1 Introduction
The Madsen–Weiss theorem [Reference Madsen and Weiss14] can be formulated as follows: let $\mathfrak {M}_{g,1}$ denote the moduli space of Riemann surfaces of genus $g\geqslant 0$ with one parametrised boundary curve. By [Reference Miller19] and [Reference Bödigheimer4], the collection
admits the structure of an $E_2$ algebra, more precisely an algebra over the little $2$ discs operad ${\mathscr {D}}_2$ . Madsen and Weiss identify the group completion $\Omega B\mathfrak {M}_{*,1}$ with the infinite loop space $\Omega ^{\infty }\mathbf {MTSO}(2)$ , where $\mathbf {MTSO}(2)$ is the twodimensional oriented tangential Thom spectrum [Reference Galatius, Madsen, Tillmann and Weiss8].
One can consider the analogous problem with $\mathfrak {M}_{*,1}$ replaced by the mapping space ${\mathrm {map}}(X,\mathfrak {M}_{*,1})$ . This space is again a ${\mathscr {D}}_2$ algebra by pointwise composition, and it is our goal to understand its group completion $\Omega B\,{\mathrm {map}}(X,\mathfrak {M}_{*,1})$ . Note that the ${\mathscr {D}}_2$ algebra structure extends to an algebra structure over Tillmann’s surface operad built out of $\mathfrak {M}_{*,1}$ , so the main theorem of [Reference Tillmann26] implies the homotopy type we wish to understand is an infinite loop space.
In this article, we will focus on the simplest nontrivial case $X=S^1$ : i.e., we consider the free loop space $\Lambda \mathfrak {M}_{*,1}:={\mathrm {map}}(S^1,\mathfrak {M}_{*,1})$ ; we will briefly discuss in the appendix the general case, which is very similar.
For any discrete group $\Gamma $ , one can identify $\Lambda B\Gamma \simeq \coprod _{[\gamma ] \in {\mathrm {Conj}}(\Gamma )} BZ(\gamma ,\Gamma )$ , where ${\mathrm {Conj}}(\Gamma )$ denotes the set of conjugacy classes of $\Gamma $ , and $Z(\gamma ,\Gamma )$ is the centraliser of $\gamma \in \Gamma $ . Note also that the isomorphism type of the group $Z(\gamma ,\Gamma )$ only depends on the conjugacy class of $\gamma \in \Gamma $ .
The problem we address in this paper is strongly related to analysing the structure of centralisers of elements of mapping class groups: indeed, recall that $\mathfrak {M}_{g,1}$ is a classifying space for the mapping class group $\Gamma _{g,1}$ of a smooth oriented surface of genus g with one parametrised boundary curve; we then have a homotopy equivalence
Results
The free loop space of the moduli space of surfaces of genus g with n parametrised boundary circles, $\Lambda \mathfrak {M}_{g,n}$ , admits an action by the isometry group of the disjoint union of n oriented circles: that is, by $T^n \rtimes \mathfrak {S}_n=(S^1)^n\rtimes \mathfrak {S}_n$ .
We introduce an irreducibility criterion for mapping classes that is invariant under conjugation. We then consider, for any $n\geqslant 1$ and $g \geqslant 0$ , the subspace $\mathfrak C_{g,n} \subseteq \Lambda \mathfrak {M}_{g,n}$ of free loops whose corresponding conjugacy classes of elements in $\pi _0 (\mathfrak {M}_{g,n}) \cong \Gamma _{g,n}$ are irreducible. The pointwise action of $T^n \rtimes \mathfrak {S}_n$ on $\Lambda \mathfrak {M}_{g,n}$ restricts to an action on $\mathfrak C_{g,n}$ , and the main result of this work is the following identification, where stands for homotopy quotient.
Theorem 1.1. There is a weak homotopy equivalence
The key to proving this theorem is a good understanding of mapping class groups of surfaces (also with more than one boundary component) as well as an extension of classical operadic techniques to a coloured setting. As a first step, we prove a structure result for centralisers of mapping classes in $\Gamma _{g,n}$ , which might be of independent interest: see Proposition 3.8.
Second, we develop a machinery for Ncoloured operads with homological stability ${\mathscr {O}}$ containing a suboperad ${\mathscr {P}}$ , such as a family of topological groups: the group completion of the derived relatively free algebra $\tilde {F}_{{\mathscr {P}}}^{{\mathscr {O}}}(\boldsymbol {X})$ over a ${\mathscr {P}}$ algebra $\boldsymbol {X}$ is computed colourwise as an infinite loop space, under suitable assumptions on ${\mathscr {O}}$ and ${\mathscr {P}}$ ; see Theorem 5.9. This part of the work is a generalisation of [Reference Tillmann26] and [Reference Basterra, Bobkova, Ponto, Tillmann and Yeakel1] to the coloured and relative setting.
The two ingredients are put together by proving that $\Lambda \mathfrak {M}_{*,1}$ is the colour $1$ part of a relatively free algebra over a coloured version ${\mathscr {M}}$ of Tillmann’s surface operad, relative to a suboperad built out of $T^n \rtimes \mathfrak {S}_n$ ; the ‘relative generators’ are precisely the spaces $\mathfrak C_{g,n}$ mentioned above: see Theorem 6.5.
Related work
One approach to studying classifying spaces of diffeomorphism groups pertains to the notion of cobordism categories. It was pioneered with the breakthrough theorem by Madsen and Weiss and refined by Galatius, Madsen, Tillmann and Weiss [Reference Galatius, Madsen, Tillmann and Weiss8].
Recall that, in the orientable setting, the cobordism category $\mathrm {Cob}_d$ is a topological category, with object space given by the union of all moduli spaces of closed, oriented $(d1)$ manifolds, and morphism space given by the union of all moduli spaces of compact, oriented dmanifolds with incoming and outgoing boundary. It is natural to study two related generalisations of $\mathrm {Cob}_d$ , in an equivariant and a parametrised direction:

1. for a (topological) group G, we can consider the Gequivariant cobordism category $\mathrm {Cob}_d^G$ : objects and morphisms are, respectively, $(d1)$  and dmanifolds endowed with an (continuous) action of G by orientationpreserving diffeomorphisms;

2. for a topological space Y, we can consider the Yparametrised cobordism category $\mathrm {Cob}_d(Y)$ : objects and morphisms are, respectively, orientable $(d1)$  and dmanifold bundles over Y.
In the case $G=\mathbb {Z}$ and $Y=S^1$ , there is a continuous functor $\mathrm {Cob}_d^{\mathbb {Z}}\to \mathrm {Cob}_d(S^1)$ , given by taking mapping tori: using that every smooth bundle over $S^1$ is induced from a diffeomorphism, this functor is in fact a levelwise equivalence. For more general groups G and $Y = BG$ , the analogous argument pertaining to Gactions and bundles over $BG$ can fail: see [Reference Reinhold22] for a discussion of this phenomenon and counterexamples in case $G = \text {SU}(2)$ .
Our work can be seen as a contribution toward understanding the homotopy type of ${\mathrm {Cob}_d^{\mathbb {Z}}\simeq \mathrm {Cob}_d(S^1)}$ : gluing a pair of pants gives a map of monoids $\Lambda \mathfrak {M}_{*,1}\to \mathrm {Cob}_2(S^1)_{S^1}$ , where $\mathrm {Cob}_2(S^1)_{S^1}$ denotes the full subcategory of $\mathrm {Cob}_2(S^1)$ on a single object represented by a trivial $S^1$ bundle over $S^1$ .
In the nonparametrised setting, the composition $\mathfrak {M}_{*,1}\to \mathrm {Cob}_2_{S^1}\to \mathrm {Cob}_2$ is known to induce an equivalence after taking classifying spaces; we hope that in a similar way, the understanding of $B\Lambda \mathfrak {M}_{*,1}$ can shed some light on the homotopy type of $B\mathrm {Cob}_2^{\mathbb {Z}} \simeq B\mathrm {Cob}_2(S^1)$ in future work.
In the case of a finite group G, the homotopy type of $\mathrm {Cob}_d^{G}$ was recently determined by Szűcs and Galatius [Reference Galatius and Szűcs9]. In work by Raptis and Steimle [Reference Raptis and Steimle20], parametrised cobordism categories $\mathrm {Cob}_d(Y)$ featured as a tool to prove index theorems; however, it was not necessary for the scopes of that work to describe the homotopy type of the classifying spaces of these categories. The much older work of Kreck on bordisms of diffeomorphisms [Reference Kreck13] can be seen as a description of $\pi _0\left (\mathrm {Cob}_d(S^1)\right )$ .
Outline
In Section 2, we recall Alexander’s method concerning arc systems. We apply these concepts to associate with a mapping class $\varphi \in \Gamma _{g,n}$ a canonical decomposition of $\Sigma _{g,n}$ along a system of simple closed curves, called the cut locus of $\varphi $ . The goal of Section 3 is a detailed understanding of centralisers of mapping classes: this uses the canonical decomposition described in the previous section in a crucial way.
In Section 4, we recall some basic definitions and constructions related to coloured operads and introduce the coloured surface operad ${\mathscr {M}}$ . In Section 5, we introduce the notion of a coloured operad with homological stability and prove a levelwise splitting result in the spirit of [Reference Basterra, Bobkova, Ponto, Tillmann and Yeakel1], which applies in particular to ${\mathscr {M}}$ . Finally, in Section 6, we show that $\Lambda \mathfrak {M}_{*,1}$ is the colour $1$ part of a relatively free ${\mathscr {M}}$ algebra, which in combination with the splitting result concludes the proof of Theorem 1.1.
We briefly discuss in Appendix A the analogue of Theorem 1.1 for a general parametrising space X (see Theorem A.2) as well as a weak form of naturality in X of the equivalence (see Theorem A.7); in Appendix B, we address two similar problems concerning group completion of free loop spaces, related to braid groups and symmetric groups, respectively.
2 Arc systems and the cut locus
The aim of this and the next section is to study centralisers in mapping class groups of surfaces. This interest is motivated by the following observation: for $g\geqslant 1$ , the space $\Lambda \mathfrak {M}_{g,1}\simeq \Lambda B\Gamma _{g,1}$ has one connected component for each conjugacy class $[\varphi ]\in {\mathrm {Conj}}(\Gamma _{g,1})$ ; this component is homotopy equivalent to $B Z(\varphi ,\Gamma _{g,1})$ , where we denote by $Z(\varphi ,\Gamma _{g,1})\subset \Gamma _{g,1}$ the centraliser of $\varphi $ in $\Gamma _{g,1}$ : that is, the subgroup of all mapping classes $\psi \in \Gamma _{g,1}$ commuting with $\varphi $ .
In this section, we will first introduce some notation for surfaces and mapping class groups and then define the cut locus of a mapping class.
2.1 Surfaces and mapping class groups
We work in the entire article with smooth, oriented surfaces and orientationpreserving 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 nonempty 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.
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 $\vartheta ^{\text {out}}$ .
Definition 2.2. Let $\Phi \colon (\mathcal {S},\vartheta )\to (\mathcal {S}',\vartheta ')$ be an orientationpreserving diffeomorphism of surfaces. We say that $\Phi $ preserves the boundary parametrisation if the following conditions hold:

○ $\Phi $ restricts to diffeomorphisms $\Phi \colon \partial ^{\text {in}}\mathcal {S}\overset {\cong }{\to }\partial ^{\text {in}}\mathcal {S}'$ and $\Phi \colon \partial ^{\text {out}}\mathcal {S}\overset {\cong }{\to }\partial ^{\text {out}}\mathcal {S}'$ .

○ If $n:= \#\pi _0(\partial ^{\text {in}}\mathcal {S})=\#\pi _0(\partial ^{\text {in}}\mathcal {S}')$ and $n':=\#\pi _0(\partial ^{\text {out}}\mathcal {S})=\#\pi _0(\partial ^{\text {out}}\mathcal {S}')$ , then there exist permutations $\sigma ^{\text {in}}\in \mathfrak {S}_n$ and $\sigma ^{\text {out}}\in \mathfrak {S}_{n'}$ such that

– $(\Phi \circ \vartheta ^{\text {in}})(j,\zeta ) =(\vartheta ')^{\text {in}}(\sigma ^{\text {in}}(j),\zeta )$ for each $1\leqslant j\leqslant n$ and $\zeta \in S^1$ ;

– $(\Phi \circ \vartheta ^{\text {out}})(j,\zeta ) =(\vartheta ')^{\text {out}}(\sigma ^{\text {out}}(j),\zeta )$ for each $1\leqslant j\leqslant n'$ and $\zeta \in S^1$ .

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 nontrivial. 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.
Notation 2.3. For all $g\geqslant 0$ and $n\geqslant 1$ , we fix a model surface $\Sigma _{g,n}$ : it is a connected surface of genus g with n outgoing and no incoming boundary components. We say that $\mathcal {S}$ is of type $\Sigma _{g,n}$ if there exists a diffeomorphism $\mathcal {S}\to \Sigma _{g,n}$ preserving the boundary parametrisation.
Definition 2.4. The mapping class group $\Gamma (\mathcal {S},\partial \mathcal {S})$ is the group of isotopy classes of diffeomorphisms $\Phi \colon \mathcal {S}\to \mathcal {S}$ that fix the boundary pointwise: that is, $\Phi \circ \vartheta =\vartheta $ . Such a $\Phi $ is called a diffeomorphism of $(\mathcal {S},\partial \mathcal {S})$ . For $\mathcal {S}=\Sigma _{g,n}$ , we also write $\Gamma _{g,n}$ for $\Gamma (\mathcal {S},\partial \mathcal {S})$ . We usually denote isotopy classes by small Greek letters $\varphi $ and use capital Greek letters for diffeomorphisms.
Remark 2.5. Note that any diffeomorphism $\Xi \colon \mathcal {S}\to \mathcal {S}'$ induces an identification of the groups $\Gamma (\mathcal {S},\partial \mathcal {S})\cong \Gamma (\mathcal {S}',\partial \mathcal {S}')$ by conjugation with $\Xi $ : the mapping class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , represented by the diffeomorphism $\Phi $ , corresponds to the mapping class $\varphi ^\Xi \in \Gamma (\mathcal {S}',\partial \mathcal {S}')$ , represented by $\Xi \circ \Phi \circ \Xi ^{1}$ .
Definition 2.6. Let $\mathfrak {H}\subset \mathfrak {S}_{\pi _0(\partial ^{\text {out}}\mathcal {S})}\times \mathfrak {S}_{\pi _0(\partial ^{\text {in}}\mathcal {S})}$ be a subgroup, where ‘ $\mathfrak {S}$ ’ denotes the symmetric group on the finite set given as index. We define the extended mapping class group $\Gamma ^{\mathfrak {H}}(\mathcal {S})$ as the group of isotopy classes of diffeomorphisms $\Phi \colon \mathcal {S}\to \mathcal {S}$ that preserve the orientation of $\mathcal {S}$ and the boundary parametrisation, and permute the boundary components of $\partial ^{\text {out}}\mathcal {S}$ and $\partial ^{\text {in}}\mathcal {S}$ according to a pair of permutations in $\mathfrak {H}$ .
If we take $\mathfrak {H}=\mathfrak {S}_{\pi _0(\partial ^{\text {out}}\mathcal {S})}\times \mathfrak {S}_{\pi _0(\partial ^{\text {in}}\mathcal {S})}$ , we also write $\Gamma (\mathcal {S})$ for the extended mapping class group. If $\mathcal {S}=\Sigma _{g,n}$ , we also write $\Gamma _{g,n}^{\mathfrak {H}}=\Gamma ^{\mathfrak {H}}(\mathcal {S})$ and $\Gamma _{g,(n)}=\Gamma (\mathcal {S})$ for the extended mapping class groups.
Note that we have an extension $1\to \Gamma (\mathcal {S},\partial \mathcal {S})\to \Gamma ^{\mathfrak {H}}(\mathcal {S})\to \mathfrak {H}\to 1$ .
Definition 2.7. If G is a group, we denote by ${\mathrm {Conj}}(G)$ the set of conjugacy classes of G. For a group element $\gamma \in G$ , we denote by $Z(\gamma ,G)\subseteq G$ the centraliser of $\gamma $ : that is, the subgroup of all elements $\gamma '\in G$ that commute with $\gamma $ .
Notation 2.8. We fix, once and for all, for all conjugacy classes in ${\mathrm {Conj}}(\Gamma _{g,n})$ , a representative of the class. We denote by $\mathfrak {g}\colon \Gamma _{g,n}\to \Gamma _{g,n}$ the function of sets assigning to each element of $\Gamma _{g,n}$ the representative of its class.
Definition 2.9. Let $\mathcal {S}$ be a surface. We denote by $\mathfrak {M}(\mathcal {S})$ the moduli space of Riemann structures on $\mathcal {S}$ ; two Riemann structures on $\mathcal {S}$ are considered equivalent if there is a diffeomorphism $\Psi \colon \mathcal {S}\to \mathcal {S}$ fixing $\partial \mathcal {S}$ pointwise and pulling back one Riemann structure to the other. If $\mathcal {S}=\Sigma _{g,n}$ , we also write $\mathfrak {M}_{g,n}$ for the moduli space $\mathfrak {M}(\Sigma _{g,n})$ .
The hypothesis that every connected component of $\mathcal {S}$ has nonempty 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].
Definition 2.10. An arc in $\mathcal {S}$ is a smooth embedding $\alpha \colon [0;1]\hookrightarrow \mathcal {S}$ such that $\alpha ^{1}(\partial \mathcal {S})=\{0,1\}$ and $\alpha $ is transverse to $\partial \mathcal {S}$ . Two arcs are disjoint if their images are disjoint (also at the endpoints). An arc is essential if it does not cut $\mathcal {S}$ in two parts, one of which is a disc.
Two arcs $\alpha $ and $\alpha '$ are directly isotopic if $\alpha (0)=\alpha '(0)$ , $\alpha (1)=\alpha '(1)$ , and there is an isotopy of embeddings $[0;1]\to \mathcal {S}$ that is stationary on $\left \{0,1\right \}$ and connects $\alpha $ to $\alpha '$ . Two arcs are inversely isotopic if the previous holds after reparametrising one of the two arcs in the opposite direction. Two arcs are isotopic if they are directly or inversely isotopic; we write $\alpha \sim \alpha '$ if $\alpha $ and $\alpha '$ are isotopic.
Two arcs $\alpha $ and $\beta $ are in minimal position if they are disjoint at their endpoints, they intersect transversely, and the number of intersection points in $\alpha \cup \beta $ is minimal among all choices of $\alpha '\sim \alpha $ and $\beta '\sim \beta $ with $\alpha '$ and $\beta '$ transverse.
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 $\chi (\mathcal {S})=22gn\leqslant 0$ , 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 nonisotopic, 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 nonpositively curved metric, so that geodesic representatives are unique; moreover, geodesic representatives are automatically pairwise in minimal position.
Among all connected surfaces with nonempty 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].
Proposition 2.11. Let $\alpha _0,\dots ,\alpha _k$ and $\beta $ be a collection of essential arcs in $\mathcal {S}$ , and assume the following:

○ all arcs are pairwise in minimal position;

○ the arcs $\alpha _0,\dots ,\alpha _k$ are pairwise disjoint.
Let $\Phi $ be a diffeomorphism of $(\mathcal {S},\partial \mathcal {S})$ , and suppose that $\Phi $ fixes each of $\alpha _0,\dots ,\alpha _k$ and $\beta $ up to isotopy relative to the endpoints. Then $\Phi $ can be isotoped to a diffeomorphism $\Phi '$ of $\mathcal {S}$ that fixes $\alpha _0\cup \dotsb \cup \alpha _k\cup \beta $ pointwise.
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 $\alpha _0\cup \dotsb \cup \alpha _k\cup \beta \cup \partial \mathcal {S}$ .
Remark 2.12. The Alexander method, as stated in [Reference Farb and Margalit7], only applies under the additional hypothesis that the arcs $\alpha _0,\dots ,\alpha _k$ and $\beta $ are pairwise nonisotopic. We remark, however, that this hypothesis is not essential.
To see this, suppose that $\alpha _0,\dots ,\alpha _k$ and $\beta $ is a collection of arcs as in Proposition 2.11: up to reordering the arcs $\alpha _i$ , we can assume that there is $0\leqslant k'\leqslant k$ such that the arcs $\alpha _0,\dots ,\alpha _{k'}$ and $\beta $ are pairwise nonisotopic, and, moreover, each $\alpha _i$ with $i\geqslant k'+1$ is isotopic to some $\alpha _j$ with $j\leqslant k'$ .
We can then apply the Alexander method to the collection $\alpha _0,\dots ,\alpha _{k'}$ and $\beta $ , obtaining a diffeomorphism $\Phi '$ that fixes a small neighbourhood U of $\alpha _0\cup \dotsb \cup \alpha _{k'}\cup \beta $ pointwise. We then argue as follows: for each index $i\geqslant k'+1$ , there is an index $j\leqslant k'$ such that $\alpha _i$ and $\alpha _j$ are isotopic and in minimal position: this implies that they cobound (together with two segments in $\partial \mathcal {S}$ ) a rectangle in $\mathcal {S}$ , and, up to shrinking, we can assume that this rectangle lies already inside U: that is, we can assume that $\alpha _i$ is fixed pointwise by $\Phi '$ as well.
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.
Definition 2.13. Two arcs $\alpha $ and $\alpha '$ in $\mathcal {S}$ are directly parallel if they are disjoint and there is an embedding $[0;1]\times [0;1]\hookrightarrow \mathcal {S}$ restricting to $\alpha $ on $[0;1]\times \left \{0\right \}$ and to $\alpha '$ on $[0;1]\times \left \{1\right \}$ , and restricting to an embedding $\left \{0,1\right \}\times [0;1]\hookrightarrow \partial \mathcal {S}$ .
Two arcs $\alpha $ and $\alpha '$ are inversely parallel if the previous holds after reparametrising one of the two arcs in the opposite direction. Two arcs $\alpha $ and $\alpha '$ are parallel if they are directly or inversely parallel.
Note that in the previous definition, we do not insist that the embedding $[0;1]\times [0;1]\hookrightarrow \mathcal {S}$ is orientationpreserving; see Figure 2.
Definition 2.14. The fixedarc complex of $\varphi $ is an abstract simplicial complex whose vertices are all isotopy classes of essential arcs $\alpha $ in $\mathcal {S}$ fixed by $\varphi $ . A collection of isotopy classes of arcs $\alpha _0,\dotsc ,\alpha _k$ spans a ksimplex if the arcs $\alpha _0,\dotsc ,\alpha _k$ can be isotoped to disjoint, pairwise nonparallel arcs $\alpha ^{\prime }_0,\dotsc ,\alpha ^{\prime }_k$ .
The mapping class $\varphi $ is called $\partial $ irreducible if its fixedarc complex is empty and if $\mathcal {S}$ is not of type $\Sigma _{0,1}$ .
Example 2.15. Every isotopy class of essential arcs in $\mathcal {S}$ is fixed (up to isotopy) by the identity . Therefore is not $\partial $ irreducible, provided that $\mathcal {S}$ admits some essential arc; if $\mathcal {S}$ does not admit essential arcs, then $\mathcal {S}$ is a disc $\Sigma _{0,1}$ and we have prescribed, also in this case, that is not $\partial $ irreducible. The reason to regard as not being $\partial $ irreducible will become clear later; for the moment we make a simple comparison and say that, in a similar way, $1\in \mathbb {Z}$ is not considered a prime number.
Example 2.16. The fixedarc complex of $\varphi \in \Gamma _{0,2}\cong \mathbb {Z}$ is empty if and consists of uncountably many vertices, joined by no higher simplex if . Therefore every nontrivial element in $\Gamma _{0,2}$ is $\partial $ irreducible.
Example 2.17. For $g\geqslant 1$ the boundary Dehn twist $T_{\partial }\in \Gamma _{g,1}$ is $\partial $ irreducible, though there are plenty of isotopy classes of simple closed curves in $\Sigma _{g,1}$ that are fixed by $T_{\partial }$ : in fact, all simple closed curves are fixed, up to isotopy, by $T_{\partial }$ . Nevertheless, no isotopy class of essential arcs is fixed by $\varphi $ ; here it is crucial to consider isotopy classes of arcs relative to the endpoints.
Note that a simplex in the fixedarc 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 nonparallel 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 $3\chi (\mathcal {S})=3\ell 3(k+1)=k1$ .
Construction 2.18. Let $\mathcal {S}$ be not of type $\Sigma _{0,1}$ , let $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , and let $\alpha _0,\dots, \alpha _k$ be disjoint, essential, pairwise nonparallel arcs in $\mathcal {S}$ , representing a maximal simplex in the fixedarc complex of $\varphi $ . Let U be a closed, small neighbourhood of the union $\alpha _0\cup \dotsb \cup \alpha _k\cup \partial \mathcal {S}$ . The complement $\mathcal {S}\smallsetminus U$ consists of many regions, some of which may be discs: let $W\subset \mathcal {S}$ denote the union of U and all discs in $\mathcal {S}\smallsetminus U$ . Then W is a closed, possibly disconnected subsurface of $\mathcal {S}$ ; we denote by Y the closure of $\mathcal {S}\smallsetminus W$ . If $\partial W$ denotes the union of all boundary components of W, and $\partial Y$ denotes the union of all boundary components of Y, then $\partial W$ takes the form $\partial \mathcal {S}\cup c_1\cup \dotsb \cup c_h$ , for some $h\geqslant 0$ and some curves $c_1,\dots ,c_h\subset \mathcal {S}$ ; similarly $\partial Y=c_1\cup \dotsb \cup c_h$ . The curves $c_1,\dots ,c_h$ inherit a canonical boundary orientation from Y, which is oriented as subsurface of $\mathcal {S}$ .
Definition 2.19. For $\varphi $ and $\alpha _0,\dots, \alpha _k$ as above, we define the cut locus of $\varphi $ , relative to the simplex $\alpha _0,\dots ,\alpha _k$ , as the isotopy class of the multicurve $c_1,\dots ,c_h$ , denoted $[c_1,\dots ,c_h]$ . Here and in the following, a multicurve is an unordered collection of disjoint and oriented simple closed curves, and two multicurves are considered isotopic if there is an ambient isotopy bringing the first to the second.
The two regions W and Y are called the associated white and yellow regions or subsurfaces of $\mathcal {S}$ , and they depend, as subsets of $\mathcal {S}$ , on a choice of a multicurve representing the cut locus. If , we declare the cut locus to be empty and W to be the entire surface $\Sigma _{0,1}$ .
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 nonoriented simple closed curves, but the isotopy bringing $c_i$ to $c_j$ , spanned by the cylinder of Y, ends with an orientationreversing 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.
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 nontrivial mapping class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ is always nonempty.
Definition 2.19 depends a priori on a choice of a maximal simplex in the fixedarc complex of $\varphi $ ; there is, moreover, a subtle detail that we should check to guarantee that Definition 2.19 is wellposed: 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 fixedarc 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 fixedarc complex of $\varphi $ .
Lemma 2.20. Let $\varphi $ and $\alpha _0,\dots ,\alpha _k$ be as in Definition 2.19, and let $\beta $ be an arc whose endpoints are disjoint from the endpoints of $\alpha _0,\dots ,\alpha _k$ . Suppose that the isotopy class of $\beta $ is fixed by $\varphi $ ; then $\beta $ can be isotoped, relative to its endpoints, to an arc $\hat \beta $ lying in a small neighbourhood U of $\alpha _0\cup \dotsb \cup \alpha _k\cup \partial \mathcal {S}$ .
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 fixedarc 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 $\alpha _i(0)$ .
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 halfbigons 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 halfbigons). 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.
Proposition 2.21. Let $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ , and let $\alpha _0,\dots ,\alpha _k$ and $\beta _0,\dots ,\beta _{k'}$ be two sequences of arcs representing two different maximal simplices in the fixedarc complex of $\varphi $ ; we assume that all arcs $\alpha _0,\dots ,\alpha _k,\beta _0,\dots ,\beta _{k'}$ are in minimal position. Then the associated cut loci constitute the same isotopy class of an oriented multicurve in $\mathcal {S}$ .
Proof. Let $U_{\alpha }$ be a closed small neighbourhood of $\alpha _0\cup \dotsb \cup \alpha _k\cup \partial \mathcal {S}$ and $U_{\beta }$ be a closed small neighbourhood of $\beta _0\cup \dotsb \cup \beta _{k'}\cup \partial \mathcal {S}$ ; let $W_{\alpha }$ and $W_{\beta }$ be obtained from $U_{\alpha }$ and $U_{\beta }$ by adjoining the disc components of $\mathcal {S}\smallsetminus U_{\alpha }$ and $\mathcal {S}\smallsetminus U_{\beta }$ , respectively (see Definition 2.19).
By Lemma 2.20, we can find an isotopy of the identity of $(\mathcal {S},\partial \mathcal {S})$ bringing $U_{\alpha }$ in the interior of $U_{\beta }$ , and hence in the interior of $W_{\beta }$ : without loss of generality, in the following assume $U_{\alpha }\subseteq U_{\beta }\subseteq W_{\beta }$ . If D is a disc component of $\mathcal {S}\smallsetminus U_{\alpha }$ , then $D\smallsetminus U_{\beta }$ is a union of discs contained in $\mathcal {S}\smallsetminus U_{\beta }$ , and therefore $D\subset W_{\beta }$ . It follows that $W_{\alpha }\subseteq W_{\beta }$ . Since every component of $W_{\beta }$ touches $\partial \mathcal {S}$ , the map $\pi _0(\partial \mathcal {S})\to \pi _0(W_\beta )$ is surjective. This map factors through the canonical map $\pi _0(W_{\alpha })\to \pi _0(W_{\beta })$ , as $\partial \mathcal {S}\subset W_\alpha $ . We conclude that $\pi _0(W_{\alpha })\to \pi _0(W_{\beta })$ is surjective.
By the same argument we can find an isotopy of the identity of $\mathcal {S}$ bringing $U_{\beta }$ in the interior of $U_{\alpha }$ , and hence $W_{\beta }$ in the interior of $W_{\alpha }$ : as a consequence we can obtain a surjection $\pi _0(W_{\beta })\to \pi _0(W_{\alpha })$ , showing that $\#\pi _0(W_{\beta })=\#\pi _0(W_{\alpha })$ and that the map $\pi _0(W_{\alpha })\to \pi _0(W_{\beta })$ induced by the inclusion is in fact a bijection.
We next prove that each component V of $W_{\beta }\smallsetminus W_{\alpha }$ is a cylinder with one boundary curve equal to some $c_i$ and one boundary curve equal to some $\smash {c^{\prime }_j}$ . Fix a component $\smash {\bar W_{\beta }\subset W_{\beta }}$ , and let $\bar W_{\alpha }\subset W_{\alpha }$ be the unique component of $W_{\alpha }$ contained in $\bar W_{\beta }$ . We know that, conversely, $\bar W_{\beta }$ can be embedded in $\bar W_{\alpha }$ : since the genus is weakly increasing along embeddings of orientable surfaces with boundary, the surfaces $\bar W_{\alpha }$ and $\bar W_{\beta }$ must have the same genus; this implies in particular every component V of $\bar W_{\beta }\smallsetminus \bar W_{\alpha }$ has genus $0$ and touches at most one curve $c_i\in \partial \bar W_{\alpha }$ .
Since $\bar W_{\alpha }$ and $\bar W_{\beta }$ are connected, we have that a component V of $\bar W_{\beta }\smallsetminus \bar W_{\alpha }$ touches exactly one curve $c_i\subset \partial \bar W_{\alpha }$ , and since V cannot be a disc, we obtain that V is a surface of genus $0$ with at least two boundary components; more precisely, one boundary component of V is a curve $c_i\subset \partial \bar W_{\alpha }$ , and all other boundary components are curves $c^{\prime }_j\subset \partial \bar W_{\beta }$ .
This proves in particular that the number of boundary components of $\bar W_{\beta }$ is greater or equal to the number of boundary components of $\bar W_{\alpha }$ ; we can again reverse the rôles of $\alpha $ and $\beta $ and conclude that $\bar W_{\alpha }$ and $\bar W_{\beta }$ have the same number of boundary component; this in turn implies that every connected component V of $\bar W_{\beta }\smallsetminus \bar W_{\alpha }$ is a cylinder with one boundary curve equal to some $c_i$ and one boundary curve equal to some $c^{\prime }_j$ .
Thus, we obtain a bijection between the curves $c_1,\dots ,c_h$ and the curves $c^{\prime }_1,\dots ,c^{\prime }_{h'}$ , showing in particular that $h=h'$ ; note also that the bijection associates with each curve $c_i$ a curve $c^{\prime }_j$ in the same isotopy class of oriented curves.
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 fixedarc complex of $\varphi $ .
The cut locus of a mapping class behaves well under conjugation of the mapping class, as explained in the following Lemma.
Lemma 2.22. Let $\varphi ,\psi \in \Gamma (\mathcal {S},\partial \mathcal {S})$ be mapping classes, let $\Psi $ be a diffeomorphism representing $\psi $ , and let $[c_1,\dots ,c_h]$ be the cut locus of $\varphi $ ; then $[\Psi (c_1),\dots ,\Psi (c_h)]$ is the cut locus of $\psi \varphi \psi ^{1}$ . In particular, if $\varphi $ and $\psi $ commute, then $\psi $ preserves the cut locus of $\varphi $ as an unordered collection of isotopy classes of oriented simple closed curves.
Proof. Let $\alpha _0,\dots ,\alpha _k$ be disjoint arcs representing a maximal simplex in the fixedarc 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 fixedarc 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 fixedarc 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
are automatically curves representing the cut locus of $\psi \varphi \psi ^{1}$ , according to Construction 2.18.
3 Centralisers of mapping classes
We fix a surface $\mathcal {S}\cong \Sigma _{g,n}$ and a mapping class $\varphi \in \Gamma (\mathcal {S},\partial \mathcal {S})\cong \Gamma _{g,n}$ as in the previous section. In this section, we prove a structural result for the centraliser $Z(\varphi ,\Gamma (\mathcal {S},\partial \mathcal {S}))$ of $\varphi $ in $\Gamma (\mathcal {S},\partial \mathcal {S})$ ; see Proposition 3.8.
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 orientationcompatible 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 fixedarc 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
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 nonempty, 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 $\varphi _P\in \Gamma (P,\partial P)$ .
Definition 3.1. Two path components P and $P'$ of Y are similar if there is a diffeomorphism $\Xi \colon P\to P'$ preserving the boundary parametrisation and such that $\smash {\varphi _P=(\varphi _{P'})^\Xi }$ . Note that the path components of $\partial P$ are not equipped with a preferred order, as well as the path components of $\partial P'$ ; yet Definition 2.2 is meaningful here. See also Remark 2.5.
Notation 3.2. We write $Y=\coprod _{i=1}^r\coprod _{j=1}^{s_i}Y_{i,j}$ , where $Y_{1,1},\dotsc ,Y_{r,s_r}\subseteq Y$ are the connected components of Y and $Y_{i,j}$ is similar to $Y_{i',j'}$ if and only if $i=i'$ . We also let $Y_i:= \coprod _{j=1}^{s_i}Y_{i,j}$ . Here $r\geqslant 0$ is the number of similarity classes of components of Y (it can be $0$ if Y is empty, i.e., if $\varphi $ is the identity mapping class), whereas $s_i\geqslant 1$ is the number of components of Y belonging to the i ^{th} similarity class.
For each $1\leqslant i\leqslant r$ , there are unique $g_i\geqslant 0$ and $n_i\geqslant 1$ such that $Y_{i,j}$ is of type $\Sigma _{g_i,n_i}$ . We denote by $\varphi _{i,j}\in \Gamma (Y_{i,j},\partial Y_{i,j})$ the class represented by the restriction $\Phi _{Y_{i,j}}$ , Moreover, we fix diffeomorphisms $\Xi _{i,j}\colon Y_{i,j}\to \Sigma _{g_i,n_i}$ and assume that $\Xi _{i,j}$ preserves the boundary parametrisation.
The conjugation by $\Xi _{i,j}$ induces an identification $\Gamma (Y_{i,j},\partial Y_{i,j})\to \Gamma _{g_i,n_i}$ , under which $\varphi _{i,j}$ corresponds to some element $\smash {\bar \varphi _{i,j}:=\smash {(\varphi _{i,j})}^{\smash {\Xi _{i,j}}}}\in \Gamma _{g_i,n_i}$ . Up to replacing $\Xi _{i,j}$ by another diffeomorphism $Y_{i,j}\to \Sigma _{g_i,n_i}$ , we can assume that $\bar \varphi _{i,j}\in \Gamma _{g,n}$ coincides with $\mathfrak {g}(\bar \varphi _{i,j})$ : that is, it is the representative of its own conjugacy class (see Notation 2.8). Note that the diffeomorphism replacing $\Xi _{i,j}$ is not required to induce the same bijection of sets of boundary components as $\Xi _{i,j}$ . It can be helpful to remark that $\pi _0(\partial Y_{i,j})$ is not equipped a priori with a preferred order, and only after choosing $\Xi _{i,j}$ , we obtain an order on $\pi _0(\partial Y_{i,j})$ by pulling back the canonical order on $\pi _0(\partial \Sigma _{g_i,n_i})$ . Under the assumption that the diffeomorphisms $\Xi _{i,j}$ are wellchosen, we also have $\bar \varphi _{i,j}=\bar \varphi _{i,j'}$ for all $1\leqslant i\leqslant r$ and $1\leqslant j,j'\leqslant s_i$ . We therefore write $\bar \varphi _i:=\bar \varphi _{i,j}$ .
Note that, counting the components of $\partial Y$ , we obtain $h=\sum _{i=1}^r n_i\cdot s_i$ .
Lemma 3.3. In the situation above, for each $1\leqslant i\leqslant r$ and $1\leqslant j\leqslant s_i$ , we have that $\varphi _{i,j} \in \Gamma (Y_{i,j},\partial Y_{i,j})$ is $\partial $ irreducible.
Proof. Suppose that there is an essential arc $\beta \subset Y_{i,j}$ (that is, the endpoints of $\beta $ are on $\partial Y_{i,j}$ ) that is fixed up to isotopy by $\varphi _{i,j}$ . Then we can isotope $\Phi $ relative to $\mathcal {S}\smallsetminus \smash {\mathring {Y}_{i,j}}$ so that $\Phi $ fixes $\beta $ pointwise; without loss of generality, we assume that $\Phi $ already fixes $\beta $ pointwise.
We can extend $\beta $ to an arc $\alpha \subset \mathcal {S}$ with endpoints on $\partial \mathcal {S}$ by joining the endpoints of $\beta $ inside W with $\partial \mathcal {S}$ . Then $\Phi $ fixes $\alpha $ pointwise. By Lemma 2.20, we can isotope $\alpha $ into the region W. This implies that $\alpha $ is not in minimal position with respect to $\partial Y_{i,j}$ , and therefore $\alpha $ must form a bigon with the multicurve $\partial Y_{i,j}$ . Since $\alpha $ intersects $\partial Y_{i,j}$ in exactly two points, namely the endpoints of $\beta $ , there must be a bigon with one side equal to $\beta $ and the other contained in $\partial Y_{i,j}$ . This bigon is contained in $Y_{i,j}$ , contradicting the assumption that $\beta $ is essential in $Y_{i,j}$ .
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
Gluing Y and W along $\partial Y=\partial ^{\text {in}} W$ yields a map of groups
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).
Lemma 3.5. The restriction $\varepsilon =\hat \varepsilon _{\tilde Z(\varphi )}$ has image inside $Z(\varphi ,\Gamma (\mathcal {S},\partial \mathcal {S}))\subset \Gamma (\mathcal {S},\partial \mathcal {S})$ .
Proof. Note that is a central element of $\tilde Z(\varphi )$ : indeed, given a pair $(\psi _W,\psi _Y)\in \tilde Z(\varphi )$ , we have that $\psi _Y\in Z(\varphi _Y)$ , so $\psi _Y$ commutes with $\varphi _Y$ , and clearly commutes with $\psi _W$ in $\Gamma ^{\mathfrak {S}_h}(W)$ . Applying $\varepsilon $ , we obtain that $\varepsilon (\psi _W,\psi _Y)$ commutes with .
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
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
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
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 $Z(\varphi _Y)\to \mathfrak {S}_h$ .
Notation 3.7. We denote by $\mathfrak {H}_i\subset \mathfrak {S}_{n_i}$ the image of $Z(\bar \varphi _i)$ under the natural map $\Gamma _{g_i,(n_i)}\to \mathfrak {S}_{n_i}$ .
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
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