2.1 Surfaces and mapping class groups

We work in the entire article with smooth, oriented surfaces and orientation-preserving diffeomorphisms of surfaces.

Notation 2.1. We usually denote by $\mathcal {S}$ a smooth, compact, oriented, possibly disconnected surface, such that each component of $\mathcal {S}$ has non-empty boundary; we denote the boundary of $\mathcal {S}$ by $\partial \mathcal {S}\subset \mathcal {S}$ .

The boundary $\partial \mathcal {S}$ is equipped with a decomposition $\partial \mathcal {S}=\partial ^{\text {in}}\mathcal {S}\sqcup \partial ^{\text {out}}\mathcal {S}$ , into unions of connected components: the incoming boundary $\partial ^{\text {in}}\mathcal {S}$ is allowed to be empty, whereas each component of $\mathcal {S}$ is required to intersect the outgoing boundary in at least one curve; see Figure 1 for an example.

Figure 1 A surface $\mathcal {S}$ with $5$ incoming and $4$ outgoing boundary curves.

Both parts of the boundary are equipped with an ordering and a parametrisation: that is, there are preferred diffeomorphisms $\vartheta ^{\text {in}}\colon \{1,\dotsc ,n\}\times S^1\to \partial ^{\text {in}} \mathcal {S}$ and $\vartheta ^{\text {out}}\colon \{1,\dotsc ,n'\}\times S^1\to \partial ^{\text {out}} \mathcal {S}$ , where $n=\#\pi _0(\partial ^{\text {in}}\mathcal {S})$ and $n'=\#\pi _0(\partial ^{\text {out}}\mathcal {S})$ .

Note that each boundary component $c\subset \partial \mathcal {S}$ is endowed with two natural orientations: the first is induced from the orientation of $\mathcal {S}$ , that is, it is the unique orientation of c that, concatenated with a vector field along c pointing out of $\mathcal {S}$ , returns the orientation of $\mathcal {S}$ ; the second orientation comes from the parametrisation of c. For an incoming boundary component $c\subset \partial ^{\text {in}}\mathcal {S}$ , we require that these two orientations coincide, whereas for an outgoing boundary component $c\subset \partial ^{\text {out}}\mathcal {S}$ , we require that these two orientations differ.

We usually denote a surface by $(\mathcal {S},\vartheta )$ , or shortly by $\mathcal {S}$ when it is not necessary to mention the parametrisation of the boundary; here $\vartheta $ is the map $\left \{1,\dots ,n+n'\right \}\times S^1\to \partial \mathcal {S}$ obtained by concatenation of $\vartheta ^{\text {in}}$ and $\vartheta ^{\text {out}}$ .

Note that in the previous definition, we do not require that $\Phi $ also preserves the orderings of the incoming and outgoing components of $\partial \mathcal {S}$ and $\partial \mathcal {S}'$ : that is, the permutations $\sigma ^{\text {in}}\in \mathfrak {S}_n$ and $\sigma ^{\text {out}}\in \mathfrak {S}_{n'}$ may be non-trivial. To emphasise this, we distinguish between the words ‘ordering’ and ‘parametrisation’. In Section 4, when introducing the coloured operad ${\mathscr {M}}$ , we will also consider surfaces equipped with a parametrisation of collar neighbourhoods of the incoming and the outgoing boundary.

Note that we have an extension $1\to \Gamma (\mathcal {S},\partial \mathcal {S})\to \Gamma ^{\mathfrak {H}}(\mathcal {S})\to \mathfrak {H}\to 1$ .

The hypothesis that every connected component of $\mathcal {S}$ has non-empty boundary implies that $\mathfrak {M}(\mathcal {S})$ is a classifying space for the group $\Gamma (\mathcal {S},\partial \mathcal {S})$ .

2.2 Arcs and the Alexander method

For the rest of this section, we fix a connected surface $\mathcal {S}$ of type $\Sigma _{g,n}$ , with $g\geqslant 0$ and $n\geqslant 1$ , and focus on the mapping class group $\Gamma (\mathcal {S},\partial \mathcal {S})$ . Given a mapping class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , we construct a system of simple closed curves on $\mathcal {S}$ cutting $\mathcal {S}$ into two subsurfaces W and Y: the subsurface $W\subset \mathcal {S}$ is the white subsurface and is, up to isotopy, the maximal subsurface of $\mathcal {S}$ satisfying the following conditions:

• ○ all connected components of W touch $\partial \mathcal {S}$ ;

• ○ $\varphi $ can be represented by a diffeomorphism of $\mathcal {S}$ fixing W pointwise.

We start by recalling some standard facts about embedded arcs in surfaces. The material of this subsection is taken, up to minor changes, from [Reference Farb and Margalit7]. For the following definition see [Reference Farb and Margalit7, § 1.2.7].

Note that we only consider isotopy classes of arcs relative to their endpoints; two arcs sharing one endpoint are never considered in minimal position (and, by convention, cannot be isotoped to be in minimal position). In particular, unless $\mathcal {S}$ is a disc, there are more than countably many isotopy classes of essential arcs in $\mathcal {S}$ .

If $\chi (\mathcal {S})=2-2g-n\leqslant 0$ , then according to [Reference Farb and Margalit7, § 1.2.7], the following statement holds: given a collection of essential arcs $\alpha _1,\dots ,\alpha _k$ in $\mathcal {S}$ , which have all distinct endpoints and are pairwise non-isotopic, one can replace each $\alpha _i$ with an arc $\alpha ^{\prime }_i\sim \alpha _i$ so that $\alpha ^{\prime }_1,\dots ,\alpha ^{\prime }_k$ are pairwise in minimal position. In fact, it suffices to choose a Riemannian metric of constant curvature on $\mathcal {S}$ such that $\partial \mathcal {S}$ is geodesic, and replace each $\alpha _i$ with its geodesic representative relative to the endpoints: the hypothesis on $\chi (\mathcal {S})$ ensures that we get a non-positively curved metric, so that geodesic representatives are unique; moreover, geodesic representatives are automatically pairwise in minimal position.

Among all connected surfaces with non-empty boundary, the only one with positive Euler characteristic is $\Sigma _{0,1}$ , that is, the disc: note that the statement holds vacuously also for the disc, which contains no essential arc. The following is a special case of the Alexander method [Reference Farb and Margalit7, Prop. 2.8].

In Proposition 2.11, one can enhance the requirement on $\Phi '$ to be the following: the map $\Phi '$ fixes pointwise a small neighbourhood $U\subset \mathcal {S}$ of the union $\alpha _0\cup \dotsb \cup \alpha _k\cup \beta \cup \partial \mathcal {S}$ . Here and in the following, a small neighbourhood is required to deformation retract onto $\alpha _0\cup \dotsb \cup \alpha _k\cup \beta \cup \partial \mathcal {S}$ by restriction of an ambient homotopy, defined on $\mathcal {S}$ and stationary on $\alpha _0\cup \dotsb \cup \alpha _k\cup \beta \cup \partial \mathcal {S}$ .

2.3 The cut locus of a mapping class

In this subsection, we fix a class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , represented by a diffeomorphism $\Phi $ , and study the isotopy classes of arcs and curves that it fixes.

Note that in the previous definition, we do not insist that the embedding $[0;1]\times [0;1]\hookrightarrow \mathcal {S}$ is orientation-preserving; see Figure 2.

Figure 2 A maximal collection of six pairwise non-parallel arcs $\alpha _0,\dotsc ,\alpha _5$ on a surface of type $\Sigma _{1,2}$ .

Note that a simplex in the fixed-arc complex of a class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ has dimension at most $-3\chi (\mathcal {S})-1$ if $\chi (\mathcal {S})<0$ , and at most $0$ if $\mathcal {S}$ is of type $\Sigma _{0,2}$ . In the second case, just note that each two disjoint essential arcs in $\Sigma _{0,2}$ are parallel. In the first case, let $\alpha _0,\dots ,\alpha _k$ be disjoint, pairwise non-parallel and essential arcs in $\mathcal {S}$ ; then cutting $\mathcal {S}$ along the arcs $\alpha _i$ yields a surface whose connected components are either hexagons or connected surfaces of negative Euler characteristic. Up to adjoining more arcs (and thus increase k), we can assume to have only hexagons. Each hexagon has 3 sides coming from $\partial \mathcal {S}$ and 3 sides coming from the cuts. If $\ell \geqslant 1$ denotes the number of hexagons, we have $3\ell =2(k+1)$ , as each arc contributes to 2 hexagons; moreover $\chi (\mathcal {S})=\ell -(k+1)$ , implying $3\chi (\mathcal {S})=3\ell -3(k+1)=-k-1$ .

See Figure 3 for an example of the cut locus of a mapping class obtained as a simple product of Dehn twists. Note that it is possible that two curves $c_i$ and $c_j$ cobound a cylinder in Y: in this case, the two curves are isotopic as non-oriented simple closed curves, but the isotopy bringing $c_i$ to $c_j$ , spanned by the cylinder of Y, ends with an orientation-reversing diffeomorphism $c_i\cong c_j$ , as the two curves inherit their orientation from Y while being on opposite sides of the cylinder contained in Y. Hence the isotopy classes of the curves $c_1,\dots ,c_h$ , considered as oriented curves, are all distinct.

Figure 3 Two examples of a decomposition into the ‘yellow’ and the ‘white’ region according to a fixed mapping class $\varphi $ . In the first case, $\varphi $ is given by the product of the Dehn twists along the curves $d_1,\dotsc ,d_7$ , and in the second case, it is just the Dehn twist along the single green curve d. In the second case, the mapping class $\varphi $ is $\partial $ -irreducible, the cut locus consists of the only isotopy class of d, oriented as a boundary of the yellow region, and the white region is just a collar neighbourhood of $\partial \mathcal {S}$ .

Note that the cut locus of the identity is empty, and the white and yellow decomposition consists of a white region $W=\mathcal {S}$ and an empty yellow region. Vice versa, the cut locus of a non-trivial mapping class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ is always non-empty.

Definition 2.19 depends a priori on a choice of a maximal simplex in the fixed-arc complex of $\varphi $ ; there is, moreover, a subtle detail that we should check to guarantee that Definition 2.19 is well-posed: suppose that the disjoint arcs $\alpha ^{\prime }_0,\dotsc ,\alpha ^{\prime }_k$ are isotopic to the disjoint arcs $\alpha _0,\dotsc ,\alpha _k$ (that is, $\alpha ^{\prime }_i\sim \alpha _i$ for all $0\leqslant i\leqslant k$ ), so that the two collections of arcs represent the same maximal simplex in the fixed-arc complex of $\varphi $ ; then we need to check that the two collections of arcs give rise to the same collection of isotopy classes of oriented, disjoint simple closed curves $c_1,\dots , c_h$ .

We will prove directly that the cut locus only depends on $\varphi $ , and not on the chosen maximal simplex (and its representative) in the fixed-arc complex of $\varphi $ .

Proof. Up to isotoping $\beta $ to another arc, we can assume that $\beta $ is in minimal position with respect to the arcs $\alpha _0,\dots ,\alpha _k$ . By Proposition 2.11, we can represent $\varphi $ by a diffeomorphism $\Phi $ that fixes a closed neighbourhood $U'$ of the union $\alpha _0\cup \dotsb \cup \alpha _k\cup \beta \cup \partial \mathcal {S}$ . Let $U\subset U'$ be a small neighbourhood of $\alpha _0\cup \dotsb \cup \alpha _k\cup \partial \mathcal {S}$ .

If $\beta $ is disjoint from the arcs $\alpha _i$ , by maximality of the simplex $\alpha _0,\dots ,\alpha _k$ in the fixed-arc complex of $\varphi $ , we obtain that either $\beta $ is not essential (and can then be isotoped inside a small neighbourhood of $\partial \mathcal {S}$ , hence inside U) or $\beta $ is parallel to one of the arcs $\alpha _i$ (and can then be isotoped to a small neighbourhood of $\partial \mathcal {S}\cup \alpha _i$ , hence inside U).

If $\beta $ is not disjoint from the arcs $\alpha _i$ , let $\ell \geqslant 1$ be the number of transverse intersections of $\beta $ with $\alpha _0\cup \dotsb \cup \alpha _k$ : by induction, let us suppose that the statement of the lemma holds whenever $\beta $ is replaced by an arc that can be isotoped so as to have at most $\ell -1$ intersection points with the arcs $\alpha _i$ . Suppose that p is one intersection point of $\beta $ with one arc $\alpha _i$ : suppose further that the segment $[\alpha _i(0);p]\subset \alpha _i$ contains no other point of $\alpha _i\cap \beta $ in its interior (this means that p is an outermost point of $\alpha _i\cap \beta $ along $\alpha _i$ ).

We can operate a surgery on $\beta $ and produce two arcs $\beta '$ and $\beta ''$ also contained in $U'$ and transverse to $\alpha _0\cup \dotsb \cup \alpha _k$ ; see Figure 4: the arc $\beta '$ is obtained by smoothing the concatenation of the segments $[\beta (0);p]\subset \beta $ and $[p;\alpha _i(0)]\subset \alpha _i$ , whereas the arc $\beta ''$ is obtained by smoothing the concatenation of the segments $[\alpha _i(0);p]\subset \alpha _i$ and $[p;\beta (1)]\subset \beta $ . We assume that $\beta '$ and $\beta ''$ are disjoint and have an endpoint on a small interval of $\partial \mathcal {S}$ centred at $\alpha _i(0)$ , on opposite sides with respect to $\alpha _i(0)$ .

Figure 4 If $\varphi $ is the Dehn twist along the curve d, then the blue arcs $\alpha _0,\dotsc ,\alpha _3$ constitute a maximal simplex in the fixed-arc complex of $\varphi $ ; the subset U is a small neighbourhood of the union of the blue arcs and the black boundary curve. The red arc $\beta $ intersects $\alpha _2$ transversally, and the surgery produces the yellow and violet arcs $\beta '$ and $\beta ''$ .

Both $\beta '$ and $\beta ''$ are contained in $U'$ , hence they are fixed pointwise by $\Phi $ : in particular the isotopy classes of $\beta '$ and $\beta ''$ are fixed by $\Phi $ . Moreover, each of $\beta '$ and $\beta ''$ has strictly less than $\ell $ intersections with $\alpha _0\cup \dotsb \cup \alpha _k$ , and hence, by inductive hypothesis, each of $\beta '$ and $\beta ''$ can be isotoped to lie in U, relative to its endpoints.

Let $\hat \beta '$ and $\hat \beta ''$ be the two arcs obtained in this way, and assume that $\hat \beta '$ and $\hat \beta ''$ are transverse. If $\hat \beta '$ and $\hat \beta ''$ are not in minimal position, they must form some bigon in $\mathcal {S}$ ; the possibility of half-bigons in the sense of [Reference Farb and Margalit7, § 1.2.7] is irrelevant, since we consider arcs up to isotopy relative to the endpoints. Clearly, we can simplify all bigons formed by $\hat \beta '$ and $\hat \beta ''$ without losing that these two arcs are contained in U.

Suppose therefore that $\hat \beta '$ and $\hat \beta ''$ are in minimal position: then they are disjoint because $\beta '$ and $\beta ''$ were disjoint (in particular, it is automatic that $\hat \beta '$ and $\hat \beta ''$ do not form half-bigons). The arc $\beta $ is homotopic, relative to its endpoints, to the concatenation of the arcs $\hat \beta '$ and $\hat \beta ''$ , which can be connected using a small segment near $\alpha _i(0)$ . Note that this concatenation gives an embedded arc $\hat \beta $ with the same endpoints as $\beta $ ; since homotopic arcs relative to their endpoints are also isotopic relative to their endpoints, we have that $\beta $ is isotopic to $\hat \beta $ ; moreover, $\hat \beta $ lies in U.

When referring to the cut locus of a mapping class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , we will henceforth mean the cut locus of $\varphi $ with respect to any maximal simplex in the fixed-arc complex of $\varphi $ .

The cut locus of a mapping class behaves well under conjugation of the mapping class, as explained in the following Lemma.

Proof. Let $\alpha _0,\dots ,\alpha _k$ be disjoint arcs representing a maximal simplex in the fixed-arc complex of $\varphi $ , and represent $\varphi $ by a diffeomorphism $\Phi $ fixing pointwise a small neighbourhood U of $\alpha _0\cup \dots \cup \alpha _k\cup \partial \mathcal {S}$ ; finally, represent the cut locus of $\varphi $ by curves $c_1,\dots ,c_h$ contained in U. Then $\Psi $ induces an isomorphism from the fixed-arc complex of $\varphi $ to the fixed arc complex of $\psi \varphi \psi ^{-1}$ ; in particular $\Psi (\alpha _0),\dots ,\Psi (\alpha _k)$ are disjoint arcs representing a maximal simplex in the fixed-arc complex of $\psi \varphi \psi ^{-1}$ . Moreover, $\Psi (U)$ is a small neighbourhood of $\Psi (\alpha _0)\cup \dots \cup \Psi (\alpha _k)\cup \partial \mathcal {S}$ , which is fixed pointwise by the representative $\Psi \Phi \Psi ^{-1}$ of $\psi \varphi \psi ^{-1}$ . We can represent the cut locus of $\varphi $ by curves $c_1,\dots ,c_h\subset \partial U$ ; then

$$\begin{align*}\Psi(c_1),\dots,\Psi(c_h)\subset\partial\Psi(U)\end{align*}$$

are automatically curves representing the cut locus of $\psi \varphi \psi ^{-1}$ , according to Construction 2.18.

3.1 Yellow components, similarity and irreducibility

We fix oriented simple closed curves $c_1,\dots ,c_h\subset \mathcal {S}$ representing the cut locus of $\varphi $ , and we let $W\cup Y$ be the associated decomposition of $\mathcal {S}$ into its white and yellow regions.

Recall from Construction 2.18 that the curves $c_1,\dots ,c_h$ inherit an orientation from Y: we fix an orientation-compatible parametrisation of each curve $c_i$ : that is, an identification with $S^1$ . Note that in this way $c_1,\dots ,c_h$ are incoming curves for W and outgoing for Y. In fact, we have $\partial Y=\partial ^{\text {out}}Y=c_1\cup \dotsb \cup c_h =\partial ^{\text {in}}W$ , whereas $\partial ^{\text {out}}W=\partial ^{\text {out}}\mathcal {S}=\partial \mathcal {S}$ .

We fix a representative $\Phi \colon \mathcal {S}\to \mathcal {S}$ of $\varphi $ , which fixes the white region W pointwise: to see that this is possible, choose arcs $\alpha _0,\dots ,\alpha _k$ in a maximal simplex of the fixed-arc complex of $\varphi $ , choose a first representative $\Phi '$ of $\varphi $ fixing pointwise a small neighbourhood U of $\alpha _0\cup \dots \cup \alpha _k\cup \partial \mathcal {S}$ , construct W and Y starting from U, and use that the group $\operatorname {\mathrm {Diff}}(D^2,\partial D^2)$ of diffeomorphisms of a disc relative to its boundary is contractible, and in particular connected, in order to isotope $\Phi '$ relative to U to a diffeomorphism $\Phi $ fixing W pointwise.

We note that this representative $\Phi $ is unique up to an isotopy that is stationary on W: to see this, first note that there is a fibration

$$\begin{align*}\operatorname{\mathrm{Diff}}(Y,\partial Y)\hookrightarrow \operatorname{\mathrm{Diff}}(\mathcal{S},\partial\mathcal{S})\overset{\mathfrak{p}}{\to} \mathrm{Emb}_{\partial^{\text{out}}}(W,\mathcal{S}),\end{align*}$$

where $\mathrm {Emb}_{\partial ^{\text {out}}}(W,\mathcal {S})$ denotes the space of embeddings of W into $\mathcal {S}$ restricting to the identity on the boundary $\partial ^{\text {out}}W=\partial \mathcal {S}$ . A result of Earle–Schatz [Reference Earle and Schatz5] ensures that $\operatorname {\mathrm {Diff}}(\mathcal {S},\partial \mathcal {S})$ has contractible components for every compact orientable surface $\mathcal {S}$ such that every connected component of $\mathcal {S}$ is connected; in particular, $\operatorname {\mathrm {Diff}}(Y,\partial Y)$ also has contractible components. A result of Gramain [Reference Gramain10, Thm. 5] ensures that for disjoint, properly embedded arcs $\alpha _0,\dots ,\alpha _k\subset \mathcal {S}$ , the space $\mathrm {Emb}_{\coprod \partial \alpha _i}(\coprod \alpha _i,\mathcal {S})$ has also contractible components; this, together with contractibility of $\operatorname {\mathrm {Diff}}(D^2,\partial D^2)$ , implies that also $\mathrm {Emb}_{\partial ^{\text {out}}}(W,\mathcal {S})$ has contractible connected components. Thus, all spaces involved in the above fibration have contractible connected components; in particular the component $\operatorname {\mathrm {Diff}}(\mathcal {S},\partial \mathcal {S})_{\varphi }$ intersects the fibre $\mathfrak {p}^{-1}(W\subset \mathcal {S})\cong \operatorname {\mathrm {Diff}}(Y,\partial Y)$ in a connected or empty subspace, and the representative $\Phi $ of $\varphi $ witnesses that this intersection is non-empty, hence contractible, in particular connected.

For each path component $P\subseteq Y$ , the diffeomorphism $\Phi $ restricts to $\Phi |_P\colon P\to P$ , giving an element $\varphi _P\in \Gamma (P,\partial P)$ .

Note that, counting the components of $\partial Y$ , we obtain $h=\sum _{i=1}^r n_i\cdot s_i$ .

3.2 The group $\tilde Z(\varphi )$ and its relation to $Z(\varphi ,\Gamma (\mathcal {S},\partial \mathcal {S}\,))$

In this subsection, we introduce a certain group $\tilde Z(\varphi )$ built out of small mapping class groups and symmetric groups. The peculiarity of $\tilde Z(\varphi )$ is that it admits a natural map $\varepsilon \colon \tilde Z(\varphi )\to Z(\varphi ,\Gamma (\mathcal {S},\partial \mathcal {S}))\subset \Gamma (\mathcal {S},\partial \mathcal {S})$ . In the next subsection, we will identify the kernel of $\varepsilon $ , and we will prove in the final subsection that $\varepsilon $ is surjective.

Recall that $\varphi _Y\in \Gamma (Y,\partial Y)$ denotes the mapping class represented by $\Phi |_Y$ . We consider the centraliser $Z(\varphi _Y)\subset \Gamma (Y)$ of $\varphi _Y$ in the extended mapping class group $\Gamma (Y)$ using the natural inclusion $\Gamma (Y,\partial Y)\subset \Gamma (Y)$ . Note that $\Gamma (Y)$ admits a natural map to $\mathfrak {S}_h\cong \mathfrak {S}_{\smash {\pi _0(\partial Y)}}$ given by the action of mapping classes on boundary components. This map restricts to a map $Z(\varphi _Y)\to \mathfrak {S}_h$ .

Similarly, we can consider the extended mapping class group $\Gamma ^{\mathfrak {S}_h}(W)$ that contains mapping classes of W that fix $\partial ^{\text {out}}W=\partial \mathcal {S}$ pointwise but may permute the h incoming boundary components of W: here we identify $\mathfrak {S}_h\cong \mathfrak {S}_{\pi _0(\partial ^{\text {in}} W)}\subset \mathfrak {S}_{\pi _0(\partial ^{\text {out}} W)}\times \mathfrak {S}_{\pi _0(\partial ^{\text {in}} W)}$ .

Definition 3.4. We define $\tilde Z(\varphi )$ as the fibre product

$$\begin{align*}\tilde Z(\varphi):= \Gamma^{\mathfrak{S}_h}(W) \times^{\mathfrak{S}_h} Z(\varphi_Y).\end{align*}$$

Gluing Y and W along $\partial Y=\partial ^{\text {in}} W$ yields a map of groups

$$\begin{align*}\hat\varepsilon\colon \Gamma^{\mathfrak{S}_h}(W) \times^{\mathfrak{S}_h} \Gamma(Y) \to\Gamma(\mathcal{S},\partial\mathcal{S}).\end{align*}$$

Explicitly, for a couple of mapping classes $(\psi _W,\psi _Y)$ , we choose representatives $\Psi _W\colon W\to W$ and $\Psi _Y\colon Y\to Y$ . The fact that $\psi _W$ and $\psi _Y$ project to the same permutation of $\pi _0(\partial ^{\text {in}}W)=\pi _0(\partial Y)\cong \left \{1,\dots ,h\right \}$ , together with the fact that both $\Psi _W$ and $\Psi _Y$ preserve the boundary parametrisation, implies that $\Psi _Y|_{\partial Y}=\Psi _W|_{\partial ^{\text {in}}W}$ , and hence we can glue the two diffeomorphisms to a diffeomorphism of $\mathcal {S}$ (we skip all details about smoothing the output homeomorphism near the gluing curves).

In the following lemma, we decompose $Z(\varphi _Y)$ , which is the second factor appearing in the formula for $\tilde Z(\varphi )$ .

Lemma 3.6. There is an isomorphism of groups

$$\begin{align*}Z(\varphi_Y)\cong\prod_{i=1}^r Z(\bar\varphi_i)\wr \mathfrak{S}_{s_i},\end{align*}$$

where $Z(\bar \varphi _i)\subset \Gamma _{g_i,(n_i)}$ is the centraliser in the extended mapping class group, and where $\smash {Z(\bar \varphi _i)\wr \mathfrak {S}_{s_i}=\big (Z(\bar \varphi _i)\big )^{s_i}\rtimes \mathfrak {S}_{s_i}}$ denotes the wreath product.

Proof. Let $\psi _Y\in Z(\varphi _Y)$ be a centralising mapping class, and represent $\psi _Y$ by a diffeomorphism $\Psi _Y$ preserving the boundary parametrisation. Furthermore, let $P,P'\subset Y$ be two connected components with $\Psi _Y(P)=P'$ ; then restricting the commutativity of $\psi _Y$ and $\varphi _Y$ to these two components, we obtain the equality $\smash {\smash {\varphi _Y|}_{P}^{\smash {\Psi _Y}}=\varphi _Y|_{P'}}$ in $\Gamma (P',\partial P')$ . This implies that P and $P'$ are similar, and using Notation 3.2, we have that each $Y_i$ is $\Psi _Y$ -invariant and therefore

$$\begin{align*}Z(\varphi_Y)=\prod_{i=1}^r Z(\varphi|_{Y_i}),\end{align*}$$

where $\varphi |_{Y_i}$ is defined using that $Y_i$ is a $\varphi $ -invariant union of connected components of Y. Now fix $1\leqslant i\leqslant r$ ; using the diffeomorphisms $\Xi _{i,j}$ for varying $1\leqslant j\leqslant s_i$ , we can identify the surface $Y_i$ with $\coprod _{1\leqslant j\leqslant s_i} \Sigma _{g_i,n_i}$ and thus identify $\Gamma (Y_i)$ with $\Gamma _{g_i,(n_i)}\wr \mathfrak {S}_{s_i}$ . Thus, $\varphi |_{Y_i}$ corresponds to the element

$$\begin{align*}(\bar\varphi_i,\dots,\bar\varphi_i)\in (\Gamma_{g_i,(n_i)})^{s_i} \subset\Gamma_{g_i,(n_i)}\wr \mathfrak{S}_{s_i}.\end{align*}$$

It follows that $Z(\varphi |_{Y_i})$ is isomorphic to $Z(\bar \varphi _i)\wr \mathfrak {S}_{s_i}$ .

We conclude the subsection by analysing the actual subgroup of $\mathfrak {S}_h$ over which the fibre product $\tilde Z(\varphi )$ lives. Now we will focus on the case in which W is connected because the exposition is a bit easier: indeed, the natural map $\Gamma ^{\mathfrak {S}_h}(W)\to \mathfrak {S}_h$ is surjective under this hypothesis on W, so we just have to describe the image of the map $Z(\varphi _Y)\to \mathfrak {S}_h$ .

The proof of Lemma 3.6 shows that the image of $Z(\varphi _Y)\to \mathfrak {S}_h\cong \mathfrak {S}_{\pi _0(\partial Y)}$ is the subgroup $\prod _i\mathfrak {H}_i\wr \mathfrak {S}_{s_i}$ consisting of those permutations of the set $\pi _0(\partial Y)$ that preserve each subset $\pi _0(\partial Y_i)$ for each $1\leqslant i\leqslant r$ , and send each subset $\pi _0(\partial Y_{i,j})$ to some subset $\pi _0(\partial Y_{i,j'})$ in a way that, under the identifications $\pi _0(\partial Y_{i,j})\cong \{1,\dots ,n_i\}\cong \pi _0(\partial Y_{i,j'})$ , gives a permutation in $\mathfrak {S}_{n_i}$ , which can also be attained by projecting an element $\bar \psi _i\in Z(\bar \varphi _i)$ .

3.3 The kernel of $\varepsilon $

Recall the gluing homomorphism $\hat \varepsilon \colon \Gamma ^{\mathfrak {S}_h}(W)\times ^{\mathfrak {S}_h}\Gamma (Y)\to \Gamma (\mathcal {S},\partial \mathcal {S})$ from the previous subsection. We proceed by identifying the kernel of $\hat \varepsilon $ . Note that $\hat \varepsilon $ has its image in the subgroup $\smash {\Gamma (\mathcal {S},\partial \mathcal {S})_{[c_1,\dots ,c_h]}}$ of $\Gamma (\mathcal {S},\partial \mathcal {S})$ containing all mapping classes $\psi $ that preserve the cut locus $[c_1,\dots ,c_h]$ of $\varphi $ : that is, send each oriented homotopy class of a curve $c_i$ to the oriented homotopy class of some (possibly different) curve $c_j$ . If $(\psi _W,\psi _Y)\in \Gamma ^{\mathfrak {S}_h}(W)\times ^{\mathfrak {S}_h}\Gamma (Y)$ belongs to the kernel of $\hat \varepsilon $ , then in particular $\hat \varepsilon (\psi _W,\psi _Y)$ acts trivially on the components of the cut locus. It follows that both $\psi _W$ and $\psi _Y$ project to the identity element in $\mathfrak {S}_h$ : that is, $(\psi _W,\psi _Y)$ is in fact contained in the subgroup $\Gamma (W,\partial W)\times \Gamma (Y,\partial Y)$ of $\Gamma ^{\mathfrak {S}_h}(W)\times ^{\mathfrak {S}_h}\Gamma (Y)$ . Hence the kernel of $\hat \varepsilon $ coincides with the kernel of the restriction of $\hat \varepsilon $ to $\Gamma (W,\partial W)\times \Gamma (Y,\partial Y)$ .

We can now use [Reference Farb and Margalit7, Thm. 3.18], in the version for disconnected surfaces: since no component of W or Y is a disc, the kernel of

$$\begin{align*}\hat\varepsilon\colon\Gamma(W,\partial W)\times\Gamma(Y,\partial Y)\to\Gamma(\mathcal{S},\partial\mathcal{S})\end{align*}$$

is generated by the couples $(D_{c_i},D_{c_i}^{-1})$ , where $D_{c_i}$ denotes the Dehn twist about the curve $c_i$ .

Since each component of W has at least one outgoing boundary, whereas the curves $c_i$ are incoming for W, we can apply [Reference Farb and Margalit7, Lem. 3.17] to the first coordinates of the elements $(D_{c_i},D_{c_i}^{-1})$ and conclude that they generate a subgroup of $\Gamma (W,\partial W)\times \Gamma (Y,\partial Y)$ isomorphic to $\mathbb {Z}^h$ . Finally, we note that the elements $(D_{c_i},D_{c_i}^{-1})$ belong to the subgroup $\tilde Z(\varphi )$ , as $D_{c_i}^{-1}\in \Gamma (Y,\partial Y)$ , being the inverse of a boundary twist, is a central element and in particular it commutes with $\varphi _Y$ . It follows that the kernel of $\varepsilon $ is the free abelian group of rank h generated by the elements $(D_{c_i},D_{c_i}^{-1})$ .

3.4 Surjectivity of $\varepsilon $

We now prove that the map $\varepsilon \colon \tilde Z(\varphi )\to Z(\varphi ,\Gamma (\mathcal {S},\partial \mathcal {S}))$ is surjective. In order to do so, let $\psi \in Z(\varphi ,\Gamma (\mathcal {S},\partial \mathcal {S})) \subset \Gamma (\mathcal {S},\partial \mathcal {S})$ be a centralising mapping class (see Definition 2.7). Then, by Lemma 2.22, $\psi $ preserves the cut locus of $\varphi $ .

We can fix a representative $\Psi \colon \mathcal {S}\to \mathcal {S}$ of $\psi $ that permutes the curves $c_1,\dots ,c_h$ preserving their parametrisation. In particular, $\Psi $ restricts to a diffeomorphism of W and of Y, respectively. Moreover, $\Psi $ fixes pointwise $\partial \mathcal {S}=\partial ^{\text {out}}W$ , and both $\Psi |_W$ and $\Psi |_Y$ are diffeomorphisms preserving the boundary parametrisation of W and Y, respectively. Consider now the mapping class $\varphi _Y\in \Gamma (Y,\partial Y)$ represented by $\Phi |_Y$ , and note that also $(\Psi |_Y)\circ (\Phi |_Y)\circ (\Psi ^{-1}|_Y)$ represents a mapping class in $\Gamma (Y,\partial Y)$ , which we denote by $\smash {\varphi _Y^{\smash {\Psi |_Y}}}$ .

We claim that $\varphi _Y=\varphi _Y^{\smash {\Psi |_Y}}$ holds in $\Gamma (Y,\partial Y)$ . To see this, note that gluing with the identity in $\Gamma (W,\partial W)$ gives a map $\smash {\lambda _Y^{\mathcal {S}}\colon \Gamma (Y,\partial Y)\to \Gamma (\mathcal {S},\partial \mathcal {S})}$ , which is injective by [Reference Farb and Margalit7, Thm. 3.18]; the claim follows from the observation that $\smash {\lambda _Y^{\mathcal {S}}(\varphi _Y^{\smash {\Psi |_Y}})=\varphi ^\Psi }$ , which by the hypothesis on $\Psi $ is equal to $\varphi =\lambda _Y^{\mathcal {S}}(\varphi _Y)$ .

It follows that $\Psi |_Y$ represents a mapping class $\psi _Y\in \Gamma (Y)$ that belongs to $Z(\varphi _Y)$ . Similarly, $\Psi |_W$ represents a class in $\Gamma ^{\mathfrak {S}_h}(W)$ , and it is clear that $\psi _W$ and $\psi _Y$ project to the same element of $\mathfrak {S}_h$ : that is, the couple $(\psi _W,\psi _Y)$ gives an element of $\tilde Z(\varphi )$ . It is also evident that $\varepsilon (\psi _W,\psi _Y)=\psi $ . This implies that $\varepsilon $ is surjective. Putting together the discussion of this and the previous subsections, we conclude the following.

Proposition 3.8. Let $g\geqslant 0, n\geqslant 1$ , let $\mathcal {S}$ be a surface of type $\Sigma _{g,n}$ , let $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , let $c_1,\dots ,c_h$ be a system of oriented curves representing the cut locus of $\varphi $ , and let W and Y be the corresponding white and yellow regions of $\mathcal {S}$ , using Notation 3.2. Then there is an isomorphism of groups induced by $\varepsilon $

where $\mathbb {Z}^h$ is the free abelian group generated by the elements $(D_{c_i},D_{c_i}^{-1})$ . If, moreover, W is connected, we can use Notation 3.7 and rewrite the isomorphism as

4.1 Notation and diagram categories

By ‘space’, we mean a topological space that is compactly generated and has the weak Hausdorff property. Let $\mathbf {Top}$ be the category of spaces; it is Cartesian closed, complete, and cocomplete.

For a topologically enriched category $\mathbf {I}$ and two objects $k,n\in {\mathrm {ob}}(\mathbf {I})$ , we denote by $\mathbf {I}\tbinom {k}{n}$ the space of morphisms from k to n, and we denote the identity of n by .

Example 4.4. If $\boldsymbol {X}=(X_n)_{n\in N}$ is a sequence of spaces, then we obtain a functor

$$\begin{align*}\boldsymbol{X}\colon N\wr\mathbf{\Sigma}\to \mathbf{Top}, \quad K\mapsto \boldsymbol{X}(K)=X_{k_1}\times\dotsb\times X_{k_r}.\end{align*}$$

Construction 4.5. Let $\boldsymbol {X}:= (X_n)_{n\in N}$ be a family of based spaces, together with based left actions of $G_n$ on $X_n$ for each $n\in N$ . Then the functor from Example 4.4 can be extended to a functor $\boldsymbol {G}\wr \mathbf {Inj}\to \mathbf {Top}$ as follows. For each injective map $u\colon \underline {r}\hookrightarrow \underline {r}'$ , each fibre has at most one element; therefore we obtain an extension $N\wr \mathbf {Inj}\to \mathbf {Top}$ by

$$\begin{align*}u_*(x_1,\dotsc,x_r):= (x_{\smash{u^{-1}(1)}},\dotsc,x_{\smash{u^{-1}(r')}}),\end{align*}$$

where we define $x_\varnothing $ to be the basepoint.

Moreover, $\boldsymbol {G}(K)$ acts on $\boldsymbol {X}(K)$ component-wise, so for a morphism $(\boldsymbol {\gamma },u)$ in $\boldsymbol {G}\wr \mathbf {Inj}$ , we can define $(\boldsymbol {\gamma },u)_*(x) := u_*(\boldsymbol {\gamma }\cdot x)$ .

Definition 4.6. Let $\mathbf {I}$ be a small and topologically enriched category and let $H\colon \mathbf {I}^{\operatorname {op}}\times \mathbf {I}\to {\large \mathbf {Top}}$ be a functor. Then we define the coend to be the coequaliser

4.2 Coloured operads

We assume that the reader is familiar with the classical notion of an operad, as it is for example presented in [Reference Markl, Shnider and Stasheff15]; in particular, the visualisation of operations by trees is taken for granted.

We will give a brief introduction to the notion of a coloured operad mostly for the purpose of fixing the notation we will use later. For a detailed introduction to coloured operads, we refer the reader to [Reference Yau28].

Definition 4.7. Let N be a fixed set. An N-coloured operad is a family of functors ${\mathscr {O}}\tbinom {-}{n}\colon (N\wr \mathbf {\Sigma })^{\operatorname {op}}\to \mathbf {Top}$ for each $n\in N$ , together with:

1. 1. choices of identities ;

2. 2. composition maps for each $n,k_i,l_{ij}\in N$ , which are of the form

   $$\begin{align*}{\mathscr{O}}\tbinom{k_1,\dotsc,k_r}{n}\times\prod_{i=1}^r {\mathscr{O}}\tbinom{l_{i1},\dotsc,l_{is_i}}{k_i}\to {\mathscr{O}}\tbinom{l_{11},\dotsc,l_{rs_r}}{n},\quad\! (\mu;\mu^{\prime}_1,\dotsc,\mu^{\prime}_r)\mapsto \mu\circ (\mu^{\prime}_1,\dotsc,\mu^{\prime}_r);\end{align*}$$

such that the usual coherence axioms from [Reference Yau28, § 11.2] are satisfied. For $\mu \in {\mathscr {O}}\tbinom {k_1,\dotsc ,k_r}{n}$ , we will call n the output, $(k_1,\dotsc ,k_r)$ the input profile, and $\# \mu := r$ the arity of $\mu $ . For the first few values of r, we call $\mu $ nullary, unary, respectively binary if $\mu $ has arity $0$ , $1$ , respectively $2$ . For the empty tuple, we will write ${\mathscr {O}}\tbinom {}{n}$ for the space of nullaries. We say that ${\mathscr {O}}$ is $\mathfrak {S}$ -free if for each $K=(k_1,\dotsc ,k_r)$ , the subgroup $\mathfrak {S}_K\subseteq \mathfrak {S}_r$ , which fixes the tuple K acts freely on ${\mathscr {O}}\tbinom {K}{n}$ .

We call ${\mathscr {O}}$ monochromatic if $N=*$ is just a singleton. In this case, we also write ${\mathscr {O}}(r):= {\mathscr {O}}\tbinom {*,\dotsc ,*}{*}$ for the space of r-ary operations. For an N-coloured operad ${\mathscr {O}}$ and a fixed colour $n\in N$ , we also consider the monochromatic operad ${\mathscr {O}}|_n$ with operation spaces $({\mathscr {O}}|_n)(r) := {\mathscr {O}}\tbinom {n,\dotsc ,n}{n}$ .

Additionally, we will use the short notation for ‘partial’ composition: for $\mu \in {\mathscr {O}}\tbinom {k_1,\dotsc ,k_r}{n}$ and $\mu '\in {\mathscr {O}}\tbinom {l_1,\dotsc ,l_s}{k_i}$ , we will write

Definition 4.9. Let ${\mathscr {O}}$ be an N-coloured operad. An ${\mathscr {O}}$ -algebra is an N-indexed family $\boldsymbol {X}:= (X_n)_{n\in N}$ of spaces, together with maps

$$\begin{align*}{\mathscr{O}}\tbinom{K}{n}\times \boldsymbol{X}(K)\to X_n,\quad (\mu;x_1,\dotsc,x_r)\mapsto \mu(x_1,\dotsc,x_r)\end{align*}$$

such that the usual coherence axioms from [Reference Yau28, § 13] are satisfied.

A morphism $f\colon \boldsymbol {X}\to \boldsymbol {X}'$ of ${\mathscr {O}}$ -algebras is a family $(f_n\colon X_n\to X^{\prime }_n)_{n\in N}$ of maps satisfying $f_n(\mu (x_1,\dotsc ,x_r)) = \mu (f_{k_1}(x_1),\dotsc ,f_{k_r}(x_r))$ for all operations $\mu $ and all elements $x_i$ . This gives rise to the category ${\mathscr {O}}\text {-}\mathbf {Alg}$ .

Definition 4.11. For a fixed colour set N, a morphism $\rho \colon {\mathscr {P}}\to {\mathscr {O}}$ of N-coloured operads is a family $\rho ^-_n\colon {\mathscr {P}}\tbinom {-}{n}\Rightarrow {\mathscr {O}}\tbinom {-}{n}$ of transformations such that we have, abbreviating $\rho :=\rho ^K_n$ for all K and n,

1. 1. for each $n\in N$ ;

2. 2. $\rho (\mu )\circ (\rho (\mu ^{\prime }_1),\dotsc ,\rho (\mu ^{\prime }_r)) = \rho (\mu \circ (\mu ^{\prime }_1,\dotsc ,\mu ^{\prime }_r))$ .

Each operad morphism $\rho \colon {\mathscr {P}}\to {\mathscr {O}}$ gives rise to a base-change adjunction

as follows: each ${\mathscr {O}}$ -algebra is a ${\mathscr {P}}$ -algebra by restriction. For the converse, we consider the absolute adjunction

, where $U^{{\mathscr {O}}}$ just forgets the action, and where for each N-indexed family ${\boldsymbol {X}}$ of spaces, we define $F^{{\mathscr {O}}}({\boldsymbol {X}})_n := \smash {\int ^{K\in N\wr \mathbf {\Sigma }}{\mathscr {O}}\tbinom {K}{n}\times \boldsymbol {X}(K)}$ . Then each ${\mathscr {P}}$ -algebra ${\boldsymbol {X}}$ can be presented as the reflexive coequaliser of $F^{\mathscr {P}} U^{\mathscr {P}} F^{\mathscr {P}} U^{\mathscr {P}} {\boldsymbol {X}} \rightrightarrows F^{\mathscr {P}} U^{\mathscr {P}}{\boldsymbol {X}}$ , whence the induced ${\mathscr {O}}$ -algebra $\rho _!{\boldsymbol {X}}$ is the reflexive coequaliser of $F^{\mathscr {O}} U^{\mathscr {P}} F^{\mathscr {P}} U^{\mathscr {P}} {\boldsymbol {X}} \rightrightarrows F^{\mathscr {O}} U^{\mathscr {P}}{\boldsymbol {X}}$ , compare [Reference Berger and Moerdijk2, § 4].

Intuitively, $\rho _!{\boldsymbol {X}}$ is a quotient of the free ${\mathscr {O}}$ -algebra over ${\boldsymbol {X}}$ by the existing ${\mathscr {P}}$ -action on ${\boldsymbol {X}}$ . This adjunction clearly respects compositions: if $\rho \colon {\mathscr {Q}}\to {\mathscr {P}}$ and $\rho '\colon {\mathscr {P}}\to {\mathscr {O}}$ are two morphisms of N-coloured operads, then we clearly have $(\rho '\circ \rho )^* = \rho ^*\circ \rho '{}^*$ , so by the uniqueness of left adjoints, we also have $(\rho '\circ \rho )_! \cong \rho ^{\prime }_!\circ \rho _!$ . When $\rho $ is clear from the context, we also write for the base-change adjunction.

4.3 The coloured surface operad

We define an $\mathbb {N}_{\geqslant 1}$ -coloured operad ${\mathscr {M}}$ , which is a ‘coloured’ version of Tillmann’s surface operad [Reference Tillmann26]; see Figure 5.

Figure 5 An element in . Note that the colours green, yellow and red only indicate which inputs belong together, while the actual ‘colours’ of the inputs are $2$ , $1$ and $2$ , respectively.

We first recall Segal’s cobordism category $\mathbf {M}$ [Reference Segal25], which is a topologically enriched category: objects of $\mathbf {M}$ are non-negative integers $n\geqslant 0$ ; a morphism from n to $n'$ is represented by a (possibly disconnected) Riemann surface W with n incoming and $n'$ outgoing boundary components; the surface W is equipped with a choice of collar neighbourhoods $U^{\text {in}}_{\partial W}$ and $U^{\text {in}}_{\partial W}$ of $\partial ^{\text {in}}W$ and $\partial ^{\text {out}}W$ , respectively; these neighbourhoods are equipped with:

1. 1. a holomorphic parametrisation $\tilde \vartheta ^{\text {in}}\colon \left \{1,\dots ,n\right \} \times S^1\times [0;1)\to U^{\text {in}}_{\partial W}$ ; and

2. 2. an antiholomorphic parametrisation $\tilde \vartheta ^{\text {out}}\colon \left \{1,\dots ,n'\right \} \times S^1\times [0;1)\to U^{\text {out}}_{\partial W}$ .

Note that restricting $\tilde \vartheta ^{\text {in}}$ and $\tilde \vartheta ^{\text {out}}$ to $\left \{1,\dots ,n\right \}\times S^1\times 0$ and $\left \{1,\dots ,n'\right \}\times S^1\times 0$ , respectively, we obtain parametrisations of $\partial ^{\text {in}}W$ and $\partial ^{\text {out}}W$ , respectively, as in Notation 2.1.

The space of morphisms $\mathbf {M}\tbinom {n\,}{n'}$ is the moduli space of conformal classes of such Riemann surfaces W, considered up to biholomorphism compatible with the choice of parametrised collar neighbourhoods of the incoming and outgoing boundary. We usually denote by $(W,\tilde \vartheta )$ a morphism, or shortly by W when it is not necessary to mention the parametrisation of the collar neighbourhood of the boundary; here $\smash {\tilde \vartheta \colon \left \{1,\dots ,n+n'\right \} \times S^1\times [0;1)\to W}$ is obtained by concatenation of $\tilde \vartheta ^{\text {in}}$ and $\tilde \vartheta ^{\text {out}}$ .

The composition of two morphisms $(W,\tilde \vartheta )\colon n\to n'$ and $(W',\tilde \vartheta ')\colon n'\to n''$ is given by gluing the Riemann surfaces $W\smallsetminus \partial ^{\text {out}}W$ and $W'\smallsetminus \partial ^{\text {in}}W'$ , using the identification $U^{\text {out}}_{\partial W}\smallsetminus \partial ^{\text {out}}W \cong U^{\text {in}}_{\partial W'}\smallsetminus \partial ^{\text {in}}W'$ given by

$$\begin{align*}\tilde\vartheta^{\text{out}}(j,\zeta,t)\equiv(\tilde\vartheta')^{\text{in}}(j,\zeta,1-t),\end{align*}$$

for all $1\leqslant j\leqslant n'$ , $\zeta \in S^1$ and $0<t<1$ . The resulting surface $W''$ is also endowed with collar neighbourhoods of the incoming and outgoing boundaries whose parametrisations are given by $\tilde \vartheta ^{\text {in}}$ and $(\tilde \vartheta ')^{\text {out}}$ , respectively. The identity of $n\in \mathbf {M}$ is described in Construction 4.13.

We can regard $R_n$ as a kind of twisted torus; it is the isometry group of $\coprod _n S^1$ .

Construction 4.13. We can embed $R_n$ in the endomorphism space $\mathbf {M}\tbinom {n}{n}$ ; see Figure 6: given an element $(z_1,\dotsc ,z_n,\sigma )\in R_n$ , we consider the morphism $(W,\tilde \vartheta )\colon n\to n$ given by the following:

1. 1. as a Riemann surface, W is $\left \{1,\dots ,n\right \}\times S^1\times [0;1]$ , with the canonical Riemann structure;

2. 2. we let $U_{\partial W}^{\text {in}}=W\smallsetminus \partial ^{\text {out}}W$ , and $\tilde \vartheta ^{\text {in}}\colon \left \{1,\dots ,n\right \}\times S^1\times [0;1)\hookrightarrow W$ is the canonical inclusion;

3. 3. we let $U_{\partial W}^{\text {out}}=W\smallsetminus \partial ^{\text {in}}W$ , and let $\tilde \vartheta ^{\text {out}}\colon \left \{1,\dots ,n\right \}\times S^1\times [0;1)\hookrightarrow W$ be

   $$\begin{align*}\tilde\vartheta^{\text{out}}(j,\zeta,t)=\left(\sigma^{-1}(j),z_j\cdot \zeta,1-t\right).\end{align*}$$

Figure 6 An instance of $R_n\hookrightarrow \mathbf {M}\tbinom {n}{n}$ .

This assignment embeds in fact $R_n$ as a group into the automorphisms of $n\in \mathbf {M}$ ; in particular, the identity of n can be described as the image of the unit of $R_n$ along the embedding; see Figure 6.

Consider now the subcategory $\mathbf {M}_\partial \subseteq \mathbf {M}$ containing all objects and those cobordisms W whose components have non-empty outgoing boundary. In detail, for fixed $k,n\geqslant 0$ , the morphism space $\mathbf {M}_\partial (k,n)$ looks as follows: $\pi _0\left (\mathbf {M}_\partial (k,n)\right )$ is indexed by the number $1\leqslant l\leqslant n$ of path components, an unordered partition $\{1,\dotsc ,n\}=\boldsymbol {n}_1\sqcup \dotsb \sqcup \boldsymbol {n}_l$ into non-empty subsets with $\min (\boldsymbol {n}_j)<\min (\boldsymbol {n}_{j+1})$ , an ordered partition $\{1,\dotsc ,k\} = \boldsymbol {k}_1\sqcup \dotsb \sqcup \boldsymbol {k}_l$ into possibly empty sets, and genera $g_1,\dotsc ,g_l\geqslant 0$ . If we let $n_j:=\#\boldsymbol {n}_j$ and $k_j:= \# \boldsymbol {k}_j$ , then the corresponding path component is homotopy equivalent to $\mathfrak {M}_{g_1,k_1+n_1}\times \dotsb \times \mathfrak {M}_{g_l,k_l+n_l}$ , by restricting collar parametrisations to boundary parametrisations.

Let $n,n'\in \mathbf {M}_\partial $ , and note that the embedding $R_n\subset \mathbf {M}\tbinom {n}{n}$ has image inside $\mathbf {M}_\partial \tbinom {n}{n}$ . In the following lemma, we consider the right action of $R_n$ on $\mathbf {M}_\partial \tbinom {n}{n'}$ by precomposition.

Definition 4.15. We consider $\mathbf {M}_\partial $ as a symmetric monoidal category with monoidal sum being the disjoint union; since the monoidal sum behaves as the usual sum of natural numbers on objects, we have an associated coloured operad ${\mathscr {M}}$ with colours $\mathbb {N}_{\geqslant 1}=\{1,2,\dotsc \}$ and

$$\begin{align*}{\mathscr{M}}\tbinom{k_1,\dotsc,k_r}{n} := \mathbf{M}_\partial\tbinom{k_1+\dotsb+k_r}{n}.\end{align*}$$

Note that the restriction ${\mathscr {M}}|_1$ to the colour $1$ is exactly Tillmann’s surface operad [Reference Tillmann26]. For each ${\mathscr {M}}$ -algebra $\boldsymbol {X}=(X_n)_{n\geqslant 1}$ , the space $X_1$ is an algebra over the classical surface operad ${\mathscr {M}}|_1$ .

Example 4.16. In contrast to the little disc operads, the initial ${\mathscr {M}}$ -algebra is non-trivial: for instance, its colour- $1$ part ${\mathscr {M}}\tbinom {}{1}$ homotopy equivalent to the familiar collection of moduli spaces

$$\begin{align*}{\mathscr{M}}\tbinom{}{1} = \mathbf{M}_\partial\tbinom{0}{1} \simeq \coprod_{g\geqslant 0}\mathfrak{M}_{g,1}.\end{align*}$$

4.4 Tensor products and based operads

Definition 4.17. In [Reference Boardman and Vogt3, § II.3], Boardman and Vogt constructed a tensor product for operads. We are only interested in the following special case: let $\mathscr {A}$ be a monochromatic operad and $\mathbf {I}$ be a small, topologically enriched category with object set N. Then $\mathscr {A}\otimes \mathbf {I}$ is an N-coloured operad with operation spaces

$$\begin{align*}(\mathscr{A}\otimes\mathbf{I})\tbinom{k_1,\dotsc,k_r}{n} = \mathscr{A}(r)\times \prod_{i=1}^r \mathbf{I}\tbinom{k_i}{n},\end{align*}$$

together with the following structure, where we denote operations in $\mathscr {A}\otimes \mathbf {I}$ by $\mu \otimes (\nu _1,\dotsc ,\nu _r)$ with $\mu \in \mathscr {A}(r)$ and $\nu _i\in \mathbf {I}\tbinom {k_i}{n}$ :

1. 1. symmetric actions $\tau ^*(\mu \otimes (\nu _1,\dotsc ,\nu _r)) = (\tau ^*\mu )\otimes (\nu _{\tau (1)},\dotsc ,\nu _{\tau (r)})$ ;

2. 2. identities ;

3. 3. compositions

   $$ \begin{align*} & \left(\mu\otimes (\nu_1,\dotsc,\nu_r)\right)\circ \left(\mu_1\otimes (\nu_{1,1},\dotsc,\nu_{1,s_1}),\dotsc,\mu_r\otimes (\nu_{r,1},\dotsc,\nu_{r,s_r})\right) \\ &:= \left(\mu\circ (\mu_1,\dotsc,\mu_r)\right)\otimes \left(\nu_1\circ\nu_{1,1},\dotsc,\nu_1\circ\nu_{1,s_1},\dotsc, \nu_r\circ\nu_{r,1},\dotsc,\nu_r\circ\nu_{r,s_r}\right). \end{align*} $$

For $n\in N$ , we also abbreviate . Note that $(\mathscr {A}\otimes \mathbf {I})$ -algebras are the same as enriched functors $\mathbf {I}\to \mathscr {A}\text {-}\mathbf {Alg}$ .

This construction is bifunctorial: if $\rho _1\colon \mathscr {A}\to \mathscr {A}'$ is a morphism of operads and $\rho _2\colon \mathbf {I}\to \mathbf {I}'$ is a functor that is the identity on objects, then we get a morphism $\rho _1\otimes \rho _2\colon \mathscr {A}\otimes \mathbf {I}\to {\mathscr {A}'}\otimes {\mathbf {I}'}$ .

Example 4.18. Regard N as the discrete category with objects N. Then we get

$$\begin{align*}(\mathscr{A}\otimes N)\tbinom{k_1,\dotsc,k_r}{n} = \begin{cases}\mathscr{A}(r) & \text{for}\ k_1=\dotsb=k_r=n\\\varnothing & \text{else,}\end{cases}\end{align*}$$

and $(\mathscr {A}\otimes N)$ -algebras are just N-indexed families of $\mathscr {A}$ -algebras. One example that will be of particular importance for us later is the operad ${\mathscr {D}}_1\otimes N$ , which has a copy of the little $1$ -discs operad ${\mathscr {D}}_1$ in each colour $n\in N$ .

5.1 Coloured operads with homological stability

The notation ‘ ’ is pictorially inspired by the following example:

Example 5.3. Recall Definition 4.15 and Example 4.18. We have a map of operads $\imath _{\mathscr {M}}\colon {\mathscr {D}}_2\otimes \mathbb {N}_{\geqslant 1}\to {\mathscr {M}}$ given by applying the classical inclusion of ${\mathscr {D}}_2$ into Tillmann’s surface operad level-wise; see Figure 7. This morphism restricts to a map ${\mathscr {D}}_1\otimes \mathbb {N}_{\geqslant 1}\to {\mathscr {M}}$ of operads satisfying the weak homotopy commutativity condition and thus turns ${\mathscr {M}}$ into an operad under ${\mathscr {D}}_1$ .

Figure 7 An instance of .

The input blocking maps $\beta \colon {\mathscr {M}}\tbinom {K}{n}\to {\mathscr {M}}\tbinom {}{n}$ are induced by capping each ingoing boundary curve with a disc, and the abelian monoid $\pi _0({\mathscr {M}}\tbinom {}{n})$ contains all isomorphism types of surfaces $\mathcal {S}$ with n ordered outgoing boundary curves and no incoming boundary curve with $\mathcal {S}$ possibly disconnected, such that each path component of $\mathcal {S}$ has non-empty boundary. The addition on $\pi _0({\mathscr {M}}\tbinom {}{n})$ is given by gluing n pairs of pants; the neutral element is given by an ordered collection of n discs. The abelian monoid $\pi _0({\mathscr {M}}\tbinom {}{n})$ is finitely generated: for instance, it can be generated by the following elements $e_n^i$ and $\smash {e_n^{\smash {i,j}}}$ ; see Figure 8:

1. 1. For $1\leqslant i\leqslant n$ , we let $e_n^i$ be the isomorphism type of surfaces with n path components, such that the component carrying the i ^th boundary curve has genus $1$ , whereas all others components are discs.

2. 2. For each $1\leqslant i<j\leqslant n$ , we let $e_n^{\smash {i,j}}$ be the isomorphism type of surfaces with $n-1$ path components, all of genus $0$ , such that one component is a cylinder carrying the i ^th and the j ^th boundary curve.

Figure 8 The three generators $e_2^1$ , $e_2^2$ and $e_2^{1,2}$ of $\pi _0({\mathscr {M}}\tbinom {}{2})$ .