Finiteness and infiniteness results for Torelli groups of (hyper-)K\"ahler manifolds

The Torelli group $\mathcal T(X)$ of a closed smooth manifold $X$ is the subgroup of the mapping class group $\pi_0(\mathrm{Diff}^+(X))$ consisting of elements which act trivially on the integral cohomology of $X$. In this note we give counterexamples to Theorem 3.4 of Verbitsky's paper"Mapping class group and a global Torelli theorem for hyperk\"ahler manifolds"(Duke Math.~J.~162 (2013), no.~15, 2929-2986) which states that the Torelli group of simply connected K\"ahler manifolds of complex dimension $\ge 3$ is finite. This is done by constructing under some mild conditions homomorphisms $J: \mathcal T(X) \to H^3(X;\mathbb Q)$ and showing that for certain K\"ahler manifolds this map is non-trivial. We also give a counterexample to Theorem 3.5 (iv) in this paper where Verbitsky claims that the Torelli group of hyperk\"ahler manifolds are finite. These examples are detected by the action of diffeomorphsims on $\pi_4(X)$. Finally we confirm the finiteness result for the special case of the hyperk\"ahler manifold $K^{[2]}$.


Introduction
Let X be a closed connected oriented manifold. The mapping class group M(X) is the group of connected components of the orientation preserving self diffeomorphisms of X. The subgroup consisting of diffeomorphisms which act trivially on the integral cohomology ring is called the Torelli group of X, denote by T (X). In this note we consider two classes of manifolds, the first is characterized by the following assumptions: Assumption (*) The manifold X has trivial first rational homology and the first Pontrjagin class (considered in rational cohomology) p 1 (X) ∈ H 4 (X; Q) is a linear combination of products of 2-dimensional classes, i.e. there exist classes z i ∈ H 2 (X; Q) and rational numbers a ij such that p 1 (X) = a ij z i ∪ z j . We denote the set consisting of the classes z i and of the rational numbers a ij by D, like definition data.
We construct a homomorphism which we consider as a sort of Johnson homomorphism since it is landing in an abelian group and constructed via the mapping torus. The construction is as follows. Consider the mapping torus X f of f . Since f acts trivially on the cohomology groups the Wang sequence looks like: Since H 1 (X; Q) = 0 this implies that the inclusion induces an ismomorphism H 2 (X f ; Q) → H 2 (X; Q) and so for each class z ∈ H 2 (X; Q) there is a unique class z ∈ H 2 (X f ; Q) restricting to z. Next we look at p 1 (X f )− a ijzi ∪z j ∈ H 4 (X f ; Q). By construction it's restriction to H 4 (X; Q) is zero. Thus there exists a unique class denoted by J D (f ) ∈ H 3 (X; Q) mapping to this class in the Wang sequence. is a homomorphism.
Next we are looking for Kähler manifolds where this invariant is non-trivial. For this we prove a purely topological result. Theorem 1.2. Let X be a simply connected 6-manifold with H 2 (X; Z) ∼ = Z and nontrivial cohomology product H 2 (X; Z) × H 2 (X; Z) → H 4 (X; Z). Then the Assumptions (*) are fulfilled and the invariant is independent of any choices and is denoted by J. The image of J is a lattice in H 3 (X; Q), i.e. a finitely generated abelian group of same rank as the dimension of H 3 (X; Q).
Considering hypersurfaces X(d) of degree d > 1 in CP 4 we obtain complex 3dimensional Kähler manifolds with infinite Torelli group, since d > 1 implies that H 3 (X(d); Q) = 0. Taking the product with copies of CP 1 one obtains examples in all complex dimensions ≥ 3. Namely if f ∈ T (X(d)) then by definition of J D we have for f × id : Thus we obtain the Corollary contradicting Verbitsky's theorem: 3. For all n ≥ 3 there are Kähler manifolds of complex dimension n with infinite Torelli group, actually containing elements of infinite order. In particular taking for example the quintic in CP 4 we obtain Calabi-Yau manifolds with infinite Torelli group.
Remark 1.4. For simply connected 6-manifolds X with H 2 (X; Z) ∼ = Z we compute in [7] the full mapping class group as well as the Torelli group and give generators of these groups. In particular we prove that an element f in the Torelli group has finite order if and only if J(f ) = 0.
Remark 1.5. In general the invariant J D depends on the choice of the data. So, in general there are many homomorphisms depending on the choice of the data. The simplest example is X = S 2 × S 4 ♯S 3 × S 3 , where for z = 0 and z the generator of H 2 (X; Z) the difference of the J-invariants with these data is non-trivial, as shown in [7]. Now we come to the second class of manifolds which is characterized by the following assumptions: Assumption (**) The manifolds X are simply connected and 8-dimensional, whose Betti numbers satisfy one of the following conditions Theorem 1.6. If X satisfies the assumptions (**), then the Torelli group T (X) is infinite.
We apply this result to find a counterexample to Theorem 3.5 (iv) in [12]. For this we consider the complex 4-dimensional hyperkähler manifold K 2 (T ). For the construction and the following information we refer to [10] (where it is called K 2 ). The manifold is simply connected and the Betti numbers are: b 2 (K 2 (T )) = 7, b 3 (K 2 (T )) = 8 and b 4 (K 2 (T )) = 108. Thus our theorem implies: Corollary 1.7. The Torelli group of the hyperkähler manifold K 2 (T ) is not finite.
There is another complex 4-dimensional hyperkähler manifold denoted by K [2] , where K is a K 3 -surface (for the construction see again for example [10]). In this case we show that the Torelli group is finite. This follows from: Theorem 1.8. Let X be a simply-connected closed 8-manifold satisfying the following conditions (1) H 4 (X; Q) isomorphic to the second symmetric power of H 2 (X; Q).
Corollary 1.9. The Torelli group of K [2] , where K is a K 3 -surface, is finite.
Proof. By [10] the Betti numbers are b 2 = 23, b 3 = 0 and b 4 = 276, which is the same as the dimension of the second symmetric product of Q 23 . Since the second symmetric power of the second cohomology of a hyperkähler manifold always injects into the 4-th cohomology [2], this implies (1). That p 1 (K [2] ) is non-zero follows from [1]. Remark 1.10. This implies that Verbitsky's Theorem 3.5 is true for K [2] and so his main result about the moduli space of hyperkähler manifolds holds for K [2] .
We would like to thank Daniel Huybrechts for bringing Verbitsky's paper [12] to our attention when we told him about our paper [7] where we give a complete computation of the mapping class group of certain 6-manifolds. Since counterexamples to Verbitsky's theorem might be of separate interest we wrote this note. We have sent this note to several people and received a paper by Richard Hain [4] confirming our result that Theorem 3.4 in [12] is incorrect by showing that the Torelli group of certain complex 3-dimensional Kähler manifolds M have an abelian quotient of infinite rank. These examples are different from our examples, they are detected by the induced map , and so the Torelli groups acts trivial on π 3 (X) ⊗ Q.

Proofs
Proof of Proposition 1.1. To prove that J D is a homomorphism we first note that if X is a fibre in X f then we have an exact sequence with rational coefficients and the term on the left is by suspension isomorphism isomorphic to H 3 (X) and under this isomorphism J D (f ) corresponds to the class in H 4 (X f , X) which restricts to p 1 (X f ) − a ijzi ∪z j . Now we construct X f g from X f by cutting I × X along 1/2 × X and regluing it via g from the left to the right. Then we consider the two fibres over 1/4 and 3/4 and denote them by X 1 and X 2 . We consider the restriction map (with rational coefficients) given by the sum, and the element Next we consider the 6-manifolds in Proposition 1.2. Let z ∈ H 2 (X) be a generator, by Poincaré duality H 4 (X) ∼ = Z and thus z 2 equals to d times a generator of H 4 (X). The condition that the cohomology product is non-trivial is equivalent to saying that d is nonzero. It's shown in [13] and [5] that for every d = 0 there exist such X 6 . Also there is a connected sum decomposition We introduce a construction of diffeomorphisms, which is a generalization of Dehn twists. Let S 3 × D 3 ⊂ X, α ∈ π 3 (SO(4)), choose a smooth map ϕ : (D 3 , ∂) → (SO(4), I) representing α, such that a neighborhood of ∂D 3 is mapped to the identity. Define f : X → X by We call f a Dehn twist in S 3 × D 3 with parameter α.
Proof of Theorem 1.2. The main ingredients of Theorem 1.2 already appeared in [6]. The proof of a slightly different situation will be given in [7]. For convenience of the reader we repeat the proof. By the assumptions the rational Pontrjagin class In the decomposition X = N ♯g(S 3 × S 3 ) we number the standard embedded spheres They represent a symplectic basis {e 1 , · · · , e 2g } of H 3 (X). Let α ∈ π 3 (SO(3)) ∼ = Z be a generator, i * (α) ∈ π 3 (SO(4)) be its image induced by the inclusion i : SO(3) → SO(4). Then the 4-dimensional vector bundle ξ over S 4 corresponding to i * (α) has trivial Euler class and p 1 (ξ) equals 4 times a generator of H 4 (S 4 ) (see [9,Lemma 20.10]). Now let f : X → X be the Dehn twist in S 3 i × D 3 with parameter i * (α). Since the Euler class of i * (α) is trivial, it's easy to see that f acts trivially on homology, hence f ∈ T (X). We claim that J(f ) equals 4 times the Poincaré dual of e i in H 3 (X). This implies that the image of J is a subgroup of H 3 (X) of the same rank, hence it is a lattice in H 3 (X; Q).
It is enough to show the claim for i = 1. By the adjunction formula this is equivalent to J(f ), e i = 0 for i = 2 and J(f ), e 2 = 4.
Now notice that f is the identity on a tubular neighborhood of S 3 i 's for i = 2, therefore the normal bundle of S 3 i × S 1 in X f is trivial, and we have be the quotient map, from the geometric construction it's easy to see that the normal bundle of S 3 2 × S 1 in X f equals c * ξ. Therefore The proofs of Theorem 1.6 and Theorem 1.8 are based on modified surgery theory [8]. We recall some basic definitions here. The normal k-type of a smooth oriented manifold X is a fibration p : B → BSO, which is characterized by the assumptions that there is a lift of the normal Gauss map ν : X → BO by a (k + 1)-equivalencē ν : X → B and that the homotopy groups of the homotopy fibre F are trivial in degree ≥ k + 1 (for details see [8]).
To show Theorem 1.6 we will construct infinitely many mapping classes in T (X), whose actions on π 4 (X) are pairwisely distinct. Let p : B → BSO be the normal 4-type of X, denote the homotopy fiber of p by F , and the kernel of the Hurewicz homomorphism π 4 (F ) → H 4 (F ) by Kπ 4 (F ).
Proof. We use Sullivan's rational homotopy theory [11] to show the assumptions (**) imply the conclusions in this lemma. For this we construct the minimal model from information of the cohomology ring. Let x 1 , · · · , x b2 be free generators of the minimal model in degree 2 with differential 0. Let y 1 , · · · , y b3 be a part of the free generators in degree 3, which generate H 3 (X; Q). They have trivial differentials. The elements x i y j generate a space of dimension b 2 b 3 , on which the differential is trivial. Since by Poincaré duality the 5-th Betti number is b 3 , there must be at least k = (b 2 − 1)b 3 linearly independent indecomposable elements z 1 , · · · , z k in degree 4, which by the differential are mapped injectively to a subspace of the vector space generated by the x i y j . In degree 4 the dimension of the subspace generated by x i x j (i ≤ j) is b 2 (b 2 + 1)/2, whereas the 4-th Betti number is b 4 , there are additional (to the z i ) free generators u 1 , · · · , u l (where l ≥ b 4 − b 2 (b 1 + 1)/2 ≥ 1) with trivial differential in degree 4. There might be more generators in degree 4 depending on the product H 2 (X; Q) ⊗ H 3 (X; Q) → H 5 (X; Q). But alone from this information the two conclusions in the lemma follow. Namely Hom(π 4 (X), Q) is isomorphic to the degree 4 subspace modulo the decomposable elements and so contains the subspace with basis u 1 , ...., u l , z 1 , ..., z k . The dual of the rational Hurewicz homomorphism maps u i injectively. Dualizing we see that the dual of the u i are mapped injectively showing (1) and the dual of the z i are mapped to 0 in rational homology, so the rank of Consider the following commutative diagram This show that under the assumption dim Q Kπ 4 (F ) ⊗ Q ≥ 1.
Let Z 4 (B; Kπ 4 (F )) be the group of 4-cocycles with coefficients in Kπ 4 (F ), Aut(B) be the group of fiber homotopy classes of fiber homotopy equivalences of p : B → BSO. We will first define a map Φ : Z 4 (B; Kπ 4 (F )) → Aut(B) as follows.
By the construction h is compatible with the fiber projection p, i. e. the following diagram commutes up to homotopy Now we show that we can extend h to a map on the 5-skeleton of B compatible with p. Let e 5 be a 5-cell, with attaching map f : S 4 → B (4) . By Blakers-Massey theorem we have an isomorphism Since α is a cocyle, α, ∂e 5 = δα, e 5 = 0, therefore h extends to a map h : B (5) → B, compatible with the fiber projection p. (The extension may not be unique, we choose one extension.) Since π n (F ) = 0 for n ≥ 5, we can extend h further to a fiber map h : B → B.
Proof. It's clear from the construction that h * is the identity on H i (B) for i ≤ 3 and h * is the identity on H i (B) for i ≤ 3. On the chain level h * is i ) is a boundary by definition. Therefore h * is the identity on H 4 (B). By the Hurewicz theorem we see that h * : π i (B) → π i (B) is an isomorphism for i ≤ 3 and is a surjection for i = 4. Notice that π 4 (B) is a finitely generated abelian group, a surjective endomorphism of a finitely generated abelian group must also be injective, therefore h * is an isomorphism on π 4 (B). Since p : B → BSO is 4coconnected, h induces isomorphisms on π i (B) for all i. Therefore h is a fiber homotopy equivalence.
Since π n (F ) = 0 for n ≥ 5, by rational homotopy theory, there are no indecomposable cohomology classes in H n (F ; Q) for n ≥ 5. Therefore elements in Proof of Theorem 1.6. By Lemma 2.1 the image of π 4 (B) → H 4 (B) has rank k > 0. Let x 1 , · · · , x k be homology classes in the image which generate a free abelian subgroup of rank k in H 4 (B), y i ∈ π 4 (B) be a pre-image of x i , represented by a map S 4 → B. Then we may take B (4) of the form where X is a 4-complex, and [S 4 i ] = y i for i = 1, · · · , k. Let ϕ ∈ Hom(H 4 (B), Kπ 4 (F )) be a homomorphism, since there are surjective homomorphisms we may pick a pre-image of ϕ, say α ∈ Z 4 (B, Kπ 4 (F )). Let h : B → B be the image of α under Φ. Consider the action of h on π 4 (B), especially the image of y i under h * : let ι : S 4 → B be the inclusion of S 4 i , clearly the homotopy class of the composition h • ι : By Lemma 2.1 there are infinitely many ϕ ∈ Hom(H 4 (B), Kπ 4 (F )) such that {ϕ(x i )| i = 1, · · · , k} are pairwisely distinct. Therefore we have constructed infinitely many h ∈ Aut(B) with properties in Lemma 2.2.
Let Ω 8 (B, p) be the 8 dimensional B-bordism group. It's isomorphic to the 8dimensional stable homotopy group of the corresponding Thom spectrum. By the Atiyah-Hirzebruch spectral sequence there is an isomorphism where the image of a bordism class [ Sinceν is a 5-equivalence, we see that f i * is the identity on H i (X) for i ≤ 4, and f * i is the identity on H i (X) for i ≤ 3. Therefore by Poincaré duality, f i * is the identity on H * (X) and hence f ∈ T (X). But f i * on π 4 (X) are pairwisely distinct. Therefore we have constructed infinitely many elements in T (X).
Proof of Theorem 1.8. Let p : B → BSO be the normal 4-type of X, F be the homotopy fiber of p. Fix a normal 4-smoothingν : X → B.
We claim that π 3 (X) and π 4 (X) are finite. For this we consider the minimal model of X. It has generators x 1 , ..., x r in degree 2, where r is the second Betti number. By assumption x i ∪ x j for i ≤ j is a basis of H 4 (X; Q). This and the fact that H 3 (X; Q) = 0 implies that there are no generators in degree 3 and hence π 3 (X) is finite. This implies that there are no decomposable elements in degree 5. Thus the differential on elements of degree 4 is zero. But then there are no indecomposable elements in degree 4, since they would produce indecomposable cohomology classes. Thus π 4 (X) is finite. Finally we conclude that also π 3 (F ) and π 4 (F ) are finite. This follows from the homotopy sequence of the fibration F → B → BSO since π i (X) ∼ = π i (B) for i ≤ 4 and p 1 (X) = 0 implying that π 4 (B) → π 4 (BSO) is non-trivial.
Let f ∈ T (X) be a self-diffeomorphism, thenν • f is also a normal 4-smoothing. Let T 0 (X) be the subset of T (X) consisting of self-diffeomorphisms f such that ν • f andν are homotopic as liftings of ν. Proof. It suffices to show that there are finitely many homotopy classes ofν • f over p, which in turn follows if there are finitely many lifts of the Gauss map ν over p. This follows by induction over the skeleta and the Puppe sequence and the fact that f induces the identity on π 2 (X) = H 2 (X) from the fact that the homotopy groups π 3 (F ) and π 4 (F ) are finite and π i (F ) = 0 for i ≥ 5.
We finish the proof of the theorem by showing that T 0 (X) is finite. Given f ∈ T 0 (X), choosing a homotopy h : X × [0, 1] → B betweenν andν • f we obtain a normal B-structure ϕ : X f → B, where X f is the mapping torus. This represents