Linear representations of hyperelliptic mapping class groups
LLINEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS
MARCO BOGGIA
BSTRACT . Let p : S → S g be a finite G -covering of a closed surface of genus g ≥ and let B its branch locus. To this data, it is associated a representation of a finite index subgroupof the mapping class group Mod( S g (cid:114) B ) in the centralizer of the group G in the symplecticgroup Sp( H ( S ; Q )) . They are called virtual linear representations of the mapping classgroup and are related, via a conjecture of Putman and Wieland, to a question of Kirby andIvanov on the abelianization of finite index subgroups of the mapping class group. Thepurpose of this paper is to study the restriction of such representations to the hyperellipticmapping class group Mod( S g , B ) ι , which is a subgroup of Mod( S g (cid:114) B ) associated to agiven hyperelliptic involution ι on S g . We extend to hyperelliptic mapping class groupssome previous results on virtual linear representations of the mapping class group. Wethen show that, for all g ≥ , there are virtual linear representations of hyperellipticmapping class groups with nontrivial finite orbits. In particular, we show that there issuch a representation associated to an unramified G -covering S → S , thus providing acounterexample to the genus case of the Putman-Wieland conjecture.
1. A
CONJECTURE BY P UTMAN AND W IELAND
Let S g be an oriented closed surface of genus g and let P , P , P , . . . be a sequencein S g of pairwise distinct points. We put S g,n := S g (cid:114) { P , . . . , P n } . We always assumethat S g,n has negative Euler characteristic: g − n > . We abbreviate π ( S g,n , P n +1 ) by Π g,n , denote by Mod( S g,n ) the mapping class group of self-homeomorphisms of S g,n and by PMod( S g,n ) the subgroup consisting of mapping classes of self-homeomorphismswhich preserve the order of the punctures. Let us recall that these groups act by outerautomorphisms on Π g,n and PMod( S g,n ) (if n > ) acts in a conventional fashion on Π g,n − . A fundamental open question on the mapping class group of a surface is thefollowing (cf. Problem 2.11.A in [15] and Problem 7 in [14]): Problem 1.1.
Assume g ≥ and n ≥ . Is it true that H (Γ) = 0 for every finite indexsubgroup Γ of Mod( S g,n ) ? A positive answer has been given for all finite index subgroups of
Mod( S g,n ) containingthe Johnson subgroup, i.e. the normal subgroup of Mod( S g,n ) generated by Dehn twistsabout separating simple closed curves (cf. [3] and [20]). More recently, Ershov and He(cf. [8]) showed that this question has a positive answer also for finite index subgroups Mathematics Subject Classification.
Primary: 14H10; 14H37; 57M60, secondary: 14H40.
Key words and phrases.
Mapping class group, Monodromy.Research partially supported by CNPq, grant n. 305716/2018-2. a r X i v : . [ m a t h . A T ] A p r MARCO BOGGI of Mod( S g,n ) , for n ≤ , which contain the k -th term of the lower central series of theTorelli subgroup of Mod( S g,n ) , where k ≤ g +12 .Let S be an oriented closed surface of genus ≥ and G a finite group acting faithfullyon S . The finite group G then embeds in the mapping class group Mod( S ) of the surface S . Let S G be the orbifold quotient of S by the action of G . The natural map S → S G thendetermines a finite index normal subgroup K ∼ = π ( S ) of Π G := π ( S G ) (we omit basepoints since they are not relevant here). Let Mod( S G ) be the mapping class group of theorbifold S G . This identifies with the colored mapping class group of the surface S G (cid:114) B ,where B is the inertia locus of the orbifold S G and isomorphic inertia points are assignedthe same color. Let us denote by Mod( S ) G the centralizer of G in Mod( S ) . There is thena short exact sequence:(1) → Z ( G ) → Mod( S ) G → Mod( S G )[ K ] → , where Z ( G ) is the center of the group G and Mod( S G )[ K ] is the subgroup of Mod( S G ) consisting of mapping classes of self-homeomorphisms which preserve the subgroup K of Π G and act by inner automorphisms on the quotient group Π G /K ∼ = G . We call Mod( S G )[ K ] the geometric level associated to the normal subgroup K of Π G . If K isa characteristic subgroup of Π G and all points of B have the same inertia, then thesubgroup Mod( S G )[ K ] identifies with the usual geometric level of Mod( S G (cid:114) B ) definedas the kernel of the natural representation Mod( S G (cid:114) B ) → Out(Π G /K ) .In [21], Putman and Wieland made a conjecture which, with the above notations, canbe formulated as follows: Conjecture 1.2 (Putman-Wieland Conjecture) . Suppose that g ( S G ) ≥ . Then every non-trivial Mod( S ) G -orbit in H ( S ; Q ) is infinite. A well-known result of Serre asserts that if H is a free abelian group of finite rank, thenany finite group of automorphisms of H acts faithfully on H/mH when m ≥ . So, if Γ isa finite index subgroup of Mod( S ) G which, for some m ≥ , acts trivially on H ( S, Z /m ) ,the assertion of the Putman-Wieland Conjecture is equivalent to H ( S ; Q ) Γ = 0 .Putman and Wieland proved that Conjecture 1.2 is essentially equivalent to a positiveanswer to Problem 1.1. Their precise statement is as follows: Theorem 1.3 (Putman-Wieland) . Let us consider the following statements for g ≥ and n ≥ : ( A g,n ) For every subgroup Γ ⊂ Mod( S g,n ) of finite index, we have H (Γ; Q ) = 0 . ( B g,n ) For every orbifold quotient S → S G as above such that g ( S G ) = g and | B | ≤ n , theassociated action of Mod( S ) G on H ( S ; Q ) has no finite nontrivial orbit.Then ( A g +1 ,n +1 ) ⇒ ( B g +1 ,n ) and ( B g,n +1 ) ⇒ ( A g +1 ,n ) . In particular, if ( A g,n ) holds for some g ≥ and n ≥ , then it holds for all g ≥ g and n ≥ . Remark 1.4.
Note that if one of the listed properties holds for ( g, n ) , then it holdsfor ( g, m ) , m ≥ n . This follows by using the natural epimorpisms Π g,m → Π g,n and PMod( S g,m ) → PMod( S g,n ) or the fact that a covering of S g,n defines one of S g,m . INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 3
In Section 2 (cf. Corollary 3.15), we will show that the Putman-Wieland conjecture isfalse for g = 2 . This however leaves the conjecture unsettled in genus ≥ , since, in theimplication ¬ ( B g,n ) ⇒ ¬ ( A g,n +1 ) , the genus does not increase.2. T HE HYPERELLIPTIC MAPPING CLASS GROUP A hyperelliptic (topological) surface ( S, ι ) is the data consisting of an oriented con-nected, not necessarily closed, surface S of genus ≥ and negative Euler characteristictogether with a hyperelliptic involution ι on S (that is to say a self-homeomorphism ι of S such that ι = id S and the quotient surface S/ι has genus ). Then, the hyperellipticmapping class group Mod( S ) ι of ( S, ι ) is the centralizer of ι in the mapping class group Mod( S ) . For { P , . . . , P n } a set of distinct points on S , we define the pointed hyperellipticmapping class group Mod(
S, P , . . . , P n ) ι (which we will usually denote by Mod(
S, n ) ι ) tobe the inverse image of Mod( S ) ι under the natural epimorphism Mod(
S, P , . . . , P n ) → Mod( S ) . We denote by PMod(
S, P , . . . , P n ) ι (or just by PMod(
S, n ) ι ) the subgroup of Mod(
S, P , . . . , P n ) ι which consists of mapping classes of self-homeomorphisms whichpreserve the order of the set { P , . . . , P n } . The main reason to adopt this definition ofthe pointed hyperelliptic mapping class group is that, by restriction, we get the Birmanshort exact sequence: → π ( S (cid:114) { P , . . . , P n } , P n +1 ) → PMod(
S, P , . . . , P n +1 ) ι → PMod(
S, P , . . . , P n ) ι → . Remarks 2.1.
The following remarks can also help to clarify the above definitions. Let W be the set of points fixed by the hyperelliptic involution ι on S .(i) There is a natural isomorphism Mod( S (cid:114) W , n ) ι ∼ = Mod( S, n ) ι , for all n ≥ .(ii) There is a natural action of Mod(
S, n ) ι on the set W and so a representation ρ W : Mod( S, n ) ι → Σ W , where Σ W is the permutation group of the set W .(iii) ( S (cid:114) P, ι ) , for P ∈ W , is a hyperelliptic surface and, for S closed, Mod( S (cid:114) P, n ) ι identifies with the stabilizer Mod(
S, n ) ιP of P for the action of Mod(
S, n ) ι on W .(iv) For ( S, ι ) a closed hyperelliptic surface of genus ≥ and Q ∈ S (cid:114) W , the opensurface S (cid:114) { Q, ι ( Q ) } , together with the involution ι , is a hyperelliptic surface.In (iv) of Theorem 2.4, we will show that there is a natural epimorphism, withnontrivial kernel, Mod( S (cid:114) { Q, ι ( Q ) } ) ι (cid:16) Mod(
S, Q ) ι . Definition 2.2.
A simple closed curve γ on a hyperelliptic surface ( S, ι ) is symmetric if ι ( γ ) is isotopic to γ . A bounding pair { γ , γ } on ( S, ι ) is symmetric if ι ( γ ) is isotopic to γ . We say that a simple closed curve γ (resp. a bounding pair { γ , γ } ) on the puncturedsurface S (cid:114) { P , . . . , P n } is symmetric if its image in ( S, ι ) is a symmetric simple closedcurve (resp. a symmetric bounding pair). Proposition 2.3.
Let ( S, ι ) be either a closed hyperelliptic surface of genus ≥ or a one-punctured hyperelliptic surface of genus . Then, the pure hyperelliptic mapping class group PMod(
S, n ) ι is generated by Dehn twists about nonseparating symmetric simple closedcurves on the surface S (cid:114) { P , . . . , P n } . MARCO BOGGI
Proof.
For n = 0 , the result is well known (for g ( S ) ≥ , see Theorem 3 in [1]). For n ≥ ,we proceed by induction. By the Birman short exact sequence, it is enough to show thatthe image of the push map π ( S (cid:114) { P , . . . , P n } , P n +1 ) → PMod(
S, P , . . . , P n +1 ) ι is gener-ated by Dehn twists about nonseparating symmetric simple closed curves, for all n ≥ .For γ ∈ π ( S (cid:114) { P , . . . , P n } , P n +1 ) a simple loop, its image in PMod(
S, P , . . . , P n +1 ) ι is the product of Dehn twists τ γ τ − γ , where γ and γ are simple closed curves on S (cid:114) { P , . . . , P n +1 } which are freely homotopic to γ on S (cid:114) { P , . . . , P n } (cf. Section 4.2.2 in[9]). We then conclude observing that the fundamental group π ( S (cid:114) { P , . . . , P n } , P n +1 ) is generated by nonseparating loops on the surface S (cid:114) { P , . . . , P n } whose images on thehyperelliptic surface ( S, ι ) are freely homotopic to symmetric simple closed curves. (cid:3) The above definitions and remarks easily generalize to the case when S is not con-nected. We will need this generalization to describe stabilizers of symmetric simpleclosed curves for the action of the hyperelliptic mapping class group. Let us observethat, for ( S, ι ) a hyperelliptic surface and γ a nonperipheral simple closed curve on S preserved by ι , the pair ( S (cid:114) γ, ι ) is also a (not necessarily connected) hyperelliptic sur-face. The stabilizer Mod(
S, n ) ιγ for the action of the hyperelliptic mapping class group Mod(
S, n ) ι on the isotopy class of a symmetric simple closed curve γ on S (cid:114) { P , . . . , P n } is then described by the short exact sequence:(2) → τ Z γ → Mod(
S, n ) ιγ → Mod( S (cid:114) γ, n ) ι → . We can make the description (2) more precise if the hyperelliptic surface ( S, ι ) is closedof genus g ( S ) = g ≥ or ( S, ι ) is a hyperelliptic one-punctured genus surface. For this,we will need the following geometric interpretation of the corresponding hyperellipticmapping class group Mod(
S, n ) ι . Let M g, [ n ] , for g − n > , be the moduli stackof genus g smooth complex projective curves endowed with n unordered marked pointsand let H g, [ n ] be its closed substack parametrizing hyperelliptic curves. Observe that, for g = 1 , , all curves are hyperelliptic, i.e. admit a degree morphism onto P .Let ( C, x , . . . , x n ) be an n -pointed genus g smooth complex projective curve and C top the closed topological surface underlying C . The choice of a pointed homeomorphism ( S, P , . . . , P n ) ∼ → ( C top , x , . . . , x n ) determines an isomorphism between the fundamentalgroup π ( M g, [ n ] , [ C ]) and the mapping class group Mod(
S, P , . . . , P n ) .If we choose as base point for the fundamental group a hyperelliptic curve C , then theembedding H g, [ n ] ⊂ M g, [ n ] induces a monomorphism π ( H g, [ n ] , [ C ]) (cid:44) → π ( M g, [ n ] , [ C ]) .Via the identification of π ( M g, [ n ] , [ C ]) with Mod(
S, P , . . . , P n ) , this inclusion identi-fies the fundamental group π ( H g, [ n ] , [ C ]) with the hyperelliptic mapping class group Mod(
S, P , . . . , P n ) ι , where ι is the involution induced on S by the hyperelliptic invo-lution of C via the fixed homeomorphism S → C top . Observe that, for g = 1 , , we have Mod(
S, P , . . . , P n ) ι = Mod( S, P , . . . , P n ) .For us, the most important feature of the moduli stacks H g and M , is their relationwith the moduli stacks of pointed genus curves. More precisely, there is, for g ≥ , anatural Z / -gerbe H g → M , [2 g +2] defined by assigning to a genus g hyperelliptic curve C , the genus zero curve C/ι , where ι is the hyperelliptic involution of C , labeled by INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 5 the branch points of the covering C → C/ι . In the genus case, there is a Z / -gerbe M , → M , [3]+1 , where by the notation ” [3] + 1 ” we mean that there are labels, onedistinguished and the others unordered. There are natural isomorphisms H g (cid:40) (cid:104) ι (cid:105) ∼ = M , [2 g +2] and M , (cid:40) (cid:104) ι (cid:105) ∼ = M , [3]+1 , where by (cid:40) we denote the operation of erasing thegeneric group of automorphisms from an algebraic stack.As it is clear from the definition, there is a natural smooth morphism H g,n +1 → H g,n (defined by forgetting the ( n +1) -th marked point) which can be regarded as the universalpunctured curve over H g,n . This fibration determines the Birman short exact sequence: → Π g,n → PMod( S g , n + 1) ι → PMod( S g , n ) ι → . For m ≥ , the abelian hyperelliptic level Mod( S g , n ) ι ( m ) of order m is defined asthe kernel of the representation ρ ( m ) : Mod( S g , n ) ι → Sp g ( Z /m ) and the correspond-ing ´etale Galois covering H g, [ n ] ( m ) → H g, [ n ] is the hyperelliptic abelian level structure oforder ( m ) . By [2] Proposition 3.3, there are natural isomorphisms H g (2) (cid:40) (cid:104) ι (cid:105) ∼ = M , g +2 and M , (2) (cid:40) (cid:104) ι (cid:105) ∼ = M , , which, at the level of fundamental groups, identify squaresof Dehn twists about ι -invariant nonseparating simple closed curves in S g,n with Dehntwists about simple closed curves bounding a -punctured disc in S ,n .Let us now describe more precisely the stabilizer Mod(
S, n ) ιγ introduced above. For g ≤ , the hyperelliptic mapping class group Mod(
S, n ) ι coincides with the mapping classgroup Mod( S g,n ) and the usual description applies. For g ≥ , we have: Theorem 2.4.
Let ( S, ι ) be a hyperelliptic closed surface of genus g ≥ and let γ be asymmetric simple closed curve on S (cid:114) { P , . . . , P n } . Let then PMod(
S, n ) ι(cid:126)γ be the subgroupof PMod(
S, n ) ι consisting of mapping classes which preserve the isotopy class of γ and afixed orientation on it. (i) For γ separating, let ( S (cid:114) γ, P , . . . , P n ) = ( S , P i , . . . , P i n ) (cid:113) ( S , P j , . . . , P j n ) .The stabilizer PMod(
S, n ) ι(cid:126)γ fits in the short exact sequence → τ Z γ → PMod(
S, n ) ι(cid:126)γ → PMod( S , n ) ι × PMod( S , n ) ι → . (ii) Moreover, if S i , for i = 1 , , denotes the closed surface obtained by filling in thepuncture on S i with a point Q i and PMod( S i , n i ) ιQ i is the stabilizer of Q i for theaction of PMod( S i , n i ) ι on the set of Weierstrass points W i of S i through the repre-sentation ρ W i (cf. (ii) of Remarks 2.1), then there is a natural isomorphism PMod( S i , n i ) ι ∼ = PMod( S i , n i ) ιQ i . (iii) For γ nonseparating, let S γ := S (cid:114) γ . The stabilizer PMod(
S, n ) ι(cid:126)γ is described bythe short exact sequence → τ Z γ → PMod(
S, n ) ι(cid:126)γ → PMod( S γ , n ) ι ◦ → , where PMod( S γ , n ) ι ◦ is the index subgroup of PMod( S γ , n ) ι consisting of thoseelements which do not swap the two punctures of S γ . (iv) Moreover, if we denote by S γ the closed surface obtained by filling in the twopunctures on S γ with the points Q and Q , by W γ the set of the g Weierstrass
MARCO BOGGI points on S γ and by N W γ the kernel of the natural epimorphism of fundamentalgroups π ( S γ (cid:114) {{ P , . . . , P n } ∪ W γ } ) → π ( S γ (cid:114) { P , . . . , P n } ) , then the group PMod( S γ , n ) ι ◦ fits in the short exact sequence: → N W γ → PMod( S γ , n ) ι ◦ → PMod( S γ , n, Q ) ι ≡ PMod( S γ , n + 1) ι → . Proof.
The proof is based on the interpretation of
PMod(
S, n ) ι as the topological fun-damental group of the moduli stack H g,n of n -pointed, genus g , hyperelliptic complexsmooth projective curves.A non-peripheral simple closed curve γ on S g,n determines a boundary component ofthe D–M compactification H g,n of H g,n . Let us denote by δ γ its normalization and by ˙ δ γ its open substack parameterizing curves with just one singular point. Let (cid:98) H g,n be the realoriented blow-up of H g,n along its D–M boundary, let (cid:98) ∆ γ → δ γ be the pull-back of thenatural map (cid:98) H g,n → H g,n along δ γ → H g,n and let ˆ δ γ be the real oriented blow-up of δ γ along the divisor δ γ (cid:114) ˙ δ γ . Then, the natural map (cid:98) ∆ γ → ˆ δ γ is an S -bundle.The natural inclusions H g,n (cid:44) → (cid:98) H g,n and ˙ δ γ (cid:44) → ˆ δ γ are homotopy equivalences. Thestabilizer PMod(
S, n ) ιγ of the simple closed curve γ identifies with the fundamental group π ( (cid:98) ∆ γ ) of (cid:98) ∆ γ , which is described by the short exact sequence: → µ → π ( (cid:98) ∆ γ ) → π (ˆ δ γ ) → , where µ is a free cyclic group and π (ˆ δ γ ) = π ( ˙ δ γ ) . It is easy to see that the identifica-tion of π ( (cid:98) ∆ γ ) with the stabilizer PMod(
S, n ) ιγ identifies µ with the cyclic subgroup τ Z γ generated by the Dehn twist τ γ .For [ C, P , . . . , P n ] ∈ ˙ δ γ , let ˜ C be the normalization of C and let Q and Q be theinverse images in ˜ C of the singular point of C . Let then ˙ δ (cid:126)γ → ˙ δ γ be the (possibly triv-ial) ´etale covering induced by ordering the set of points { Q , Q } and let δ (cid:126)γ be its D–Mcompactification over δ γ . Let ˆ δ (cid:126)γ be the real oriented blow-up of δ (cid:126)γ in the divisor param-eterizing curves with more than one singularity. As above, the embedding ˙ δ (cid:126)γ ⊂ ˆ δ (cid:126)γ is ahomotopy equivalence. Let (cid:98) ∆ (cid:126)γ = (cid:98) ∆ γ × ˆ δ γ ˆ δ (cid:126)γ . The natural map (cid:98) ∆ (cid:126)γ → ˆ δ (cid:126)γ is an S -bundleand the stabilizer PMod(
S, n ) ι(cid:126)γ identifies with the fundamental group of (cid:98) ∆ (cid:126)γ . There isthen a short exact sequence:(3) → τ Z γ → PMod(
S, n ) ι(cid:126)γ → π ( ˙ δ (cid:126)γ ) → . The proof of the theorem will follow from a careful description of the locus parame-terized by ˙ δ γ in the moduli stack H g,n . We need to distinguish between the separatingand the nonseparating case.In the separating case, ˙ δ γ parameterizes reducible n -pointed, genus g , hyperellipticcurves with a single node. For [ C, P , . . . , P n ] ∈ ˙ δ γ , let C and C be the two irreduciblecomponents of the normalization of C , of genus g and g , respectively, and let us assumethe markings P i , . . . , P i n lie on C while the markings P j , . . . , P j n lie on C . Let also Q i be the inverse image of the node of C in C i , for i = 1 , . The curves C and C are INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 7 hyperelliptic, with hyperelliptic involutions ι and ι and sets of Weistrass points W and W , respectively, and we have Q ∈ W , Q ∈ W .Let PMod( S g i , n i ) ιQ i be the stabilizer of the Weirestrass point Q i ∈ W i , for the naturalrepresentation PMod( S g i , n i ) ι → Σ W i , for i = 1 , , and let H (cid:48) g i ,n i → H g i ,n i be the associ-ated ´etale Galois covering. Then, it is clear that there is a natural isomorphism of stacks H (cid:48) g ,n × H (cid:48) g ,n → ˙ δ (cid:126)γ . Part (i) and (ii) of the theorem then follow.In the non separating case, ˙ δ γ parameterizes n -pointed, genus g , hyperelliptic curveswith a single node which are irreducible. For [ C, P , . . . , P n ] ∈ ˙ δ γ , the normalization ˜ C of C is hyperelliptic of genus g − with a hyperelliptic involution ι such that ι ( Q ) = Q .Then, ˙ δ (cid:126)γ → ˙ δ γ is the double ´etale covering induced by ordering the set of points { Q , Q } and the short exact sequence (3) yields item (iii) of the theorem.Let H W g − ,n +1 be the moduli stack of smooth projective hyperelliptic complex curves ofgenus g − endowed with a hyperelliptic involution ι , a set of ordered marked points P , . . . , P n and an extra-marking Q lying outside the set of Weierstrass points of thecurve. It is clear that the assignment [ C, P , . . . , P n , Q ] (cid:55)→ [ C, P , . . . , P n , Q , ι ( Q )] de-fines an isomorphism of moduli stacks: H W g − ,n +1 ∼ −→ ˙ δ (cid:126)γ . The locus H g − ,n +1 (cid:114) H W g − ,n +1 is a divisor in the moduli stack H g − ,n +1 and the naturalmap H W g − ,n +1 → H g − ,n is a fibration in (2 g + n ) -punctured smooth curves of genus g − (the fibers of the map H g − ,n +1 → H g − ,n minus their (2 g − g Weierstrass points).So, there is a short exact sequence → Π g − , g + n → π ( ˙ δ (cid:126)γ ) ≡ π ( H W g − ,n +1 ) → π ( H g − ,n ) ≡ PMod( S g − , n ) ι → and the open embedding H W g − ,n +1 ⊂ H g − ,n +1 induces an epimorphism π ( ˙ δ (cid:126)γ ) ≡ π ( H W g − ,n +1 ) (cid:16) π ( H g − ,n +1 ) ≡ PMod( S g − , n + 1) ι , whose kernel equals that of the epimorphism Π g − , g + n (cid:16) Π g − ,n induced filling in thepunctures corresponding to the Weierstrass points of a member of the family of curves H W g − ,n +1 → H g − ,n . The short exact sequence of item (iv) then follows. (cid:3) Remark 2.5.
The reason for which Theorem 2.4 fails for g = 1 , (this case falls indeedin the usual description of stabilizers for the mapping class group) is that, by removing anontrivial symmetric simple closed curve from a closed surface of genus ≤ , we obtaina surface whose connected components have genus ≤ and a closed Riemann surface ofgenus ≤ has infinitely many hyperelliptic involutions, while, if the genus is ≥ , thereis at most one. The complex of symmetric nonseparating curves.
Let us define a version of the non-separating curve complex C ns ( S g,n ) for the hyperelliptic mapping class group Mod( S g , n ) ι .Let us recall that the latter is the simplicial complex whose k -simplices consist of un-ordered ( k + 1) -tuples { γ , . . . , γ k } of distinct isotopy classes of pairwise disjoint simple MARCO BOGGI closed curves on S g,n such that S g,n (cid:114) { γ , . . . , γ k } is connected. The hyperelliptic nonsep-arating curve complex H C ns ( S g,n ) is then defined to be the full subcomplex of C ns ( S g,n ) generated by isotopy classes of symmetric nonseparating simple closed curves.Let us observe that filling in a puncture induces surjective maps C ns ( S g,n +1 ) → C ns ( S g,n ) and H C ns ( S g,n +1 ) → H C ns ( S g,n ) . We will need the fact that the latter map, or, better, itsgeometric realization, is a homotopy equivalence. For the first map, this follows fromHarer’s results (cf. Theorem 1.1 in [12]) but we want to give here a more direct proofwhich can be easily adapted to the hyperelliptic case: Proposition 2.6.
There is a natural surjective map C ns ( S g,n +1 ) → C ns ( S g,n ) , induced byfilling in the last puncture, which is a homotopy equivalence, in the sense that the geometricrealization of this map is a homotopy equivalence. In particular, the homotopy type of C ns ( S g,n ) is independent of n .Proof. Let
Teich g,n be the Teichm¨uller space associated to the surface S g,n and let (cid:91) Teich g,n be the
Harvey cuspidal bordification of Teich g,n (cf. [13]) which can be explicitly describedas follows. Let (cid:99) M g,n be the real oriented blow-up of the D–M compactification M g,n ofthe stack M g,n along its D–M boundary. This is a real analytic D–M stack with cornerswhose boundary ∂ (cid:99) M g,n := (cid:99) M g,n (cid:114) M g,n is homotopic to a deleted tubular neighborhoodof the D–M boundary of M g,n . For an intrinsic construction of (cid:99) M g,n , we apply Weilrestriction of scalars Res C / R to the complex D–M stack M g,n and blow up Res C / R M g,n along the codimension two substack Res C / R ∂ M g,n . The real oriented blow-up (cid:99) M g,n is then obtained taking the set of real points of this blow-up and cutting it along theexceptional divisor (which has codimension ). The natural projection (cid:99) M g,n → M g,n restricts over each codimension k stratum to a bundle in k -dimensional tori.The Harvey cuspidal bordification (cid:91) Teich g,n is then the universal cover of the real an-alytic stack (cid:99) M g,n . It can be shown that (cid:91) Teich g,n is representable and thus a real an-alytic manifold with corners (cf. Section 4 in [4]). The inclusion M g,n (cid:44) → (cid:99) M g,n is ahomotopy equivalence and then induces an inclusion of the respective universal covers Teich g,n (cid:44) → (cid:91) Teich g,n , which is also a homotopy equivalence. The ideal boundary of theTeichm¨uller space
Teich g,n is defined to be ∂ (cid:91) Teich g,n := (cid:91) Teich g,n (cid:114)
Teich g,n .The Harvey cuspidal bordification (cid:91)
Teich g,n is endowed with a natural action of themapping class group
Mod( S g,n ) and has the property that the nerve of the cover of theideal boundary ∂ (cid:91) Teich g,n by (analytically) irreducible components is described by thecurve complex C ( S g,n ) (cf. [13] or [4] for more details). Since such irreducible compo-nents are all contractible, there is a Mod( S g,n ) -equivariant weak homotopy equivalencebetween ∂ (cid:91) Teich g,n and the geometric realization of C ( S g,n ) (cf. Theorem 2 in [13]).Let ∂ ns (cid:91) Teich g,n be the locus of the ideal boundary which parameterizes irreduciblenodal Riemann surfaces, that is to say ∂ ns (cid:91) Teich g,n is the inverse image in (cid:91)
Teich g,n , viathe natural map (cid:91)
Teich g,n → M g,n , of the locus in ∂ M g,n which parameterizes irreducible INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 9 nodal complex algebraic curves. Then, the nonseparating curve complex C ns ( S g,n ) natu-rally identifies with the nerve of the covering of ∂ ns (cid:91) Teich g,n by irreducible components,which are also contractible.The natural map (cid:99) M g,n +1 → (cid:99) M g,n , induced by filling in the last puncture, lifts to a mapof universal covers (cid:91) Teich g,n +1 → (cid:91) Teich g,n whose restriction ∂ ns (cid:91) Teich g,n +1 → ∂ ns (cid:91) Teich g,n ishomotopy equivalent to the geometric realization of the map C ns ( S g,n +1 ) → C ns ( S g,n ) .The fibers of the map (cid:99) M g,n +1 → (cid:99) M g,n are homeomorphic to the compact surfacewith boundary (cid:98) S g,n obtained from S g,n replacing punctures with circles. The boundaryof a fiber of the map (cid:99) M g,n +1 → (cid:99) M g,n is in ∂ (cid:99) M g,n +1 and parameterizes nodal reducible curves. It follows that the map ∂ ns (cid:91) Teich g,n +1 → ∂ ns (cid:91) Teich g,n is a Serre fibration withfibers consisting of Poincar´e discs (without boundary) and, in particular, is a homotopyequivalence. Hence, the proposition follows. (cid:3)
Remark 2.7.
Let C ( S g,n ) (resp. C nb ( S g,n ) ) be the simplicial complex whose k -simplicesconsist of unordered ( k + 1) -tuples { γ , . . . , γ k } of distinct isotopy classes of pairwisedisjoint nonperipheral simple closed curves on S g,n (resp. and which, moreover, do notbound a -punctured disc). A statement similar to Proposition 2.6 then also holds forthe simplicial sets C nb ( S g,n +1 ) • and C ( S g,n ) • associated, respectively, to C nb ( S g,n +1 ) and C ( S g,n ) . There is still indeed a natural map C nb ( S g,n +1 ) • → C ( S g,n ) • , induced by fillingin the last puncture (but observe that this is not anymore true, for n ≥ , if we replace C nb ( S g,n +1 ) with the full curve complex C ( S g,n +1 ) ) and essentially the same argument ofProposition 2.6 applies. In particular, this implies the well-known fact that C ( S g, ) and C ( S g ) have the same homotopy type. Proposition 2.8.
The geometric realization of the map H C ns ( S g,n +1 ) → H C ns ( S g,n ) , in-duced by filling in the last puncture, is a homotopy equivalence. In particular, the homotopytype of H C ns ( S g,n ) is independent of n .Proof. The fixed point set
Teich ιg of the hyperelliptic involution ι is contractible and iden-tifies with the universal cover of the moduli stack of hyperelliptic curves H g and thefixed point set (cid:91) Teich ιg is a bordification of Teich ιg . For n ≥ , we then let Teich ιg,n and (cid:91)
Teich ιg,n be, respectively, the inverse images of
Teich ιg and (cid:91) Teich ιg via the natural map (cid:91) Teich g,n → (cid:91) Teich g . Clearly, (cid:91) Teich ιg,n is a bordification of
Teich ιg,n for n ≥ as well.Let then ∂ (cid:91) Teich ιg,n := (cid:91) Teich ιg,n (cid:114)
Teich ιg,n , for all n ≥ and let ∂ ns (cid:91) Teich ιg,n be the portionof ∂ (cid:91) Teich ιg,n which parameterizes irreducible nodal Riemann surfaces. The hyperellipticnonseparating curve complex H C ns ( S g,n ) naturally identifies with the nerve of the cov-ering of ∂ ns (cid:91) Teich ιg,n by irreducible components, which are contractible. Therefore, thegeometric realization of the natural map H C ns ( S g,n +1 ) → H C ns ( S g,n ) is homotopy equiv-alent to the natural map ∂ ns (cid:91) Teich ιg,n +1 → ∂ ns (cid:91) Teich ιg,n . In the proof of Proposition 2.6, wesaw that the latter map is a Serre fibration with fibers consisting of Poincar´e discs and,in particular, is a homotopy equivalence. Hence, the proposition follows. (cid:3)
For n ≥ , let us define the full sub-complex C b ( S ,n ) of the complex of curves C ( S ,n ) generated by the vertices corresponding to simple closed curves on S ,n which bound a -punctured disc. The quotient map S g → S g /ι = S , g +2 then identifies H C ns ( S g ) withthe curve complex C b ( S , g +2 ) . Theorem 2.9.
For n ≥ , the curve complex C b ( S ,n ) is of dimension [ n/ − and, for n ≥ , is ([( n + 1) / − -connected (here [ ] denotes the integral part).Proof. The -connectivity of the curve complex C b ( S ,n ) , for n ≥ , can be proved bythe same argument which proves the connectivity of standard curve complexes (see, forinstance, Section 4.1.1 and 4.1.2 in [9]).The ([( n + 1) / − -connectivity of C b ( S ,n ) , for n ≥ , is derived from the fact that C ( S ,n ) , for n ≥ , is ( n − -connected (cf. Theorem 1.2 in [12]) and the same argumentused in the proof of Theorem 1.1 [12]. Let us observe first that the curve complex C b ( S ,n ) , for n = 4 , , coincides with C ( S ,n ) and thus is ( n − -connected. The case n = 6 of the theorem is then also clear. Let us proceed by induction on n . So let usassume that the theorem holds for all k such that ≤ k < n and let us prove it for k = n .We almost follow word by word Harer’s argument in the proof of Theorem 1.1 [12].Let S m be an m -dimensional piecewise linear simplicial sphere, for m < [( n + 1) / − ,and f : S m → C b ( S ,n ) ⊆ C ( S ,n ) be a simplicial map. Since C ( S ,n ) is ( n − -connectedand m < n − , there is an ( m + 1) -dimensional piecewise linear simplicial disc B suchthat ∂B = S m and a simplicial map ˆ f : B → C ( S ,n ) which extends f .We say that a simplex σ of B is pure if ¯ f ( σ ) = { γ , . . . , γ s } , where no γ i bounds a discwith two punctures, and define the complexity c ( ¯ f ) of ¯ f to be the largest k for whichsome k -simplex of B is pure.Let then σ be a pure k -simplex of B with k = c ( ¯ f ) , and let ¯ f ( σ ) = { γ , . . . , γ h } ,with h ≤ k ≤ [ n + 1 / − . Let S , . . . , S h +1 be the connected components of S ,n (cid:114) { γ , . . . , γ h } . The join ∆ σ = C b ( S ) ∗ . . . ∗ C b ( S h +1 ) identifies with a subcomplex of C b ( S ,n ) and, by hypothesis, ¯ f (Link B ( σ )) lies in ∆ σ . Moreover, if S i has n i punctures, there holds (cid:80) h +1 i =1 n i = n + 2 h + 2 .By the induction hypotheses, the connectivity of ∆ σ is described by the following for-mula. The join ∆ σ is µ -connected, where: µ : = (cid:80) h +1 i =1 (cid:18)(cid:20) n i + 12 (cid:21) − (cid:19) + h + 1 = (cid:80) h +1 i =1 (cid:20) n i + 12 (cid:21) − h − ≥≥ (cid:20) n + 2 h + 22 (cid:21) − h − (cid:104) n (cid:105) − h − ≥ (cid:20) n + 12 (cid:21) − − h ≥ (cid:20) n + 12 (cid:21) − − k. Then, the rest of the argument proceeds exactly as in the last two paragraphs of the proofof Theorem 1.1 in [12]. (cid:3)
By Proposition 2.8 and Theorem 2.9, we then have:
Corollary 2.10.
For g ≥ , the hyperelliptic nonseparating curve complex H C ns ( S g,n ) isspherical of dimension g − . INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 11
For the proof of Theorem 3.13, we will just need the fact that the complex H C ns ( S g,n ) is connected for g ≥ and the following result: Proposition 2.11.
Let α and β be nonseparating symmetric simple closed curves on S g,n ,for g ≥ . There is then a chain of symmetric simple closed curves γ , . . . , γ k on S g,n suchthat γ = α , γ k = β and γ i intersects γ i +1 transversally in a single point, for i = 0 , . . . , k − .Proof. It is easy to see that, for two disjoint nonseparating symmetric simple closed curves α and β on S g,n , there is a a symmetric simple closed curve γ which intersects both α and β transversally in a single point. The claim of the proposition then follows from theconnectedness of the curve complex H C ns ( S g,n ) (cf. Corollary 2.10 above). (cid:3)
3. V
IRTUAL LINEAR REPRESENTATION OF THE HYPERELLIPTIC MAPPING CLASS GROUP
As in Section 1, let us consider a finite group G acting faithfully on an oriented closedsurface S of genus ≥ . To the map S → S G , we have associated the short exact se-quence (1): → Z ( G ) → Mod( S ) G → Mod( S G )[ K ] → , where K is the kernel of thenatural epimorphism π ( S G ) → G .This short exact sequence has the following geometric meaning (cf. Section 3 in [5]).Let S G (cid:114) B ∼ = S g,n , where B is the set of inertia points of S G . Then, Mod( S G )[ K ] iden-tifies with a finite index subgroup of Mod( S g,n ) and so determines an ´etale covering M [ K ] → M g, [ n ] (the so called geometric level structure associated to the finite index nor-mal subgroup K of Π G ) and is isomorphic to the fundamental group of M [ K ] . On theother hand, the natural inclusion φ : G (cid:44) → Mod( S ) determines the moduli stack M φ ofcomplex G -curves of topological type φ (cf. Section 1 in [5]) whose fundamental groupis isomorphic to Mod( S ) G . There is a natural morphism M φ → M [ K ] which is a Z ( G ) -gerbe and whose associated short exact sequence of fundamental groups identifies withthe sequence (1). Let Sp( H ( S ; Q )) G be the centralizer of the group G in the symplecticgroup Sp( H ( S ; Q )) . There is then a natural representation ρ G : Mod( S ) G → Sp( H ( S ; Q )) G , which, by the short exact sequence (1), is indeed a virtual linear representation of themapping class group Mod( S g,n ) and identifies with the monodromy representation asso-ciated to the universal family of G -curves over M φ . In [5], this fact is used to deduceseveral properties of the representation ρ G (cf. Theorem 3.2 and Corollary 3.5 ibid.). Letus recall briefly these results.For an irreducible Q G -module V χ , with character χ , let D χ be the opposite (skew)algebra of End Q G ( V χ ) and regard V χ as a left Q G -module and a right D χ -module. Thesimple component of Q G acting nontrivially on V χ naturally identifies with the simplealgebra End D χ ( V χ ) . Let then H ( S ; Q )[ χ ] := Hom Q G ( V χ , H ( S ; Q )) . This is a left D χ -module and the χ -primary component H ( S ; Q ) χ of H ( S ; Q ) is naturally isomorphic to V χ ⊗ D χ H ( S ; Q )[ χ ] . Let X( Q G ) be the set of rational irreducible characters of G . We then have a series of natural isomorphisms: Sp( H ( S ; Q )) G ∼ = (cid:89) χ ∈ X( Q G ) Sp( H ( S ; Q ) χ ) G ∼ = (cid:89) χ ∈ X( Q G ) U( H ( S ; Q )[ χ ]) , where U( H ( S ; Q )[ χ ]) denotes the group of D χ -linear automorphisms of H ( S ; Q )[ χ ] which preserve a skew-hermitian form (cid:104) , (cid:105) χ which is induced by the intersection pairingon H ( S ; Q ) (cf. Section 2 of [5] for details).Let M on ◦ ( S ) G be the identity component of the Zariski closure of the image of therepresentation ρ G in Sp( H ( S ; Q )) G and M on ◦ ( S ) Gχ the projection of M on ◦ ( S ) G to thefactor Sp( H ( S ; Q ) χ ) G ∼ = U( H ( S ; Q )[ χ ]) . Theorem 3.2 in [5] states that M on ◦ ( S ) Gχ oper-ates Q -isotipically on the D χ -module H ( S ; Q )[ χ ] . Moreover, if the Q -rank of M on ◦ ( S ) Gχ is positive (e.g. M on ◦ ( S ) Gχ contains a nontrivial unipotent element), then it is an almostsimple Q -algebraic group acting irreducibly on H ( S ; Q )[ χ ] .The Putman-Wieland conjecture (1.2) is then equivalent to the statement that, for allcharacters χ ∈ X( Q G K ) afforded by the Q G -module H ( S ; Q ) , we have M on ◦ ( S ) Gχ (cid:54) = 1 .The above results have also interesting application to simple loop homology : Definition 3.1.
Let S be an oriented closed surface endowed with a faithful action of afinite group G . The simple loop homology H s ( S ; R ) of S , for some commutative ring R ,is the RG -submodule of H ( S ; R ) generated by cycles whose support covers some simpleclosed curve on S G (cid:114) B via the covering S → S G .In [16], Koberda and Santharoubane proved that there exist coverings S → S G suchthat H s ( S ; Z ) (cid:54) = H ( S ; Z ) , for all g ( S G ) ≥ and | B | ≥ , and, in [17], Malestein andPutman showed that there exist coverings S → S G such that H s ( S ; Q ) (cid:54) = H ( S ; Q ) , forall g ( S G ) ≥ and | B | ≥ ( | B | ≥ for g ( S G ) = 0 ). In Corollary 3.15, we will show thatthere exist coverings S → S G with B = ∅ and g ( S G ) = 2 such that H s ( S ; Q ) (cid:54) = H ( S ; Q ) .The problem remains open for R = Q , B = ∅ and g ( S G ) > .By Corollary 3.5 in [5], for g ( S G ) ≥ , the simple loop homology H s ( S ; Q ) is a sumof isotypical components. This immediately yields many nontrivial examples where infact H s ( S ; Q ) = H ( S ; Q ) (cf. Remark 3.6 in [5]). This corollary also states that, for all χ ∈ X( Q G K ) afforded by H s ( S ; Q ) , the algebraic monodromy group M on ◦ ( S ) Gχ is analmost simple Q -algebraic group acting irreducibly on H ( S ; Q )[ χ ] . The hyperelliptic connected monodromy group.
Let, as above, S G (cid:114) B = S g,n , let usassume that g ≥ and fix a hyperelliptic involution ι on S g = S/G . The hyperelliptic geo-metric level associated to the finite index normal subgroup K of π ( S G ) is the intersection Mod( S G ) ι [ K ] := Mod( S G )[ K ] ∩ Mod( S g , n ) ι and we define Mod( S ) G,ι to be the inverseimage of this subgroup in
Mod( S ) G . There is then a short exact sequence:(4) → Z ( G ) → Mod( S ) G,ι → Mod( S G ) ι [ K ] → , whose geometric interpretation is similar to the one given above. The finite index sub-group Mod( S G ) ι [ K ] of the hyperelliptic mapping class group Mod( S g , n ) ι determines a hyperelliptic geometric level structure H [ K ] over H g, [ n ] and identifies with its fundamental INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 13 group. The inclusion φ : G (cid:44) → Mod( S ) determines the moduli stack H φ of complex G -curves of topological type φ whose G -quotient is a hyperelliptic curve. The fundamentalgroup of H φ then identifies with Mod( S ) G,ι . There is a natural morphism H φ → H [ K ] which is a Z ( G ) -gerbe and whose associated short exact sequence of fundamental groupsis precisely the sequence (4). The restriction ρ G,ι : Mod( S ) G,ι → Sp( H ( S ; Q )) G of ρ G to Mod( S ) G,ι is then a virtual linear representation of the hyperelliptic mappingclass group
Mod( S g , n ) ι which identifies with the monodromy representation associatedto the universal family of G -curves over H φ .Let then M on ◦ ( S ) G,ι be the connected component of the identity of the Zariski closureof the image of the representation ρ G,ι defined above and let us denote by M on ◦ ( S ) G,ιχ its projection to the factor
Sp( H ( S ; Q ) χ ) G ∼ = U( H ( S ; Q )[ χ ]) of Sp( H ( S ; Q )) G . Let also H g ( S ) G,ι be the generic Hodge group of the polarized variation of Hodge structuresassociated to the universal G -curve over H φ and denote by H g ( S ) G,ιχ its projection to thefactor
Sp( H ( S ; Q ) χ ) G ∼ = U( H ( S ; Q )[ χ ]) of Sp( H ( S ; Q )) G . For a given algebraic group G , let us denote by D G its derived group. We then have:
Theorem 3.2.
Let S be an oriented closed surface endowed with a faithful and free action ofa finite group G such that the quotient surface S G has genus ≥ and let ι be a hyperellipticinvolution on S G : (i) The group M on ◦ ( S ) G,ιχ (resp. H g ( S ) G,ιχ ) acts Q -isotypically (resp. Q -irreducibly) onthe D χ -module H ( S ; Q )[ χ ] . (ii) If M on ◦ ( S ) G,ιχ has positive Q -rank, then M on ◦ ( S ) G,ιχ = D H g ( S ) G,ιχ is an almostsimple Q -algebraic group acting irreducibly on H ( S ; Q )[ χ ] . (iii) If M on ◦ ( S ) G,ιχ is trivial, then
Mod( S ) G,ι acts through a finite group on H ( S ; Q )[ χ ] and all G -invariant conformal structures on S which induce on S G the structureof a hyperelliptic Riemann surface define the same indecomposable Hodge structureon H ( S ; Q )[ χ ] .Proof. Possibly after replacing the covering S → S G with a G (cid:48) -covering S (cid:48) → S G (cid:48) ≡ S G which factors through it, we can assume that the finite index normal subgroup K ofthe fundamental group π ( S G ) associated to the covering S → S G is left invariant bythe hyperelliptic involution ι on S G . This implies that ι lifts to an involution ˜ ι on S which normalizes G . Let then ˜ G := (cid:104) ˜ ι (cid:105) (cid:110) G . The pair ( S, ˜ G ) satisfies the hypothesesof Theorem 1.11 in [5] and then also of Theorem 3.2 Ibid.. It is clear that we have M on ◦ ( S ) ˜ G = M on ◦ ( S ) G,ι and H g ( S ) ˜ G = H g ( S ) G,ι . The theorem then follows from acareful analysis of the relation between the Q ˜ G - and Q G -isotopy spaces associated to H ( S ; Q ) . Now, it is easy to see that Proposition 5.1 in [10] also holds with rationalcoefficients. The same is then true for (i) and (ii) of Theorem 1.11 with H ( S ; Q ) insteadof H ( C, Ω C ) (except for the multiplicity in (ii), which must be multiplied by ).There are two cases to consider: (a) ˜ χ is an irreducible Q ˜ G -character afforded by H ( S ; Q ) which remains irreducible after restriction to Q G (in this case, we denote the corresponding Q G -character by χ ); (b) ˜ χ is an irreducible Q ˜ G -character afforded by H ( S ; Q ) which, after restriction to G , splits as the sum of two irreducible Q G -characters χ (cid:48) and χ (cid:48)(cid:48) . In the first case, the Q ˜ G -isotypic and the Q G -isotypic components of H ( S ; Q ) associated to ˜ χ and χ , respectively, coincide. Let V ˜ χ be an irreducible Q ˜ G -module. Then V ˜ χ = V χ is also an irreducible Q G -module with character χ and there are a naturalisomorphism End Q ˜ G ( V ˜ χ ) ∼ = End Q G ( V χ ) := D χ and a natural isomorphism of D χ -modules H ( S ; Q )[ ˜ χ ] ∼ = H ( S ; Q )[ χ ] . In the second case, the Q ˜ G -isotypic component of H ( S ; Q ) associated to ˜ χ splits in the direct sum of the Q G -isotypic components associated to χ (cid:48) and χ (cid:48)(cid:48) . Let V ˜ χ be an irreducible Q ˜ G -module. Then V ˜ χ = V χ (cid:48) ⊕ V χ (cid:48)(cid:48) as a Q G -module, where V χ (cid:48) and V χ (cid:48)(cid:48) are irreducible with characters χ (cid:48) and χ (cid:48)(cid:48) , respectively. There are naturalisomorphisms D ˜ χ := End Q ˜ G ( V ˜ χ ) ∼ = End Q G ( V χ (cid:48) ) ∼ = End Q G ( V χ (cid:48)(cid:48) ) and natural isomorphismsof D ˜ χ -modules H ( S ; Q )[ ˜ χ ] ∼ = H ( S ; Q )[ χ (cid:48) ] ∼ = H ( S ; Q )[ χ (cid:48)(cid:48) ] . Theorem 3.2 then followsfrom Theorem 3.2 in [5]. (cid:3) Remark 3.3.
Let us observe that, in the case (b) of the proof of Theorem 3.2, thegeneric hyperelliptic Hodge group H g ( S ) G,ι (and so the hyperelliptic monodromy group M on ◦ ( S ) G,ι ) projects to the diagonal of the factor U( H ( S ; Q )[ χ (cid:48) ]) × U( H ( S ; Q )[ χ (cid:48)(cid:48) ]) of Sp( H ( S ; Q )) G . On the other hand, since ˜ ι is not an automorphism of a general G -curveof topological type ( S, G ) , the generic Hodge group H g ( S ) G does not project to suchdiagonal. In particular, we have a strict inclusion H g ( S ) G,ι (cid:40) H g ( S ) G . A Putman-Wieland type problem for the hyperelliptic mapping class group.
A nat-ural question for the hyperelliptic monodromy group is also whether, for all characters χ ∈ X( Q G K ) afforded by the Q G -module H ( S ; Q ) , we have M on ◦ ( S ) G,ιχ (cid:54) = 1 . Thisamounts to the following hyperelliptic version of the Putman-Wieland conjecture:
Problem 3.4 (Hyperelliptic Putman-Wieland Problem) . Let S be an oriented closed surfaceendowed with a faithful action of a finite group G such that g ( S/G ) ≥ and let ι be ahyperelliptic involution on S/G . Is every nontrivial
Mod( S ) G,ι -orbit in H ( S ; Q ) infinite? In Theorem 3.13, we will show that, in general, this question has a negative answer.This, in particular, will provide a negative answer to the genus case of the Putman-Wieland conjecture. Hyperelliptic simple loop homology.Definition 3.5.
Let S be an oriented closed surface endowed with a faithful action of afinite group G such that g ( S/G ) ≥ and let ι be a hyperelliptic involution on S/G . The hyperelliptic simple loop homology H s,ι ( S ; R ) of S , for some commutative ring R , is the RG -submodule of H ( S ; R ) generated by cycles whose support covers, via the covering S → S G , a symmetric simple closed curve or a connected component of a symmetricbounding pair on S G (cid:114) B (cf. Definition 2.2).For a simple closed curve α on S G (cid:114) B , let us denote by A the set of connected com-ponents of its inverse image in S . The group G acts transitively on A . In particular, themultitwist (cid:81) γ ∈ A τ γ ∈ Mod( S ) is centralized by G . We then denote by T A ∈ Sp( H ( S ; Z )) G INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 15 its image via the representation ρ G . Let us define the Dehn group D ehn ( S ) G to be theZariski closure of the (normal) subgroup of M on ◦ ( S ) G generated by all such T A for α asimple closed curve on S G (cid:114) B and let us denote by D ehn ( S ) Gχ the projection of this groupto the factor Sp( H ( S ; Q ) χ ) G ∼ = U( H ( S ; Q )[ χ ]) of Sp( H ( S ; Q )) G . Let the hyperellipticDehn group D ehn ( S ) G,ι be the Zariski closure of the (normal) subgroup of M on ◦ ( S ) G,ι generated by the T A such that α is a symmetric simple closed curve on S G (cid:114) B and bythe products T A · T B such that the simple closed curves α and β on S G (cid:114) B , which define T A and T B , respectively, form a symmetric bounding pair. We denote by D ehn ( S ) G,ιχ theprojection of this group to the factor
Sp( H ( S ; Q ) χ ) G ∼ = U( H ( S ; Q )[ χ ]) of Sp( H ( S ; Q )) G .From the discussion which precedes Corollary 3.5 in [5], it follows that H s ( S ; Q ) ⊥ and H s,ι ( S ; Q ) ⊥ can be characterized as the subspaces of fixed points in H ( S ; Q ) for the ac-tions of D ehn ( S ) G and D ehn ( S ) G,ι , respectively. In particular H s ( S ; Q ) and H s,ι ( S ; Q ) are both nondegenerate subspaces for the intersection pairing. We have proved the firstitem of the following theorem: Theorem 3.6. (i)
The subspace H s ( S ; Q ) (resp. H s,ι ( S ; Q ) ) of H ( S ; Q ) is nondegen-erate for the intersection pairing and is generated by the infinite orbits of D ehn ( S ) G (resp. D ehn ( S ) G,ι ). (ii) For g ( S G ) ≥ , the subspace H s ( S ; Q ) is a direct sum of Q G -isotypic componentsof H ( S ; Q ) . Moreover, if χ is an irreducible Q G -character afforded by H s ( S ; Q ) ,we have D ehn ( S ) Gχ = M on ◦ ( S ) Gχ and this is an almost simple Q -algebraic groupacting irreducibly on H ( S ; Q )[ χ ] . (iii) For g ( S G ) ≥ and B = ∅ , the subspace H s,ι ( S ; Q ) is a direct sum of Q G -isotypiccomponents of H ( S ; Q ) . Moreover, if χ is an irreducible Q G -character affordedby H s,ι ( S ; Q ) , we have D ehn ( S ) G,ιχ = M on ◦ ( S ) G,ιχ and this is an almost simple Q -algebraic group acting irreducibly on H ( S ; Q )[ χ ] .Proof. The first statement of item (i) was proved above. The second statement of item(i) also follows from the discussion preceding Corollary 3.5 in [5]. Item (ii) is essentiallya reformulation of Corollary 3.5 in [5] and item (iii) follows from the same argument ofthis corollary and Theorem 3.2. (cid:3)
Remark 3.7.
With the hypotheses of (iii) of Theorem 3.6, assume moreover that S G contains a symmetric simple closed curve γ over which the covering is trivial and is suchthat its preimage ˜ γ in S has the property that S (cid:114) ˜ γ is still connected. The same argumentof Remark 3.6 in [5] shows that, in this case, H s,ι ( S ; Q ) = H ( S ; Q ) . The pure cohomology of the hyperelliptic mapping class group.
The homology andcohomology of the hyperelliptic mapping class group
Mod( S g , n ) ι with rational coeffi-cients identify with the singular homology and cohomology of a smooth D–M stack, assuch they are endowed with a natural mixed Hodge structure. The same holds for anyfinite index subgroup of Mod( S g , n ) ι . Proposition 3.8.
For g ≥ , there holds W − H (PMod( S g , n ) ι ; Q ) = 0 . Proof.
For g ≥ , by Proposition 2.3, PMod( S g , n ) ι is generated by Dehn twists aboutsymmetric simple closed curves, whose classes in the first homology group have weight − , from which the conclusion follows. (cid:3) Proposition 3.9.
For g ≥ , the hyperelliptic mapping class group Mod( S g , n ) ι contains anormal finite index subgroup Υ such that W H (Υ) (cid:54) = 0 .Proof. It is enough to prove the proposition for g = 1 , n = 1 and g ≥ and n = 0 , since, if p : Mod( S g , n ) ι → Mod( S g ) ι is the natural epimorphism, then, for a finite index subgroup Υ of Mod( S g , n ) ι , the natural homomorphism W H ( p (Υ)) → W H (Υ) is injective.For Mod( S , ) = SL ( Z ) , the claim is well-known since W H (Υ) is just the first coho-mology group of the projective modular curve associated to Υ .Let us assume then n = 0 and g ≥ . Let H Teich g be the universal cover of themoduli stack of hyperelliptic curves H g . For g = 2 , this is just the ordinary genus Teichm¨uller space
Teich , while, for g ≥ , it can be identified with a closed contractiblesubmanifold of the genus g Teichm¨uller space
Teich g . Since there is a natural ´etalecovering H g → M , [2 g +2] , we have a natural isomorphism H Teich g ∼ = Teich , g +2 . Thereis also a natural epimorphism p : PMod( S , g +2 ) → PMod( S , ) and a corresponding p -equivariant surjective map φ : Teich , g +2 → Teich , .Let us assume that Υ is contained in the abelian level Mod( S g ) ι (2) . By Proposition 3.3in [2], there is a natural epimorphism r : Mod( S g ) ι (2) → PMod( S , g +2 ) . It followsthat the map φ induces a surjective map H Teich g / Υ → Teich , /p ( r (Υ)) , which inducesan epimorphism on fundamental groups. Therefore, we have a natural injective map W H (Teich , /p ( q (Υ))) (cid:44) → W H ( H Teich g / Υ) = W H (Υ) .It is easy to check that, for a suitable normal finite index subgroup Υ of Mod( S g ) ι , thecurve Teich , /p ( q (Υ)) has genus ≥ . Then, W H (Teich , /p ( r (Υ))) (cid:54) = 0 and the claimof the proposition follows. (cid:3) Let us observe that, in the proof of Proposition 3.9, we can choose the finite indexsubgroup Υ of Mod( S g , n ) ι in a way that, for some power m k of an even integer m > ,every symmetric simple closed curve γ and symmetric bounding pair γ , γ , we have that τ m k γ and τ m k γ τ m k γ belong to Υ . Let us denote by N the normal subgroup of Υ gener-ated by these elements and let H Υ be the D–M compactification of the level structure H Υ over H g, [ n ] associated to Υ . There is an epimorphism Υ /N (cid:16) H ( H Υ , Z ) and thereholds H ( H Υ , Z ) ⊗ Q ∼ = W H (Υ) (cid:54) = 0 . A consequence of independent interest of Propo-sition 3.9 is a slight improvement of Corollary 1.1 in [22] (cf. [11] and [18] for themapping class group version of this result): Corollary 3.10.
For every even integer m and for k (cid:29) , the normal subgroup of thehyperelliptic mapping class group Mod( S g , n ) ι generated by the m k -powers of symmetricmultitwists is of infinite index. For γ a non-peripheral symmetric simple closed curve on S g,n = S g (cid:114) { P , . . . , P n } ,as we observed in Section 2 (cf. (2)), the stabilizer Mod( S g , n ) ιγ of γ in Mod( S g , n ) ι is INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 17 described by the short exact sequence → τ Z γ → Mod( S g , n ) ιγ q → Mod( S g (cid:114) γ, n ) ι → .We then have: Lemma 3.11.
Given a non-peripheral simple closed curve γ on S g,n and a finite indexsubgroup Υ γ ⊆ Mod( S g , n ) ιγ , then the restriction map W H ( q (Υ γ )) → W H (Υ γ ) is anisomorphism. Let us now consider the hyperelliptic mapping class group version of the Birman exactsequence → Π g,n → PMod( S g , n + 1) ι p → PMod( S g , n ) ι → . For S a closed surface endowed with a faithful action of a group G such that g := g ( S/G ) ≥ and ι a hyperelliptic involution on S g = S/G , let
Ann( H s,ι ( S ; Q )) be theannihilator in H ( S ; Q ) for the cap product with elements of H s,ι ( S ; Q ) . We then have: Lemma 3.12.
For Υ ⊆ PMod( S g , n + 1) ι a finite index normal subgroup, put K := Π g,n ∩ Υ and let S → S g be the associated, possibly ramified, covering with Galois group G = Π g,n /K .Then, we have Ann( H s,ι ( S ; Q )) p (Υ) = H ( S ; Q ) p (Υ) and there is a natural exact sequence ofpolarized Hodge structures of weight one → W H ( p (Υ)) → W H (Υ) → Ann( H s,ι ( S ; Q )) p (Υ) → . Proof.
As above, for a finite index subgroup Υ of PMod( S g , n ) ι , we let H Υ g,n → H g,n bethe associated ´etale covering of D–M stacks which extends to a finite and flat morphism H Υ g,n → H g,n of the respective D–M compactifications. There is then a projective flatmorphism ¯ f : H Υ g,n +1 → H p (Υ) , with fibers semistable G -curves, such that the restriction f : C Υ → H p (Υ) of this morphism over H p (Υ) is a smooth projective G -curve, whose fiberscan be identified to the G -surface S . By a theorem of Deligne (cf. [6]), the associatedLeray spectral sequence degenerates at the E level and there is a short exact sequence: → H ( H p (Υ) ; Q ) f ∗ → H ( C Υ ; Q ) → H ( H p (Υ) ; R f ∗ Q ) → , where H ( H p (Υ) ; R f ∗ Q ) identifies with the monodromy invariants H ( f − ( z ); Q ) p (Υ) , forall z ∈ H p (Υ) .According to the global invariant cycle theorem (cf. [7]), H ( f − ( z ); Q ) p (Υ) equalsthe image of the restriction map H ( H Υ g,n +1 ; Q ) → H ( f − ( z ); Q ) and, in particular, is aHodge substructure of H ( f − ( z ); Q ) . Any simple closed curve in f − ( z ) , which covers asymmetric simple closed curve or a connected component of a symmetric bounding pair,can be collapsed inside H Υ g,n +1 . It follows that, after identifying H ( f − ( z ); Q ) p (Υ) with H ( S ; Q ) p (Υ) , the image of H ( H Υ g,n +1 ; Q ) in H ( S ; Q ) p (Υ) is contained in the annihilatorof H s,ι ( S ; Q ) . Therefore, we have a short exact sequence: → H ( H p (Υ) ; Q ) f ∗ → H ( C Υ ; Q ) → Ann( H s,ι ( S ; Q )) p (Υ) → . This sequence remains exact passing to its weight part: → W H ( H p (Υ) ; Q ) f ∗ → W H ( C Υ ; Q ) → Ann( H s,ι ( S ; Q )) p (Υ) → . The conclusion of the lemma then follows observing that H ( H p (Υ) ; Q ) = H ( p (Υ); Q ) and that W H ( C Υ ; Q ) = W H ( H Υ ; Q ) = W H (Υ; Q ) . (cid:3) Solution of the hyperelliptic Putman-Wieland problem 3.4.Theorem 3.13.
For all integers g > and n ≥ ( n > for g = 1 ), there is an orientedclosed surface S endowed with a faithful action of a finite group G and a hyperellipticinvolution ι on the quotient S/G , such that g ( S/G ) = g and the branch locus B of thecovering S → S/G has cardinality at most n , with the property that the associated action of Mod( S ) G,ι on H ( S ; Q ) has a finite nontrivial orbit. In particular, H s,ι ( S ; Q ) (cid:40) H ( S ; Q ) .Proof. Thanks to Proposition 3.9 and (i) of Theorem 3.6 (for the last statement of thetheorem), it is enough to prove the following lemma which is a reformulation of thePutman-Wieland Theorem 1.3 for the pure cohomology of finite index subgroups of thehyperelliptic mapping class group:
Lemma 3.14.
Let us consider the following statements for a pair of nonnegative integers ( g, n ) with g ≥ ( n > for g = 1 ): ( A (cid:48) g +1 ,n ) For every finite index subgroup Υ ⊆ Mod( S g +1 , n ) ι , we have W H (Υ) = 0 . ( B (cid:48) g,n ) Theorem 3.13 does not hold for the given pair of integers ( g, n ) .Then we have the implications: ( B (cid:48) ,n +1 ) ⇒ ( A (cid:48) ,n ) and, for g ≥ , ( B (cid:48) g,n ) ⇒ ( A (cid:48) g +1 ,n ) .Proof. The discrepancy between the indices in the genus case of the lemma and inthe case of genus ≥ is determined by the different descriptions of the stabilizers of asymmetric nonseparating simple closed curve in the cases g = 1 , and g ≥ (cf. Theo-rem 2.4 and Remark 2.5). The proof of the implication ( B (cid:48) ,n +1 ) ⇒ ( A (cid:48) ,n ) is essentiallyPutman-Wieland’s argument except that the cohomology functor H ( ) is replaced by thepure cohomology functor W H ( ) . For brevity of exposition, we will then only prove theimplication ( B (cid:48) g − ,n ) ⇒ ( A (cid:48) g,n ) , where we assume g ≥ .In the statement ( A (cid:48) g,n ) , we can clearly replace Mod( S g , n ) ι with PMod( S g , n ) ι . Letthen Υ be a normal finite index subgroup of PMod( S g , n ) ι . By Proposition 3.8, for g ≥ ,we have W H (Υ) PMod( S g ,n ) ι = W H (PMod( S g , n ) ι ) = 0 . So it is enough to prove that ( B (cid:48) g − ,n ) implies that PMod( S g , n ) ι acts trivially on W H (Υ) , for g ≥ . By Proposi-tion 2.3, the hyperelliptic mapping class group PMod( S g , n ) ι is generated by symmetricnonseparating Dehn twists for g ≥ . Therefore, we just need to show that, for anysymmetric nonseparating simple closed curve γ on S g,n , the associated Dehn twist τ γ actstrivially on W H (Υ) .Without loss of generality, we may assume that Υ is contained in the abelian level PMod( S g , n ) ι ( m ) for some m ≥ so that Υ γ = Υ (cid:126)γ . Note that S g,n (cid:114) γ is homeomor-phic to S g − ,n +2 with the two extra punctures (which we shall denote by Q and Q )corresponding to the two orientations of γ . The image of Υ γ = Υ (cid:126)γ under the map q : PMod( S g , n ) ι(cid:126)γ → PMod( S g (cid:114) γ, n ) ι ◦ (cf. (iii) of Theorem 2.4) is a finite index subgroupand, according to Lemma 3.11, this map induces an isomorphism:(5) W H (Υ γ ) ∼ → W H ( q (Υ γ )) . INEAR REPRESENTATIONS OF HYPERELLIPTIC MAPPING CLASS GROUPS 19
Let S γ be obtained from S g (cid:114) γ by filling in the two punctures bounded by γ , so that S γ ∼ = S g − . By (iv) of Theorem 2.4, there is an epimorphism φ : PMod( S g (cid:114) γ, n ) ι ◦ → PMod( S γ , n + 1) ι with kernel the subgroup N W of π ( S γ (cid:114) {{ P , . . . , P n } ∪ W } ) normally generated bythe loops around the punctures corresponding to the Weierstrass points of S γ , whichdetermine elements of weight − in H ( S γ (cid:114) {{ P , . . . , P n } ∪ W } ; Q ) . Therefore, thismap induces an isomorphism:(6) W H ( q (Υ γ )) ∼ → W H ( φ ( q (Υ γ ))) . Let then p : PMod( S γ , n + 1) ι → PMod( S γ , n ) ι be the natural map induced forgettingthe marked point Q . The subgroup p ( φ ( q (Υ γ ))) is of finite index in PMod( S γ , n ) ι and ( B (cid:48) g − ,n ) together with Lemma 3.12 imply that there is an isomorphism:(7) W H ( φ ( q (Υ γ ))) ∼ → W H ( p ( φ ( q (Υ γ )))) . Let δ be a simple closed curve on S g,n with the property that it intersects γ transversallyin a single point and let α be the boundary of a tubular neighborhood of δ ∪ γ in S g,n .Then, α is a separating simple closed curve on S g,n and we let S α be the component of S g (cid:114) α of genus > . The natural embedding S α ⊂ S γ is homotopic to the embedding S α ⊂ S α , where S α is the closed surface obtained from S α filling in the hole bounded by α with a point R . Therefore, we can identify PMod( S α , n ) ι with PMod( S γ , n ) ι and, by(ii) of Theorem 2.4, there is a natural identification of PMod( S α , n ) ι with the finite indexstabilizer PMod( S α , n ) ιR in PMod( S α , n ) ι of the Weierstrass point R ∈ S α .By (i) of Theorem 2.4, there is also an epimorphism onto PMod( S α , n ) ι from the sta-bilizer PMod( S g , n ) ια = PMod( S g , n ) ι(cid:126)α in PMod( S g , n ) ι of the isotopy class of α . Let usdenote by r : PMod( S g , n ) ια → PMod( S γ , n ) ι the homomorphism with finite cokernel ob-tained from this epimorphism and the identification of PMod( S α , n ) ι with the subgroup PMod( S α , n ) ιR of PMod( S α , n ) ι and of PMod( S α , n ) ι with PMod( S γ , n ) ι .In conclusion, if Υ α is the stabilizer in Υ of the isotopy class of α , its image r (Υ α ) isa finite index subgroup of p ( φ ( q (Υ γ ))) . By restriction, we then obtain a monomorphism W H ( p ( φ ( q (Υ γ )))) (cid:44) → W H ( r (Υ α )) and, by composing this map with the isomorphisms(5), (6), (7), we obtain a natural monomorphism: W H (Υ γ ) (cid:44) → W H ( r (Υ α )) . Since we may interchange γ and δ , we also obtain a natural monomorphism: W H (Υ δ ) (cid:44) → W H ( r (Υ α )) . The compositions W H (Υ) → W H (Υ γ ) (cid:44) → W H ( r (Υ α )) and W H (Υ) → W H (Υ δ ) (cid:44) → W H ( r (Υ α )) define the same map. Therefore, W H (Υ) → W H (Υ γ ) and W H (Υ) → W H (Υ δ ) have the same kernel. Since, by Proposition 2.11, any two symmetric nonseparatingclosed curves on S g,n can be connected by a sequence of such curves whose successiveterms meet transversally in a single point, we see that this kernel is independent of γ .By Corollary 2.10, the hyperelliptic nonseparating curve complex H C ns ( S g,n ) is con-nected for g ≥ . Hence, the low terms exact sequence associated to the first spectralsequence of equivariant cohomology yields the exact sequence → H ( H C ns ( S g,n ) / Υ; Q ) → H (Υ; Q ) → (cid:77) γ H (Υ γ ; Q ) , where γ runs over a set of representatives of the Υ -orbits of nonseparating simple closedcurves. Since the natural monomorphism H ( H C ns ( S g,n ) / Υ; Q ) (cid:44) → H (Υ; Q ) has imagecontained in the weight zero subspace of H ( H Υ ; Q ) = H (Υ; Q ) , we then get the exactsequence: → W H (Υ) → (cid:77) γ W H (Υ γ ) . Above we proved that, for all symmetric nonseparating simple closed curves, the maps W H (Υ) → W H (Υ γ ) have the same kernel. It follows that W H (Υ) → W H (Υ γ ) isinjective. Since the Dehn twist τ γ acts trivially on W H (Υ γ ) , it does so on W H (Υ) too,which completes the proof of the lemma. (cid:3)(cid:3) Corollary 3.15.
For a closed surface S of genus , there is a finite unramified G -covering S → S such that the corresponding action of Mod( S ) G on H ( S ; Q ) has a finite nontrivialorbit. In particular, we have H s ( S ; Q ) (cid:40) H ( S ; Q ) . Remark 3.16.
In a blog post (cf. Theorem 2 and 3 in [19]), Carlos Matheus claimsthat a certain cyclic covering of S , ramified over points, provides a counterexampleto Putman-Wieland conjecture. However, his proof of Theorem 3 is flawed where heclaims that the pure mapping class group PMod( S , ) is centralized by some hyperellipticinvolution ι ∈ Mod( S , ) (there are several conjugacy classes of hyperelliptic involutionsin Mod( S , ) while PMod( S , ) does not contain any). But this is not true. In fact, itis not difficult to see that the centralizer of a hyperelliptic involution in Mod( S , ) is ofinfinite index. Note also that the explicit computations mentioned in the introduction of[17] seem to rule out that a counterexample to the Putman-Wieland conjecture may beobtained by means of a covering S → S of low degree. Acknowledgements.
I thank Eduard Looijenga. This paper is a spinoff of my collabo-ration with him and grew out of the many conversations we had in the last years. Inparticular, the Hodge theoretic proof of Lemma 3.12 was suggested by him. I thank thereferee for pointing out a mistake in the previous version of Theorem 2.4 and helping toimprove the exposition. I also thank Louis Funar for suggesting the reference [22].
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