Nariya Kawazumi

Arthur Soulié

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

Keywords: 2020 Mathematics Subject Classification: Primary: 20J06; Secondary: 55R20, 57R20 stable (co)homology with twisted coefficients, mapping class groups, unit tangent bundle (co)homology representations

We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary with twisted coefficients given by the first homology of the unit tangent bundle of the surface. The stable twisted cohomology is not free as a module over the stable cohomology algebra with constant coefficients. In fact, it is out of the scope of the traditional framework for twisted homological stability, since these twisted coefficients define a covariant functor over the classical category associated to mapping class groups to study homological stability, rather than a contravariant one. For comparison, we also compute the stable cohomology group with coefficients in the first cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

Introduction

Considering a smooth compact connected orientable surface of genus g ≥ 0 and with one boundary component Σ g,1 , we denote its mapping class group (that is the isotopy classes of its diffeomorphisms restricting to the identity on the boundary) by Γ g,1 . The study of the (co)homology of the groups {Γ g,1 , g ∈ N} has been a very active research topic over the past decades. In particular, the full computations of their cohomology groups with twisted coefficients remain an active research topic; see [GKR19, Section 5.5] and [START_REF] Hain | Johnson homomorphisms[END_REF]Part 4]. In this paper, we compute the stable cohomology groups of the mapping class groups with twisted coefficients defined from the first (co)homology group of the unit tangent bundles of the considered surfaces; see Theorems A and B. This is the first example of stable twisted cohomology for that mapping class groups is not free over the rational stable cohomology of these groups.

Background on stable (twisted) (co)homology.

A key step towards the computations of the (co)homology of the mapping class groups is their homological stability properties. Namely, we consider canonical injections Γ g,1 ֒→ Γ g+1,1 induced by viewing Σ g,1 as a subsurface of Σ g+1,1 extending the diffeomorphisms of Σ g,1 by the identity on the complement Σ 1,1 , and a set of Γ g,1modules {F (g), g ∈ N} with Γ g,1 -equivariant morphisms F (g) → F (g + 1). This data defines morphisms for the homology groups H i (Γ g,1 ; F (g)) → H i (Γ g+1 ; F (g + 1)). for any i ≥ 0. For each i ≥ 0, if this canonical morphism is an isomorphism for N (i, F ) ≤ g with some N (i, F ) ∈ N depending on i and F , then the mapping class groups are said to satisfy homological stability (with (twisted) coefficient in F ). The homological stability property with constant coefficients (i.e. for F (g) = Z for all g ≥ 0) is due to Harer [START_REF] Harer | Stability of the homology of the mapping class groups of orientable surfaces[END_REF]. The range N (i, Z) is improved by Boldsen [START_REF] Boldsen | Improved homological stability for the mapping class group with integral or twisted coefficients[END_REF] and Randal-Williams [START_REF] Randal-Williams | Resolutions of moduli spaces and homological stability[END_REF]. Furthermore, Ivanov [START_REF] Ivanov | On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients[END_REF] proves homological stability property with twisted coefficients given Γ g,1 -modules forming a functor F : UM 2 → Ab (where UM 2 is defined in §2.1 and Ab the category of abelian groups) satisfying some polynomiality conditions; we refer to §2 for further details. Finally, Randal-Williams and Wahl [START_REF] Wahl | Homological stability for automorphism groups[END_REF] and Galatius, Kupers and Randal-Williams [GKR19, Section 5.5] extract the general framework for proving homological stability properties (both for constant and twisted coefficients) for general families of groups, subsuming these previous studies and recovering their results.

For a fixed i, the colimit Colim g∈(N,≤) (H i (Γ g,1 ; F (g))) is denoted by H i (Γ ∞,1 ; F ). When the homological stability property is satisfied, the value of the twisted homology group H i (Γ g,1 , F (g)) for g ≥ N (i, d) corresponds to the colimit H i (Γ ∞,1 ; F ) and is called the stable homology of the mapping class groups. All the stable homology results obtained so far in the literature have been carried out using a field as ground ring, generally the rationals Q. We will also mainly use this field although some of our results still hold over Z. In particular, using the rationals allows us to equivalently consider the stable cohomology with twisted coefficients (thanks to the universal coefficient theorem with twisted coefficients, see Lemma 2.6).

By proving Mumford's conjecture [START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves[END_REF], Madsen and Weiss [START_REF] Madsen | The stable moduli space of Riemann surfaces: Mumford's conjecture[END_REF] compute the rational stable homology (i.e. for F (g) = Q for all g ≥ 0) of the mapping class groups. Namely, they use the standard cohomology classes defined by Mumford [START_REF] Mumford | Towards an enumerative geometry of the moduli space of curves[END_REF], Morita [Mor84;[START_REF] Morita | Characteristic classes of surface bundles[END_REF] and Miller [START_REF] Miller | The homology of the mapping class group[END_REF]

{e i ∈ H 2i (Γ ∞,1 ; Q); i ≥ 1}, called the classical Mumford-Morita-Miller classes, to describe the algebra H * (Γ ∞,1 ; Q) as follows: H * (Γ ∞,1 ; Q) ∼ = Q[{e i , i ≥ 1}]. (0.1)
Denoting the free Q-vector space ∞ i=1 Qe i by E and by Sym Q (E) its symmetric algebra, Madsen-Weiss theorem (0.1) may be reframed as that there is an isomorphism

H * (Γ ∞,1 ; Q) ∼ = Sym Q (E).
Furthermore, considering the twisted coefficient given by the first homology of the surface Σ g,1 denoted by H(g), the stable twisted H * (Γ ∞,1 ; H) was first computed by Harer [Har91, Section 7]. Also, considering the closed oriented surfaces Σ g analogues (obtained from Σ g,1 by capping the boundary component with a disc), Looijenga [START_REF] Looijenga | Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map[END_REF] computed the stable cohomology of the associated mapping class groups {Γ g , g ∈ N} with coefficients in any irreducible representation of the rational symplectic group. Looijenga did not use the stability result of [START_REF] Ivanov | On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients[END_REF] but only that of [START_REF] Harer | Stability of the homology of the mapping class groups of orientable surfaces[END_REF]. Independently from these previous works, the first author [START_REF] Kawazumi | A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces[END_REF] introduced a series of twisted cohomology classes on the mapping class group Γ g,1 , called the twisted Mumford-Morita-Miller classes {m i,j ; i ≥ 0, j ≥ 1}. Based on Looijenga's idea [START_REF] Looijenga | Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map[END_REF], one can prove that some algebraic combinations of the twisted Mumford-Morita-Miller classes define a free basis of the stable cohomology group of with coefficients in the tensor products of H(g); see [START_REF] Kawazumi | On the stable cohomology algebra of extended mapping class groups for surfaces[END_REF]. These computations may also be done via other methods; see Randal-Williams [Ran18, Appendix B] and [Sou20, Section 2.3.1].

The unit tangent bundle homology representations. The representation theory of the mapping class groups of surface is wild and remains an active research topic; see Margalit's expository paper [START_REF] Margalit | Problems, questions, and conjectures about mapping class groups[END_REF]. In particular, there are few known representations of the mapping class groups Γ g,1 apart from the first homology of the surface H(g). However, other representations which appear naturally are those given by homology and cohomology of the unit tangent bundle of the surface Σ g,1 . We denote by H(g) the first integral homology group H 1 (U T Σ g,1 ; Z). Its dual representation is denoted by H∨ (g), while we denote its corresponding rational homology by HQ (g). These representations have been first studied by Trapp [Tra92, Theorem 2.2] and we refer the reader to §1.2 for further details.

In particular, the representation H(g) is a non-trivial extension of H(g) by Z; see (1.3). This extension corresponds to the twisted Mumford-Morita-Miller class m 1,1 in the stable cohomology algebra of each cohomology class H * (Γ ∞,1 ; H). It is the image of the cohomology class introduced Earle [START_REF] Earle | Families of Riemann surfaces and Jacobi varieties[END_REF] which generates H 1 (Γ g,1 ; H(g)) for g ≥ 2. We refer the reader to §1.2 and §2.2.1 for further details.

Results.

In the present paper, we consider the cohomology of the mapping class groups with twisted coefficients given by HQ (g) and H∨ Q (g). We consider cohomology rather than homology because of the key usefulness of the cup product structure. The pathway to make these computations is based on the short exact sequences (1.3) and (1.3) defining these modules, on the determination of the cohomology long exact sequence connecting homomorphisms (see Lemmas 3.2 and 3.5) and the Contraction formula (2.3) between stable twisted cohomology classes. We denote by Qθ the trivial Sym Q (E)-module generated by the integral stable 0th-cohomology θ ∈ H 0 (Γ ∞,1 ; H) defined by the fiber of the unit tangent bundle.

First, the computation for the dual representations H∨ Q (g) is the least difficult. Indeed, these define a covariant functor H∨ Q : UM 2 → Ab and therefore we already know from [Sou20, Theorem C] that the stable cohomology algebra H * (Γ ∞,1 ; H∨ Q ) is free over Sym Q (E). We prove: Theorem A [Theorem 3.3] The stable cohomology algebra H * (Γ ∞,1 ; H∨ Q ) is isomorphic to the free Sym Q (E)-module with basis {m i,1 , i ≥ 2}. In particular, it is concentrated in odd degrees.

On the contrary, the representations HQ (g) induce a contravariant functor HQ : UM 2 → Ab. As far as the authors know, any qualitative general result and computations for such coefficients have not been realised yet.

Theorem B [Theorem 3.9] The stable cohomology algebra H * (Γ ∞,1 ; HQ ) is isomorphic to direct sum Qθ M, where M is the Sym Q (E)-module generated by the classes M i,j := e i m j,1 -e j m i,1 for i, j ≥ 1 and with relations

e i M j,k + e j M k,i + e k M i,j ∼ 0 for all i, j, k ≥ 1.
In particular, the stable twisted cohomology H * (Γ ∞,1 ; HQ ) is not free as Sym Q (E)-module; see also Theorem 3.8.

Perspectives.

The key steps of the results of Theorems A and B are proved with integral coefficients. Namely, the contraction formula and the connecting homomorphisms still hold for integral coefficients. Therefore, we might in principle be able to do the computations with Z as ground ring. However, although the stable twisted cohomology module H * (H) is free over H * (Z) (see [Kaw08, Theorem 1.B]), the stable cohomology H * (Z) is still poorly known.

On another note, a natural extension of the results of Theorems A and B consists in considering the exterior powers of the representations HQ (g) and H∨ Q (g) respectively. This is the aim of the forthcoming work [KS]. In particular, the stable twisted cohomology algebras H * (Γ ∞,1 ; Λ d H∨ Q ) (for all d ≥ 2) and H * (Γ ∞,1 ; Λ d HQ ) (for 2 ≤ d ≤ 5) are thoroughly studied.

Outline. The paper is organised as follows. In §1, we make recollections on the representation theory of mapping class groups and on the classical and twisted Mumford-Morita-Miller cohomology classes. In §2, we recall the framework and properties for twisted homological stability of mapping class groups. In §3, we make the full computations of the mapping class groups stable twisted cohomology with coefficients in the first homology and cohomology of the unit tangent bundle.

Conventions and notations.

For a ring R, we denote by R-Mod the category of left R-modules.

For R = Z, the category of Z-modules is also denoted by Ab. For a map f , we generically (when everything is clear from the context) denote by f * the induced map in homology and by f * the induced map in cohomology. For G a group, R a commutative ring and

V a R[G]-module, we denote by V ∨ the dual R[G]-module Hom R (V, R).
For K a field and V a K-vector space, we denote by Sym K (V ) the symmetric algebra on V over K.

Considering a G-module M , if there is no risk of confusion, we generally denote the twisted cohomology groups H * (G; M ) by H * (M ) for sake of simplicity. We denote the cup product •, but also often abuse the notation denoting it by an empty space for simplicity when there is no risk of confusion. The first integral homology group H 1 (Σ g,1 ; Z) is generally denoted by H(g) all along the paper, and we denote by H Q (g) the first rational homology group H 1 (Σ g,1 , Q).

Acknowledgements. The authors would like to thank Oscar Randal-Williams, Christine Vespa and Shun Wakatsuki for illuminating discussions and questions. The authors were supported by the PRC CNRS-JSPS French-Japanese Project "Cohomological study of MCG and related topics".

Representations and cohomological structures

In this section, we review some representations of the mapping class groups, in particular the unit tangent bundle homology representations (see §1.2) for which we make the connection with the Earle class (see §1.1).

Let us first consider the first integral homology group of the surface Σ g,1 , denoted by H(g). Since the Poincaré-Lefschetz duality is the cap product with the fundamental class, there is a Γ g,1 -module isomorphism between H 1 (Σ g,1 ; Z) and H(g). By the universal coefficient theorem for cohomology of spaces (see [Wei94, Example 3.6.7]), we know that H 1 (Σ g,1 ; Z) is isomorphic to H ∨ (g) as Γ g,1modules. A fortiori, we have a Γ g,1 -module isomorphism H ∨ (g) ∼ = H(g).

Let us now move on to the unit tangent bundle homology representations. We first need to recall the notion of framings of the unit tangent bundle. We denote by T Σ g,1 the tangent bundle of the surface Σ g,1 and fix a Riemannnian metric • on it. By definition, the unit tangent bundle U T Σ g,1 is the set of elements of T Σ g,1 whose length is 1 with respect to • . It can be regarded as the quotient of the complement of the zero section in T Σ g,1 by the action of the positive real numbers R + by scalar multiplication. So, for any diffeomorphism ϕ of Σ g,1 , its differential dϕ acts on the unit tangent bundle U T Σ g,1 . The canonical projection of the unit tangent bundle U T Σ g,1 onto the surface defines the locally trivial fibration

S 1 ι ֒→ U T Σ g,1 ̟ → Σ g,1 .
Definition 1.1 A framing of U T Σ g,1 is an orientation-preserving isomorphism of vector bundles U T Σ g,1 ∼ = Σ g,1 ×S 1 , which can be regarded as a continuous map ξ : U T Σ g,1 → S 1 whose restriction to each fiber is an orientation-preserving homeomorphism. Since Σ g,1 has non-empty boundary, there exist framings of U T Σ g,1 .

For α : S 1 → Σ g,1 an immersed loop, its rotation number rot ξ (α) ∈ Z with respect to the framing ξ is the mapping degree of the composite of

• α/ • α : S 1 → U T Σ g,1 and ξ : U T Σ g,1 → S 1 .
The set F(Σ g,1 ) of homotopy classes (without fixing the boundary) of framings of U T Σ g,1 is an affine set modeled in the cohomology group H 1 (Σ g,1 ; Z). More precisely, the difference of two framings ξ and ξ ′ is given by a cohomology class u ∈ H

1 (Σ g,1 ; Z) if and only if rot ξ ′ (α) -rot ξ (α) = u([α]) ∈ Z, where [α] ∈ H 1 (Σ g,1 ; Z)
is the homology class of the immersed loop α. The mapping class group Γ g,1 acts on the set F(Σ g,1 ) by

ϕ • ξ = ξ • dϕ -1 : U T Σ g,1 dϕ -1 -→ U T Σ g,1 ξ -→ S 1
for ϕ ∈ Γ g,1 and ξ ∈ F(Σ g,1 ). For an immersed loop α on Σ g,1 , we have rot ϕ•ξ (α) = rot ξ (ϕ -1 • α).

Earle class

We recall here some classical facts about the first cohomology of mapping class groups with twisted coefficients in H(g). For any g ≥ 1, we recall that gluing a disc with a marked point Σ 1 0,1 on the boundary component of Σ g,1 induces the following short exact sequence:

1 -→ Z -→ Γ g,1 Cap -→ Γ 1 g -→ 1, (1.1)
where Γ 1 g denotes the mapping class group of the punctured surface Σ 1 g . This is called the forgetful exact sequence or the capping short exact sequence; we refer the reader to [FM12, §4.2.5] for more details. In particular, considering the Lyndon-Hochschild-Serre spectral sequence associated to the short exact sequence (1.1), the fact that H(g) Γ 1 g = 0 implies that the forgetful homomorphism Cap provides an isomorphism

H 1 (Γ 1 g ; H(g)) ∼ = H 1 (Γ g,1 ; H(g)).
We denote by e ∈ H 2 (Γ 1 g ; Z) the Euler class of the short exact sequence (1.1) seen as a central extension.

Furthermore, we make the following computation for the twisted first cohomology of the mapping class group of the torus with one boundary. This fact is probably known to the experts (see [START_REF] Callegaro | The cohomology of the braid group B3 and of SL2(Z) with coefficients in a geometric representation[END_REF] for instance), but we give a short proof for the convenience of the reader.

Lemma 1.2 The cohomology group

H 1 (Γ 1,1 ; H(1)) is trivial.
Proof. It is a classical fact that the mapping class group Γ 1,1 is isomorphic to the braid group on three strands B 3 , and that the standard generators σ 1 and σ 2 of B 3 act on

H 1 (Σ 1,1 ; Z) ∼ = Z 2 through the isomorphism by σ 1 → 1 1 0 1 and σ 2 → 1 0 -1 1 respectively; see for instance [Mil71, Theorem 10.5]. Let f be a cocycle of B 3 with values in H 1 (Σ 1,1 ; Z) ∼ = Z 2 . It is determined by the values f (σ 1 ) = a 1 b 1 and f (σ 2 ) = a 2 b 2 . We deduce from the braid relation σ 1 σ 2 σ 1 = σ 2 σ 1 σ 2 that b 1 = a 2 and b 1 = -a 2 . Hence we have a 2 = b 1 = 0, and thus f = d 1 -b 2 a 1 is a coboundary.
Now, we assume that g ≥ 2. Morita [Mor89a, Proposition 6.4] proves that H 1 (Γ 1 g ; H(g)) ∼ = Z, and that the map induced by the push map π 1 (Σ g , p 0 ) → Γ 1 g maps its generators to ±(2 -2g)1 H(g) in H 1 (Σ g ; H(g)). Before this result, Earle [START_REF] Earle | Families of Riemann surfaces and Jacobi varieties[END_REF] constructed the generator by using theta constants, and so we call it the Earle class. The push map π 1 (U T Σ g , p 0 ) → Γ 1 g maps the generators of the latter group to ±(2 -2g) times the projection homomorphism H 1 (U T Σ g,1 ; Z) → H(g). Following Mikio Furuta [START_REF] Morita | Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles[END_REF], one of the generators is constructed in the following explicit way. We fix a framing ξ of U T Σ g,1 . The map k ξ (g) : Γ g,1 → H 1 (Σ g,1 ; Z) defined by

k ξ (g, ϕ) = ϕ • ξ -ξ ∈ H 1 (Σ g,1 ; Z) (1.2)
is a 1-cocycle of Γ g,1 . Kuno [START_REF] Kuno | A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group Mg[END_REF] gives a combinatorial formula for the cocycle k ξ . Under the Poincaré-Lefschetz duality, we have:

Theorem 1.3 ([Mor97, §4]) For each g ≥ 2, the cohomology class k(g) := [k ξ (g)]
does not depend on the choice of ξ and generates the infinite cyclic group H 1 (Γ g,1 ; H(g)).

Proof. By the computation of [START_REF] Morita | Families of Jacobian manifolds and characteristic classes of surface bundles[END_REF], it suffices to compute the value of k ξ (g) at a push map. Such a computation was carried out in the original proof of [START_REF] Morita | Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles[END_REF]. However, for sake of completeness, we give another explicit computation. Let T C ∈ Γ g,1 be the right-handed Dehn twist along an oriented simple closed curve C of Σ g,1 . Then, it follows from some elementary considerations that rot ξ (T

C (α)) -rot ξ (α) = ([α] • [C])rot ξ (C),
where

k ξ (g, T C ) = ξ • dT -1 C -ξ = -(rot ξ•dT -1 C C)[C] = -(rot ξ (C))[C] ∈ H 1 (Σ g,1 ; Z).
Assume that the curve C passes near the boundary of Σ g,1 . Then, fattening the union of the boundary and the curve C, we obtain a pair of pants embedded in Σ g,1 whose 3 boundary components are given as follows: one is parallel to the boundary of Σ g,1 , and the other two simple closed curves C -1 1 and C 2 are parallel to C except near the boundary. Then the push map along C is equal to T -1 C1 T C2 ∈ Γ g,1 . By the above formula, we obtain k ξ (g, T -1

C1 T C2 ) = •(-rot ξ (C 2 ) + rot ξ (C 1 ))[C].
From the Poincaré-Hopf theorem, -rot ξ (C 1 ) + rot ξ (C 2 ) + rot ξ (∂Σ g,1 ) is equal to -1, the Euler characteristic of the pair of pants, and rot ξ (∂Σ g,1 ) = χ(Σ g,1 ) = 1 -2g. Hence we have

k ξ (g, T -1 C1 T C2 ) = •(2 -2g)[C]
, which ends the proof.

Notation 1.4 By Smale's theorem [Sma59, Theorem B], we have H 1 (Γ 0,1 , H 1 (Σ 0,1 ; Z)) = 0 and we know from Lemma 1.2 that H 1 (Γ 1,1 , H 1 (Σ 1,1 ; Z)) is also trivial. We thus assign k(0) : Γ 0,1 → H 1 (Σ 0,1 ; Z) and k(1) : Γ 1,1 → H 1 (Σ 1,1 ; Z) to be the trivial cohomology classes.

The unit tangent bundle homology representations

We consider the first integral homology group H(g). The dual H∨ (g) is isomorphic to the first integral cohomology group H 1 (U T Σ g,1 ; Z) by the universal coefficient theorem for cohomology of spaces (see [Wei94, Example 3.6.7]). Then the Serre spectral sequence of the locally trivial fibration S 1 ι ֒→ U T Σ g,1 ̟ → Σ g,1 provides the following Γ g,1 -equivariant short exact sequences:

0 / / Z ι * / / H(g) ̟ * / / H(g) / / 0, (1.3) 0 / / H(g) ̟ *
/ / H∨ (g)

ι * / / Z / / 0. (1.4)
We also have the analogue short exact sequences to (1.3) and (1.4) with the rational homologies H Q (g), HQ (g) and H∨ Q (g). We denote by ∈ H(g) the homology class of the fiber of the unit tangent bundle.

In addition, Trapp [Tra92, Theorem 2.2] describes more precisely the Γ g,1 -module structure. Namely, for an element ϕ ∈ Γ g,1 , the action of ϕ on H(g) is given by the matrix

id Z k(g)(ϕ) (0) H(ϕ) (1.5)
where H(ϕ) denotes the action of ϕ on H and k(g) is Earle's cohomology class of Notation 1.4.

In particular, the crossed homomorphism k(g) is the extension class of the short exact sequences (1.3) and (1.4). We note that the Γ g,1 -action on Z is trivial representation. Also, although the sequences (1.3) and (1.4) split as abelian groups short exact sequences, they do not as Γ g,1 -modules. Hence, contrary to the homology and cohomology of the surface (where Poincaré-Lefschetz duality applies), the dual H∨ (g) is not isomorphic to H(g) as a Γ g,1 -module.

Homological stability framework and tools

In this section, we review the notions of coefficient systems, polynomiality and homological stability with twisted coefficients with respect to the framework of the present paper. These highlight the interest in the work and mainspring of the results of §3.

Twisted coefficient systems

First of all, we present the suitable categories to encodes compatible representations of the mapping class groups originally introduced in [RW17, §5.6.1]. We consider the groupoid defined by the smooth compact connected orientable surfaces S with one boundary component along with a parametrized interval in the boundary, and the isotopy classes of diffeomorphisms restricting to the identity on a neighbourhood of the parametrized interval for the morphisms. We fix a 2-disc D 2 and a torus with one boundary component that we denote by Σ 1,1 . Let M 2 be the skeleton of that groupoid defined as the full subgroupoid of the monoidal sums on the objects D 2 and Σ 1,1 . The groupoid M 2 has a (strict) braided monoidal ♮ structure induced by gluing on half of the marked interval; see [RW17, §5.6.1] for further details. Let UM 2 be the category, called the Quillen's bracket construction over M 2 , with the same objects as M 2 and UM 2 (Σ g,1 , Σ g ′ ,1 ) is the set Colim M2 [M 2 (-♮Σ g,1 , Σ g ′ ,1 )]. This definition is a particular output of a general construction of [START_REF] Grayson | Higher algebraic K-theory[END_REF]; we refer the reader to [RW17, §1.1] if additional details are required. As we will see in §2.2, this type of category is very useful to deal with homological stability questions.

We may now encode compatible representations of mapping class groups by considering functors with the category UM 2 as source and a module category as target. We distinguish two types of such functors because of their distinct qualitive properties with respect to homological stability detailed in §2.2. A covariant system (resp. contravariant system) over UM 2 is a functor F : UM 2 → Ab (resp. F ∨ : UM op 2 → Ab). We now recursively define the notion of polynomiality for covariant systems as follows:

• the constant functors UM 2 → Ab are the polynomial covariant systems of degree 0;

• for a natural number d ≥ 1, the functor F : UM 2 → Ab is a polynomial covariant system of degree less than or equal to

d if the morphism F ([Σ 1,1 , id Σ1,1♮S ]
) is injective for each surface S of UM 2 , and the induced functor δ(F ) : UM 2 → Ab defined by S → (Coker(F (S) → F (Σ 1,1 ♮S)) is a polynomial covariant system of degree less than or equal to d -1.

A first example of polynomial covariant system is given by the first homology group of the surfaces. Namely, assigning the first integral homology group to each surface define functor

H : (UM 2 , ♮, D 2 ) → (Ab, ⊕, 0), which is strong monoidal in the sense that H(Σ g ′ +g,1 ) ∼ = H(Σ g ′ ,1 )⊕ H(Σ g,1 ) and H(ϕ ′ ♮ϕ) ∼ = H(ϕ ′ ) ⊕ H(ϕ) for all g, g ′ ≥ 1, ϕ ∈ Γ g,1 and ϕ ′ ∈ Γ g ′ ,1
. In particular, H : UM 2 → Ab is a polynomial covariant system of degree one. Furthermore, the first homology groups of the unit tangent bundle of the surfaces along with the natural action of the mapping class groups (see §1.2) define a functor H : M 2 → Ab. We note the following key property of the crossed homomorphisms of Theorem 1.3:

Lemma 2.1 Let g and g ′ be two natural numbers. For all ϕ ∈ Γ g,1 and ϕ ′ ∈ Γ g ′ ,1 , we have

k(g ′ + g, ϕ♮ϕ ′ ) = k(g ′ , ϕ ′ ) + k(g, ϕ).
Proof. Let ξ and ξ ′ be framings on Σ g,1 and Σ g ′ ,1 . Then the cocycles k ξ (g, ϕ) = ϕ • ξ -ξ and

k ξ ′ (ϕ ′ ) = ϕ ′ • ξ ′ -ξ ′ represent k(g)
and k(g ′ ) respectively; see (1.2). Let N and N ′ be neighbourhood s of the parametrized intervals I + Σg,1 and I - Σ g ′ ,1 respectively. Since each of the intervals is contractible, we can choose the framings ξ and ξ ′ such that ξ ′ |I -

Σ g ′ ,1 = ξ |I + Σ g,1
and their union define a framing of N ′ ♮N . Hence we define an appropriate framing ξ ′ ♮ξ on Σ g ′ +g,1 , which thus induces a cocyle. We thus deduce from (1.2) that

k(g ′ + g, ϕ♮ϕ ′ ) = [(ϕ ′ • ξ ′ -ξ ′ )♮(ϕ • ξ -ξ)] = k(g ′ , ϕ ′ ) + k(g, ϕ).
Hence we can lift the functor H(g) to UM 2 :

Proposition 2.2 Assigning the canonical injection id Z ⊕ H([Σ g ′ ,1 , id Σ g ′ +g,1 ]) : Z ⊕ H(g) ֒→ Z ⊕ H(g ′ + g) for the morphism H([Σ g ′ ,1 , id Σ g ′ +g,1 ]) for each g, g ′ ≥ 1, defines a covariant system H : UM 2 → Ab that lifts H : M 2 → Ab
and which is polynomial of degree 1.

Proof. For two natural numbers for g, g ′ ≥ 1, we consider ϕ ∈ Γ g,1 and ϕ ′ ∈ Γ g ′ ,1 . It follows from the action of the mapping class group on H(g) of (1.5), from Lemma 2.1 and from the fact that H is a strong monoidal functor that:

H([Σ g ′ ,1 , id Σ g ′ +g,1 ]) • H(ϕ) =   id Z (0) k(g, ϕ) (0) id Σ g ′ ,1 (0) (0) (0) H(ϕ)   ; H(ϕ ′ ♮ϕ) =   id Z k(g ′ , ϕ ′ ) k(g, ϕ) (0) H(ϕ ′ ) (0) (0) (0) H(ϕ)   . Therefore H([Σ g ′ ,1 , id Σ g ′ +g,1 ])• H(ϕ) = H(ϕ ′ ♮ϕ)• H([Σ g ′ ,1 , id Σ g ′ +g,1 ]
). That the functor H : M 2 → Ab lifts to a functor with UM 2 as source category thus follows from the general criterion [Sou22,

In particular, the stable twisted cohomology algebra

H * (Γ ∞,1 ; F ∨ ∞ ) is a free Sym Q (E)-module.
Remark 2.8 The result of Theorem 2.7 does not depend on any polynomiality condition, and more generally on whether there is homological stability or not: the formula holds in general for the colimit of the homology groups which always exists.

Twisted cohomology classes with coefficients in H Q . We now review the stable twisted cohomology group with coefficients in the homology group of the surface. For the reference [KM96], we will rather quote the preprint version [START_REF] Kawazumi | The primary approximation to the cohomology of the moduli space of curves and cocycles for the Mumford-Morita-Miller classes[END_REF] as it contains more content and details.

We recall the short exact sequence (1.1) and its Euler class e ∈ H 2 (Γ 1 g ; Z). Based on the Harer stability [START_REF] Harer | Stability of the homology of the mapping class groups of orientable surfaces[END_REF], Harer [Har91, Theorem 7.1(b)] proves that in the stable range

H k-1 (Γ 1 g ; H Q ) ∼ = i≥1 H k-2i (Γ 1 g ; Q) and H k-1 (Γ g,1 ; H Q ) ∼ = i≥1 H k-2i (Γ g,1 ; Q).
The first isomorphism says that there is an element in H 2l-1 (Γ 1 g ; H Q ) for each l ≥ 1, such that these elements form a free basis of the free

H * (Γ 1 g ; Q)-module H * (Γ 1 g ; H Q ). The element is written by m l,1 ∈ H 2l-1 (Γ 1 g ; H Q ), l ≥ 1, in the terminology of the first author [Kaw98; KM96]. So we have H * (Γ 1 g ; H Q ) ∼ = l≥1 H * (Γ 1 g ; Q)m l,1
in the stable range. Similarly the pullback of m l,1 , l ≥ 1, on Γ g,1 gives an isomorphism in the stable range:

H * (Γ g,1 ; H Q ) ∼ = l≥1 H * (Γ g,1 ; Q)m l,1 (2.1)
For further details, see also [Kaw08, Theorem 1.B].

In the original definition [START_REF] Kawazumi | A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces[END_REF] the cohomology class m l,1 was constructed on the group Γ g,1 .

Later the following definition of m l,1 on Γ 1 g was introduced in [START_REF] Kawazumi | The primary approximation to the cohomology of the moduli space of curves and cocycles for the stable characteristic classes[END_REF]. Let π ′ : Γ 1 g ։ Γ g be the forgetful map of the puncture. Let Γ 1 g be the pullback Γ 1 g × Γg Γ 1 g . More precisely, there is a defining fiber square

Γ 1 g / / Γ 1 g π ′ Γ 1 g π ′ / / σ T T Γ g ,
where the section σ : Γ 1 g → Γ 1 g is given by σ(φ) = (φ, φ). We deduce that there is an isomorphism Γ

1 g ∼ = π 1 (Σ g ) ⋊ Γ 1
g defined by (φ, ψ) → (ψφ -1 , φ). Under this isomorphism, σ is given by σ(φ) = (1, φ). Similarly to [START_REF] Morita | Families of Jacobian manifolds and characteristic classes of surface bundles[END_REF]§7], this semi-direct product decomposition gives rise to a cocycle ω ∈ Z 1 (Γ 1 g ; H(g)) given by ω((x, φ)) = [x] for all x ∈ π 1 (Σ g ) and φ ∈ Γ 1 g . By an abuse of notation, we also use ω to denote the associated element of H 1 (Γ 1 g ; H(g)). We denote by ē ∈ H 2 (Γ 1 g ; Z) the pullback of the Euler class e ∈ H 2 (Γ 1 g ; Z) by the second projection Γ

1 g → Γ 1 g , (φ, ψ) → ψ. The the class m l,1 is defined by m l,1 := π ! (ē i • ω) ∈ H 2l-1 (Γ 1 g ; H). (2.2)
Contraction formula. Finally, we recall a classical operation on the twisted Mumford-Morita-Miller class induced by the contraction of the twisted coefficients. Let µ : H(g) ⊗ H(g) → Z be the intersection pairing associated to Poincaré-Lefschetz duality. Let M and M ′ be two Γ g,1 -modules. The contraction map is the abelian group morphism id

M ⊗ µ ⊗ id M ′ : M ⊗ H(g) ⊗ H(g) ⊗ M ′ → M ⊗ M ′ .
The induced map for the cohomology groups is generically denoted by (id

M ⊗ µ ⊗ id M ′ ) * .
Proposition 2.9 ([KM01, Theorem 6.2]) For all l, l ′ ≥ 1, we have

µ * (m l,1 m l ′ ,1 ) = -e l+l ′ -1 ∈ H * (Γ g,1 ; Q).
(2.3) Sketch of proof. We consider the short exact sequence

1 -→ π 1 (Σ g,1 , x) ι -→ Γ 1 g,1 π -→ Γ g,1 -→ 1 (2.4)
known as the forgetful exact sequence or the Birman short exact sequence; we refer the reader to [FM12, §4.2.1] for more details. As is proved in [KM01, Theorem 5.3], the Lyndon-Hochschild-Serre spectral sequence for the group extension (2.4) induces a canonical decomposition

H * (Γ 1 g ; M ) ∼ = H * (Γ 1 g ; M ) ⊕ H * -1 (Γ 1 g ; H(g) ⊗ M ) ⊕ H * -2 (Γ 1 g ; M )
for any Γ 1 g -module M . It is multiplicative and described explicitly by using the cohomology classes ω and e. The following formula is then deduced from a direct computation based on the decomposition: for m ∈ H * (Γ 1 g ; M ) and m ′ ∈ H * (Γ 1 g ; M ′ ), we have

(id M ⊗µ⊗id M ′ ) * (π ! (m⊗ē•ω)•π ! (ē•ω⊗m ′ )) = -π ! (m⊗m ′ )+s * (m)π ! (m ′ )+π ! (m)s * (m ′ )-eπ ! (m)π ! (m ′ ).
(2.5) Here s : Γ 1 g → Γ 1 g , φ → (φ, φ), is the diagonal map. In particular, we have µ * (m l,1 m l ′ ,1 ) = -e l+l ′ -1 + e l e l ′ -1 + e l ′ e l-1 -ee l-1 e l ′ -1 . Since the Euler class e vanishes on Γ g,1 , we deduce formula (2.3) Remark 2.10 Analogous formulas to (2.5) are computed in [KR19, Proposition 3.10] for mapping class groups of surfaces and higher even-dimensional manifolds. With the conventions of [KR19, §3], there is no -1 sign.

Exotic situations

On the basis of current knowledge, contrary to the covariant cases, there is no general framework to deal with cohomological stability with twisted coefficients by covariant systems. However, the functor H : UM 2 → Ab defined by the groups {H 1 (U T Σ g,1 ; Z), g ∈ N} define satisfies that property. Indeed, we consider the short exact sequences 0 → Q → H(g) → H(g) → 0 and 0 → Q → H(g + 1) → H(g + 1) → 0 obtained from (1.3). Then, the two induced long exact sequences in cohomology are connected by the images of the images by H, H and the constant functor at Q of the canonical arrow [Σ 1,1 , id Σg+1,1 ], which thus define a ladder of commutative diagrams. We deduce the homological stability with twisted coefficients in H from Example 2.5 with a straightforward induction on the cohomological degree and the use of the five lemma.

Twisted stable cohomology computations

In this section, we compute the stable twisted cohomology given by the first cohomology group of its unit tangent bundle H∨ (g) (see §3.1) and by the first homology group of its unit tangent bundle H(g) (see §3.2); see Theorems 3.3 and 3.9. In both cases, the work relies on the determination of the connecting morphisms of the long exact sequences associated to the short exact sequences(1.4) and (1.3) respectively. We fix the following conventions for notations for all the remainder of the paper.

Convention 3.1 Considering a functor M : UM 2 → Ab, we denote the stable twisted cohomology groups H * (Γ ∞,1 ; M ) by H * (M ). From now on, we implicitly assume that g ≥ 2i + 5 each time we consider a cohomological degree i for H i (Γ g,1 ; M (g)) where M (g) = H(g) or H∨ (g), for the homological stability bound of Theorem 2.4 to be reached. Also, we denote by H odd (M ) and H even (M ) the N-graded submodules of H * (M ) defined by {H 2i+1 (M ), i ∈ N} and {H 2i (M ), i ∈ N} respectively.

First cohomology group system

We start by studying the stable cohomology groups of the mapping class groups Γ g,1 with twisted coefficient given by H∨ (g). Let δ i : H i (Γ g,1 ; Z) → H i+1 (Γ g,1 ; H(g)) be the i th connecting homomorphism of the cohomology long exact sequence associated with the extension (1.4).

Lemma 3.2 The connecting homomorphism of the cohomology long exact sequence associated

(1.4) is equal to m 1,1 • -.
Proof. Let [z] be a cohomology class of H i (Γ g,1 ; Z). We use the normalized cochain complex and generically denote by ∂ its differentials. We recall from §1.2 that ∈ H(g) denotes the homology class of the fiber of the unit tangent bundle, and from Definition 1.1 that ξ denotes the homotopy class of a framing of the tangent bundle U T Σ g,1 . We deduce from the action of Γ g,1 on H∨ (g) (see dual of the matrix (1.5)) that the map s = s ξ : Z ֒→ H∨ (g) defined by u → u + k ξ (g, -) ∨ defines a splitting (as an abelian group morphism) to the surjection H∨ (g) ։ Z, v → v( ). Since z is a cocyle, we have ∂z = 0 and therefore that

-sϕ 0 (z([ϕ 1 | • • • | ϕ i ])) = i j=1 (-1) j s(z([ϕ 0 | • • • | ϕ j ϕ j+1 | • • • | ϕ i ]))+(-1) i+1 s(z([ϕ 0 | • • • | ϕ i-1 ])). (3.1) Since δ i [z] = [∂(s(z))
] by the formal definition of δ i , it follows from (3.1) and from (2.2) on the extension classes that

δ i ([z])([ϕ 0 | • • • | ϕ i ]) = (ϕ 0 sϕ -1 0 -s)ϕ 0 (z([ϕ 1 | • • • | ϕ i ])) = k ξ (g, ϕ 0 )ϕ 0 (z([ϕ 1 | • • • | ϕ i ])) = (m 1,1 • z)([ϕ 0 | • • • | ϕ i ]),
which ends the proof.

We consider the long exact sequence in cohomology applied to the rational version of (1.4) to make further computations. We thus obtain the exact sequence for all natural numbers i ≥ 0:

H 2i ( H∨ Q (g)) / / H 2i (Q) m1,1•- / / H 2i+1 (H Q (g)) / / / / H 2i+1 ( H∨ Q (g)), (3.2)
since H odd (Q) = 0 by (0.1) and H even (H Q (g)) = 0 by (2.1). Since m 1,1 • e α = e α m 1,1 for all α ≥ 1, we deduce that:

Theorem 3.3 The map m 1,1 • -is injective. Therefore, the Sym Q (E)-module H even ( H∨ Q ) is null and H odd ( H∨ Q ) is isomorphic to Sym Q (E){m a,1 , a ≥ 2}.
Remark 3.4 (Interpretation in terms of functor homology.) The modules H∨ Q (g) define a contravariant twisted coefficient system; see Proposition 2.2. It follows from Lemma 2.6 and Theorem 2.7 that H * ( H∨

Q (g)) ∼ = Sym Q (E) ⊗ Q H * (UM 2 ; HQ (g)).
In particular, this explains why the stable cohomology is a free Sym Q (E)-module. Then the long exact sequence for the homology of categories (see [START_REF] Franjou | Stable K-theory is bifunctor homology (after A. Scorichenko). Rational representations, the Steenrod algebra and functor homology[END_REF]§2] for instance) associated with the short exact sequence (1.4) directly gives that H i (UM 2 ; HQ (g)) ∼ = H i (UM 2 ; H Q (g)) if i ≥ 2. However, we need the above reasoning to compute that H 0 (UM 2 ; HQ (g)) = H 1 (UM 2 ; HQ (g)) = 0.

First homology group system

Constrasting with the situation of §3.1, we now study the stable cohomology groups of the mapping class groups Γ g,1 with twisted coefficient in the first cohomology group of the unit tangent bundle of the surface.

Determination of the connecting homomorphism

Let δ i : H i (Γ g,1 ; H(g)) → H i+1 (Γ g,1 ; Z) be the i th connecting homomorphism of the cohomology long exact sequence associated with the extension (1.3).

Lemma 3.5

The morphism δ i is equal to µ(m 1,1 , -).

Proof. Let [z] be a cohomology class of H i (Γ g,1 ; H(g)). We use the normalized cochain complex and generically denote by ∂ its differentials. We deduce from the action of Γ g,1 on H(g) (see the matrix (1.5)) that the homotopy class of a framing ξ of the tangent bundle U T Σ g,1 induces a map s ′ = s ′ ξ : H(g) ֒→ H(g) defined by h → h + k ξ (g, -), which is a splitting (as an abelian group morphism) to the surjection ̟ * : H(g) ։ H(g). Since z is a cocyle, we have ∂z = 0 and therefore

-s ′ ϕ 0 (z([ϕ 1 | • • • | ϕ i ])) = i j=1 (-1) j s ′ (z([ϕ 0 | • • • | ϕ j ϕ j+1 | • • • | ϕ i ]))+(-1) i+1 s ′ (z([ϕ 0 | • • • | ϕ i-1 ])).
(3.3) Since δ i [z] = [∂(s ′ (z))] by the formal definition of δ i , it follows from (3.3) that

δ i ([z])([ϕ 0 | • • • | ϕ i ]) = (ϕ 0 s ′ -s ′ ϕ 0 )(z([ϕ 1 | • • • | ϕ i ])).
On the other hand, we recall that the extension class of the short exact sequences (1.3) is Earle's cohomology class m 1,1 = k(g) = [k ξ (g, -)]; see (2.2). We thus compute from the definition of the contraction morphism that

µ(m 1,1 , z)([ϕ 0 | • • • | ϕ i ]) = µ(k ξ (g, ϕ 0 )ϕ 0 ⊗ z([ϕ 1 | • • • | ϕ i ])) = (ϕ s ′ -s ′ ϕ 0 )(z([ϕ 1 | • • • | ϕ i ]))
which ends the proof.

Computation of the stable twisted cohomology

To make further computations, we must restrict to considering the rational homology representations. We recall from (0.1) that H odd (Q) = 0, and from (2.1) that thus H even (H Q ) = 0. Then we deduce from the long exact sequence in cohomology applied to (1.3) that H 0 (Q) ∼ = H 0 ( HQ ) ∼ = Qθ and obtain the exact sequence for all natural numbers i ≥ 0: is stable under the action of the algebra Sym Q (E).

H 2i+1 ( HQ ) / / H 2i+1 (H Q ) µ(m1
Computations of Tor-groups. In order to give some qualitative properties of the stable twisted cohomology groups that we study, we may consider these twisted cohomology groups as forming an algebra H * ( HQ ) over the stable cohomology algebra Sym Q (E) as follows.

We may thus compute the homology of that algebra:

  [α] • [C] denotes the (algebraic) intersection number of the homology classes [α] and [C]. We deduce that ξ • dT C -ξ = •(rot ξ (C))[C] and in particular that

  The first author was supported in part by the grants JSPS KAKENHI 15H03617, 18K03283, 18KK0071, 19H01784, 20H00115 and 22H01120. The second author was partially supported by a Rankin-Sneddon Research Fellowship of the University of Glasgow, by IBS-R003-D1 and by the ANR Project AlMaRe ANR-19-CE40-0001-01.

	Contents
	§1. Representations and cohomological structures	4
	§1.1 Earle class	5
	§1.2 The unit tangent bundle homology representations	6
	§2. Homological stability framework and tools	6
	§2.1 Twisted coefficient systems	6
	§2.2 Twisted homological stability and stable (co)homology	8
	§3. Twisted stable cohomology computations	10
	§3.1 First cohomology group system	10
	§3.2 First homology group system	11
	References	14

  Moreover, using the injection H odd ( HQ ) ֒→ H odd (H Q ) and the surjection Sym Q (E) ։ Q and noting that there is a compatibility with respect to the grading because H odd (Γ ∞,1 ; Q) = 0, we deduce that: The cohomology groups H * ( HQ ) inherits a Sym Q (E)-module structure from H * (H Q ) and Sym Q (E). Moreover, the decomposition H * ( HQ ) = H even ( HQ ) ⊕ H odd ( HQ ) (3.6)

	/ / 0.	(3.5)
	Lemma 3.7

,1,-) / / H 2i+2 (Q) / / / / H 2i+2 ( HQ ).

(3.4)

From the contraction formula (2.3), we then compute that µ(m 1,1 , m α,1 ) = -e α for all α ≥ 1, and deduce that:

Proposition 3.6 The map µ(m 1,1 , -) is surjective. Therefore, the Sym Q (E)-module H even ( HQ ) is isomorphic to the trivial Sym Q (E)-module H 0 ( HQ ) ∼ = Qθ and H odd ( HQ ) is isomorphic to the kernel of the graded morphism µ odd (m 1,1 , -)

:= µ(m 1,1 , -) : H odd (H Q ) → Sym Q (E).

Therefore, we have the following exact sequence of Sym Q (E)-modules

H odd ( HQ ) / / H odd (H Q ) µ odd (m1,1,-) / / Sym Q (E) aug / / Q

Lemma 1.2]. Finally, the morphism H([Σ 1,1 , id Σ1+g,1 ]) is by definition an injection for each g ≥ 0, and the induced functor δ( H(g)) : UM 2 → Ab is the constant covariant system at Z. Remark 2.3 By Proposition 2.2, the first cohomology groups of the unit tangent bundle of the surfaces define a contravariant system H∨ : UM op 2 → Ab. However, as far as the authors know, there does not exist any notion of polynomiality for such object.

Twisted homological stability and stable (co)homology

In this section, we review some classical results on homological stability with twisted coefficients for mapping class groups. In particular, these prove that all the twisted coefficient systems we consider in this paper satisfy the homological stability property and thus motivate the computations of §3. Also, we recall some computations and results on the stable homology of mapping class groups with twisted coefficients, which will be used for the work of §3.

Classical framework

Both homological stability and stable twisted homology are already well-studied for covariant systems, and we recollect the corresponding classical results in the following paragraphs.

Stable homology general framework. The following classical result illustrates how polynomial covariant systems turn out to be very useful for (co)homological stability problems.

Theorem 2.4 ([Iva93, Theorem 4.1], [RW17, Theorem 5.26]) Let F : UM 2 → Ab be polynomial covariant system of degree d. Then, for each g, i ≥ 0 such that g ≥ 2i + 2d + 3, the injection

Example 2.5 Since the first homology groups {H 1 (Σ g,1 ; Z), g ∈ N} define a polynomial covariant system of degree one H : UM 2 → Ab, there is an isomorphism H i (Γ g,1 ; H(g)) ∼ = H i (Γ g+1,1 ; H(g + 1)) for g ≥ 2i + 5. Furthermore, the second author proves in [START_REF] Soulié | Some computations of stable twisted homology for mapping class groups[END_REF] a general decomposition for the stable homology of the mapping class groups with twisted coefficients givrn by a covariant system. In order the rephrase this result in terms of cohomology groups, we take this opportunity to recall and prove the following version of the Universal Coefficient theorem (for which it is difficult to find a reference), in order to make the connection between the Γ g,1 -modules H(g) and H∨ (g).

Lemma 2.6

Let G be a group, R be a principal ideal domain, and M a left R [G]-module which is free as a R-module. We denote by M ∨ the dual right R [G]-module Hom R (M, R). Then there is a natural short exact sequence admitting a non-canonical splitting:

Since a submodule of a free module over a principal ideal domain is free, all the terms of the resolution

Hence, the result follows from applying the universal coefficient theorem for chain complexes over a principal ideal domain (see [Wei94, Theorem 3.6.5] for instance) on the right-hand side of the isomorphism.

Then, using Lemma 2.6, we deduce from [Sou20, Theorem C]:

Theorem 2.7 For K a field and any covariant system F : UM 2 → K-Mod, for each g, i ≥ 0 such that g ≥ 2i + 2d + 3, we have a natural isomorphism of K-modules

Theorem 3.8 For any j > 0, we have Tor

In particular, the Sym Q (E)-module H * ( HQ ) is not free. Furthermore, we have Tor

Proof. Using the general decomposition (3.6), the results follow from the respective computations of the left derived functors Tor

-modules H even ( HQ ) and H odd ( HQ ). It straightforwardly follows from the Koszul resolution (see [START_REF] Loday | Grundlehren der Mathematischen Wissenschaften[END_REF]§3.4.6] for instance) that Tor

We note that the projection map Sym Q (E) → Qθ induces isomorphisms Tor

(Q, Qθ). We thus deduce from the long exact sequence for the left derived functor Tor Sym Q (E) , associated with the short exact sequence Im(µ odd (m 1,1 , -)) ֒→ Sym Q (E) ։ Qθ, that for any j ≥ 0 Tor

We recall that

because Im(µ odd (m 1,1 , -)) equals the augmentation ideal of Sym Q (E). Therefore, the long exact sequence for the left derived functor Tor Sym Q (E) associated with the short exact sequence deduced from Proposition 3.6

(Q, Q) for all j ≥ 0, which ends the proof.

In our forthcoming paper [KS], we will compute Tor

In particular, this group is non-trivial for each degree j ≥ 0, except for the case d = 2.

Computation of the stable cohomology. Actually, we may explicitly describe the generators and relations of the module H odd ( HQ ) as follows. We consider Ω n

Sym Q (E)|Q ; see [START_REF] Loday | Grundlehren der Mathematischen Wissenschaften[END_REF]. We consider the Euler vector field

with respect to D is a Sym Q (E)-linear derivation satisfying p D (m i,1 ) = e i for all i ≥ 1. Here we identify each m i,1 with the 1-form de i . Hence we deduce from Proposition 3.6 that

Using this description, we can prove:

Theorem 3.9 The Sym Q (E)-module H odd ( HQ ) is generated by M i,j := e i m j,1 -e j m i,1 for i, j ≥ 1, with relations e i M j,k + e j M k,i + e k M i,j ∼ 0 for all i, j, k ≥ 1.

Proof. The Lie derivative L D with respect to D satisfies

Hence each of Ω n Sym Q (E)|Q for n > 0 and Ker(aug : Sym Q (E) → Q) is the direct sum of eigenspaces of L D with positive eigenvalues. Then the Cartan formula L D = dp D + p D d implies that the following sequence of Sym Q (E)-modules is exact:

Therefore the truncation of that sequence

defines a free resolution of the module H odd ( HQ ). The result thus follows from the definition of the interior product p D .

Remark 3.10 (Interpretation in terms of functor homology.) Contrary to the case of H∨ Q (g) (see Remark 3.4), Lemma 2.6 and Theorem 2.7 do not apply since the modules HQ (g) define a covariant twisted coefficient system.


Garcia Ruiz
email: rgarciar@mit.edu

T Miyagi

Cory L Binnersley

J Billowes

Mark L Bissell

Jeremy Bonnard

Thomas E Cocolios

Jacek Dobaczewski

Kieran T Flanagan

Adam R Vernon
email: vernona@mit.edu

Ronald Fernando

George Georgiev

Wouter Gins

Reinhart Heinke

Jason D Holt

Ágota Koszorús

David Leimbach

Kara M Lynch

Gerda Neyens

Steven R Stroberg

Xiaofei F Yang

Deyan T Yordanov

Gregory J Farooq-Smith

R F Garcia Ruiz

R P De Groote

J Hustings

Á Koszor Ús

S G Wilkins

Nuclear moments put a new spin on the structure of 131In

Keywords: nuclear physics, electromagnetism, 131In

Introduction

The atomic nucleus is formed by strongly-interacting nucleons (protons, Z, and neutrons, N), packed tightly into a volume around a trillion times smaller than that of atoms. These highdensity systems are bound by the strong nuclear force which, in contrast to the well understood electromagnetic force that dominates the properties of atoms and molecules, is exceedingly more complex and not yet fully understood. Hence, describing the atomic nuclei and predicting their properties at extreme values of mass and charge are major long-standing challenges for nuclear science.

Similar to electrons in an atom, the nucleons (protons and neutrons) in the atomic nucleus occupy quantum 'shells'. Thus, nuclei with a single valence particle or hole around a nuclear closed shell provide important foundations for our understanding of the atomic nucleus. Their simpler structure vastly reduces the complexity of the quantum many-body problem, providing critical guidance for the development of nuclear theory.

Recent advances in our understanding of the strong interaction and the development of many-body methods, combined with escalation in computer power, have enabled theoretical (color online) Evolution of nuclear electromagnetic properties for the 9/2 + ground states of In isotopes: a) electric quadrupole moments; and b) magnetic dipole moments. The horizontal dotted line indicates the single-particle value (Schmidt line). Experimental results are compared with theoretical calculations from ab initio (VS-IMSRG) and Density Functional Theory (DFT). Literature experimental values for In were taken from Ref. 12 . The evolution of single-particle properties of these isotopes are illustrated at the bottom of the figure. At neutron magic number N = 82, the electromagnetic properties approach the pure single-particle value (see text for more details). descriptions of increasingly complex nuclei. Isotopes around the proton closed shell Z = 50, are now the frontier of ab initio calculations 13,14 . The properties of these nuclei can be described by complementary many-body methods such as configuration-interaction methods 4 and nuclear Density Functional Theory (DFT) 15 . This has lead to an increased focus on studying this region of the nuclear chart (around Z = 50, N = 50, 82) over the last decade 2,7,[16][17][18] .

Here we present measurements of two fundamental properties of indium isotopes using precision laser spectroscopy: the (spectroscopic) magnetic dipole moment, µ, and electric quadrupole moment, Q. Measurements were performed for the neutron-rich In (Z = 49) isotopes, reaching up to 131 In, which possesses a magic number of N = 82 neutrons (see Methods section for details). With a proton hole in the closedshell of Z = 50, the low-energy structure of the odd-mass In isotopes is expected to be governed by a single-hole configuration in the proton orbit π1g 9/2 .

As we show here, the nuclear magnetic dipole moments of odd-mass In isotopes are determined by the total spin distribution of the nucleus induced by the unpaired valence proton hole. This is schematically illustrated at the bottom right of Figure 1. The nuclear electric quadrupole moment provides a complementary measurement of the nuclear charge distribution, and is highly sensitive to the collective motion of all nucleons 19 . These observables together therefore probe distinct aspects of the nucleon distribution and measuring them across a large range of neutron numbers allows a unique insight into the evolution of the interplay between single-particle and collective nuclear phenomena.

Previously, the magnetic and quadrupole moments of the ground state I π = 9/2 -of indium isotopes were known to exhibit remarkably little variation over 22 isotopes, from A = 105 to 127 (see Figure 1b)) 12 . The constant value of the magnetic moment over such long range of isotopes has been presented as an archetypal example of the independentparticle behaviour of single-particle states near a proton shell closure 6,12 . 'How do these seemingly simple patterns emerge from complex interactions among protons and neutrons?' and 'Do they prevail at extreme number of neutrons?' are two major open questions that we address in this work.

In addition to the ground state, the indium isotopes can also exist in excited nuclear configurations with relatively long lifetimes -isomers -with spin I π = 1/2 -. These isomeric states

2/12

provide additional insight and are expected to be described by a single-hole configuration based on a different proton orbital (π2p 1/2 ). However, in contrast to the I π = 9/2 + states, the µ values of these isomeric states exhibit significant variations, posing a three-decades-long puzzle in our description of these nuclei 12 .

To unravel the microscopic origin of the electromagnetic properties of these isotopes, we compare our experimental results with two complementary state-of-the-art theoretical methods: i. ab initio valence-space in-medium similarity renormalisation group (VS-IMSRG) calculations 13,20 , which start from nucleon-nucleon interactions derived from chiral effective field theory, constrained by the properties of up to only four nucleon systems 21,22 ; and ii. symmetry-breaking nuclear Density Functional Theory (DFT) 15,23 . The latter assumes nucleons moving within their own self-consistently-generated spin-dependent broken-symmetry-confining potential. DFT provides a satisfactory description of bulk nuclear properties such as radii and binding energies across the whole nuclear chart [24][25][26] . Here, we have developed its symmetry-restored version 27 to be able to provide accurate calculations of spectroscopic µ and Q moments.

Experimental and Theoretical developments

Our measurements were performed using the collinear resonance ionisation spectroscopy (CRIS) technique at the ISOLDE facility of CERN 28 (see Methods section for details). From the hyperfine structure, we extracted the magnetic dipole and electric quadrupole parameters, A h f , B h f of the probed atomic states, corresponding to the two long-lived nuclear states, 9/2 + and 1/2 -, present in each isotope. Recent improvements in the sensitivity of the technique allowed us to achieve high-resolution spectroscopy measurements, despite production of the indium isotopes at rates below 1000 atoms/s in the presence of large isobaric contamination. Prior to the study of these exotic isotopes, sensitive ionisation schemes and atomic theory calculations had to be developed to accurately extract the nuclear properties from spectroscopy measurements 29,30 .

We performed ab initio VS-IMSRG calculations (see Methods for further details) using two different sets of initial two-(NN) and three-nucleon (3N) forces derived from chiral effective field theory 31,[START_REF] Machleidt | Chiral effective field theory and nuclear forces[END_REF] ; the 1.8/2.0(EM) 21,22 and the more recent N 2 LO GO [START_REF] Jiang | Accurate bulk properties of nuclei from A = 2 to ∞ from potentials with ∆ isobars[END_REF] . The 1.8/2.0(EM) set is constrained only by fitting to properties of two-, three-, and four-nucleon systems. N 2 LO GO was recently developed to include ∆-isobar degrees of freedom and is additionally fit to reproduce saturation properties of infinite nuclear matter [START_REF] Jiang | Accurate bulk properties of nuclei from A = 2 to ∞ from potentials with ∆ isobars[END_REF] .

We performed DFT calculations using both Hartree-Fock (HF) or Hartree-fock-Bogilibov (HFB) approaches, corresponding to configurations of the nucleus constructed using single-nucleon (HF), or nucleon-hole pair excitations (HFB) as basis states, with HFB calculations introducing pairing correlations. The electromagnetic moments of semi-magic ± 1 nucleon systems, such as the indium isotopes, are well suited to study DFT time-symmetry-breaking (time-odd) contributions to the mean field [START_REF] Engel | Time-dependent Hartree-Fock theory with Skyrme's interaction[END_REF][START_REF] Perlińska | Local density approximation for protonneutron pairing correlations: Formalism[END_REF] , which vary with the time-reversal operator. These fields are predominantly generated by the two-body spin-spin interaction terms, and up until now were poorly constrained within DFT theory. However, they are of particular interest as our understanding of time-reversalviolating mean fields is critical to tackle open problems of modern physics, e.g. the search for new physics [8][9][10][11] and dark matter searches [START_REF] Engel | Nuclear Physics of Dark Matter Detection[END_REF] . Our experimental results presented an excellent opportunity to perform and test these developments. To investigate the relative importance of time-odd fields and pairing correlations, DFT calculations were performed by turning "on" and "off" of each effect. The results of both ab initio and DFT types of calculation are shown alongside the experimental Q and µ moments in Figure 1a) and 1b), respectively, for the 9/2 + states of In. The µ moments of the 1/2 -states are shown in Figure 2. All experimental data are presented in Tables 1 and2 of Methods, and compared with literature values that exist for In.

Results and discussion

In the single-particle picture, a proton hole induces an intrinsic prolate deformation of the whole nucleus, as indicated schematically in Figure 1 (bottom left). A gradual decrease in the Q values of the 9/2 + states was previously observed up to N = 78. Our measurements reveal a notably larger decrease at N = 82, indicating a significant decrease in deformation (Figure 1a). The VS-IMSRG calculations reproduce the experimental trends, i.e local variations in neutron number, a dip around N = 64, and a gradual decrease towards N = 82. However, the magnitude of the Q is underestimated. The reproduction of the magnitude of the quadrupole moments is a known challenge for ab initio nuclear theory, as the Q moments are a highly collective emergent property of the nucleus which can require the inclusion of extensive many-body correlations [START_REF] Henderson | Testing microscopically derived descriptions of nuclear collectivity: Coulomb excitation of 22 Mg[END_REF] . Traditionally, phenomenological calculations are fitted to match the experimental data by assigning empirical 'effective' charges to valence neutrons [START_REF] Yordanov | Spins, electromagnetic moments, and isomers of 107-129 Cd[END_REF] . Neither effective charges nor effective g-factors were used in our ab initio or DFT calculations. This work therefore provides an important insight into how a correct description of these nuclei and single-particle behaviour emerges intrinsically from our calculations without the use of commonly used effective factors.

Conversely, our DFT calculations are able to closely reproduce the overall magnitude of the Q moments. As N = 82 is approached, the agreement with the calculations without pairing (HF), shows that describing individual neutron orbitals becomes important. However, due to effects induced by occupying individual neutron orbitals, an inaccurate staggering with neutron number is also produced, compared to HFB. As shown in Figure 1a), time-odd contributions have a negligible effect on Q moments.

In contrast to the Q moments, the µ moments of the 9/2 + states were known to exhibit little variation 12 , which is continued up to 129 In in our observation. However, we observe an abrupt increase at 131 In, see Figure 1b), and the atomic spectra in Figure 3. The extreme single-particle magnetic moment of a proton in the πg 9/2 orbit, the so-called 'Schmidt' limit [START_REF] Schmidt | Über die magnetischen Momente der Atomkerne[END_REF] , is also indicated in Figure 1b). Deviations from the extreme single-particle value have usually been attributed to a mean effect of the other nucleons, and are typically taken into account using an empirical 'quenched' or 'effective' gfactor 12 . Our results now reveal a value much closer to the extreme single-particle limit at N = 82, indicating the µ moments for N < 82 in fact depart from the single-particle picture. This illustrates how the use of 'effective' operators can lead to contradictory conclusions and obscure critical details and changes in the underlying nuclear structure. In contrast to the quadrupole moments, we find that the inclusion of time-odd fields proves to be essential for an accurate description of magnetic moments. We observe that without the time-odd mean fields, the DFT value obtained for 131 In is close to the single-particle value. Thus, the quadrupole deformation of this nucleus cannot explain the experimental value. However, the addition of a spin-spin parameter corresponding to the isovector Landau parameter [START_REF] Bender | Gamow-Teller strength and the spin-isospin coupling constants of the Skyrme energy functional[END_REF] of g ′ 0 = 0.82 generates time-odd mean fields that result in a perfect agreement between the DFT and the experimental 131 In magnetic moment. Similarly as for the Q moments, we find that the HF-variant of the DFT better matches experiment around N = 82, reproducing the magnitude of the jump in µ well in contrast to those including pairing correlations (HFB). In order to further investigate the abrupt change that was observed at N = 82 within the DFT framework, we traced back the properties of µ in indium directly to those of their deformed and polarised cores. To this end, we performed symmetry-restoration calculations by removing the deformed odd-proton orbitals from the self-consistent DFT states in indium or by filling in the corresponding odd-proton holes. In this way, we gained access to the corresponding 0 + , 1 + , 2 + , 3 + , 4 + , ... states of the cadmium or tin cores. At this point, it is essential to recall that without the time-odd mean fields the core is not spin-polarised and therefore its states conserve signature symmetry and odd-angular-momentum components vanish. The effect of the time-odd mean fields on magnetic moments thus proceeds through the coupling to the 1 + , 3 + , . . . , 9 + states of the polarised core. The DFT calculations indicate that the change in the 9/2 + moments can be attributed to the dominance of the effect of the spin-distribution at the N = 82 shell closure and the charge distribution for N < 82. Predictions for N > 82 using these DFT calculations are shown in Methods (Figure 4), supporting the suggestion of a possible new magic number at N = 90 [START_REF] Sarkar | New shell closure for neutron-rich Sn isotopes[END_REF] .

Previously, the decreasing µ values of the 1/2 -isomeric states with increasing neutron number (see Figure 2) were not explained by nuclear theory. Our new experimental results for 129,131 In show a sudden increase, which crosses the extreme single-particle value at N = 82. The results are compared with literature values of [115][116][117][118][119][120][121][122][123][124][125][126][127] In in Table 1. We find that the DFT calculations without pairing correlations also provide a better description than those with pairing correlations (HFB). The introduction of time-odd fields improves the agreement with experiment. However, the pronounced increase at N = 82 is not reproduced within this framework, with predictions close to the single-particle limit. This suggests the 1/2 -isomers are highly collective in nature even at N = 82. Our VS-IMSRG calculations provide some insight here. Overall, these calculations provide a good description of the previously unexplained trend observed for the magnetic moments of the 1/2 -states. Analysis of the VS-IMSRG calculations indicates that the 1/2 -state contains a significant component resulting from the coupling between the 9/2 + single-particle configuration and the 5 -(anti-aligned neutrons in the h 11/2 and s 1/2 orbitals) state of the core. This therefore suggests that further refinement of the present DFT calculations would be needed to explicitly include other core configurations e.g. negative-parity states, to accurately describe the 1/2 -states.

Summary and Outlook

The indium isotopes have been considered a textbook example for the dominance of single-particle properties in heavy nuclei. Here, we show that their ground-state electromagnetic properties significantly differ at N = 82 compared to N < 82, despite the single unpaired proton (πg 9/2 proton orbit) dominating the behaviour of this complex many-body system.

Our new experimental results for 129,131 In show that upon reaching the neutron-rich magic number N = 82, abrupt changes of their electromagnetic properties are observed, and only at the neutron magic number the single-particle structure is almost fully recovered. This challenges our previous un-

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derstanding of these isotopes, which were assumed to have a single-particle description by invoking the use of empirical effective operators.

We presented two complementary nuclear models to investigate how these seemingly simple structures emerge from the complex interactions among nucleons. Both DFT and ab initio calculations provide a good description of the experimental trends. Within the DFT framework, the intricate isotopic dependence of the indium magnetic and electric ground-state moments turns out to require correct treatment of spin and shape polarization exerted by the proton-hole on the Z = 50 magic core. The inclusion of time-odd mean fields was shown to be essential to reproduce the experimental findings.

Indium nuclei can coexist in long-lived exited states with I π = 1/2 -. The structure of these states was suggested to be dominated by a single unpaired proton in the π p 1/2 orbit. However, in contrast to the simple trends observed for the ground states, the magnetic moments of the 1/2 -states exhibit a large variation with neutron number, previously considered a puzzle in the region 12 . These features and the new experimental observations are well described by our ab initio calculations. Our new measurements show that in addition to previous observations, the trend of their magnetic moments changes, with a sudden uptick at N = 82.

Although our ab initio calculations provide a good description of the experimental trends, they fall short in reproducing the magnitude of the electromagnetic moments. This may be attributed to the lack of inclusion of many-body currents, e.g. meson-exchange currents, which are known to be essential to describe the electromagnetic properties of light nuclei [START_REF] Pastore | Quantum Monte Carlo calculations of electromagnetic moments and transitions in A ≤ 9 nuclei with meson-exchange currents derived from chiral effective field theory[END_REF] . Future theoretical developments would be needed to include many-body currents and clarify the role that they play in heavy nuclei. On the other hand, a theoretical description of the nuclear quadrupole moments requires the extensive inclusion of neglected many-body correlations. Thus, our new experimental results at N = 82 provide critical data for future developments of both ab initio and DFT theory.

Our experimental results for 131 In allowed an investigation of the strength of the time-odd mean fields, which until now have been poorly constrained in DFT theories. Such time-odd channels are essential for a correct description of numerous nuclear properties, such as double beta-decay rates 8,9 , permanent electric dipole moment measurements 10,11 , and dark matter searches [START_REF] Engel | Nuclear Physics of Dark Matter Detection[END_REF] . This provides strong motivation to extend experiments to other isotopes possessing single-hole (particle) with respect to suggested nuclear closed shells at extreme proton-to-neutron ratios, and to further pursue theoretical studies of these systems.

Methods

The collinear resonance ionisation setup The indium isotopes were produced at CERN's on-line isotope separator facility, ISOLDE, by impinging 1.4-GeV protons onto the proton-to-neutron converter of a thick uranium-carbide target 8 . The converter suppressed nearby caesium mass contamination 9 . The indium isotopes diffused through the target material and their ionisation was significantly enhanced by the use of the resonance laser ion source, RILIS 10 . The produced indium ions were then accelerated to 40 keV and mass separated using the ISOLDE high-resolution mass separator before being cooled and bunched using a gas-filled linear Paul trap (ISCOOL) 11,12 . After a trapping time of up to 10 ms, ion bunches with a temporal width of 2 µs, were then re-accelerated to 40034 (1) eV and deflected into the CRIS beamline 13,14 . The indium ions were then neutralised with a sodium-filled charge-exchange cell with an efficiency of up to 60% and predicted relative atomic populations of 57% and 37% respectively for the 5p 2 P 3/2 metastable state and 5p 2 P 1/2 ground state 15 . The remaining ion fraction was removed by electrostatic deflectors, and the neutralized atom bunch was collinearly overlapped with two pulsed lasers, one for excitation and another for non-resonant ionisation. The atoms were resonantly excited using two different UV transitions in separate measurements. The first using 246.8-nm laser light for the 5p 2 P 3/2 → 9s 2 S 1/2 atomic transition. The second using 246.0-nm laser light for the 5p 2 P 1/2 → 8s 2 S 1/2 atomic transition, which is more sensitive to nuclear magnetic moments, µ, but alone does not give the nuclear electric quadrupole moment, Q. The resonant laser light was produced by frequency tripling the light from an injection-locked Ti:Sapphire laser system 16 . This laser was seeded using a narrowband SolsTiS continuous-wave Ti:Sapphire laser, and pumped using a Lee Laser LDP-100MQ Nd:YAG laser, producing pulsed narrowband 740(738)-nm laser light at 1 kHz repetition rate. This light was then frequency tripled to 246.8(246.0)-nm light by the use of two non-linear crystals 17 . About 3 µJ of laser light was used to saturate both transitions. The excited atoms were then ionized by a final nonresonant 1064-nm step, provided by a Litron LPY 601 50-100 PIV Nd:YAG laser at 100 Hz repetition rate. The frequency of the resonant first step was scanned and the resulting ions were deflected onto a detector, producing the hyperfine structure containing spectra from which the hyperfine parameters were obtained. The wavelengths were measured using a HighFinesse WSU-2 wavemeter, which was 7/12 drift stabilized by simultaneous measurement of a Toptica DLC DL PRO 780 diode laser locked to the 5s 2 S 1/2 → 5p 2 P 3/2 F = 2 → 3 transition of 87 Rb using a saturated absorption spectroscopy unit.

Evaluation of nuclear magnetic and quadrupole moments

The µ values were determined using a reference NMR value of µ re f = +5.5408(2) µ N for 115 In 4 , and the relation

µ = µ re f IA I re f , A re f (1 + ∆), (1)
where the differential hyperfine anomaly, ∆, is negligible for these atomic states of indium 18 . Here A re f are our experimentally determined values for the 5p 2 P 3/2 , 9s 2 S 1/2 , 5p 2 P 1/2 and 8s 2 S 1/2 states of stable 115 In, and A are those of short-lived isotopes. The final µ values presented in Table 1 are a weighted average of the µ values from each atomic state, which were self-consistent within 2σ . The spectroscopic nuclear electric quadrupole moments, Q, were extracted using the relation

B h f = e QV ZZ , (2)
where a value of B h f ( 2 P 3/2 )/Q = 576(4) MHz/b was used, obtained from relativistic coupled-cluster atomic calculations 7 . Here V ZZ is the electric field gradient produced by the electrons at the nucleus. The ratio of A h f factors is depends dominantly on atomic structure and therefore remains constant when with the correct nuclear spin assignment is used for fitting of spectra resulting from hyperfine structure 19 . Although reduced χ 2 fitting of the spectra (as shown in Figure 3) resulted in minima when we use I = 9/2; the spin assignment from ratio of upper and lower atomic state A h f factors (of the 5p 2 P 3/2 → 9s 2 S 1/2 transition) could not rule out nuclear spins 7/2, 11/2, outside of 1σ uncertainty. The spin of the I = 9/2 states are tentatively assigned by experiment and strongly supported by nuclear theory. The spin assignments of the I = 1/2 states were confirmed unambiguously from the number of peaks in their spectra.

VS-IMSRG calculations

The VS-IMSRG calculations start from the underlying NN and 3N interactions expressed in the harmonic-oscillator basis. We then construct an effective Hamiltonian designed for a particular valence space where the exact diagonalization is feasible 20 . The effective Hamiltonian was decoupled from the full A-body Hamiltonian through the application of an approximate unitary transformation derived from the Magnus expansion method 20 . Using the same transformation, the effective M1 and E2 valence-space operators were then decoupled consistently with the Hamiltonian. During the calculation, all of operators arising in nested commutator expansions were truncated at the two-body level, the IMSRG(2) approximation.

We begin in a spherical harmonic-oscillator basis with frequency of èω = 16 MeV and e = 2n + l ≤ e max = 14 for 1.8/2.0 (EM) (and 12 for N 2 LO GO ) with an additional cut of e 1 + e 2 + e 3 ≤ E 3max = 16 imposed due to storage limitations of 3N matrix elements. The valence space was taken as the proton {1p 1/2 , 1p 3/2 , 0 f 5/2 , 0g 9/2 } and neutron {2s 1/2 , 1d 3/2 , 1d 5/2 , 0g 7/2 , 0h 11/2 } single-particle orbits above 78 Ni core. The final valence-space diagonalization and evaluation of electromagnetic moments were performed with the KSHELL code 21 . The effect of fully accounting for translation invariance in the M1 operator was found to be less than 1%, largely due to the lack of radial dependence of the M1 operator.

DFT calculations

We performed the DFT calculations using code HFODD 22,23 version (3.01m) and a standard Skyrme density functional UNEDF1 24 . The deformed DFT single-particle wave functions were expanded on the spherical harmonic-oscillator basis up to N 0 = 16 quanta. An A-dependent harmonic-oscillator frequency èω was fixed according to the prescription defined in Ref. 25 . For the state 9/2 + , the proton configuration (with no pairing correlations) was fixed by occupying 49 lowest axially deformed prolate orbitals, that is, the Nilsson orbital 26 [404] 9 2 with the angular-momentum projection on the symmetry axis Ω = -9/2 was left empty. Similarly, for the state 1/2 -, the hole was kept in the oblate Nilsson orbital [301] 1 2 with Ω = -1/2. It was essential to pick the proton-hole configurations with angular 9/12 momenta properly aligned along the axial-symmetry axis. Indeed, as discussed in Refs. 27,28 , the time-odd mean fields and core spin polarization depend on the relative orientation of the intrinsic angular momentum and shape.

For both proton configurations, the open-shell neutron configurations were constructed either with (HFB results) or without (HF results) pairing correlations included. For all deformed mean-field configurations, the angular-momentum symmetry was restored 27 , and in addition, for paired configurations the particle-number symmetry was restored as well 27 . Neither effective charges nor effective g-factors were used in the ab initio or DFT calculations.

Figure 4 extends Figure 1, showing the DFT predictions of the magnetic dipole and electric quadrupole moments in indium, in an as-yet experimentally inaccessible region beyond the N = 82 shell closure. The confirmation of the moments at N = 90 would give strong evidence to support the existence of a new magic number of N = 90 in this neutron-rich tin region, for which theoretical and experimental evidence is beginning to hint towards [29][30][31] .

Figure 1 .

 1 Figure 1. (color online) Evolution of nuclear electromagnetic properties for the 9/2 + ground states of 105-131 In isotopes: a) electric quadrupole moments; and b) magnetic dipole moments. The horizontal dotted line indicates the single-particle value (Schmidt line). Experimental results are compared with theoretical calculations from ab initio (VS-IMSRG) and Density Functional Theory (DFT). Literature experimental values for In were taken from Ref.12 . The evolution of single-particle properties of these isotopes are illustrated at the bottom of the figure. At neutron magic number N = 82, the electromagnetic properties approach the pure single-particle value (see text for more details).

Figure 2 .

 2 Figure 2. Nuclear magnetic moments for the 1/2 -isomeric states of 113-131 In isotopes. Results are compared with ab initio and Density Functional Theory calculations. The horizontal dotted line indicates the single-particle value (Schmidt line). See text for more details

Figure 3 .

 3 Figure 3. (color online) a) Example hyperfine spectra of the 129 In and 131 In isotopes, measured using the 246.8-nm (5p 2 P 3/2 → 9s 2 S 1/2 ) transition. The 9/2 + ground and 1/2 -isomer states are indicated. b) Example spectra of the 1/2 -isomer structure measured with the 246.0-nm (5p 2 P 1/2 → 8s 2 S 1/2 ) transition.

Figure 4 .

 4 Figure 4. (color online) The a) nuclear magnetic dipole moments and b) electric quadrupole moments, respectively, of the 9/2 + states extracted by our experiment. Shown alongside our ab initio and DFT calculations, including predictions for N>82

Table 1 .

 1 The magnetic hyperfine structure parameters, A hf , measured in this work for the odd-mass[113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128][129][130][131] In isotopes and corresponding extracted magnetic dipole moment values.

	A 105 107 109 111 113 113 115 115 117 117 119 119 121 121 123	I 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 1/2 -9/2 + 1/2 -9/2 + 1/2 -9/2 + 1/2 -9/2 + 1/2 -9/2 +	A h f (MHz) 9s 2 S 1/2 5p 2 P 1/2 +241.8(8) +130(1) 5p 2 P 3/2 +2276.0(8) +242.7(8) +5.5264(19) +5.5289(2) 2 8s 2 S 1/2 µ ‡ (µ N ) µ Lit. (µ N ) +5.675(5) 1 +5.585(8) 1 +5.538(4) 1 +5.503(7) 1 -87(10) -38(20) -774(50) -90(50) -0.21(1) -0.21074(2) 3 +239.9(5) +130.3(8) +2281.9(8) +243.3(6) +5.5408(2) ‡ +5.5408(2) 4 -96(3) -48(9) -903(20) -66(10) -0.2405(38) -0.24398(5) 5 +241(4) +130(5) +2277(2) +244(2) +5.5286(43) +5.519(4) 1 -106(4) -49(10) -1028(10) -113(8) -0.276(27) -0.25174(3) 1 +240(4) +130(5) +5.499(62) +5.515(1) 1 -132(7) -70(10) -0.342(12) -0.319(5) 1 +243(4) 133(5) +5.575(62) +5.502(5) 1 -140(2) -85(6) -0.3600(41) -0.355(4) 1 +238(4) +129(5) +5.442(61) +5.491(7) 1
	123 1/2 (-) 125 9/2 +	-160(2) +240.3(6) +129.9(8) -80(5)			-0.4047(54) -0.400(4) 1 +5.496(24) +5.502(9) 1
	125 1/2 (-) 127 9/2 +	-176(7) +241.8(7) +130(1) -90(10)	-0.450(17) +2278.3(6) +243.8(4) +5.5321(14) +5.522(8) 1 -0.433(4) 1
	127 1/2 (-)	-171(3)	-91(10)	-1613(9)	-174(8)	-0.4355(24)
	129 (9/2 + ) +243.3(8) +132(1)	+2304.9(9) +244.8(7) +5.5961(23)
	129 1/2 (-)	-156(3)	-80(4)	-1434(2)	-162(10) -0.3871(6)
	131 (9/2 + ) +275.9(6) +149.3(7)			+6.312(14)
	131 1/2 (-)	-20(7)	-11(4)	-188(20)	-20(2)	-0.051(3)
			‡ These µ values were determined using a reference NMR value of µ=+5.5408(2) µ N	4 .
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Table 2 .

 2 The B hf hyperfine structure parameter (from the 2 P 3/2 state) values, measured in this work for the odd-mass[113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128][129][130][131] In isotopes and the extracted electric quadrupole moment values. + ) +177.3(57) +0.310(10) † Extracted using a value of B h f ( 2 P 3/2 )/Q = +576(4) MHz/b 7 .

	A 105 107 109 111 113 115 117 119 121 123 125 127	I 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 + 9/2 +	B h f (MHz) +441(15) +454.2(65) +0.789(13) +0.770(8) 1, 6 Q † S (b) Q Lit. S (b) +0.79(5) 1, 6 +0.77(5) 1, 6 +0.80(3) 1, 6 +0.76(2) 1, 6 +0.767(27) +0.759(8) 1, 6 +465(13) +0.807(23) +0.788(10) 1, 6 +462(13) +0.802(23) +0.812(7) 1, 6 +462(13) +0.803(23) +0.774(10) 1, 6 +424(13) +0.736(23) +0.720(9) 1, 6 +382.3(55) +0.664(11) +0.68(3) 1, 6 +338(16) +0.588(29) +0.56(3) 1, 6
	129 (9/2 + ) +280.4(73) +0.487(13)
	131 (9/2

Acknowledgements

This work was supported by ERC Consolidator Grant No.648381 (FNPMLS); STFC grants ST/L005794/1, ST/L005786/1, ST/P004423/1, ST/M006433/1, ST/P003885/1 and Ernest Rutherford Grant No. ST/L002868/1;the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under grant DE-SC0021176; GOA 15/010 from KU Leuven, BriX Research 11/12 Program No. P7/12; the FWO-Vlaanderen (Belgium); the European Unions Grant Agreement 654002 (ENSAR2); National Key R&D Program of China (Contract No: 2018YFA0404403); the National Natural Science Foundation of China (No:11875073); the Polish National Science Centre under Contract No. 2018/31/B/ST2/02220. TRIUMF receives funding via a contribution through the National Research Council of Canada. The theoretical work was further supported by NSERC and the US Department of Energy (DOE) under contract DE-FG02-97ER41014. The VS-IMSRG Computations were performed with an allocation of computing resources on Cedar at WestGrid and Compute Canada, and on the Oak Cluster at TRIUMF managed by the University of British Columbia department of Advanced Research Computing (ARC ). We would also like to thank the ISOLDE technical group for their support and assistance and the University of Jyväskylä for the use of the injection-locked cavity. We acknowledge the CSC-IT Center for Science Ltd., Finland, for the allocation of computational resources. This project was partly undertaken on the Viking Cluster, which is a high performance compute facility provided by the University of York. We are grateful for computational support from the University of York High Performance Computing service, Viking and the Research Computing team.

Author contributions statement

A.R.V. prepared the manuscript with input from all authors, especially R.F.G.R., J.Bo., J.D., J. All authors reviewed the manuscript.

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