ON THE YAGITA INVARIANT OF MAPPING CLASS GROUPS

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

March 1993

Abstract. Let Γ denote a group of finite virtual cohomological dimension and p a prime. If the cohomology ring H^{∗}(Γ; 𝔽_{p}) has Krull dimension one, the p-period of Γ is defined; it measures the periodicity of H^{∗}(Γ; 𝔽_{p}) in degrees above the virtual cohomological dimension of Γ. The Yagita invariant p(Γ) of Γ is a natural generalization of the p-period to groups with H^{∗}(Γ; 𝔽_{p}) of Krull dimension larger than one. We compute the Yagita invariant p(Γ_{g}) for the mapping class group Γ_{g} with respect to an arbitrary odd regular prime p.

Introduction

The mapping class group Γ_{g} is defined to be the group of path components of the group of orientation preserving diffeomorphisms of an oriented closed surface S_{g} of genus g. In this paper, we study for a given prime p the Yagita invariant p(Γ_{g}), which is defined as follows (see [Y] for the case of finite groups and [T] for more general groups).

Let Γ be a group of finite virtual cohomological dimension and π ⊂ Γ any subgroup of prime order p. It is well known that the image Im(H^{k}(Γ; ℤ) → H^{k}(π; ℤ)) of the restriction map in cohomology is non-zero for some degree k > 0. Because the natural map H^{∗}(π; ℤ) → H^{∗}(π; 𝔽_{p}) maps onto 𝔽_{p}[u] ⊂ H^{∗}(π; 𝔽_{p}) with u a generator in H^{2}(π; 𝔽_{p}), there exists a maximum value m = m(π) such that

Im((H^{∗}(Γ; ℤ) ⟶ H^{∗}(π; 𝔽_{p})) ⊂ 𝔽_{p}[u^{m}] ⊂ H^{∗}(π; 𝔽_{p}) .

It is easy to see (cf. Lemma 1.1) that the possible values m(π) are bounded by a number depending on Γ only. The Yagita invariant p(Γ) of Γ with respect to the prime p is then defined to be the least common multiple of values 2m(π), where π ranges over all subgroups of order p of Γ. We use the convention that p(Γ) = 1 if Γ is p-torsion free. The invariant p(Γ) agrees with the p-period of a p-periodic group (i.e., a group with p-periodic Farrell cohomology groups, see [X1] and [X2] for a discussion of that concept) and, as it is the case for the p-period, the Yagita invariant p(Γ) divides 2(p − 1)p^{k}, for some k ≥ 0 (see Section 1).

The interest in p(Γ) stems from the fact that it provides a lower bound for the dimension of a complex, which admits a certain type of action of Γ (see [Y] for the case of finite groups). For instance, one checks easily that if Γ acts properly discontinuously on ℝ^{n} × (S^{m})^{k} and trivially on H^{∗}(ℝ^{n} × (S^{m})^{k}; ℤ), in a way that the stabilizer of any point

Typeset by AMS-TEX

x ∈ ℝ^{n} × (S^{m})^{k} is a p-torsion free group, then m + 1 is a multiple of the Yagita invariant p(Γ). We will discuss a variation of this in Section 4.

The mapping class group Γ_{g} is never 2-periodic for g > 1. For an odd prime p and p-periodic Γ_{g} we completely determined the p-period in [GMX]. We recall that for an odd prime p and genus g ≢ 1 mod p, Γ_{g} is always p-periodic; thus we will only need to be concerned with the case g ≡ 1 mod p in the sequel. We have a complete result in case p is an odd regular prime, and partial results for general primes (recall that a prime p is called regular if p does not divide the class number of the cyclotomic field ℚ(exp(2π\sqrt{−1}/p)); the smallest irregular prime is 37). It is convenient for what follows to employ the following terminology.

Definition. Let p be a prime. We say that an integer g satisfies the (p)-condition if and only if g is of the form lp^{α} + 1 with l prime to p, α > 0, and 2l = p(2h − 2) + k(p − 1) for some integers h > 0, k ≥ 0 with k ≠ 1.

Our main result is the following.

Theorem 1. Let p be an odd regular prime and assume that g = lp^{α} + 1 with l prime to p and α > 0. Then the Yagita invariant p(Γ_{g}) is determined as follows.

(i) If g does not satisfy the (p)-condition, then p(Γ_{g}) equals 2(p − 1)p^{α−1}.

(ii) If g satisfies the (p)-condition, then p(Γ_{g}) equals 2(p − 1)p^{α}.

For the case of a general odd prime, we have the following partial results, which underline the role of the (p)-condition.

Theorem 2. Let p be an odd prime and g = lp^{α} + 1 with l prime to p and α > 0. Then the following holds.

(i) p(Γ_{g}) has the form 2(p − 1)p^{α} or 2(p − 1)p^{α−1}.

(ii) If g satisfies the (p)-condition, then p(Γ_{g}) = 2(p − 1)p^{α}.

(iii) If 1 < 2l < p − 1 then p(Γ_{g}) = 2(p − 1)p^{α−1}.

Remark 1. For a fixed prime p and α > 0 there are obviously only finitely many genera g of the form lp^{α} + 1 with l prime to p which do not satisfy the (p)-condition. Thus, we can think of (ii) in Theorem 2 as the generic case.

Remark 2. If the Krull dimension of H^{∗}(Γ_{g}; 𝔽_{p}) equals one so that Γ_{g} is p-periodic, and if we assume g of the form lp^{α} + 1 with l prime to p and α > 0, then the (p)-condition does not hold for g. This can be seen by comparing the (p)-condition with the formula for the Krull dimension for Γ_{g} as stated in ([B]). Moreover, α is then necessarily equal to 1 (otherwise the Krull dimension is larger than 1 by [B]), and we recover the formula p(Γ_{g}) = 2(p − 1) of [GMX] for that case. The finite set of values l prime to p for which g = lp + 1 gives rise to a p-periodic Γ_{g} was determined in an explicit way in [X3], where it is proved that Γ_{lp+1} with l prime to p is p-periodic if and only if l + 1 is prime to p and the interval [(2l + 3)/p, (2l + 2)/(p − 1)] does not contain an integer. The reader can then easily check that the smallest genus g for which there is an odd prime such that the Yagita invariant is not given either by [GMX], Theorem 1 or Theorem 2 is g = 1296 = 35 · 37 + 1.

Remark 3. From Lemma 2.2 it will become obvious that the (p)-condition has a geometric interpretation as follows. If p denotes a prime and g = lp^{α} + 1 with l prime to p and α > 0 then g satisfies the p-condition if and only if Γ_{g} contains a cyclic subgroup C of order p^{α+1} such that C lifts to an action on S_{g} with all stabilizers of order ≤ p; note also that the C action on S_{g} cannot be free, since g − 1 = lp^{α} is not divisible by p^{α+1}.

The remainder of this paper is organized as follows. In Section 1 we provide some basic facts about the Yagita invariant and establish rather crude lower and upper bounds. In Section 2 we recall the definition of the fixed point data of an element of finite order in the mapping class group Γ_{g} and develop a technique of moves, which requires the prime in question to be regular. The moves are used in Section 3 to study representations of subgroups of order p of Γ_{g} on the space of holomorphic differentials of a Riemann surface of genus g. As a result, we will obtain precise lower bounds for Γ_{g}. Using the action of Γ_{g} on a suitable submanifold of the Teichmüller space of surfaces of genus g we establish in Section 4 an upper bound for Γ_{g} and use it to determine sharp upper bounds in Section 5, completing the proofs of the Theorems 1 and 2.

We would like to thank R. Swan for providing the proof of Proposition 2.3.

Section 1 : Some basic facts concerning the Yagita invariant

Our first lemma provides an upper bound for the Yagita invariant p(Γ) .

Lemma 1.1. Let p be a prime and Γ a group of finite virtual cohomological dimension, which has p-torsion. Let G denote a finite factor group of Γ such that the kernel of the projection Γ → G is p-torsion free. Then the Yagita invariant p(Γ) divides 2(p − 1)p^{k−1}, where p^{k} denotes the largest power of p which divides the order of G.

Proof. We consider the regular representation ρ : G → Gl_{|G|}(ℂ) of the finite group G. If we restrict ρ to a subgroup π of order p in G, then ρ|π is of the form sp^{k−1}σ, where s is prime to p and σ denotes the regular representation of π. Since the total Chern class c(σ) in H^{∗}(π; ℤ) has the form 1 + c_{p−1}(σ) with c_{p−1}(σ) ≠ 0, we see that c_{(p−1)p^{k−1}}(ρ) restricts to sc_{p−1}(σ)^{p^{k−1}}, which is non-trivial in the integral cohomology of π. Therefore, the Yagita invariant of the factor group G must divide 2(p − 1)p^{k−1}. But every subgroup of order p in Γ_{g} injects into G via the projection, and therefore viewing ρ as a representation ρ̃ of Γ, its Chern class c_{(p−1)p^{k−1}}(ρ̃) will restrict non-trivially to any subgroup of order p in Γ, showing that p(Γ) divides 2(p − 1)p^{k−1} too.

It is often possible to improve on the power of p in the upper bound of p(Γ) as follows.

Lemma 1.2. Let p be a prime, Γ a group of finite virtual cohomological dimension, and assume that ρ : Γ → Gl_{d}(ℂ) is a representation of degree d such that ρ does not have any element of order p in the kernel.

(i) If d < p^{m} then p(Γ) divides 2(p − 1)p^{m−1}.

(ii) If ρ(Γ) ⊂ Gl_{d}(ℚ) and d < (p − 1)p^{m} then p(Γ) divides 2(p − 1)p^{m−1}.

Proof. We know already that p(Γ) divides 2(p − 1)p^{n} for some n. If we were not able to choose n = m − 1, then there would exist a subgroup π ⊂ Γ such that the restriction map H^{∗}(Γ; ℤ) → H^{∗}(π; ℤ) is zero for 0 < ∗ < 2p^{m}. But, assuming d < p^{m}, one would infer for the total Chern class c(ρ|π) equals 1. This is a contradiction, since ρ is faithful when restricted to π . In case ρ(Γ) ⊂ Gl_{d}(ℚ) , all non-zero Chern classes of ρ|π lie in degrees of the form 2(p − 1)p^{l}, see ([EM]), and thus it suffices to assume d < (p − 1)p^{m} to conclude that some Chern class of ρ|π is non-zero in the range 0 < ∗ < 2p^{m}.

Applying this to the case of the mapping class group Γ_{g} we obtain the following upper bound for the Yagita invariant.

Lemma 1.3. Let p be an arbitrary prime and 0 < 2g < (p − 1)p^{m}. Then p(Γ_{g}) divides 2(p − 1)p^{m−1}.

Proof. We consider the natural action of Γ_{g} on H_{1}(S_{g}; ℚ) which defines a representation ρ : Γ_{g} → Gl_{2g}(ℚ) with torsion-free kernel. The result then follows from (ii) of Lemma 1.2 .

It is plain from the definition that the Yagita invariant for a subgroup of Γ divides p(Γ). We can therefore find lower bounds for p(Γ) by looking at suitable subgroups of Γ. This leads to the following useful lemma.

Lemma 1.4. Let π be a cyclic subgroup of p-power order of Γ, with p an odd prime. If we denote by C(π) (respectively N(π)) the centralizer (respectively normalizer) of π in Γ then p(Γ_{g}) is a multiple of 2[N(π) : C(π)].

Proof. Let W be the cyclic group N(π)/C(π). The image of the restriction map in cohomology H^{∗}(Γ; ℤ) → H^{∗}(π; ℤ) maps into the subring of W -invariant elements, which is of the form ℤ[x]/(p^{k}x, x^{m}), where x ∈ H^{2}(π; ℤ) denotes a generator, m = |W | the order of W, and p^{k} the order of π. The result then follows readily.

As an application of Lemma 1.4, we deduce the lower bound (i) of Theorem 2 of the introduction.

Lemma 1.5. Let p be an odd prime and assume that g is of the form lp^{α} + 1 with l prime to p and α > 0. Then p(Γ_{g}) equals 2(p − 1)p^{β} for some β ≥ α − 1.

Proof. We know from Lemma 1.1 that p(Γ_{g}) divides 2(p−1)p^{n} for some n ≥ 0. If g = lp^{α}+1 with l prime to p, the surface S_{g} is a p^{α}-fold regular (unbranched) covering space of S_{h} where h = l+1, with cyclic covering transformation group generated by a map f : S_{g} → S_{g}. Since f acts freely, it is conjugate in Diffeo^{+}(S_{g}) to f^{j} for any j prime to p ([N], see also Lemma 2.1 below). The subgroup Δ ⊂ Γ_{g} generated by the image of f in Γ_{g} has order p^{α} and N(Δ)/C(Δ) is isomorphic to the full automorphism group of Δ ≅ ℤ/p^{α} which has order (p−1)p^{α−1} (see also Lemma 2.1 below). Therefore, using Lemma 1.1 and Lemma 1.4, the result follows.

Comparing the lower and upper bounds for the Yagita invariant of the mapping class group, we obtain part (iii) of Theorem 2 of the Introduction.

Corollary 1.6. Let p be an odd prime and g = lp^{α} + 1 with 1 < 2l < p − 1 and α > 0. Then the Yagita invariant p(Γ_{g}) equals 2(p − 1)p^{α−1}.

Proof. Since 2g = 2lp^{α} + 2 ≤ (p − 2)p^{α} + 2 < (p − 1)p^{α} we infer from Lemma 1.3 that p(Γ_{g}) divides 2(p − 1)p^{α−1}. On the other hand Lemma 1.5 shows that p(Γ_{g}) is a multiple of 2(p − 1)p^{α−1}.

Section 2 : Fixed point data and moves

A basic invariant of an orientation preserving diffeomorphism f of S_{g} of period n > 1 is its fixed point data. It is defined as follows. Since f preserves orientation, the singular set of f is necessarily discrete in S_{g}. Let {x_{i}} be a set of representatives of the singular orbits of f and write α_{i} for the order of stab_{f}(x_{i}), the stabilizer of f at x_{i} ∈ S_{g}. Then f^{n/α_{i}} generates stab_{f}(x_{i}) and, with respect to a fixed Riemannian structure, the differential of f acts faithfully by rotation on the tangent space at x_{i}. Let β_{i} be an integer such that f^{β_{i}n/α_{i}} acts by rotation through 2π/α_{i}. The number β_{i} is well defined modulo α_{i}, and β_{i} is prime to α_{i}. The fixed point data of f, denoted δ(f), is then the collection

δ(f) =< g, n | β_{1}/α_{1}, …, β_{q}/α_{q} >

where g is the genus of the surface S_{g}, n the order of f, and q the number of singular orbits of the f-action; the numbers β_{1}/α_{1}, …, β_{q}/α_{q} are unique up to order, if we consider them as elements in ℚ/ℤ.

A classical theorem of Nielsen [N] states that two diffeomorphisms of finite order are conjugate in Diffeo^{+}(S_{g}) if and only if they have the same fixed point data. Symonds [Sy] proved that the fixed point data of a diffeomorphism of finite order depends only upon its isotopy class, and thus is well defined for an element of finite order of the mapping class group Γ_{g}; we will thus write δ(x) for the fixed point data of an element of finite order x ∈ Γ_{g}. He also shows that the Nielsen Theorem is still true for the mapping class group Γ_{g}, that is, two elements of finite order in Γ_{g} are conjugate if and only if they have the same fixed point data. As an immediate consequence, one can deduce the following.

Lemma 2.1. Suppose that f ∈ Diffeo^{+}(S_{g}) has finite order n and acts freely on S_{g}. Denote by x the image of f in Γ_{g}. Then the index [N(x) : C(x)] of the centralizer of x in its normalizer equals ϕ(n), ϕ the Euler function.

Proof. Indeed, all the generators of the cyclic group < x > generated by x have the same fixed point data δ =< g, n| > and are therefore conjugate in Γ_{g}. Thus N(x) maps onto the automorphism group of < x >, with kernel C(x), and the result follows.

The well-known techniques on realizing fixed point data lead to the following result, of which we sketch the proof for the convenience of the reader.

Lemma 2.2. Let p be a prime and β_{1}, …, β_{q} integers prime to p. Then the collection

< g, p^{t} | β_{1}/p^{t_{1}}, …, β_{q}/p^{t_{q}} >

can be realized as the fixed point data of an element of order p^{t} in Γ_{g} if and only if the following three conditions are satisfied.

(i) ∑ β_{i}/p^{t_{i}} is an integer .

(ii) The Riemann-Hurwitz formula 2g − 2 = p^{t}(2h − 2) + p^{t} ∑ (1 − 1/p^{t_{i}}) holds for

some h ≥ 0. (iii) If h = 0 in (ii), then t = max(t_{1}, …, t_{q}).

Proof. Suppose the given collection is the fixed point data of some x ∈ Γ_{g}, represented by the diffeomorphism f of S_{g} of order p^{t}. Then there is a branched covering S_{g} → S_{h} with set of branch points {y_{1}, …, y_{q}} ⊂ S_{h} and group ℤ/p^{t}, giving rise to a regular covering

S_{g} \ π^{−1}{y_{1}, …, y_{q}} ⟶ S_{h} \ {y_{1}, …, y_{q}},

which induces a short exact sequence

π_{1}(S_{g} \ π^{−1}{y_{1}, …, y_{q}}) ⟶ π_{1}(S_{h} \ {y_{1}, …, y_{q}}) ⟶^{∂}< f >≅ ℤ/p^{t} .

Note that

π_{1}(S_{h} \ {y_{1}, …, y_{q}}) =< a_{1}, b_{1}, …, a_{h}, b_{h}, x_{1}, …, x_{q} | ∏_{i=1}^{h}[a_{i}, b_{i}]x_{1} … x_{q} = 1 >

and ∂(x_{i}) = f^{β_{i}p^{t−t_{i}}} for 1 ≤ i ≤ q. Therefore (i) is true since the map ∂ preserves the relation ∏_{i=1}^{h}[a_{i}, b_{i}]x_{1} … x_{q} = 1. By calculating the Euler characteristics of S_{g} \ π^{−1}{y_{1}, …, y_{q}} and S_{h} \ {y_{1}, …, y_{q}} one gets (ii), and (iii) follows from the surjectivity of the map ∂. Conversely, given (i), (ii) and (iii) we begin by constructing a surjective homomorphism

∂ : π_{1}(S_{h} \ {y_{1}, …, y_{q}}) ⟶ ℤ/p^{t}

such that ∂(x_{i}) is a suitable element of order p^{t_{i}} for 1 ≤ i ≤ q. This can be done, if h > 0, by putting ∂(a_{1}) = ∂(b_{1}) = 1, ∂(a_{i}) = ∂(b_{i}) = 0 for 2 ≤ i ≤ h, and ∂(x_{j}) = β_{j}p^{t−t_{j}} for 1 ≤ j ≤ q. In case h = 0, we still put ∂(x_{j}) = β_{j}p^{t−t_{j}} for 1 ≤ j ≤ q. The condition (iii) then guarantees that the map ∂ is surjective. The kernel of ∂ defines a p^{t}-sheeted regular covering S_{g} \{z_{1}, …, z_{r}} ⟶ S_{h} \{y_{1}, …, y_{q}}, giving rise to a branched covering S_{g} ⟶ S_{h}, with covering transformation group generated by a diffeomorphism with the desired fixed point data.

Note that given an element x ∈ Γ_{g} of order p^{t}, one can determine the fixed point data δ(x^{k}) from δ(x) =< g, p^{t}|β_{1}/p^{t_{1}}, … β_{1}/p^{t_{q}} > as follows. When k is prime to p then

δ(x^{k}) =< g, p^{t} | lβ_{1}/p^{t_{1}}, …, lβ_{q}/p^{t_{q}} >

where l is a multiplicative inverse of k mod p^{t} . When k is a multiple of p, say k = mp^{s}, with m prime to p, the subgroup generated by k in ℤ/p^{t} is naturally isomorphic to ℤ/p^{t−s} by mapping k = mp^{s} to m in ℤ/p^{t−s}. If we write n for a multiplicative inverse of m mod p^{t−s} then

δ(x^{mp^{s}}) =< g, p^{t−s} | A_{1,1}, …, A_{1,m_{1}}, …, A_{q,1}, …, A_{q,m_{q}} > , 1 ≤ i ≤ q , 1 ≤ j ≤ m_{i} ,

where the A_{i,j} correspond to β_{i} in δ(x), m_{i} = p^{min(s,t−t_{i})}, and A_{i,j} = nβ_{i}/p^{min(t−s,t_{i})} for 1 ≤ j ≤ m_{i}.

For example, if

δ(x) =< 183, 81 | 11/81, 14/81, 1/27, 1/9, 1/3 >

then

δ(x^{41}) =< 183, 81 | 22/81, 28/81, 2/27, 2/9, 2/3 >
δ(x^{3}) =< 183, 27 | 11/27, 14/27, 1/27, 1/27, 1/27, 1/9, 1/9, 1/9, 1/3, 1/3, 1/3 >
δ(x^{9}) =< 183, 9 | 2/9, 5/9, 1/9, 1/9, 1/9, 1/9, 1/9, 1/9, 1/9,
1/9, 1/9, 1/9, 1/9, 1/9, 1/3, 1/3, 1/3, 1/3, 1/3, 1/3, 1/3, 1/3, 1/3 >

and so on.

The following proposition, which will be used repeatedly later on, was suggested to us by an explicit computation with the help of a computer program. The actual proof as presented below was communicated to us by R. Swan ([Sw]). For an integer n and a fixed prime number p we will use the notation ñ to denote the unique integer satisfying 0 ≤ ñ < p and n ≡ ñ mod p. We will also write < r > for the fractional part of a number r ∈ ℚ, so that r = [r]+ < r >, with [r] the integral part of r. Note that for any n ∈ ℤ one has ñ = p < n/p >= n − p[n/p].

Proposition 2.3. Let p be an odd prime and consider the integral (p − 1)/2 × (p − 1)/2 matrix A = (a_{ij}) whose (i, j) entry a_{ij} is defined to be p if i = 1, and ĩj if 2 ≤ i ≤ (p−1)/2. Then the matrix A is non-singular. Moreover, the entries of the matrix pA^{−1} are rational numbers which have, in reduced form, no p in the denominators if and only if p is a regular prime.

Proof [Sw]. For p ≤ 5 the claim is easily checked directly. If p > 5 we proceed as follows. By subtracting each column of A from the next one we get a matrix B which, after subtracting the second column of B from the first one, takes the form

C = ( \begin{matrix}p & 0 & 0 & … & 0 \\ 0 & 2 & 2 & … & 2 \\ 0 & 3 \\ ⋮ & ⋮ & & a_{i,j} − a_{i,j−1} \\ 0 & (p − 1)/2 \end{matrix} ) .

Now a_{i,j} − a_{i,j−1} = ij − p[ij/p] − i(j − 1) + p[i(j − 1)/p] = i − pe_{i,j}, where

e_{i,j} = [ij/p] − [i(j − 1)/p] , 3 ≤ i, j ≤ (p − 1)/2

are the entries of a (p − 5)/2 × (p − 5)/2 matrix E. Subtracting the second column of C from the rest reduces C to

D = ( \begin{matrix}p & 0 & 0 & … & 0 \\ 0 & 2 & 0 & … & 0 \\ 0 & 3 \\ ⋮ & ⋮ & & −pE \\ 0 & (p − 1)/2 \end{matrix} )

with

D^{−1} = ( \begin{matrix}p^{−1} & 0 & 0 & … & 0 \\ 0 & 2^{−1} & 0 & … & 0 \\ 0 & f_{3,2} \\ 0 & f_{4,2} \\ ⋮ & ⋮ & & −\frac{1}{p}E^{−1} \\ 0 & f_{(p−1)/2,2} \end{matrix} ) .

The last (p − 5)/2 entries of the second column of D^{−1} are given by

( f_{3,2} ; f_{4,2} ; ⋮ ; f_{(p−1)/2,2} ) = −\frac{1}{2p}E^{−1} ( 3 ; 4 ; ⋮ ; (p − 1)/2 )

and pD^{−1} has no p in the denominators if and only if det E is prime to p. By [CO], det E = ±h_{1}, where h_{1} is the first factor of the class number of ℚ(exp 2π\sqrt{−1}/p). But Kummer showed p is regular if and only if p does not divide h_{1}.

The following lemma is immediate and we state it without proof.

Lemma 2.4. Suppose we are given a prime p and an integral linear system

(L)  Ax = b

such that (i) A ∈ M_{n}(ℤ) has non-zero determinant, (ii) b ≡ 0 mod p^{2}, (iii) pA^{−1} in reduced form has the property that the matrix entries have no p in their

denominators. If x denotes an integral solution of (L), then x ≡ 0 mod p.

As a consequence we will prove the following proposition, which plays a crucial role in the sequel.

Proposition 2.5. Let p be a regular prime and suppose we are given an integer n > 0 divisible by p and n integers β_{i} with 0 < β_{i} < p, where 1 ≤ i ≤ n. Write α_{j} for the number of β_{i}’s which equal j for 1 ≤ j < p. If the fractional parts < jβ_{i}/p > satisfy

(1)  ∑_{i} < jβ_{i}/p > ≡ 0 mod p,  1 ≤ j < p

then α_{j} ≡ α_{p−j} mod p for all j.

Proof. The case p = 2 is trivial, so we will assume p ≥ 3. One can then rewrite each sum in (1) in terms of the α_{i}’s as follows:

e_{1} = ∑_{i} < β_{i}/p >= ∑_{i} α_{i} < i/p >≡ 0 mod p

e_{2} = ∑_{i} < 2β_{i}/p >= ∑_{i} α_{i} < 2i/p >≡ 0 mod p

. . . . . . . . . . . .

e_{p−1} = ∑_{i} < (p − 1)β_{i}/p >= ∑_{i} α_{i} < (p − 1)i/p >≡ 0 mod p .

For (p + 1)/2 ≤ i ≤ p − 1 we have 2 < i/p > − < 2i/p >= 1, and the first two equations yield

∑_{i=(p+1)/2}^{p−1} α_{i} = 2e_{1} − e_{2} .

Noting that ∑_{i=1}^{p−1} α_{i} = n = kp , we obtain an equation which we consider as the first equation of a linear system (L) of the type Ax = b considered in Lemma 2.4, with the transpose of x the vector

(x_{1}, …, x_{(p−1)/2}) = (α_{1} − α_{p−1}, …, α_{(p−1)/2} − α_{(p+1)/2})

namely the equation

px_{1} + px_{2} + ⋯ + px_{(p−1)/2} = p(n − 4e_{1} + 2e_{2}) .

Observe that the right hand side is 0 mod p^{2}. By using the fact that < (p − j)i/p >= 1− < ji/p >, we obtain (p − 3)/2 additional equations for our system (L), which take the form

p < 2/p > x_{1} + p < 4/p > x_{2} ⋯ + p < (p − 1)/p > x_{(p−1)/2} = p(e_{2} − ∑_{i=(p+1)/2}^{p−1} α_{i})

p < 3/p > x_{1} + p < 6/p > x_{2} + ⋯ + p < 3(p − 1)/2p > x_{(p−1)/2} = p(e_{3} − ∑_{i=(p+1)/2}^{p−1} α_{i})

. . . . . . . . .

p < (p − 1)/2p > x_{1} + ⋯ + p < (p − 1)^{2}/4p > x_{(p−1)/2} = p(e_{(p−1)/2} − ∑_{i=(p+1)/2}^{p−1} α_{i})

It is easy to see that the right hand sides of all equations are 0 mod p^{2} since by assumption all e_{i}’s are 0 mod p and, as we have seen, ∑_{i=(p+1)/2}^{p−1} α_{i} ≡ 0 mod p. Observing that for any integer k one has k ≡ p < k/p > mod p, we see that the matrix A of the linear system (L) is precisely the matrix A of Proposition 2.3. Thus, by applying Lemma 2.4, we infer that α_{j} − α_{p−j} ≡ 0 mod p for 1 ≤ j ≤ p − 1, completing the proof.

The following theorem can be viewed as a purely algebraic version of our “moves” concerning fixed point data as considered in the next section.

Theorem 2.6. Suppose p is an odd regular prime and n > 0 an integer which is divisible by p. Suppose the system of p − 1 equations

∑_{i=1}^{n} < jx_{i}/p >≡ 0 mod p, 1 ≤ j ≤ p − 1

has an integral solution x = (x_{1}, …, x_{n}) with 0 < x_{i} < p for all i. Then it also has an integral solution z = (z_{1}, …, z_{n}) with 0 < z_{i} < p for all i such that for any given integer j the number of z_{i}’s which equal j is divisible by p, and ∑_{i} < jx_{i}/p >= ∑_{i} < jz_{i}/p > for all j.

Proof. Let x = (x_{1}, …, x_{n}) be a solution as above, and denote by α_{j}(x) the number of x_{i}’s which are equal to j, where 0 < j < p. Then we can rewrite our system of equations as

∑_{i=1}^{p−1} α_{i}(x) < ji/p >≡ 0 mod p , 1 ≤ j ≤ p − 1

so that for all j one has α_{j}(x) ≡ α_{p−j}(x) mod p by Proposition 2.5. We can now alter the solution x = (x_{1}, …, x_{n}) to y = (y_{1}, …, y_{n}) by applying a move of type (s, t), where 1 ≤ s, t ≤ p − 1 and s ≠ t, by which we mean the following :

(i) replace α_{s}(x) by α_{s}(y) = α_{s}(x) + 1 (ii) replace α_{p−s}(x) by α_{p−s}(y) = α_{p−s}(x) + 1 (iii) replace α_{t}(x) by α_{t}(y) = α_{t}(x) − 1 (iv) replace α_{p−t}(x) by α_{p−t}(y) = α_{p−t}(x) − 1 .

Note that for such a move to be possible we need to have a t such that α_{t}(x) > 0. If we alter x accordingly into y, we obtain a new system of equations satisfying

∑_{i=1}^{p−1} α_{i}(y) < ij/p >= ∑_{i=1}^{p−1} α_{i}(x) < ij/p >

for 0 < j < p, because < ij/p > + < (p − i)j/p >= 1. Suppose now that not all α_{i}(x)’s are already divisible by p. Then, using Proposition 2.5 and writing n as kp, we have

n = kp = ∑_{i=1}^{p−1} α_{i}(x) ≡ 2 ∑_{i=1}^{(p−1)/2} α_{i}(x) ≡ 0 mod p,

and we conclude that there must exist a pair (s, t) with 1 ≤ s, t ≤ (p − 1)/2 and s ≠ t such that α_{s}(x) and α_{t}(x) are both not divisible by p. Performing a move of type (s, t) will provide a new solution y. If α_{s}(y) and α_{t}(y) are both still not divisible by p, we repeat the move of type (s, t), and eventually α_{s} or α_{t} will be a multiple of p. By continuing in this manner, we will end up with the required solution z.

Section 3 : Chern classes of representations on holomorphic differentials

The action of Γ_{g} on the symplectic space H_{1}(S_{g}; ℝ) with its intersection pairing gives rise to a representation ρ : Γ_{g} → Sp(2g, ℝ), which we call the canonical representation of Γ_{g} in the sequel. Recall that

H^{∗}(BSp(2g, ℝ); ℤ) ≅ ℤ[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, ℝ). The images of the d_{i}’s under the induced map

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

gives rise to symplectic characteristic classes

d_{i}(ρ) := ρ^{∗}(d_{i}) ∈ H^{2i}(Γ_{g}; ℤ), 0 ≤ i ≤ g .

We are going to study the behavior of the restriction of these symplectic characteristic classes to cyclic subgroups Δ of order p^{n} in Γ_{g}. Note that ρ|Δ factors, up to conjugation, through U(g) ⊂ Sp(2g, ℝ) and we can view the classes d_{i}(ρ|Δ) as the Chern classes c_{i}(ρ̄) of a representation ρ̄: Δ → U(g). This representation can be thought of in the following way. One chooses a lift of a generator of Δ to an element of order p^{n} in Diffeo^{+}(S_{g}) and chooses a complex structure on S_{g} compatible with the orientation of S_{g} such that f acts by a holomorphic automorphism. Then f acts on the associated space of holomorphic differentials, which is a complex vector space of complex dimension g. This action defines a representation of Δ, whose dual is equivalent to the representation ρ̄ : Δ → U(G) introduced above. We will sometimes, by abuse of language, refer to ρ̄ as the representation of Δ on holomorphic differentials of S_{g}.

We will make use of the following basic property of Chern classes of representations.

Lemma 3.1. Let ρ : ℤ/p^{n} → U(g) be a representation of ℤ/p^{n}, p a fixed prime. Assume that ρ can be decomposed as ⊕_{i=0}^{p^{n}−1}n_{i}ω^{i} where ω denotes a faithful one dimensional representation and where n_{i} = n_{j} for all (i, j) which satisfy 0 < i, j < p^{n} and i ≡ j ≢ 0 mod p. Then the following are equivalent.

(i) The Chern classes c_{i}(ρ) ∈ H^{2i}(ℤ/p^{n}; ℤ) are all 0 mod p for 1 ≤ i ≤ p^{n} − 1, (ii) n_{i} ≡ 0 mod p for 0 < i < p.

Proof. We first prove that (i) implies (ii). It is convenient for this proof to assume that ρ satisfies in addition n_{0} = n_{p} = ⋯ = n_{p^{n}−p}; this will not change the mod p Chern classes of ρ. Consider the ring homomorphism

ϕ : H^{∗}(ℤ/p^{n}; ℤ) → (ℤ/p^{n})[x]

given by mapping c_{1}(ω) ∈ H^{2}(ℤ/p^{n}; ℤ) to x. Let f(x) ∈ (ℤ/p^{n})[x] be defined by ∏_{i=0}^{p^{n}−1}(1+ ix)^{n_{i}} and F (x) ∈ (ℤ/p)[x] by ∏_{i=0}^{p−1}(1 + ix)^{n_{i}}. Notice that ϕ(c(ρ)) = f(x). We then have f(x) ≡ ∏_{k=0}^{p−1}(1 + kx)^{n_{k}p^{n−1}} ≡ ∏_{k=0}^{p−1}(1 + kx^{p^{n−1}})^{n_{k}} mod p, thus f(x) ≡ F (x^{p^{n−1}}) mod p.

Now, the assumption that c_{i}(ρ) ≡ 0 mod(p) for 1 ≤ i ≤ p^{n}−1 implies that f(x) ≡ 1+pg(x) mod x^{p^{n}}, and therefore, F (x) ≡ 1 mod x^{p}. So, by Lemma 4.1 of [GMX], we conclude that n_{k} ≡ 0 mod p for all k satisfying 1 ≤ k ≤ p − 1. It remains to check that (ii) implies (i). If (ii) holds, we can write ρ in the form pkσ + τ, where σ denotes the reduced regular representation of ℤ/p^{n}, and τ a representation which factors through ℤ/p^{n−1} so that c(τ) ≡ 1 mod p. Since c(σ) = 1 + c_{(p−1)p^{n−1}}(σ) we see that

c(ρ) = c(σ)^{pk} · c(τ) ≡ (1 + c_{(p−1)p^{n−1}}(σ)^{p})^{k} mod p ,

so that c_{i}(ρ) ≡ 0 mod p for 0 < i < (p − 1)p^{n}.

Our theorem on algebraic moves leads to the following result, obtained by a geometric version of “moves” of fixed point data. It will provide a proof of case (i) of Theorem 2 of the introduction (see Corollary 3.3), and it will also provide us with an improved lower bound for Γ_{g}, needed for the proof of case (ii) of Theorem 2.

Proposition 3.2. Let p denote an odd regular prime and ρ the canonical representation of Γ_{g} in Sp(2g, ℝ). Assume g = lp^{α} + 1 with l prime to p and α > 0, and suppose g does not satisfy the (p)-condition. Then for every subgroup π ⊂ Γ_{g} of order p, at least one characteristic class d_{i}(ρ|π) ∈ H^{2i}(π; ℤ) is non-zero in the range 1 ≤ i ≤ p^{α} − 1.

Proof. Suppose there is a subgroup π of order p in Γ_{g} such that d_{i}(ρ|π) = 0 for 1 ≤ i ≤ p^{α} −1. We will show that this leads to a cyclic subgroup of Γ_{g} of order p^{α+1}, which acts on S_{g} with stabilizers of order ≤ p. The Riemann-Hurwitz equation associated with this action yields then an equation contradicting the assumtion concerning the (p)-condition. The construction of such a subgroup is by induction on its order. First, we begin by modifying π slightly, not changing the Chern classes, using our technique of “moves” as follows. Let ρ̄ : π → U(g) be the representation of π on the space of holomorphic differentials on S_{g} (with respect to a suitable complex structure) and write ρ̄ as ⊕_{i=0}^{p−1}n_{i}ω^{i}, ω a faithful irreducible one dimensional representation of π. Thus we have d_{i}(ρ|π) = c_{i}(ρ̄) = 0 for 1 ≤ i ≤ p−1. Applying Lemma 3.1, we infer n_{i} ≡ 0 mod p for 1 ≤ i ≤ p−1. Let x ∈ π be a generator and denote by < g, p | β_{1}/p, …, β_{n}/p > the fixed point data of x, normalized such that 0 < β_{i} < p. As we discussed in Proposition (4.3) of [GMX], see also [FK], one has then

(2)  n_{j} = h − 1 + n − ∑_{i=1}^{n} < jβ_{i}/p > ,  0 < j < p ,

where h denotes the genus of S_{g}/π. Note that the Riemann-Hurwitz equation 2g − 2 = p(2h − 2) + n(p − 1) shows that n ≡ 0 mod p and, as argued in [GMX], by considering n_{j} − n_{p−j} = 2 ∑_{i} < jβ_{i}/p > +n it follows, because for 0 < j < p each n_{j} is divisible by p, that

∑_{i=1}^{n} < jβ_{i}/p >≡ 0 mod p , 0 < j < p .

Equation (2) then implies that h − 1 ≡ 0 mod p, which in particular shows that the orbit genus h is greater than 0. By applying Theorem 2.6 and Lemma 2.2, there are n integers γ_{i} with 0 < γ_{i} < p, forming the fixed point data < g, p|γ_{1}, …, γ_{n} > of a generator of a subgroup π̃ of order p in Γ_{g}, which gives rise to a representation π̃ → U(g) with multiplicities n_{i} of the one dimensional components still given by (2), but such that the number of γ_{i}’s equal to a fixed j is always divisible by p. By reordering the γ’s, we may assume that γ_{s} ≤ γ_{t} for s < t, and we can then consider

(3)  δ =< g, p^{2}|γ_{1}/p, γ_{p+1}/p, …, γ_{(k−1)p+1}/p >

where n = kp. Note that

∑_{i=0}^{k−1} γ_{ip+1}/p = ∑_{i=1}^{n} β_{i}/p^{2} ∈ ℤ .

Also, with h − 1 = p(s − 1) and n = pk, we get

(4)  2g − 2 = p(2h − 2) + n(p − 1) = p^{2}(2s − 2) + kp(p − 1)

so that δ is the fixed point data of an element of order p^{2} in Γ_{g} by Lemma 2.2. Furthermore, s > 0 since h > 0, and k ≠ 1, because ∑_{i=0}^{k−1} γ_{ip+1}/p is an integer. Thus, if α = 1 equation (4) implies that g satisfies the (p)-condition, a contradiction, and we are done. If α ≥ 2, we repeat our construction in the following manner. Let π(2) denote the subgroup of Γ_{g} generated by an element whose fixed point data is given by equation (3). By construction, the associated representation ρ(2) : π(2) → U(g) on holomorphic differentials (with, perhaps, a different complex structure on S_{g}), is such that again c_{i}(ρ(2)) ≡ 0 mod p for 0 < i < p^{α}. Decomposing ρ(2) as ⊕_{i=0}^{p^{2}−1}n_{i}ω^{i}, with ω faithful one dimensional, yields equations of the form

(5)  n_{j} = h(2) − 1 + k(2) − ∑_{i=1}^{k(2)} < jβ_{i}(2)/p >≡ 0 mod p ,  1 < j < p ,

with h(2) the genus of S_{g}/π(2), β_{i}(2) = γ_{(i−1)p+1}, and k(2) = k as before. Also (see [FK])), one has n_{j} = n_{j+p} if 0 < j < j + p < p^{2} and j prime to p, so that by Lemma 3.1 n_{i} ≡ 0 mod p for 0 < i < p. The Riemann-Hurwitz equation for the π(2) action is 2g − 2 = p^{2}(2h(2) − 2) + k(2)p(p − 1) and, as α ≥ 2, k(2) ≡ 0 mod p; in particular, k(2) ≠ 1. Again, by considering n_{j} − n_{p−j} = 2 ∑_{i=1}^{k(2)} < jβ_{i}(2)/p > +k(2) ≡ 0 mod p, we see that

∑_{i=1}^{k(2)} < jβ_{i}(2)/p >≡ 0 mod p , 0 < j < p ,

and thus h(2) ≡ 1 mod p from equation (5). This permits us to find k(2)/p = k(3) integers δ_{1}, …, δ_{k(3)} , 1 < δ_{i} < p, so that

∑_{i=1}^{k(3)} δ_{i}/p = ∑_{i=1}^{k(2)} β_{i}(2)/p^{2} ∈ ℤ

as well as

(6)  2g − 2 = p^{3}(2s(2) − 2) + k(3)p^{2}(p − 1)

As h(2) ≠ 0 we have s(2) > 0, and k(3) ≠ 1 because ∑_{i=1}^{k(3)} δ_{i}/p ∈ ℤ. If α = 2, equation (6) shows that g must satisfy the (p)-condition, a contradiction. If α > 2 we continue our construction. If g does not satisfies the (p)-condition, we will eventually arrive at a contradiction, finishing the proof. But we also note that if g satisfies the (p)-condition, we end up with a cyclic subgroup π(α + 1) ⊂ Γ_{g} of order p^{α+1}, which is generated by an element with fixed point data

< g, p^{α+1}|ϵ_{1}/p, …, ϵ_{w}/p >

and associated Riemann-Hurwitz equation of the form

2g − 2 = p^{α+1}(2s(α) − 2) + k(α + 1)p^{α}(p − 1)

with the property that on p^{α}π(α+1), the subgroup of order p of π(α+1), the characteristic classes d_{i}(ρ|p^{α}π(α + 1)) agree with those associated with the original group π; in particular, they vanish in the range 0 < i < p^{α}.

As an application, we can now prove part (i) of Theorem 1.

Corollary 3.3. Let p be an odd regular prime and assume g = lp^{α} + 1 with l prime to p and α > 0. If g does not satisfy the (p)-condition then p(Γ_{g}) = 2(p − 1)p^{α−1}.

Proof. We know from Lemma 1.5 that p(Γ_{g}) has the form 2(p − 1)p^{β} for some β ≥ α − 1. By Proposition 3.2. we can find for every subgroup π ⊂ Γ_{g} of order p a non-zero element d_{i} ∈ H^{2i}(π; ℤ) for some i > 0 of the form mp^{γ} with m prime to p and γ < α, which lies in the image of the restriction map H^{∗}(Γ_{g}; ℤ) → H^{∗}(π; ℤ). Thus the largest power of p dividing p(Γ_{g}) must be less than α and the result follows.

This proof of Proposition 3.2 reveals, as a by-product, the following.

Corollary 3.4. Let p be an odd regular prime and g = lp^{α} + 1 with l prime to p and α > 0. If Γ_{g} contains a subgroup π order p with d_{i}(ρ|π) = 0 for 0 < i < p^{α} and if g satisfies the (p)-condition, then Γ_{g} contains a cyclic subgroup of order p^{α+1} with associated action on S_{g} having stabilizers of order ≤ p.

The converse of that Corollary is also true, by a simple construction; it does not need any regularity condition on the prime involved.

Corollary 3.5. Let p be an arbitrary odd prime and assume g = lp^{α} + 1 with l prime to p and α > 0. If g satisfies the (p)-condition then Γ_{g} contains a cyclic subgroup π(α + 1) of order p^{α+1} generated by an element with fixed point data of the form

< g, p^{α+1}|β_{1}/p, …, β_{k}/p > .

Moreover, the subgroup π = p^{α}π(α + 1) of order p satisfies d_{i}(ρ|π) = 0 for 0 < i < p^{α}.

Proof. If g satisfies the (p)-condition, we have integers h > 0 and k ≥ 0 with k ≠ 1 satisfying 2g − 2 = p^{α+1}(2h − 2) + p^{α}k(p − 1). Therefore, by Lemma 2.2, S_{g} admits a diffeomorphism of order p^{α+1} with exactly k singular orbits, and fixed point data

< g, p^{α+1}|β_{1}/p, …, β_{k}/p >

where the β_{i}’s satisfy 0 < β_{i} < p and are chosen in such a way that ∑_{i} β_{i}/p is an integer; this is possible since k ≠ 1. If π(α + 1) denotes the corresponding subgroup in Γ_{g} then its action on holomorphic 1-forms will be given by a representation of the form

⊕_{i=0}^{p^{α+1}−1}n_{i}ω^{i} ,

with ω one dimensional and faithful, such that n_{i} = n_{i+p} for any i prime to p such that 0 < i < i + p < p^{α+1}; this follows easily from the general formula concerning the n_{i}’s, as presented for instance in [FK], by noting that the only nontrivial stabilizer occurring for the action on S_{g} is of order p. We can now apply Lemma 3.1 to the classes d_{i}(ρ|π(α + 1)) and infer that d_{i}(ρ|π) = 0 for 0 < i < p^{α}.

This Corollary leads to the following lower bound for p(Γ_{g}), which will be used in the proof of (ii) of Theorems 1 and 2, presented in Section 5.

Corollary 3.6. Let p be an odd prime and suppose g = lp^{α} + 1 with l prime to p and α > 0. If g satisfies the (p)-condition then p(Γ_{g}) is a multiple of 2(p − 1)p^{α}.

Proof. By Corollary 3.5 we can find a cyclic subgroup π(α + 1) in Γ_{g} of order p^{α+1} given by an element x with fixed point data

δ(x) =< g, p^{α+1}|β_{1}/p, …, β_{k}/p >

for some β_{i} with 0 < β_{i} < p. From our discussion on fixed point data we see that δ(x^{j}) = δ(x) if j ≡ 1 mod p. But this implies that p^{α} divides [N(x) : C(x)]. Applying Lemma 1.3 we see that p(Γ_{g}) is a multiple of p^{α} and, using Lemma 1.5, it follows that p(Γ_{g}) is actually a multiple of 2(p − 1)p^{α}.

Section 4 : The action on Teichmüller space

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 ℝ^{6g−6}, and T_{g} admits a complex structure such that Γ_{g} acts on T_{g} properly discontinuously by holomorphic automorphisms. For a fixed subgroup π ⊂ Γ_{g} of order p, let {π_{i}}_{i∈J} be the set of all order p subgroups which are conjugate to π in Γ_{g}. This is a countable set (in general infinite) so that we may assume 0 ∈ J ⊂ ℕ with π_{0} = π. The fixed point sets F_{i} = (T_{g})^{π_{i}} ⊂ T_{g} are closed submanifolds homeomorphic to ℝ^{6h−6+2n} where h denotes the genus of the surface S_{g}/π and n the number of fixed points of the π-action on S_{g}, see Proposition 2.1 of [GMX] (strictly speaking, it is not π which acts on S_{g}, but some lift of π to Diffeo^{+}(S_{g})). We may choose a triangulation of T_{g}, by triangulating the algebraic variety T_{g}/Γ_{g}, such that Γ_{g} acts simplicially. Thus each F_{i} is a subcomplex of T_{g}. We now choose a subset I ⊂ J containing 0 such that for i, j ∈ I with i ≠ j one has F_{i} ≠ F_{j}. Put X_{i} = T_{g} \ F_{i} and X_{∞} = ⋂ X_{i} = X_{0} ∩ X_{>0}, where X_{>0} = ⋂_{i>0} X_{i}. Note that F = ⋃ F_{i} is closed in T_{g}, since it is a subcomplex, and X_{∞} is therefore a (connected) open submanifold of T_{g}, on which Γ_{g} acts, with stabilizers not containing any subgroup conjugate to π. One checks easily using the Riemann-Hurwitz equation that the only case for which F = T_{g} (and thus X_{∞} an empty space) is the case of π being generated by the hyperelliptic involution, acting on a genus 2 surface; in our applications we are only interested in actions of groups of odd order and thus X_{∞} will not be empty. We will keep the notation introduced here through the entire section.

Lemma 4.1. Let π ⊂ Γ_{g} be a subgoup of prime order p and F = ⋃ F_{i} ⊂ T_{g} the subspace of points fixed by some conjugate of π. In case g = 2 assume that π is not generated by the hyperelliptic involution. Denote by h be the genus of S_{g}/π and n the number of fixed points of the π-action on S_{g}. Then the singular cohomology with compact supports of F satisfies

H_{cpt}^{6h−6+2n}(F; ℤ) ≅ ⊕_{i∈I}ℤ ,

and H_{cpt}^{k}(F; ℤ) = 0 for k > 6h − 6 + 2n.

Proof. Consider the natural map θ : ∐ F_{i} → F. Since the action of Γ_{g} on T_{g} is proper, no f ∈ F lies in infinitely many F_{i}’s, and therefore the map θ is proper and induces a map in cohomology with compact supports. Let Sing(F) ⊂ F be the subcomplex of points which lie in more than one F_{i}. Since for i ≠ j in I the complex submanifold F_{i}∩F_{j} is empty or has at least (real) codimension 2 in F_{i}, Sing(F) is empty, or is a subcomplex of F of codimension at least 2, and it follows that θ is a H_{cpt}^{∗}-isomorphism for ∗ ≥ 6h − 6 + 2n = dim(F). Since each F_{i} is homeomorphic to ℝ^{6h−6+2n} the result follows.

Lemma 4.2. Let F ⊂ T_{g} be as in Lemma 4.1 and put X_{∞} = T_{g} \ F. Then X_{∞} is a connected manifold of dimension 6g − 6 with singular cohomology satisfying

H^{6(g−h)−2n−1}(X_{∞}; ℤ) ≅ ∏_{i∈I} ℤ ,

and H^{k}(X_{∞}; ℤ) = 0 for 0 < k < 6(g − h) − 2n − 1. Furthermore, the natural map ν : X_{∞} → X_{0} × X_{>0} induces an isomorphism

H^{k}(X_{0} × X_{>0}; ℤ) → H^{k}(X_{∞}; ℤ)

for 0 < k ≤ 6(g − h) − 2n − 1.

Proof. Since F is closed in T_{g} we get by Alexander Duality

H_{k}(X_{∞}; ℤ) ≅ H_{cpt}^{(6g−6)−k}(T_{g}, F; ℤ)

and, since T_{g} ≅ ℝ^{6g−6} and using Lemma 4.1, H_{cpt}^{6g−6−k}(T_{g}, F; ℤ) = 0 for 0 < k < 6(g − h) − 2n − 1. Moreover, for k = 6(g − h) − 2n − 1 we have

H_{6(g−h)−2n−1}(X_{∞}; ℤ) ≅ H_{cpt}^{6h−6+2n+1}(T_{g}, F; ℤ) ≅ H_{cpt}^{6h−6+2n}(F; ℤ) ≅ ⊕_{i∈I}ℤ .

Noting that by the universal coefficient theorem

H^{k}(X_{∞}; ℤ) ≅ Hom(H_{k}(X_{∞}; ℤ))

for 0 ≤ k ≤ 6(g − h) − 2n − 1, the result on the cohomology of X_{∞} follows. For the map ν : X_{∞} → X_{0} × X_{>0} one observes that the same type of argument applied to X_{0} and X_{>0} shows that they are homologically 6(g − h) − 2n − 2 connected too, and satisfy

H_{6(g−h)−2n−1}(X_{0}; ℤ) ≅ ℤ , and H_{6(g−h)−2n−1}(X_{>0}; ℤ) ≅ ⊕_{i>0}ℤ .

Using the Künneth Formula and the universal coefficient theorem the isomorphism result follows readily.

Proposition 4.3. Let g > 1 and p an odd prime. Let π be a subgroup of order p, with associated Riemann-Hurwitz formula 2g − 2 = p(2h − 2) + n(p − 1). Then there exists a cohomology element e ∈ H^{6(g−h)−2n}(Γ_{g}; ℤ) whose restriction to H^{6(g−h)−2n}(π; ℤ) is non-trivial.

Proof. Since p is odd, 6(g − h) − 2n > 0 (it is the codimension of F_{0} in T_{g}) so that, in the notation used above, X_{∞} = T_{g} \ F is non-empty. We first study the Serre spectral sequences with ℤ-coefficients, associated with the fibrations

(A)  X_{∞} → EΓ_{g} ×_{Γ_{g}} X_{∞} → BΓ_{g}

and

(B)  X_{∞} → Eπ ×_{π} X_{∞} → Bπ .

Let us put ω = 6(g − h) − 2n − 1. Then, in the obvious notation, one has

E_{2}^{ω,0}(A) ≅ H^{ω}(X_{∞}; ℤ)^{Γ_{g}} =< a >≅ ℤ ,

since Γ_{g} acts on H^{ω}(X_{∞}; ℤ) ≅ ∏_{i∈I} ℤ by permuting the factors transitively (the action is induced by the action on the set I). Because H^{k}(X_{∞}; ℤ) = 0 for 0 < k < ω, one has

E_{2}^{ω,0}(A) = E_{ω}^{ω,0}(A) ⟶^{d_{ω}^{A}} E_{ω}^{0,ω+1}(A) = H^{ω+1}(Γ_{g}; ℤ) .

We will show that d_{ω}^{A}(a) ∈ H^{ω+1}(Γ_{g}; ℤ) restricts non-trivially to H^{ω+1}(π; ℤ). For this it suffices to show that the element a considered as an element in E_{ω}^{ω,0}(B) = H^{ω}(X_{∞}; ℤ)^{π} satisfies d_{ω}^{B}(a) ≠ 0 in H^{ω+1}(π; ℤ). We will analyze d_{ω}^{B}(a) using the π-map X_{∞} → X_{0}×X_{>0} considered in Lemma 4.2 . and the spectral sequences associated with

(C)  X_{0} × X_{>0} → Eπ ×_{π} (X_{0} × X_{>0}) → Bπ

(D)  X_{0} → Eπ ×_{π} X_{0} → Bπ

and

(E)  X_{>0} → Eπ ×_{π} X_{>0} → Bπ .

Since the π-action on X_{0} is free, Eπ ×_{π} X_{0} is homotopy equivalent to X_{0}/π, a finite dimensional complex, so that all of

E_{2}^{ω,0}(D) = H^{ω}(X_{0}; ℤ)^{π} = H^{ω}(X_{0}; ℤ)

is mapped isomorphically by d_{ω}^{D} onto E_{ω}^{0,ω+1}(D) ≅ H^{ω+1}(π; ℤ); otherwise, as H^{∗}(X_{0}; ℤ) ≅ H^{∗}(S^{ω}; ℤ), one would get a contradiction to the finite dimensionality of X_{0}/π. The fibration (E) has a section, because the π-action on X_{>0} has a fixed point; this follows from the fact that F_{0} ∩ (∪_{i>0}F_{i}) is either empty or a subcomplex of F_{0} codimension ≤ 2, and F_{0} is non-empty. Consequently, all elements in E_{2}^{0,ω+1}(E) = E_{ω}^{0,ω+1}(E) = H^{ω+1}(π; ℤ) are permanent d^{E}-cycles. But this implies that d_{ω}^{E} : E_{ω}^{ω,0}(E) → E_{ω}^{0,ω+1}(E) is the zero map. The π-map ν : X_{∞} → X_{0} × X_{>0} induces a map of spectral sequences E_{∗}^{∗,∗}(C) ⇒ E_{∗}^{∗,∗}(B). As ν^{∗} : H^{∗}(X_{0} × X_{>0}; ℤ) → H^{∗}(X_{∞}; ℤ) is an isomorphism for 0 < ∗ ≤ ω, and also H^{ω}(X_{0} × X_{>0}; ℤ) ≅ H^{ω}(X_{0}; ℤ) × H^{ω}(X_{>0}; ℤ), we see that for x = (x_{0}, x_{>0}) ∈ H^{ω}(X_{0} × X_{>0}; ℤ)^{π} one has

d_{ω}^{C}(x) = d_{ω}^{B}(ν^{∗}(x)) = d_{ω}^{D}(x_{0}) ∈ H^{ω+1}(π; ℤ) .

Thus, if we decompose the generator

a ∈ H^{ω}(X_{∞}; ℤ)^{Γ_{g}} ⊂ H^{ω}(X_{0}; ℤ) × H^{ω}(X_{>0}; ℤ)^{π}

as a = (a_{0}, a_{>0}), we have

res_{π}^{Γ_{g}}(d_{ω}^{A}(a)) = d_{ω}^{D}(a_{0}) ∈ H^{ω+1}(π; ℤ) ,

and we need only show that a_{0} ∈ H^{ω}(X_{0}; ℤ) is non-zero, to ensure that d_{ω}^{D}(a_{0}) ≠ 0. For this we view a as a Γ_{g} invariant function on H_{ω}(X_{∞}; ℤ) = H_{ω}(X_{0}; ℤ) ⊕ H_{ω}(X_{>0}; ℤ) But every Γ_{g}-invariant function on H_{ω}(X_{∞}; ℤ), which is constant on H_{ω}(X_{0}; ℤ) ≅ ℤ, must be trivial, because H_{ω}(X_{∞}; ℤ) is isomorphic to ⊕_{i∈I}ℤ, with Γ_{g} acting by permuting transitively the summands of ⊕_{i∈I}ℤ. It follows that a_{0} ≠ 0 and we are done.

Corollary 4.4. Let p be an odd prime and g = lp^{α} + 1 with l prime to p and α > 0. If π denotes a subgroup of order p of Γ_{g} such that the Riemann-Hurwitz equation of π has the form

2g − 2 = p(2h − 2) + n(p − 1)

with h − 1 = p^{α}(s − 1) and n = (kp + i)p^{α} where 0 < i < p, then 2l has necessarily the form mp − i, and the restriction map

H^{[(3m+k)p−3(m+k)−2i]p^{α}}(Γ_{g}; ℤ) → H^{[(3m+k)p−3(m+k)−2i]p^{α}}(π; ℤ)

is non-trivial.

Proof. Since 2g − 2 = 2lp^{α}, the Riemann-Hurwitz equation shows that 2l ≡ −i mod p so that we can write 2l in the form mp − i for a unique m > 0. Thus 2g − 2 = (mp − i)p^{α} and 2h − 2 = [m − k(p − 1) − i]p^{α} so that

6(g − h) − 2n = [(3m + k)p − 3(m + k) − 2i]p^{α} .

Our claim then follows from Proposition 4.3 .

Section 5 : The sharp upper bounds for Γ_{g}

We have already established that if g = lp^{α} + 1 with l prime to p and α > 0, then, p(Γ_{g}) = 2(p − 1)p^{β} for some β ≥ α − 1, see Lemma 1.5. To show that necessarily β ≤ α, we need only verify that for every subgroup π ⊂ Γ_{g} there exists a number j(π) prime to p such that the restriction map

(7)  H^{2j(π)p^{α}}(Γ_{g}; ℤ) → H^{2j(π)p^{α}}(π; ℤ)

is non-trivial. This will be established in the next theorem, which proves (i) of Theorem 2 of the introduction.

Theorem 5.1. Let p be an odd prime and g = lp^{α} + 1 with l prime to p and α > 0. Then the Yagita invariant Γ_{g} equals 2(p − 1)p^{α} or 2(p − 1)p^{α−1}.

Proof. As explained above, we need to study restriction maps of the type (7). Let π be a subgroup of Γ_{g} of order p and consider the associated Riemann-Hurwitz equation

2g − 2 = p(2h − 2) + n(p − 1) ,

where n ≥ 0 denotes the number of fixed points of the π action on S_{g}. Write θ : Γ_{g} → GL_{2g}(ℂ) for the representation given by the action of Γ_{g} on H_{1}(S_{g}; ℂ). Note that θ factors through Gl_{2g}(ℚ) so that θ|π = aτ + bσ, with τ the trivial one-dimensional representation and σ the reduced regular one of π. Thus

dim_{ℂ}H_{1}(S_{g}; ℂ) = 2g = a + (p − 1)b = 2lp^{α} + 2

and, for a generator ϕ of the π-action on H_{1}(S_{g}; ℂ),

trace(ϕ : H_{1}(S_{g}; ℂ) → H_{1}(S_{g}; ℂ)) = a − b = 2 − n .

Thus pb = 2g − a + b = 2g − 2 + n = 2lp^{α} + n so that the following possibilities arise.

(a) n ≢ 0 mod p^{α}. Then b = b_{0}p^{γ} with γ < α − 1 and b_{0} prime to p. But then the Chern class c_{(p−1)p^{γ}} (θ|π) = b_{0}c_{p−1}(σ)^{p^{γ}} is non-zero, providing a j(π) for which the restriction map (7) is non-zero.

(b) n = (kp + i)p^{α} with 0 ≤ i < p. Since then pb = 2lp^{α} + n = (2l + kp + i)p^{α}, we see that in case 2l+i ≢ 0 mod p, b is not zero mod p^{α}, so that the same argument as before shows that c_{(p−1)p^{α−1}}(θ) has a non-trivial restriction under the map (7). It remains to consider the case where 2l + i = mp. This yields pb = (m + k)p^{α+1} and we find again two sub-cases. Firstly, if m + k is prime to p, then we conclude again that the restriction map (7) is non-zero, with j(π) = p − 1. Secondly, if m + k ≡ 0 mod p, we observe that in the notation above 2h = a = 2g −(p−1)b, which implies that 2h − 2 is divisible by p^{α}. Also, we cannot have i = 0 in that case, because otherwise 2g − 2 were divisible by p^{α+1}. Thus 0 < i < p and we are precisely in the situation of Corollary 4.4, which shows that the restriction map (7) is non-trivial if one chooses

2j(π) = [(3m + k)p − 3(m + k) − 2i] ,

and j(π) ≢ 0 mod p since we assume that m + k is divisible by p.

This completes the proof of the theorem.

As a corollary, we obtain part (ii) of Theorem 2, by combining Theorem 5.1 with Corollary 3.6. Of course, this also implies (ii) of Theorem 1, and therefore we have completed all proofs of the theorems stated in the introduction.

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H.H. Glover, Ohio State University, Columbus, Ohio

E-mail address: glover@math.ohio-state.edu

G. Mislin, ETH Zürich, Switzerland; and Ohio State University, Columbus, Ohio E-mail address: mislin@math.ethz.ch and mislin@math.ohio-state.edu

Y. Xia, Northwestern University, Evanston, Illinois

E-mail address: xia@math.nwu.edu