟶ Sp(2g, ) which factors uniquely (up to a conjugation) through a maximal compact subgroup U(g) < Sp(2g, ). We can therefore think of the classes d_{i}(ρ|π) as Chern classes c_{i}(ρ̄) of the representation ρ̄: π ⟶ U(g) obtained in this way. We want to show that if we had d_{i}(ρ|π) = 0 for 1 ≤ i ≤ p − 1, then Γ_{g}would contain a subgroup of the form /p × /p, contradicting the assumption that Γ_{g} be p-periodic. The representation ρ̄: π ⟶ U(g) can be realized by choosing a complex structure and a Hermitian metric on H^{1}(S_{g}; ) compatible with the π-action and symplectic structure. This can be done by choosing a complex structure on S_{g} such that π acts by holomorphic automorphisms on S_{g}; the induced action on the space W^{1} of holomorphic 1-forms of S_{g} is then a model for the representation ρ̄. If we decompose ρ̄ as ⊕_{i=0}^{p−1}n_{i}ω^{i}, ω a faithful irreducible one-dimensional representation then, assuming d_{i}(ρ|π) = c_{i}(ρ̄) = 0 for 1 ≤ i ≤ p − 1, we infer n_{i} ≡ 0 mod(p) for 1 ≤ i ≤ p − 1 (cf. (4.2)). Let x ∈ π be a generator and denote by (β_{1}, …, β_{n(π)}) the fixed point datum of the x-action on S_{g}; thus 1 ≤ β_{i} ≤ p − 1, and for some numbering of the fixed points of x and in a suitable local coordinate system about the j’th fixed point, the action of x^{−1} is given by z ↦ exp(2π\sqrt{−1}β_{j}/p). We can think of the numbers n_{i} in the decomposition of ρ̄ as dimensions of eigenspaces of the x-action on W^{1}, the space of holomorphic 1-forms. According to [F-K; Chapter V, 2.2.3 and 2.5.4] one has then

n_{0} = h(π)

where h(π) denotes the genus of S_{g}/π, and for 1 ≤ j ≤ p − 1,

n_{j} = h(π) − 1 + n(π) − ∑_{i=1}^{n(π)} < δ(j)β_{i}/p >  (4.4)

for a suitable permutation δ of {1, 2, …, p − 1}; we use the notation < q > to denote the fractional part of a rational number q. By renumbering the fixed points of π suitably, we may assume that δ is the identity permutation. The equations (4.4) imply then, because n_{j} ≡ 0 mod(p) for 1 ≤ j ≤ p − 1, that

n_{p−1} − n_{1} = ∑ β_{i}/p − ∑ < (p − 1)β_{i}/p >= 2 ∑ β_{i}/p − n(π) ≡ 0 mod(p) , (4.5)

and

n_{p−1} + n_{1} = 2((h(π) − 1 + n(π)) − n(π) ≡ 0 mod(p) .  (4.6)

Moreover, the Riemann-Hurwitz equation 2 − 2g = p(2 − 2h(π)) − n(π)(p − 1) shows that, because g ≡ 1 mod(p), one has n(π) ≡ 0 mod(p) so that (4.5) and (4.6) imply

h(π) ≡ 1 mod(p) ,  (4.7)

and

∑ β_{i} ≡ 0 mod(p^{2}) .  (4.8)

With g = kp + 1, h(π) = tp + 1 and n(π) = sp, the Riemann-Hurwitz equation can now be written in the form

(2 − 2g)/p^{2} = (2 − 2(t + 1)) − s(1 − 1/p) .  (4.9)

Because of (4.8) we see that either n(π) = 0, or else n(π) > p, so that n(π) = sp with s = 0 or s > 1. We wish to prove that this implies that /p× /p < Γ_{g}. For this, it suffices in view of (4.9) to show that /p× /p admits a generating set (a_{1}, …, a_{t+1}, b_{1}, …, b_{t+1}, c_{1}, …, c_{s}) satisfying the relations ∏[a_{i}, b_{i}] ∏ c_{j} = 1 with each c_{j} of order p (see for instance [T]). By choosing a generating set {a, b} for /p × /p, we can proceed as follows. In case s = 0 (that is, there are no c_{j} ’s), we choose a_{i} = a and b_{i} = b for 1 ≤ i ≤ t + 1. In case s ≥ 2, if s ≡ 1 mod(p), choose a_{i} = a and b_{i} = b (1 ≤ i ≤ t + 1), and c_{1} = c_{2} = … = c_{s−2} = a, c_{s−1} = a^{2}, c_{s} = a^{−1}; if s ≢ 1 mod(p), choose a_{i} = a, b_{i} = b (1 ≤ i ≤ t + 1) and c_{1} = c_{2} = … = c_{s−1} = a, c_{s} = a^{1−s}. We conclude that the assumption that d_{i}(ρ|π) = 0 for 1 ≤ i ≤ p − 1 yields a contradiction. Thus, for some k(π) with 1 ≤ k(π) ≤ p − 1, we infer that d_{k(π)}(ρ|π) ∈ H^{2k(π)}(π; ) is a non-zero element. But then d_{k(π)}(ρ|π) ∈ Ĥ^{∗}(π; ) is necessarily a unit, because

Ĥ^{∗}(π; ) ≅ /p[w, w^{−1}]

with deg w = 2, finishing the proof of the Proposition.

5. An explicit formula for the p-period of Γ_{g}

We first give a proof of Theorem 3 of the introduction. It is clear from covering space theory that if g ≡ 1 mod(p), say g = kp + 1, then the surface S_{h} of genus h = k + 1 will have S_{g} as p-fold regular covering space, and therefore there exists a subgroup π < Γ_{g} of order p, which acts freely on S_{g}. By a classical result of Nielsen [N], all fixed point free homeomorphisms of S_{g}, having the same finite order, are conjugate in Γ_{g} so that we conclude that N(π)/C(π) ≅ /(p − 1). Theorem 1 then implies immediately that the p-period of Γ_{g} equals 2(p − 1), which proves Theorem 3.

For the general case, we use of the following formula.

(5.1) LEMMA [X2; 3.1]. Let g > 1 and write S for the set of conjugacy classes of subgroups π < Γ_{g} of order p. Then

lcm{[N(π) : C(π)] |π ∈ S} = lcm{gcd(p − 1, n(π)) |π ∈ S}

where n(π) denotes the number of fixed points of the π-action of S_{g}.

It is therefore possible to compute the p-period of a p-periodic Γ_{g}, if one knows the possible fixed point numbers n(π) for homeomorphisms of order p on S_{g}. These numbers were determined in [X1] and look as follows. If we write 2g − 2 in the form mp − i with 0 ≤ i ≤ p − 1, define for an odd prime p

B_{g,p} = \{ \begin{matrix}\{i, i + p, …, i + [2g/(p − 1) − m]p\}, & if i ≢ 1 mod(p); \\ \{1 + p, …, 1 + [2g/(p − 1) − m]p\}, & if i ≡ 1 mod(p). \end{matrix}

As usual, we use here the notation [x] to denote the integral part of the rational number x. According to [X1], B_{g,p} consists precisely of the set of all those numbers, which occur as cardinalities of the fixed point set for homeomorphisms of order p on S_{g}; (if, in case i ≢ 1 mod(p), one has 2g/(p − 1) < m, then B_{g,p} is empty and Γ_{g} contains no element of order p, similarly in case i ≡ 1 mod(p)). Combining Theorem 1 with (5.1), one obtains the following explicit formula for the period of a p-periodic mapping class group.

(5.2) THEOREM. Suppose p is a prime, g > 1 and that Γ_{g} is p-periodic. Then the p-period of Γ_{g} is given by

lcm{gcd(2(p − 1), 2n)|n ∈ B_{g,p}} .

For instance, (5.2) implies that for large values of g (e.g., 2g > p^{3}), the p-period of a p-periodic Γ_{g} is precisely 2(p − 1).

References.

[B] K. S. Brown, Cohomology of Groups, Graduate Texts in Math. Vol. 87, Springer Verlag, 1982.

[F-K] H. Farkas and I. Kra, Riemann Surfaces, Graduate Texts in Math. Vol. 71, Springer

Verlag, 1982. [G-M] H. Glover and G. Mislin, Torsion in the mapping class group and its cohomology,

J. Pure and Applied Alg. 44 (1987), 177-189.

[H] J. L. Harer, The virtual cohomological dimension of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985), 215-249.

[K] S. Kerckhoff, The Nielsen realization problem, Ann. of Math. 117 (1983), 235-265. [M-H] C. MacLachlan and W. J. Harvey, On mapping-class groups and Teichmüller spaces,

Proc. London Math. Soc. (3) 30 (1975), 496-512.

[N] J. Nielsen, Die Struktur periodischer Transformationen von Flächen, Danske Vid. Selsk. Mat.-Fys. Medd. 15 (1937), 1-77.

[T] T. Tucker, Finite groups acting on surfaces and the genus of a group, J. Comb. Theory B 34 (1983), 82-98.

[X1] Y. Xia, The p-periodicity of the mapping class group and the estimate of its p-period, to appear in Proc. Amer. Math. Soc.

[X2] Y. Xia, The p-period of an infinite group, to appear in Publicacions Matemàtiques.

[X3] Y. Xia, Farrell-Tate cohomology of the mapping class group, Thesis, The Ohio State University, 1990.

Ohio State University  ETH-Zentrum  Ohio State University
Columbus, Ohio  Zürich, Switzerland  Columbus, Ohio

and

Ohio State University Columbus, Ohio

September 1991, revised February 1992