Partial Torelli groups and homological stability

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.


Introduction
Let Σ b g be an oriented genus g surface with b boundary components. The mapping class group Mod(Σ b g ) is the group of isotopy classes of orientation-preserving homeomorphisms of Σ b g that fix ∂Σ b g pointwise. Harer [8] proved that Mod(Σ b g ) satisfies homological stability. More precisely, an orientation-preserving embedding Σ b g → Σ b g induces a map Mod(Σ b g ) → Mod(Σ b g ) that extends mapping classes by the identity, and Harer's theorem says that the induced map H k (Mod(Σ b g )) → H k (Mod(Σ b g )) is an isomorphism if g k.
Torelli. The group Mod(Σ b g ) acts on H 1 (Σ b g ). For b ≤ 1, the algebraic intersection pairing on H 1 (Σ b g ) is a Mod(Σ b g )-invariant symplectic form. We thus get a map Mod(Σ b g ) → Sp 2g (Z) whose kernel I(Σ b g ) is the Torelli group. The group I(Σ b g ) is not homologically stable; indeed, Johnson [11] showed that H 1 (I(Σ b g )) does not stabilize. Church-Farb's work on representation stability [3] connects this to the Sp 2g (Z)-action on H k (I(Σ b g )) induced by the conjugation action of Mod(Σ b g ). Much recent work on H k (I(Σ b g )) focuses on this action; see [2,12,14].
Partial Torelli. We show that homological stability can be restored by enlarging the Torelli group to the group acting trivially on some fixed portion of homology. As an illustration of our results, we begin by describing a very special case of them. Fix a symplectic basis {a 1 , b 1 , . . . , a g , b g } for H 1 (Σ 1 g ) in the usual way: ... For 0 ≤ h ≤ g, define I h (Σ g,1 ) to be the subgroup of Mod(Σ g,1 ) fixing all elements of {a 1 , b 1 , . . . , a h , b h }. These group interpolate between Mod(Σ 1 g ) and I(Σ 1 g ) in the sense that Mod(Σ 1 g ) = I 0 (Σ 1 g ) ⊂ I 1 (Σ 1 g ) ⊂ I 2 (Σ 1 g ) ⊂ · · · ⊂ I g (Σ 1 g ) = I(Σ 1 g ).
Homology markings. To state our more general result, we need the notion of a homology marking. A homology marking on Σ 1 g is a finitely generated abelian group A together with a homomorphism µ : H 1 (Σ 1 g ) → A. Associated to this is a partial Torelli group Example 1.1. If A = H 1 (Σ 1 g ) and µ = id, then I(Σ 1 g , µ) = I(Σ 1 g ). Example 1.2. If A = H 1 (Σ 1 g ; Z/ ) and µ : H 1 (Σ 1 g ) → A is the projection, then I(Σ 1 g , µ) is the level-subgroup of Mod(Σ 1 g ), i.e. the kernel of the action of Mod(Σ 1 g ) on H 1 (Σ 1 g ; Z/ ). Example 1.3. Let A be a symplectic subspace of H 1 (Σ 1 g ), i.e. a subspace with H 1 (Σ 1 g ) = A⊕A ⊥ , where ⊥ is defined via the intersection form. If µ : H 1 (Σ 1 g ) → A is the projection, then If A has genus h, then I(Σ 1 g , µ) ∼ = I h (Σ 1 g ).
Theorem A. Let µ : H 1 (Σ 1 g ) → A be a homology marking on Σ 1 g and let µ : H 1 (Σ 1 g+1 ) → A be its stabilization. The map H k (I(Σ 1 g , µ)) → H k (I(Σ 1 g+1 , µ )) induced by the stabilization map I(Σ 1 g , µ) → I(Σ 1 g+1 , µ ) is an isomorphism if g ≥ (2 rk(A) + 2)k + (4 rk(A) + 2). Remark 1.4. As is typical in homological stability theorems, the stabilization map is a surjection in some range before it becomes an isomorphism. In many other cases, proving this allows improvements to the range where it is an isomorphism, but in our proof it would not lead to an improvement. To avoid technicalities, we will thus not discuss this.
Multiple boundary components. We also have a theorem for surfaces with multiple boundary components. Setting this up in complete generality requires developing a category of "homology-marked surfaces" (inspired by the author's work on the Torelli group on surfaces with multiple boundary components in [15]), so we postpone it until later; see Theorem 5.2.
Remark 1.5. It would simplify our proof if we only considered Theorem A; however, the more general Theorem 5.2 is needed in the author's forthcoming work on the cohomology of the moduli space of curves with level structures [16].
Again, this reduces to our previous definition if Λ is abelian.
Nonabelian stability. Fix a basepoint * ∈ ∂Σ 1 g+1 . Define the stabilization of a Λ-marking µ : π 1 (Σ 1 g , * ) → Λ to be the following Λ-marking µ : π 1 (Σ 1 g+1 , * ) → Λ. Since the loop around ∂Σ 1 g is in ker(µ), the map µ factors as where Σ 1 g is the closed genus g surface obtained by collapsing ∂Σ 1 g to a point and * is the image of * under this collapse map. Embed Σ 1 g in Σ 1 g+1 as above, and let * ∈ ∂Σ 1 g+1 be a basepoint. Then µ is the composition where the first map is induced by the map Σ 1 g+1 → Σ 1 g that collapses Σ 1 g+1 \Int(Σ 1 g ) to a point. Just like in the abelian setting, the map Mod(Σ 1 g ) → Mod(Σ 1 g+1 ) induced by our embedding Σ 1 g → Σ 1 g+1 restricts to a map I(Σ 1 g , µ) → I(Σ 1 g+1 , µ ) that we will call the stabilization map. Our main theorem about this is as follows. It can be viewed as an analogue for the mapping class group of a theorem of Ellenberg-Venkatesh-Westerland [5, Theorem 6.1] concerning braid groups and Hurwitz spaces.
Remark 1.6. Ellenberg-Venkatesh-Westerland's main application in [5]  Proof techniques. There is an enormous literature on homological stability theorems, starting with unpublished work of Quillen on GL n (F p ). A standard proof technique has emerged that first appeared in its modern formulation in [19]. Consider a sequence of groups that we want to prove enjoys homological stability, i.e. H k (G n−1 ) ∼ = H k (G n ) for n k. To compute H k (G n ), we would need a contractible simplicial complex on which G n acts freely. Since we are only interested in the low-degree homology groups, we can weaken contractibility to high connectivity. The key insight for homological stability is that since we only want to compare H k (G n ) with the homology of previous groups in (1.2), what we want is not a free action but one whose stabilizer subgroups are related to the previous groups.
Machine. There are many variants on the above machine. For proving homological stability for the groups G n in (1.2), the easiest version requires simplicial complexes X n upon which G n acts with the following three properties: • The connectivity of X n goes to ∞ as n → ∞.
• For 0 ≤ k ≤ n − 1, the G n -stabilizer of a k-simplex of X n is conjugate to G n−k−1 .
• The group G n acts transitively on the k-simplices of X n for all k ≥ 0. Some additional technical hypotheses are needed as well; we will review these in §3.1. Hatcher-Vogtmann [9] constructed such X n for the mapping class group. Our proof of Theorem A is inspired by their work, so we start by describing a variant of it.
, is the simplicial complex whose k-simplices are sets {ι 0 , . . . , ι k } of isotopy classes of orientation-preserving embeddings ι i : g taking the endpoint of the tether to p 0 whose restriction to Σ 1 h preserves the orientation.
, is the simplicial complex whose k-simplices are collections {ι 0 , . . . , ι k } of isotopy classes of ∂-tethered genus h surfaces in Σ b g that can be realized so as to only intersect at p 0 ∈ ∂. For instance, here is a 2-simplex in T S 1 (Σ 1 5 , ∂): High connectivity. The complexes S 1 (Σ b g , ∂) and T S 1 (Σ b g , ∂) are closely related to complexes that were introduced by Hatcher-Vogtmann [9], and it follows easily from their work that they are g−3 2 -connected (see [17, proof of Theorem 6.25] for details). We generalize this as follows: Remark 1.9. Our convention is that a space is (−1)-connected if it is nonempty. Using this convention, the genus bounds for (−1)-connectivity and 0-connectivity in Theorem C are sharp. We do not know whether they are sharp for higher connectivity.
Remark 1.10. Hatcher-Vogtmann's proof in [9] that S 1 (Σ b g ) and T S 1 (Σ b g ) are g−3 2 -connected is closely connected to their proof that the separating curve complex is g−3 2 -connected. Looijenga [13] later showed that the separating curve complex is (g −3)-connected. Unfortunately, his techniques do not appear to give an improvement to Theorem C. Remark 1.11. In applications to homological stability, we will only use complexes made out of genus 1 subsurfaces. However, the more general result of Theorem C will be needed for the proof of even of the h = 1 case of Theorem D below.
Mod stability. Consider the groups The group Mod(Σ 1 g ) acts on T S 1 (Σ 1 g , ∂Σ 1 g ), and this complex has all three properties needed by the machine to prove homological stability for (1.3): • As we said above, is the mapping class group of the complement of a regular neighborhood of This complement is homeomorphic to Σ 1 g−k−1 , so this stabilizer is isomorphic to Mod(Σ 1 g−k−1 ). All such subsurface mapping class groups are conjugate; this follows from the "change of coordinates principle" from [6, §1.3.2].
• Another application of the "change of coordinates principle" shows that Mod(Σ 1 g ) acts transitively on the k-simplices of T S 1 (Σ 1 g , ∂Σ 1 g ).
Partial Torelli problem. A first idea for proving homological stability for the partial Torelli groups I(Σ 1 g , µ) is to consider their actions on T S 1 (Σ 1 g , ∂Σ 1 g ). Unfortunately, this does not work. The fundamental problem is that I(Σ 1 g , µ) does not act transitively on the k-simplices of T S 1 (Σ 1 g , ∂Σ 1 g ); indeed, it does not even act transitively on the vertices. For homology markings µ, the issue is that for a tethered torus ι and f ∈ I(Σ 1 g , µ), the compositions will be the same, but the functions µ • ι * : H 1 (τ (Σ 1 1 )) → A need not be the same for different tethered tori. A similar issue arises in the nonabelian setting. To fix this, we use a subcomplex of T S 1 (Σ 1 g , ∂Σ 1 g ) that is adapted to µ. Remark 1.12. The stabilizers are also wrong, but fixing the transitivity will also fix this.
Vanishing surfaces. For a homology marking µ : to be the full subcomplex of T S h (Σ 1 g , ∂Σ 1 g ) spanned by vertices ι such that the composition is the zero map. We will show that I(Σ 1 g , µ) acts transitively on the k-simplices of T S 1 (Σ 1 g , ∂Σ 1 g , µ) (at least for k not too large). However, there is a problem: a priori the subcomplex T S 1 (Σ 1 g , ∂Σ 1 g , µ) of T S 1 (Σ 1 g , ∂Σ 1 g ) might not be highly connected. Our third main theorem says that in fact it is g−(4 rk(A)+3) 2 rk(A)+2 -connected. More generally, we prove the following: Theorem D. Let µ : H 1 (Σ 1 g ) → A be a homology marking on Σ 1 g . Then the complex We also prove a similar theorem in the nonabelian setting.
Outline. We start in §2 by proving Theorem C. We then prove Theorem B in §3, which contains our nonabelian analogue of Theorem D. Next, in §4 we define a category of homologymarked surfaces with multiple boundary components. In §5, we use our category to state and prove Theorem 5.2, which generalizes Theorem A to surfaces with multiple boundary components. This proof depends on two stabilization results which are proved in §6. This final section also contains the proof of a generalization of Theorem D.
Conventions. Throughout this paper, A denotes a fixed finitely generated abelian group and Λ is a fixed finite group.

The complex of subsurfaces
This section is devoted to the proof of Theorem C, which asserts that S h (Σ b g ) and T S h (Σ b g , ∂) are highly connected. There are two parts: §2.1 contains a technical result about fibers of maps and §2.2 proves Theorem C.

Fibers of maps
Our proofs will require a technical tool.
Homotopy theory conventions. A space X is said to be n-connected if for k ≤ n, all maps S k → X extend to maps D k+1 → X. Since S −1 = ∅ and D 0 is a single point, a space is (−1)-connected precisely when it is nonempty. A map ψ : X → Y of spaces is an n-homotopy equivalence if for all 0 ≤ k ≤ n, the induced map [S k , X] → [S k , Y ] on unbased homotopy classes of maps out of S k is a bijection. If X and Y are connected, this is equivalent to saying that the induced maps π k (X) → π k (Y ) are isomorphisms.
Relative fibers. If ψ : X → Y is a map of simplicial complexes, σ is a simplex of Y , and σ is a face of σ, then denote by Fib ψ (σ , σ) the subcomplex of X consisting of all simplices η of X with the following properties: • ψ(η ) is a face of σ , and • there exists a simplex η of X such that η is a face of η and ψ(η) = σ.
Fiber lemma. With these definitions, we have the following lemma.
Lemma 2.1. Let ψ : X → Y be a map of simplicial complexes. For some n ≥ 0, assume the space Fib ψ (σ , σ) is n-connected for all simplices σ of Y and all faces σ of σ. Then ψ is an n-homotopy equivalence.
Proof. Replacing Y by its (n + 1)-skeleton Y n+1 and X by ψ −1 (Y n+1 ), we can assume that Y is finite-dimensional. The proof will be by induction on m = dim(Y ). The base case m = 0 is trivial since in that case Y is a discrete set of points and our assumptions imply that the fiber over each of these points is n-connected. Assume now that m ≥ 1. The key step in the proof is the following claim.
Claim. Assume that Y is obtained by adding a single m-simplex σ to a subcomplex Y . Define X = ψ −1 (Y ), and assume that ψ : X → Y restricts to an n-homotopy equivalence ψ : X → Y . Then ψ is an n-homotopy equivalence.
We now come to the key observation: the space Z is precisely the subcomplex of X consisting of the union of the subcomplexes Fib ψ (σ , σ) as σ ranges over all simplices of ∂σ. Moreover, for all simplices σ of ∂σ and all faces σ of σ , we have Fib ψ Z (σ , σ ) = Fib ψ (σ , σ), and thus by assumption Fib ψ Z (σ , σ ) is n-connected. We can therefore apply our inductive hypothesis to see that ψ Z : Z → ∂σ ∼ = S m−1 is an n-homotopy equivalence.
Summing up, we have X = X ∪ X and Y = Y ∪ σ. The map ψ restricts to n-homotopy equivalences ψ : X → Y and ψ : X → σ and ψ Z : Using Mayer-Vietoris (with local coefficients if the spaces involved are not simply-connected), we see that ψ is an n-homotopy equivalence, as desired.
Repeatedly applying this claim, we see that the lemma holds for m-dimensional Y with finitely many m-simplices. The usual compactness argument now implies that it holds for general m-dimensional Y , as desired.
Corollary 2.2. Let ψ : X → Y be a map of simplicial complexes. For some n ≥ 0, assume that the following hold.
• Y is n-connected.
• All (n + 1)-simplices of Y are in the image of ψ.
• For all simplices σ of Y whose dimension is at most n and all faces σ of σ, the space Proof. Let Y be the n-skeleton of Y and X = ψ −1 (Y ), so X contains the n-skeleton of X. Let ψ : X → Y be the restriction of ψ to X . Our assumptions allow us to apply Lemma 2.1 to ψ , so ψ is an n-homotopy equivalence. Since Y is n-connected, the space Y is (n − 1)-connected, so this implies that X and thus X are (n − 1)-connected. We also know that the induced map ψ : π n (X ) → π n (Y ) is an isomorphism. Since Y is n-connected, attaching the (n + 1)-simplices of Y to Y kills π n (Y ). By assumption, for each of these (n + 1)-simplices σ of Y there is an (n + 1)-simplex σ of X such that ψ( σ) = σ. We conclude that attaching to X the (n + 1)-simplices of X that do not already lie in X kills π n (X ), which implies that π n (X) = 0, as desired.

Subsurface complexes
We now prove Theorem C, which says that Proof of Theorem C. An argument identical to that of [9, proof of Proposition 5 . In fact, more is true. Let ∂ 1 , . . . , ∂ m be components of ∂Σ b g . Define T S h (Σ b g , ∂ 1 , . . . , ∂ m ) to be the complex of tethered genus h surfaces in Σ b g , but where the endpoints of the tethers can be fixed points on any of ∂ 1 , . . . , ∂ m (not just on a single ∂). The argument in [9, proof of Proposition 5 h+1 -connected will be by induction on h. The base case h = 1 is [17, Theorem 6.25] (which we remark shows how to derive it from a closely related result of Hatcher-Vogtmann [9]). For the inductive step, assume that Let τ (Σ 1 h , Σ 1 1 ) be the space obtained from Σ 1 h Σ 1 1 by gluing in an interval [0, 1] with 0 being attached to a point of ∂Σ 1 h and 1 being attached to a point of Σ 1 1 . The interval [0, 1] in τ (Σ 1 h , Σ 1 1 ) will be called the tether; the point 0 of the tether is the initial point and the point 1 is the endpoint.
A closed regular neighborhood of the image of an embedding τ ( In fact, there is a bijection between isotopy classes of orientation-preserving embeddings Σ 1 h+1 → Σ b g and isotopy classes of embeddings τ (Σ 1 h , Σ 1 1 ) → Σ b g whose restriction to each Σ 1 h preserves the orientation (for short, we will call these orientation-preserving embeddings of τ (Σ 1 h )). We can thus regard S h+1 (Σ b g ) as being the simplicial complex whose k-simplices are collections {ι 0 , . . . , ι k } of isotopy classes of orientation-preserving embeddings can be realized such that their images are disjoint.
We now define an auxiliary space. Let X be the simplicial complex whose k-simplices are collections {ι 0 , . . . , ι k } of isotopy classes of orientation-preserving embeddings ι i : g that can be realized such that the following hold for all 0 ≤ i < j ≤ k: , or the images under ι i and ι j of Σ 1 h are disjoint. • The images under ι i and ι j of Σ 1 1 together with the tether are disjoint except for possibly at the initial point of the tether. For instance, here is an edge of X: The next claim says that X enjoys the connectivity property we are trying to prove for S h+1 (Σ b g ).
. We will prove that the map ψ : X → S h (Σ b g ) satisfies the conditions of Corollary 2.2 for n = g−(2h+3) h+2 . Once we have done this, this corollary will show that X is n-connected, as desired.
The first condition is that S h (Σ b g ) is n-connected. In fact, our inductive hypothesis says that it is g−(2h+1) h+1 -connected, which is even stronger.
The second condition says that all (n + 1)-simplices of S h (Σ b g ) are in the image of ψ. The map ψ is Mod(Σ b g )-equivariant, and by the change of coordinates principle from [6, §1.3.2] the actions of Mod(Σ b g ) on S h+1 (Σ b g ) and S h (Σ b g ) are transitive on k-simplices for all k. To prove the second condition, therefore, it is enough to show that S h+1 (Σ b g ) contains an (n + 1)-simplex. Such a simplex contains (n + 2) disjoint copies of τ (Σ 1 h , Σ 1 1 ). Since there is enough room on Σ b g to find these (n + 2) disjoint copies of τ (Σ 1 h , Σ 1 1 ).
The final condition says that for all simplices σ of S h (Σ b g ) whose dimension is at most n and all faces σ of σ, the space Fib ψ (σ , σ) is n-connected. The space Fib ψ (σ , σ) has the following concrete description. Write As we noted in the first paragraph, the connectivity of T S 1 (Σ, ∂ 0 , . . . , ∂ m ) is at least We want to show that this is at least n = g−(2h+3)

h+2
. For this, we calculate as follows: Here the final inequality follows from the inequality h 2 /2 − h ≥ −2h, which holds for h ≥ 0.
We now use this to prove the desired connectivity property for S h+1 (Σ b g ).
, we will prove by induction on m that every map and that for all . Homotoping f , we can assume that it is simplicial with respect to a combinatorial triangulation of S m . We have -connected, we can extend f to a map to a map F : D m+1 → X that is simplicial with respect to a combinatorial triangulation of D m+1 that restricts to our given one on ∂D m+1 = S m . Our goal is to modify F such that its image lies in S h+1 (Σ b g ). We will do this via a link argument in the style of [9, §2.1].
Call a simplex σ of D m+1 a bad simplex if for all vertices v of σ, there exists another vertex v of σ such that F takes the edge {v, v } of σ to an edge of X that does not lie in S h+1 (Σ b g ). If D m+1 has no bad simplices, then its image lies in S h+1 (Σ b g ). Assume, therefore, that D m+1 has bad simplices, and let σ be a bad simplex of D m+1 whose dimension k is as large as possible. Since there are no bad vertices of D m+1 , we have 1 ≤ k ≤ m + 1.
Since no simplices of S m are bad, the simplex σ does not lie in S m . Letting L ⊂ D m+1 be the link of σ, this implies that Assume this for the moment. We can then extend F | L to a map that is simplicial with respect to some combinatorial triangulation of D m−k+1 that restricts to L on ∂D m−k+1 . The star S of σ is isomorphic to the join σ * L. We can replace S ⊂ D m+1 with ∂σ * D m−k+1 and F | S with F | ∂σ * F . Here are pictures of this operation for m + 1 = 2 and k equal to 2 and 1; on the left hand side is S, and on the right hand side is ∂σ * D m−k+1 : In doing this, we have eliminated the bad simplex σ without introducing any new bad simplices. Repeating this over and over again, we can eliminate all bad simplices, resulting in a map F : Then We must prove that S h+1 (Σ ) is (m − k)-connected. Let g be the genus of Σ . Since k ≥ 1, we have m − k < m, so our inductive hypothesis will say that This requires estimating g . The most naive such estimate of g is This is a poor estimate since it does not use the fact that σ is a bad simplex, which implies that every genus h surface contributing to this estimate is at least double-counted. Taking this into account, we see that in fact This implies that where the final inequality uses the fact that k ≥ 1.
This completes the proof of Theorem C.

Nonabelian stability
In this section, we prove Theorem B. Though the nonabelian markings considered in this theorem are more complicated than the abelian ones in Theorem A, we decided to prove this one first since it avoids the technical baggage involved in our generalization of Theorem A to surfaces with multiple boundary components.
This section has three parts. In §3.1, we discuss the homological stability machine. In §3.2, we prove the nonabelian analogue of Theorem D. Finally, in §3.3 we prove Theorem B.

The stability machine
We now introduce the standard homological stability machine. This is discussed in many places, but the account in [9, §1] is particularly convenient for our purposes.
Semisimplicial complexes. The natural setting for the machine is that of semisimplicial complexes, whose definition we now briefly recall. For more details, see [7], which calls them ∆-sets. Geometric properties. A semisimplicial complex X has a geometric realization |X| obtained by taking standard k-simplices for each element of X k and then gluing these simplices together using the boundary maps. Whenever we talk about topological properties of a semisimplicial complex, we are referring to its geometric realization. An action of a group G on a semisimplicial complex X consists of actions of G on each X n that commute with the boundary maps. This induces an action of G on |X|.
Main theorem. To study one of our stabilization maps I(Σ 1 g , µ) → I(Σ 1 g+1 , µ ), we will construct a semisimplicial complex X upon which G := I(Σ 1 g+1 , µ ) acts and where I(Σ 1 g , µ) is the stabilizer of a vertex v. The purpose of the machine is thus to provide conditions that ensure that the inclusion Our theorem is as follows.
Theorem 3.1. Let G be a group and let X be a semisimplicial complex on which G acts. For some k ≥ 1, assume that the following hold: For all 1-simplices e of X whose boundary consists of vertices v and v , there exists λ ∈ G such that λ(v) = v and such that λ commutes with all elements of G e . Then for all 0-simplices v of X, the map Proof. This is proved exactly like [9, Theorem 1.1]. The only difference is that since we do not make any assertions about stabilization maps inducing surjections on homology, we get worse bounds (but in our applications, the connectivity results we have are so poor that we would not be able to make use of better bounds).

Nonabelian vanishing surfaces
We now discuss the complexes of tethered and untethered subsurfaces.
Semisimplicial. We claim that T S h (Σ 1 g , ∂Σ 1 g , µ) is naturally a semisimplicial complex. The key point here is that its simplices {ι 0 , . . . , ι k } possess a natural ordering based on the order (going counter-clockwise) their tethers leave the basepoint in ∂: ordering.{ps,eps,pdf} not found (or no BBox) Using this ordering, a strictly increasing map [k] → [ ] naturally induces a map from -simplices to k-simplices.
Stabilizers. The Mod(Σ 1 g )-stabilizers of simplices of S h (Σ 1 g ) are poorly behaved. The issue is that mapping classes can permute their vertices arbitrarily (which is not possible for T S h (Σ 1 g , ∂Σ 1 g ) since mapping classes must preserve the counter-clockwise ordering that the tethers leave ∂Σ 1 g ). This prevents their stabilizers from being mapping class groups of subsurfaces. For T S h (Σ 1 g , ∂Σ 1 g ), however, this issue does not occur, and the Mod( We will call the complement of this open neighborhood the stabilizer subsurface of the simplex. See here: The I(Σ 1 g , µ) version of this is as follows.
Proof. Since µ vanishes on the image of π 1 (Σ 1 g \ Int(Σ ), * ) in π 1 (Σ 1 g ), the map µ factors through the fundamental group of the result of collapsing Σ ⊂ Σ 1 g to a point. This is the same as the result of collapsing ∂Σ ⊂ Σ to a point. The lemma follows.
High connectivity. The following theorem is our nonabelian analogue of Theorem D. Theorem 3.3. Let * ∈ ∂Σ 1 g be a basepoint, let Λ be a finite group, and let µ : π 1 (Σ 1 g , * ) → Λ be a Λ-marking. Then the following hold Proof. An argument identical to that of [9, proof of Proposition 5 . We thus only need to consider S h (Σ 1 g , µ).
We start by defining an auxiliary space. Let X be the simplicial complex whose vertices are the union of the vertices of the spaces S h (Σ 1 g , µ) and S h+|Λ| (Σ 1 g ) and whose simplices are collections {ι 0 , . . . , ι k } of vertices that can be isotoped such that their images are disjoint.
It is clear that this extends over the simplices of X to give a retract r : X → S h (Σ 1 g , µ). To complete the proof, therefore, it is enough to prove the following claim.

h+|Λ|+1
, every map S m → X can be homotoped such that its image lies in S h+|Λ| (Σ 1 g ). As in our proof of Theorem C, we will do this via a link argument in the style of [9, §2.1].
Homotoping f , we can assume that it is simplicial with respect to a combinatorial triangulation of S m . Call a simplex σ of S m a bad simplex if f takes all the vertices of σ to vertices of S h (Σ 1 g , µ) ⊂ X. If S m has no bad simplices, then its image lies in S h+|Λ| (Σ 1 g ). Assume, therefore, that S m has bad simplices, and let σ be a bad simplex of S m whose dimension k is as large as possible.
Letting L ⊂ S m be the link of σ, we have L ∼ = S m−k−1 . Letting L be the link of f (σ) in X, the maximality of the dimension of σ implies that f (L) ⊂ L ∩ S h+|Λ| (Σ 1 g ). Below we will prove that L ∩ S h+|Λ| (Σ 1 g ) is (m − k − 1)-connected.
Assume this for the moment. We can then extend f | L to a map that is simplicial with respect to some combinatorial triangulation of D m−k that restricts to L on ∂D m−k . The star S of σ is isomorphic to the join σ * L. We can homotope f : S m → X so as to replace S ⊂ S m with ∂σ * D m−k and f | S with f | ∂σ * F . Here are pictures of this operation for m = 2 and k equal to 2 and 1; on the left hand side is S, and on the right hand side is ∂σ * D m−k : σ σ In doing this, we have eliminated the bad simplex σ without introducing any new bad simplices. Repeating this over and over again, we can eliminate all bad simplices, resulting in a map f : S m → S h+|Λ| (Σ 1 g ), as desired.

Transitivity.
The final fact about these complexes we will need is as follows.
Proof. The proof will be by induction on k. We start with the base case k = 0.
Proof of claim. In this case, Theorem 3.3 says that T S h (Σ 1 g , ∂Σ 1 g , µ) is connected, so it is enough to prove that if ι and ι are vertices of T S h (Σ 1 g , ∂Σ 1 g , µ) that are connected by an edge, then there exists some f ∈ I(Σ 1 g , µ) taking ι to ι . Let Σ be the stabilizer subsurface of the edge {ι, ι }. Then Σ 1 g \ Int(Σ ) is a genus 2 surface with 2 boundary components containing the images of ι and ι ; see here: regular nbhd Using the change of coordinates principle from [6, §1.3.2], we can find f ∈ Mod(Σ 1 g ) taking ι to ι and acting as the identity on Σ . By the definition of T S h (Σ 1 g , ∂Σ 1 g , µ), the marking µ vanishes on all loops lying in Σ 1 g \ Int(Σ ). This immediately implies that f ∈ I(Σ 1 g , µ), as desired.

Proof of nonabelian stability
We now prove Theorem B.
We prove our result by applying Theorem 3.1 to the action of G = I(Σ 1 g+1 , µ ) on X = T S 1 (Σ 1 g+1 , ∂Σ 1 g+1 , µ ). For a vertex v = ι of T S 1 (Σ 1 g+1 , ∂Σ 1 g+1 , µ ), Lemma 3.2 says that the stabilizer G v is the subgroup of I(Σ 1 g+1 , µ ) consisting of mapping classes that are supported on the stabilizer subsurface of v. As is shown here, we can choose v such that G v = I(Σ 1 g , µ): regular nbhd Once we have verified the conditions of Theorem 3.1, it will thus imply that as desired. It remains to verify those conditions: • The first is that X = T S 1 (Σ 1 g+1 , ∂Σ 1 g+1 , µ ) is k-connected, which follows from Theorem 3.3.
• The second is that G = I(Σ 1 g+1 , µ ) acts transitively on the -simplices of X = T S 1 (Σ 1 g+1 , ∂Σ 1 g+1 , µ ) for 0 ≤ ≤ k + 2, which follows from Lemma 3.4. • The third says the following. Consider 1 ≤ i ≤ k and 0 ≤ ≤ i + 2. Let σ be an -simplex of X. We must prove that the map H k−i (G σ ) → H k−i (G) is an isomorphism. Lemma 3.2 says that the I(Σ 1 g+1 , µ )-stabilizer of σ is I(Σ , µ ), where Σ is the stabilizer subsurface of σ and µ is appropriately chosen. By its construction, we have Σ ∼ = Σ 1 g− −1 . Our inductive hypothesis now implies that the map H k−i (I(Σ , µ )) → H k−i (I(Σ 1 g+1 , µ )) is an isomorphism, which is exactly what we were trying to show. • Let e be a 1-simplex of X whose boundary consists of vertices v and v . We must construct some λ ∈ G and such that λ(v) = v and such that λ commutes with all elements of G e . Let Σ be the stabilizer subsurface of e, so by Lemma 3.2 the stabilizer G e consists of mapping classes supported on Σ . The surface Σ 1 g+1 \ Int(Σ ) is homeomorphic to Σ 2 2 ; see The key point here is that Σ 1 g+1 \ Int(Σ ) is connected. The desired λ taking one tethered torus to another can be chosen to be supported on Σ 1 g+1 \ Int(Σ ).

The category of homology-marked surfaces
We now turn to homology markings. As we said in the introduction, for applications in the author's forthcoming work on the cohomology of the moduli space of curves with level structures [16], we need to generalize Theorem A to surfaces with multiple boundary components. This section contains a categorical framework for such a generalization.
Let Surf be the category whose objects are compact connected oriented surfaces with boundary and whose morphisms are orientation-preserving embeddings. There is a functor from Surf to groups taking Σ ∈ Surf to Mod(Σ) and a morphism Σ → Σ to the map Mod(Σ) → Mod(Σ ) that extends mapping classes by the identity. In this section, we augment Surf to new categories TSurf and PTSurf(A) on which we can define Torelli and partial Torelli groups. We discuss TSurf in §4.1 and PTSurf(A) in §4.2.

The category TSurf
We start with the Torelli surface category TSurf, which was introduced in [15].
Motivation. This category captures aspects of the homology of a larger surface in which our surface is embedded. For instance, consider the following embedding of a genus 3 surface Σ with 6 boundary components into Σ  ). These classes do not lie in H 1 (Σ). The category TSurf fixes these issues by killing appropriate sums of boundary components and also considering the homology classes of certain arcs joining boundary components.
Objects. The objects of TSurf are pairs (Σ, P) as follows: • Σ is a connected compact oriented surface with boundary, and • P is a partition of the components of ∂Σ. Associated with (Σ, P) ∈ TSurf are a partitioned homology group H P 1 (Σ) and a Torelli group I(Σ, P), which we now discuss.
Torelli. Consider (Σ, P) ∈ TSurf. Enumerate P as • H is the subspace of H 1 (Σ, X) spanned by the homology classes of oriented closed curves and arcs connecting points of X lying in P-adjacent components of ∂Σ.
The group Mod(Σ) acts on H P 1 (Σ). The Torelli group associated to (Σ, P) ∈ TSurf, denoted I(Σ, P), is the kernel of the action of Mod(Σ) on H P 1 (Σ). Once we have introduced the morphisms of TSurf, this will be extended to a functor.
Morphisms. Consider (Σ, P), (Σ , P ) ∈ TSurf. A morphism (Σ, P) → (Σ , P ) is an orientation-preserving embedding Σ → Σ that is compatible with the partitions P and P in the following sense. For a component S of Σ \ Int(Σ), let B S (resp. B S ) denote the set of components of ∂S that are also components of ∂Σ (resp. ∂Σ ). In the degenerate case where S ∼ = S 1 (so S is a component of ∂Σ and ∂Σ ), our convention is that ∂S = S and thus B S = B S = {S}. Our compatibility requirements are then: • each B S is a subset of some p ∈ P, and • for all p ∈ P and all ∂ 1 , ∂ 2 ∈ p such that ∂ 1 ∈ B S 1 and ∂ 2 ∈ B S 2 with S 1 = S 2 , there exists some p ∈ P such that B S 1 ∪ B S 2 ⊂ p. Here is an illustration of these requirements: In this figure, p = {∂ 1 , ∂ 2 , ∂ 3 , ∂ 4 } ∈ P and p = {∂ 1 , ∂ 2 , ∂ 3 } ∈ P .
Torelli functoriality. The main theorem about TSurf is as follows.

The category PTSurf
We now turn to the partial Torelli surface category PTSurf(A).
Reduced partitioned homology. Consider some (Σ, P) ∈ TSurf. For the definition of PTSurf(A), it turns out to be better to consider a variant of H P 1 (Σ) that has better functoriality properties. Define H P 1 (Σ) to be the subgroup of the relative homology group H 1 (Σ, ∂Σ) spanned by the homology classes of oriented closed curves and arcs connecting P-adjacent boundary components. There is a surjection whose kernel is spanned by the homology classes of components of ∂Σ.
Functoriality. The assignment (Σ, P) → H P 1 (Σ) is a contravariant functor from TSurf to abelian groups. To see this, consider a TSurfmorphism ι : (Σ, P) → (Σ , P ). Identify Σ with its image under ι. We then have maps From the definition of a TSurf-morphism, it follows immediately that this composition restricts to a map ι * : H In this example, all boundary components of Σ (resp. Σ ) lie in a single element of P (resp. P ). We have ι * (x 1 ) = y 1 and ι * (x 2 ) = y 2 ; the element y 3 is not in the image of ι * .
Partial Torelli. The partial Torelli group associated to (Σ, P, µ) ∈ PTSurf(A) is The following lemma asserts that this is a functor from PTSurf(A) to groups.
The lemma follows.

Abelian stability
In this section, we generalize Theorem A from the introduction and reduce this generalization to two facts that will be proved in the next section. The statement is in §5.1 and the reductions are in §5.2 - §5.3.

Statement of result
For technical reasons, we will have to impose some conditions our stabilization maps.
Support. For (Σ, P, µ) ∈ PTSurf(A), we say that µ is supported on a genus h symplectic subsurface if there exists a PTSurf(A)-morphism (Σ 1 h , P , µ ) → (Σ, P, µ). If there exists some h ≥ 1 such that µ is supported on a genus h symplectic subsurface, then we will simply say that µ is supported on a symplectic subsurface.
Remark 5.1. Not all (Σ, P, µ) ∈ PTSurf(A) are supported on a symplectic subsurface. Indeed, this condition implies that for all P-adjacent boundary components ∂ 1 and ∂ 2 of Σ, there exists an oriented embedded arc α connecting ∂ 1 to ∂ 2 such that µ([α]) = 0, and it is easy to construct A-homology markings not satisfying this property; for instance, if A ∼ = H 0 (∂Σ), then we can take µ to equal Partition bijectivity. Consider a PTSurf(A)-morphism (Σ, P, µ) → (Σ , P , µ ). Identify Σ with its image in Σ . We will call this morphism partition-bijective if there exists a bijection β : P → P with the following property for all p ∈ P: • Let S p (resp. S β(p) ) be the union of the components of Σ \ Int(Σ) that intersect the boundary components lying in p (resp. β(p)). We then require that S p = S β(p) . This condition rules out two kinds of morphisms: • Ones where for some p ∈ P the union of the components of Σ \ Int(Σ) intersecting elements of p contains no components of ∂Σ . See here: Here p = {∂ 1 , ∂ 2 }.
• Ones where a single p ∈ P "splits" into multiple elements of P like this: ' ∂ Here p = {∂} and P contains both {∂ 1 } and {∂ 2 }.
Main theorem. Our main theorem is the following. It generalizes Theorem 5.2 from the introduction.
Remark 5.3. We do not know if the requirement that µ be supported on a symplectic subsurface is necessary. The condition that the morphism is partition-bijective is annoying, and we conjecture that it is not necessary. However, Theorem 5.2 is sufficiently general to be useful in our forthcoming work on the cohomology of the moduli space of curves with level structures [16].

Reduction I: open capping
In this section, we reduce Theorem 5.2 to certain kinds of PTSurf(A)-morphisms called open cappings, whose definition is below.
Remark 5.4. In [15], a capping is defined similarly to an open capping, but where Σ is closed and ∂S is simply an element of P.
Reduction. The following is a special case of Theorem 5.2.
The proof of Proposition 5.5 begins in §5.3. First, we will use it to deduce Theorem 5.2.
The proof has two cases. For the first, the discrete partition of the boundary components of a surface S is the partition {{∂} | ∂ a component of ∂S}.
Case 1. P is the discrete partition of Σ.
Case 2. P is not the discrete partition of Σ.
Since µ is supported on a symplectic subsurface, we can find a partition-bijective PTSurf(A)morphism (Σ , P , µ ) → (Σ, P, µ) such that Σ has the same genus as Σ and such that P is the discrete partition. The picture is this: In this example, P consists of two elements (the left hand boundary components and the right hand boundary components). Identify Σ with its image in Σ. We have maps H k (I(Σ , P , µ )) −→ H k (I(Σ, P, µ)) −→ H k (I(Σ , P , µ )).
Deriving Proposition 5.5. As we said above, we will prove Propositions 5.6 and 5.7 in §6. Here we will explain how to use them to prove Proposition 5.5. can be factored as The proposition follows.

The two stabilizations
It remains to prove Propositions 5.6 and 5.7. We do so in this section, closely following the proof of Theorem B in §3. We start in §6.1 with a technical lemma. We construct the complex we will use to prove Proposition 5.6 in §6.2, and we prove that proposition in §6.3. One highlight is our promised generalization of Theorem D from the introduction, which is in §6.2. Finally, we construct the complex we will use to prove Proposition 5.7 in §6.4, and we prove that proposition in §6.5.

Finding a vertex
Recall that one of the key parts of our proof of Theorem 3.3 was a theorem of Dunfield-Thurston [4,Proposition 6.16] that implies that if Λ is a finite group and µ : π 1 (Σ 1 h+|Λ| , * ) → Λ is a Λ-marking, then there exists a vertex of S h (Σ 1 h+|Λ| , µ). The following lemma is an analogous result for homology markings. Proof. Every genus h symplectic subspace U of H 1 (Σ 1 h+2 rk(A) ) can be written as U = H 1 (S) for some subsurface S of Σ 1 h+2 rk(A) satisfying S ∼ = Σ 1 h (see e.g. [10,Lemma 9]). Lemma 6.1 is thus equivalent to the purely algebraic Lemma 6.2 below; in the notation of that lemma, the desired U is U = W ⊥ . Lemma 6.2. Let V be a free abelian group equipped with a symplectic form ω(−, −) and let µ : V → A be a group homomorphism. There then exists a genus 2 rk(A) symplectic subspace W of V such that µ| W ⊥ = 0.
Proof. Without loss of generality, µ is surjective and A = 0. The proof will be by induction on rk(A). The base case is rk(A) = 1, so A is cyclic. We can factor µ as By definition, ω(−, −) identifies V with its dual Hom(V, Z). There thus exists some a ∈ V such that µ(x) = ω(a, x) for all x ∈ V . Pick b ∈ V with ω(a, b) = 1 and let W = a, b . Then W is a genus 1 symplectic subspace and W ⊥ ⊂ ker(ω(a, −)) = ker( µ) ⊂ ker(µ), as desired. Now assume that rk(A) > 1 and that the lemma is true for all smaller ranks. We can then find a short exact sequence

Vanishing subsurfaces
We now discuss the complexes of tethered and untethered subsurfaces.
Vanishing subsurfaces. Consider (Σ, P, µ) ∈ PTSurf(A). Define S h (Σ, P, µ) to be the full subcomplex of S h (Σ) spanned by vertices ι such that the composition is the zero map. The group I(Σ, P, µ) acts on S h (Σ, P, µ). Similarly, let ∂ be a component of ∂Σ, and define T S h (Σ, ∂, P, µ) to be the full subcomplex of T S h (Σ, ∂) spanned by vertices ι such that the composition is the zero map. The group I(Σ, P, µ) acts on T S h (Σ, ∂, P, µ). Just like in the nonabelian case (see §3.2), this is naturally a semisimplicial complex.
The I(Σ, P, µ) version of this is as follows.
Lemma 6.3. Consider (Σ, P, µ) ∈ PTSurf(A). Let ∂ be a component of ∂Σ, and let σ be a simplex of T S h (Σ, P, µ). Let Σ be the stabilizer subsurface of σ, and let P be the partition of the components of ∂Σ obtained by replacing ∂ in P with the component ∂ of ∂Σ that is not a component of ∂Σ. Then there exists a PTSurf(A)-morphism (Σ , P , µ ) → (Σ, P, µ) such that the I(Σ, P, µ)-stabilizer of σ is I(Σ , P , µ ). Moreover, if µ is supported on a symplectic subsurface then so is µ .
Proof. It is clear that (Σ , P ) → (Σ, P) is a TSurf-morphism. Since H 1 is a contravariant functor, we get a map H  Proof. This theorem can be derived from the fact that S h+2 rk(A) (Σ) is g−(2h+4 rk(A)+1) h+2 rk(A)+1 connected exactly like in the proof of the Theorem 3.3, which is this theorem's nonabelian analogue. The only change that must be made is that the reference to Dunfield-Thurston's paper [4] should be replaced by Lemma 6.1.
Transitivity. We will also need the following. Proof. The proof is identical to the proof of Lemma 3.4, which is this lemma's nonabelian analogue. The only change that must be made is that the reference to Theorem 3.3 should be replaced by Theorem 6.4.

Proof of Proposition 5.6
We now prove Proposition 5.6.
Proof of Proposition 5.6. We start by recalling the statement. Let (Σ, P, µ) → (Σ , P , µ ) be an open genus stabilization such that µ is supported on a symplectic subsurface. For some k ≥ 1, assume that Ind(k) holds. Also, assume that the genus of Σ is at least (2 rk(A) + 2)k + (4 rk(A) + 2). Our goal is to prove that the induced map H k (I(Σ, P, µ)) → H k (I(Σ , P , µ )) is an isomorphism.
In performing the open genus stabilization to form (Σ , P , µ ) from (Σ, P, µ), a copy of Σ 2 1 is glued to a component ∂ of ∂Σ. Let ∂ be the component of ∂Σ that lies in Σ 2 1 .
We will prove our result by applying Theorem 3.1 to the action of G := I(Σ , P , µ ) on X = T S 1 (Σ , ∂ , P , µ ). For a vertex v = ι of T S 1 (Σ , ∂ , P , µ ), Lemma 6.3 says that the stabilizer G v is the subgroup of I(Σ , P , µ ) consisting of mapping classes that are supported on the stabilizer subsurface of v. As is shown here, we can choose v such that G v = I(Σ, P, µ): Once we have verified the conditions of Theorem 3.1, it will thus imply that H k (I(Σ, P, µ)) = H k (G v ) ∼ = H k (G) = H k (I(Σ , P , µ )), as desired. It remains to verify those conditions: • The first is that X = T S 1 (Σ , ∂ , P , µ ) is k-connected. Since Σ has genus (g + 1), this follows from Theorem 6.4. • The second is that G = I(Σ , P , µ ) acts transitively on the -simplices of X = T S 1 (Σ , ∂ , P , µ ) for 0 ≤ ≤ k + 2. Again using the fact that Σ has genus (g + 1), this follows from Lemma 6.5. • The third says the following. Consider 1 ≤ i ≤ k and 0 ≤ ≤ i + 2. Let σ be an -simplex of X. We must prove that the map H k−i (G σ ) → H k−i (G) is an isomorphism. Lemma 6.3 says that the I(Σ , P , µ )-stabilizer of σ is I(Σ , P , µ ), where Σ is the stabilizer subsurface of σ and P and µ are appropriately chosen. By its construction, the stabilizer subsurface Σ of the -simplex σ has genus (g + 1) − ( + 1) = g − . Our assumption Ind(k) now implies that the map H k−i (I(Σ , P , µ )) → H k−i (I(Σ , P , µ )) is an isomorphism, which is exactly what we were trying to show. • Let e be a 1-simplex of X whose boundary consists of vertices v and v . We must construct some λ ∈ G and such that λ(v) = v and such that λ commutes with all elements of G e . Let Σ be the stabilizer subsurface of e, so by Lemma 6.3 the stabilizer G e consists of mapping classes supported on Σ . The surface Σ \ Int(Σ ) is homeomorphic to Σ 2 2 ; see regular nbhd The key point here is that Σ \ Int(Σ ) is connected. The desired λ taking one tethered torus to another can be chosen to be supported on Σ \ Int(Σ ).

Vanishing loops
We now discuss the complexes of tethered and untethered loops.
Loops. Let Σ ∈ Surf be a surface. The complex of loops on Σ, denoted L(Σ), is the simplicial complex whose k-simplices are sets {ι 0 , . . . , ι k } of isotopy classes of embeddings ι i : S 1 → Σ that can be realized such that their images are disjoint and do not separate Σ. The group Mod(Σ) acts on L(Σ). The complex L(Σ) was introduced by Harer [8], who proved that it was (g − 2)-connected.
Tethered loops. Let τ (S 1 ) be the result of gluing 0 ∈ [0, 1] to a point of S 1 . The subset [0, 1] ∈ τ (S 1 ) is the tether and 1 ∈ [0, 1] ⊂ τ (S 1 ) the endpoint of the tether. Let ∂ be a component of ∂Σ. Fix p 0 ∈ ∂. A ∂-tethered loop on Σ is an embedding ι : τ (S 1 ) → Σ taking the endpoint of the tether to p 0 . The complex of ∂-tethered loops on Σ, denoted T L(Σ, ∂), is the simplicial complex whose k-simplices are collections {ι 0 , . . . , ι k } of isotopy classes of ∂-tethered loops on Σ that can be realized so as to only intersect at p 0 ∈ ∂ and not separate Σ. Here is an example: This complex was introduced by Hatcher-Vogtmann [9], who proved it was g−3 2 -connected. Just like all of our tethered complexes, it is naturally a semisimplicial complex.
Remark 6.6. Since loops are embeddings S 1 → Σ, the orientation of S 1 gives them a natural orientation. For a tethered loop ι : τ (S 1 ) → Σ, it thus makes sense to ask whether the tether approaches the loop from the left or the right in Σ. Since we defined Ψ using a fixed ι bt : τ (S 1 ) → Σ 1 1 , all vertices of T L(Σ, ∂) that can be enveloped have the tether approach from the same side (either left or right; the arbitrary choice here does not matter).
Vanishing loops. Consider (Σ, P, µ) ∈ PTSurf(A). Define L(Σ, P, µ) to be the subcomplex of L(Σ) consisting of simplices can can be enveloped by simplices of S 1 (Σ, P, µ). This implies that the homology classes of the loops making up the simplices vanish under µ, but is a stronger condition. The group I(Σ, P, µ) acts on T L(Σ, ∂P, µ). Similarly, for a component ∂ of ∂Σ define T L(Σ, ∂, P, µ) to be the subcomplex of T L(Σ, ∂) consisting of simplices can can be enveloped by simplices of T S 1 (Σ, ∂, P, µ).
We will call this the stabilizer subsurface of σ. See here: regular nbhd = The I(Σ, P, µ) version of this is as follows.
Lemma 6.7. Consider (Σ, P, µ) ∈ PTSurf(A). Let ∂ be a component of ∂Σ, and let σ be a simplex of T L(Σ, P, µ). Let Σ be the stabilizer subsurface of σ, and let P be the partition of the components of ∂Σ obtained by replacing ∂ in P with all the components of ∂Σ that are not components of ∂Σ. Then there exists a PTSurf(A)-morphism (Σ , P , µ ) → (Σ, P, µ) such that the I(Σ, P, µ)-stabilizer of σ is I(Σ , P , µ ). Moreover, if µ is supported on a symplectic subsurface then so is µ .
Proof. It is clear that (Σ , P ) → (Σ, P) is a TSurf-morphism. Since H 1 is a contravariant functor, we get a map H P 1 (Σ) → H P 1 (Σ ). Since any arc connecting ∂ to another component of ∂Σ has to pass through some component of ∂Σ that is not a component of ∂Σ, this map is surjective. Since µ vanishes on any homology class supported on Σ \ Int(Σ ), the homomorphism µ : H P 1 (Σ) → A factors through a homomorphism µ : H P 1 (Σ ) → A. We thus have a PTSurf(A)-morphism (Σ , P , µ ) → (Σ, P, µ). That the I(Σ, P, µ)-stabilizer of σ is I(Σ , P , µ ) is clear from the construction. The assertion about symplectic subsurfaces follows from the fact that σ can be enveloped by a simplex σ of T S 1 (Σ, ∂, P, µ).
High connectivity. Our main topological theorem about these complexes is as follows: Theorem 6.8. Consider (Σ, P, µ) ∈ PTSurf(A), and let g be the genus of Σ.
Proof. This theorem can be derived from the fact that S h+2 rk(A) (Σ) is g−(2h+4 rk(A)+1) h+2 rk(A)+1 connected exactly like in the proof of the Theorem 3.3. The only change that must be made is that the reference to Dunfield-Thurston's paper [4] should be replaced by Lemma 6.1.

Proof of Proposition 5.7
We close the paper by proving Proposition 5.7.
Proof of Proposition 5.7. We start by recalling the statement. Let (Σ, P, µ) → (Σ , P , µ ) be a boundary stabilization in PTSurf(A) such that µ is supported on a symplectic subsurface. For some k ≥ 1, assume that Ind(k) holds. Also, assume that the genus of Σ is at least (2 rk(A) + 2)k + (4 rk(A) + 2). Our goal is to prove that the induced map H k (I(Σ, P, µ)) → H k (I(Σ , P , µ )) is an isomorphism.
We will prove our result by applying Theorem 3.1 to the action of G := I(Σ , P , µ ) on X = T L(Σ , ∂ , P , µ ). For a vertex v = ι of T L(Σ , ∂ , P , µ ), Lemma 6.7 says that the stabilizer G v is the subgroup of I(Σ , P , µ ) consisting of mapping classes that are supported on the stabilizer subsurface of v. As is shown here, we can choose v such that G v = I(Σ, P, µ): regular nbhd Once we have verified the conditions of Theorem 3.1, it will thus imply that H k (I(Σ, P, µ)) = H k (G v ) ∼ = H k (G) = H k (I(Σ , P , µ )), as desired. Those conditions are verified exactly like in the proof of Proposition 5.6, substituting Lemma 6.7 and Theorem 6.8 and Lemma 6.9 for Lemma 6.3 and Theorem 6.4 and Lemma 6.5.