Point-pushing actions for manifolds with boundary

Given a manifold $M$ and a point in its interior, the point-pushing map describes a diffeomorphism that pushes the point along a closed path. This defines a homomorphism from the fundamental group of $M$ to the group of isotopy classes of diffeomorphisms of $M$ that fix the basepoint. This map is well-studied in dimension $d = 2$ and is part of the Birman exact sequence. Here we study, for any $d \geqslant 3$ and $k \geqslant 1$, the map from the $k$-th braid group of $M$ to the group of homotopy classes of homotopy equivalences of the $k$-punctured manifold $M \smallsetminus z$, and analyse its injectivity. Equivalently, we describe the monodromy of the universal bundle that associates to a configuration $z$ of size $k$ in $M$ its complement, the space $M \smallsetminus z$. Furthermore, motivated by our work on the homology of configuration-mapping spaces, we describe the action of the braid group of $M$ on the fibres of configuration-mapping spaces.


Introduction
Let M be a based, connected (smooth) manifold of dimension d 2 and denote by C k (M ) the configuration space of k unordered distinct points in its interior. We may think of it as the moduli space of k distinct points in M . Its universal bundle is the fiber bundle U k (M ) that associates to each k-tuple z ∈ C k (M ) the k-punctured manifold M z: The primary goal of this paper is to describe the monodromy action (up to homotopy) of the above fibre bundle push (M,z) : π 1 (C k (M ), z) −→ π 0 (hAut(M z)) where hAut(M z) denotes the homotopy equivalences of the complement of z in M ; when M has boundary we will consider the relative homotopy equivalences, and any base point is on the boundary of M .
Let (X, * ) be a fixed connected based space and assume that M has boundary and a basepoint. Applying the continuous functor Map * ( ; X) (based maps to X) fibrewise to u defines a new fibre bundle: Our second goal is to give explicit formulas for the monodromy action for p (up to homotopy). The total space is an example of the configuration-mapping spaces studied in [EVW;PT21]. Indeed, our interest in the monodromy actions was motivated by our study of homology stability for configuration-mapping spaces.
When z is just a single point the monodromy map can be defined in terms of the point-pushing map: it sends an element [α] ∈ π 1 (M, z) to the pointed isotopy class of the diffeomorphism that pushes the point z along the curve α and is the identity outside a small neighbourhood of the image of α. It is not difficult to see that the point pushing map and more generally push (M,z) factors through the (smooth) mapping class group: push sm (M,z) : π 1 (C k (M ), z) −→ π 0 (Diff(M ; z)); here Diff(M ; z) denotes the group of (smooth) diffeomorphisms of M that permute the points in z.
If the boundary of M is non-empty we will consider those diffeomorphisms that fix the boundary.
There is a possibly more familiar alternative description of push sm For M = S a surface of negative Euler characteristic, the connected components of Diff(M ) are contractible [EE69] [ES70] and hence the fibration gives rise to the Birman exact sequence [Bir69a] 0 −→ π 1 (S, z) −→ π 0 (Diff(S; z)) −→ π 0 (Diff(S)) −→ 0. When α is represented by a simple curve that has a two-sided neighbourhood in S, its image is a product of the two Dehn twists around the two curves (oriented oppositely) that form the boundary of a tubular neighbourhood of α. On the other hand, when S = T is the torus, Diff(T ) T SL 2 (Z) [Gra73] and eval induces an isomorphism on fundamental groups: π 1 (Diff(T ); id T ) ∼ = K = π 1 (T, z) ∼ = Z 2 .
Thus the smooth point-pushing map (and hence also the non-smooth version) is well-understood when d = 2. Recently, Banks [Ban17] completely determined the kernel L also when d = 3. In particular she shows that L is trivial unless the manifold M is prime and Seifert fibered via an S 1 action. In a different direction, Tshishiku [Tsh15] studies the Nielsen realisation problem for the point-pushing map, i.e. asks when the point-pushing map can be factored through Diff(M, z). However, little seems to be known about the image of the point-pushing map in higher dimensions.
Here we give a complete description, up to homotopy, of the induced self-map of M z for any element of the fundamental group when M has non-empty boundary. As an example, in section 7, we study the manifolds M d g,1 = g (S 1 × S d−1 ) D d for d 3 and g 0 and show that the point-pushing map is injective for these examples. Inspired by our calculations in these examples, we discuss injectivity more generally in §8. We note that for these examples M d g,1 , the Nielsen realisation problem is solvable as the fundamental group is free.
Outline and results. The paper is organised as follows. Section 2 contains basic recollections about (relative) monodromy actions associated to fibrations and Section 3 discusses equivalent definitions of the point-pushing map (see Figure 3.1), and considers the induced actions for associated fibre bundles obtained from the universal bundle u by applying a continuous functor. Restricting from now on to manifolds with boundary and dimension d 3, in Section 4 we note that for a k-tuple z, up to homotopy, M z decomposes as a wedge of M with a k-fold wedge sum of spheres and π 1 (C k (M ), z) is isomorphic to the wreath product Thus the task of understanding the monodromy action is divided into understanding (on each of the terms M and W k ) the action of the symmetric group elements, which is done in Section 5, and the more complicated action of the loop elements, considered in Section 6. The elements of the symmetric group act, up to homotopy, by the identity on M and by permuting the k summands in the wedge product W k ; compare Proposition 5.1. The precise action of a loop α ∈ π 1 M is the content of Propositions 6.2 and 6.3. Roughly, when α is in the i-th factor of the wreath product, it acts on the summand W k by taking the i-th sphere S d−1 and mapping a neighbourhood of its base point around α before covering itself by a degree ±1 map depending on whether α lifts to a loop in the orientation double cover of M . The other factors of W k are mapped by the inclusion. This completely describes the monodromy action of α on W k → M ∨ W k . The action of α on M depends only on the sequence of intersections of α with the (d − 1)-cells of M , or more precisely those of an embedded CW-complex K of dimension at most d−1 such that M deformation retracts onto it; compare formula (6.8) and Figure 6.2. So, if there are no such intersections, for example when K has no (d − 1)-cells, then the action on M is simply given by the inclusion. However, if α intersects a (d − 1)-cell τ of K with intersection number (τ, α) then in addition to the inclusion of M , the monodromy action of α takes the cell τ to the i-th factor of W k by a degree (τ, α) map. These assemble to give a map: where K (d−2) denotes the (d − 2)-skeleton of K. This completely describes the monodromy action of α on M → M ∨ W k after projection to each factor M and W k . The full description of this action in Definition 6.6 takes into account the precise sequence of intersections of α and the (d − 1)-cells. We illustrate this latter more complicated action of α with examples in Section 7. In Section 8 we discuss the general question of injectivity for the point-pushing map. We show that, up to isomorphism, the kernel of the point-pushing map is independent of k regardless whether diffeomorphisms, homeomorphisms or homotopy equivalences are considered. In particular, it is always injective when the manifold has non-empty boundary. Our main result in this direction is contained in Proposition 8.2. Finally in Section 9 the induced action on the fibres of p for configuration mapping spaces is described. As a further application we compute the number of connected components for configuration mapping spaces in Corollary 9.5.

Monodromy actions
We first recall the monodromy action associated to a fibration. Let f : E → B be a continuous map and write F = f −1 (b) for a point b ∈ B. Assume that f satisfies the homotopy lifting property (covering homotopy property) (cf. [Hat02,§4.2] or [May99,§7]) with respect to the spaces F and F × [0, 1]. For example, this holds if f is a Hurewicz fibration, or if f is a Serre fibration and F is a CW-complex. In particular it holds whenever f is a fibre bundle and either B is paracompact or F is a CW-complex.
Definition 2.1 For a space F , write hAut(F ) ⊆ Map(F, F ) for the space of continuous self-maps F → F , with the compact-open topology, that admit a homotopy inverse. This is a topological monoid under composition, and grouplike, i.e. the discrete monoid π 0 (hAut(F )) is a group (it is the automorphism group of F in the homotopy category).
For a pair of spaces (F, F 0 ), we write End(F |F 0 ) for the topological monoid (with the compactopen topology) of self-maps of F that are the identity on F 0 and we write hAut(F |F 0 ) ⊆ End(F |F 0 ) for the union of those path-components of End(F |F 0 ) corresponding to the invertible elements of the discrete monoid π 0 (End(F |F 0 )). Note that hAut(F |∅) = hAut(F ). See also Remark 8.9.
There is also a relative version of this construction. Let F 0 ⊆ F be a subspace and assume that f satisfies the relative homotopy lifting property with respect to the pairs of spaces (F, F 0 ) and (F, F 0 ) × [0, 1]. For example, this holds if f is a Hurewicz fibration, or if f is a Serre fibration and (F, F 0 ) is a relative CW-complex. Also assume that we have a topological embedding i : F 0 ×B → E such that f • i is the projection onto the second factor and i(−, b) is the inclusion F 0 ⊆ F ⊆ E. (This says, essentially, that f contains the trivial fibration over B with fibre F 0 as a sub-fibration.) Definition 2.3 (Relative monodromy actions.) Under these assumptions, the relative monodromy action associated to f and F 0 is the action-up-to-homotopy

Point-pushing actions
This section defines the point-pushing action associated to a manifold M and a finite subset z ⊂M of its interior. This is given in Definition 3.2 via the monodromy action of the "universal" bundle (3.1). This may be refined (Remark 3.3) to a smooth version, and it has a simple geometric description (Lemma 3.4) for manifolds of dimension at least 3. We then describe point-pushing actions on mapping spaces and other spaces associated functorially to the complement M z (see Definitions 3.11 and 3.12).
Definition 3.1 Let U k (M ) :=C 1,k (M ) be the configuration space of k unordered green points in the interior of M and one red point in M , which may lie on the boundary. There is a fibre bundle given by forgetting the red point, whose fibres are u −1 (z) = M z. This is the universal bundle referred to in the introduction.
Definition 3.2 (The point-pushing action.) For a manifold-with-boundary M (we allow ∂M = ∅) and a finite subset z ⊆M of cardinality k, the point-pushing action of π 1 (C k (M ), z) on M z is defined as the relative monodromy action of (3.1). More precisely, we write F = u −1 (z), let F 0 = ∂M ⊆ M z and note that (M z, ∂M ) is a relative CW-complex, since it is a (smooth) manifold with boundary. There is an embedding given by colouring the point in ∂M red and the k points in the interior green, which satisfies the conditions of Definition 2.3. By Definition 2.3 and Lemma 2.4, there is therefore a well-defined relative monodromy action This is, by definition, the point-pushing action of π 1 (C k (M ), z) on M z. For [γ] ∈ π 1 (C k (M ), z), the homotopy class of maps Remark 3.3 The monodromy action (3.2) may be refined to an action by isotopy classes of diffeomorphisms, as in the following diagram: where Diff ∂ (M ) denotes the topological group of diffeomorphisms of M fixing ∂M pointwise, in the smooth Whitney topology, the topology on hAut(−) is the compact-open topology and i is induced by the continuous injection Diff ∂ (M, z) → hAut(M z|∂M ) given by ϕ → ϕ| M z .
To see this, recall (cf. [Pal60;Cer61;Lim63]) that there is a fibre bundle Diff ∂ (M ) → C k (M ) given by evaluating a diffeomorphism at a finite set of points, whose fibre over z is the subgroup Diff ∂ (M, z) of diffeomorphisms fixing z as a subset. The map push sm (M,z) in the diagram above is the connecting homomorphism in the long exact sequence of homotopy groups of this fibre bundle. We call this action the smooth point-pushing action of π 1 (C k (M )) on M z, and we call the map push sm γ = push sm If d = dim(M ) 3, there is a useful geometric description of the smooth point-pushing action, which we will use later. An element γ ∈ π 1 (C k (M ), z) is represented by a certain number of oriented loops γ 1 , . . . , γ j in M , each passing through at least one point of z, such that, for each point of z, exactly one of the loops passes through it. (The number j k of such loops is the number of cycles in the cycle decomposition of the permutation of z induced by γ.) Choose representatives of the loops γ 1 , . . . , γ j that are smoothly embedded and have pairwise disjoint images (using the fact that d 3 for disjointness). Also choose pairwise disjoint closed tubular neighbourhoods T 1 , . . . , T j of these loops, which we assume to be contained in the interior of M . Define a diffeomorphism fixing ∂M pointwise and z setwise as follows. On the complement of the tubular neighbourhoods, ϕ (T1,...,Tj ) is the identity. Suppose that the tubular neighbourhood T i contains k i of the points of z (so k 1 + · · · + k j = k) and identify See Figure 3.1 for an illustration. We record this geometric description in the following lemma. Associated point-pushing actions. We have so far described the "universal" point-pushing action of π 1 (C k (M ), z) on the complement M z, for a subset z ⊂M with |z| = k. We now discuss induced point-pushing actions associated to continuous endofunctors T : Top → Top or T : Top * → Top * (or, more generally, to a continuous functor of the form (3.7)). There is an exactly analogous construction if f is a fibre bundle in the pointed category Top * (i.e. with structure group Homeo * (F )) and T : Top * → Top * is a continuous endofunctor of the  If ∂M = ∅, the fibre bundle (3.1) admits a canonical section given by z → (z, * ), where * ∈ ∂M is a choice of basepoint, allowing us to reduce its structure group to the based homeomorphism group Homeo * (M z), where z is a basepoint of C k (M ). Thus, choosing a basepoint for X, we may also consider the fibre bundle associated to (3.1) by the continuous functor T = Map * (−, X) : Top * → Top * , which is denoted by   and its total space is the k-th based configuration-mapping space of M and X with "monodromy" or "charge" in c.
Remark 3.10 Configuration-mapping spaces are discussed in more detail in [PT21,§2], and may be generalised to configuration-section spaces, which are defined in [PT21,§3]. There are also many other natural continuous functors T : Top → Top or T : Homeo ∂M (M, z) → Top that may be used to construct interesting fibre bundles associated to the "universal" bundle (3.1). For example, one could take T to be suspension Σ k (−), symmetric powers SP k (−) or configuration spaces C k (−), each of which lead to a certain flavour of bicoloured configuration spaces.
Other interesting examples are co-representable functors, such as the based and free loop-space functors Ω(−) and L(−), which lead to spaces of configurations equipped with (based or free) continuous loops in their complement.
where the right vertical homomorphism • is defined by composition. In particular, the action up to homotopy of π 0 (hAut s (M z|∂M )) on the mapping space Map((M z, D), (X, * )) preserves the subspace Map c ((M z, D), (X, * )) for each subset c ⊆ [S d−1 , X], assuming, if M is non-orientable, that c is closed under the involution given by reflecting in S d−1 .
Remark 3.14 We have focused in this section (except in Remark 3.3 and Lemma 3.4) on monodromy actions -by homotopy automorphisms -of fibrations (as discussed abstractly in §2). This is because our main result is an explicit description of the monodromy action by homotopy automorphisms of the universal bundle (3.1) (and, as a corollary, of the configuration-mapping bundle (3.10)). However, the constructions of this section also have direct analogues for monodromy actions by homeomorphisms (diffeomorphisms) of fibre bundles (smooth fibre bundles). See also §8, where we discuss kernels of point-pushing actions in all three settings.

Formulas for point-pushing actions
Let M be a connected manifold of dimension d 3, let z ⊂M be a k-point configuration in its interior, D ⊆ ∂M an embedded (d − 1)-dimensional disc in its boundary, X a based space and c ⊆ [S d−1 , X] a non-empty set of unbased homotopy classes of maps S d−1 → X. Our goal is to give explicit formulas for the point-pushing action of π 1 (C k (M ), z) on M z (Definition 3.2). These will be given in the following two sections; in this section we first fix notation and the identifications that we will use. containing (but not equal to) D, and such that the closure of the complement (B ∩ ∂M ) D is also a disc. (In Figure 4.1, we may assume that such that each sphere intersects ∂B at the basepoint * and nowhere else, the spheres are pairwise disjoint except for * and each sphere "wraps once around each of the points of z" (this is more formally expressed by the condition that B z must deformation retract onto the union of the spheres). The union of M and the spheres is homeomorphic to the wedge sum on the right-hand side of (4.1), and there is a deformation retraction of M z onto this subspace, supported in B z, fixing the basepoint * and sending D onto { * }.

Notation 4.3
From now on, we will write π 1 (C k (M ), z) just as π 1 (C k (M )), leaving the basepoint z implicit, and using the fact that the inclusion C k (M ) → C k (M ) is a homotopy equivalence.
Notation 4.4 By the smooth version of the point-pushing action (see Remark 3.3), an element γ ∈ π 1 (C k (M )) induces (an isotopy class of) a self-diffeomorphism push sm γ : M → M , fixing ∂M pointwise and z setwise, which has an explicit geometric representative ϕ (T1,...,Tj ) given by Lemma 3.4 if dim(M ) 3. We denote its restriction to a self-diffeomorphism of M z by π γ : M z −→ M z.
By abuse of notation, we also denote by π γ the (homotopy class of a) homotopy self-equivalence of M ∨ W k fixing * induced via the deformation retraction (4.2): Recall that, for dim(M ) 3, the fundamental group π 1 (C k (M )) decomposes as the semi-direct product π 1 (M ) k Σ k . ( In the next two sections we give explicit formulas for the bottom horizontal map of (4.2) for γ = (α 1 , . . . , α k ; σ) ∈ π 1 (M ) k Σ k under this decomposition.

Notation 4.5
We collect here some additional notation that will be used in the following two sections.
• For a wedge A ∨ B, we write inc A (resp. inc B ) for the inclusion of the first (resp. second) summand, and similarly we write pr A (resp. pr B ) for the projection onto the first (resp. second) summand. • For pointed spaces A, B, C and a pointed map f : . Note that f A and f B jointly determine f , since ∨ is the coproduct in the category of pointed spaces. • Similarly, for pointed spaces A, B, C, D and a pointed map f : A ∨ B → C ∨ D, we will sometimes write f as a (2 × 2)-matrix: , so the (2 × 2)-matrix-notation loses information. (This is why we write " " instead of "=" in this case.) • As mentioned above, we have for dim(M ) 3 a splitting π 1 (C k (M )) ∼ = π 1 (M ) k Σ k . Thus, for each σ ∈ Σ k and α ∈ π 1 (M ), we have elements (1, . . . , 1; σ) and (α, 1, . . . , 1; id) ∈ π 1 (C k (M )), which we will denote simply by σ and α by abuse of notation. We will always use these letters for elements of these two subgroups of π 1 (C k (M )), and we will denote a general element of π 1 (C k (M )) by γ.
• We take the basepoint of S d−1 to be the south pole, and write pinch : for the map that collapses the equator of S d−1 to a point. The wedge sum on the right-hand side identifies the north pole of the left summand with the south pole of the right summand. We take the basepoint of S d−1 ∨S d−1 to be the south pole of the left summand (in particular, not the point at which the wedge sum is taken); with this choice, pinch is a based map.

Symmetric generators
The action of the symmetric generators of π 1 (C k (M )) on M ∨ W k is fairly easy to describe.
Proposition 5.1 For any element σ ∈ Σ k we have where σ denotes the obvious self-map of W k = k S d−1 determined by the permutation σ, and * denotes the constant map to the basepoint.
To see that ψ σ , first consider a collection of k small, unbased (d − 1)-spheres surrounding the points of z, contained in the union of tubular neighbourhoods T 1 ∪ · · · ∪ T j . It follows from its explicit description in Lemma 3.4 that ϕ (T1,...,Tj ) permutes the homotopy classes of these spheres according to σ. Since these spheres form a free basis for the the homology group is determined up to based homotopy by its effect on π d−1 (W k ); thus ψ σ .

Loop generators
For any α ∈ π 1 (M, * ), the point-pushing map π α : M z → M z may be assumed (up to basepoint-preserving homotopy) to be supported in a tubular neighbourhood of an embedded loop α in M , based at one of the points of the configuration z, in the homotopy class determined by conjugating α with a path in B from * to this point (see Figure 4.1). We may choose α and its tubular neighbourhood T to be contained in M ∪ B , so the support of π α : M z → M z is contained in M ∪ B . Under the identification (4.1), this implies the following. Lemma 6.1 For any α ∈ π 1 (M ), up to based homotopy, We therefore just have to describe the mapπ α for each α ∈ π 1 (M ). We first do this under an additional assumption on the manifold M . Recall that the handle-dimension of a manifold is the smallest i such that M may be constructed using handles of degree at most i. Using the cores of such a handle decomposition, this implies that M deformation retracts onto an embedded CWcomplex of dimension equal to the handle dimension of M . Since M , in our situation, is connected and has non-empty boundary, its handle-dimension is necessarily at most dim(M ) − 1.
Proposition 6.2 Suppose that the handle dimension of M is at most dim(M ) − 2. Then, for any element α ∈ π 1 (M ) we havē where sgn(α) : S d−1 → S d−1 has degree +1 if α lifts to a loop in the orientation double cover of M and degree −1 otherwise. The other notation is explained in Notation 4.5.
If the handle dimension of M is equal to dim(M ) − 1 (the maximum possible), the formula for π α is more complicated. The following proposition gives the general formula. Proposition 6.3 For any element α ∈ π 1 (M ) we havē where sgn(α) is as in Proposition 6.2 and the maps α and α are described in §6.2 below.
In §6.1 we prove Proposition 6.2. In §6.2 we first define the maps α and α in the statement of Proposition 6.3 (Definitions 6.5 and 6.6) and then prove Proposition 6.3.
In each case we prove the descriptions on the left-hand side of (6.1) and of (6.2), and those on the right-hand side in terms of (2 × 2) matrices follow as a consequence. We note that in each case the top-right entry of the matrix is a priori equal to α • coll : S d−1 → M , but this is nullhomotopic as a based map, so it may be replaced with * . In contrast, the appearance of α•coll in the formulas on the left-hand side of (6.1) and of (6.2) may not be replaced by * , since it is part of a description of a map S d−1 → M ∨ S d−1 where the sphere is first collapsed to [0, 1] ∨ S d−1 , so in this case the interval may not be deformation retracted to its basepoint 0, since its other endpoint 1 is attached to the sphere S d−1 , which is wrapped with sign ±1 around the S d−1 summand of M ∨ S d−1 .

Below the maximal handle dimension.
In this subsection we prove Proposition 6.2. Let us write In this notation, to prove Proposition 6.2, we need to show that We first prove the right-hand side of (6.3). This may in fact be seen purely geometrically from Figure 4.1. We need to describe the effect of π α on the loop (representing a (d−1)-sphere) pictured in the bottom-left corner of that figure. As mentioned at the beginning of this section, π α may be assumed to be supported in a tubular neighbourhood T of a loop based at the puncture z ∩ B and supported in M ∪ B , as pictured in Figure 4.1. To see the effect of point-pushing along the tube T on the (d − 1)-sphere based at * pictured in the figure, it is easier first to replace it, up to homotopy equivalence, by a (d − 1)-sphere encircling the puncture z ∩ B together with a "tether" connecting this sphere to the basepoint * (this corresponds to the pinch and collapse maps in the formula (6.3)). Point-pushing along T has the effect on the tether of sending it around a loop homotopic to α. On the (d − 1)-sphere encircling the puncture, it acts by a map of degree ±1 depending on whether the tubular neighbourhood T is orientable or not, in other words, whether or not α lifts to a loop in the orientation double cover of M , which is exactly sgn(α). Putting this all together, we obtain the desired formula on the right-hand side of (6.3).
We prove the left-hand side of (6.3) in two steps: Since the handle dimension of M is at most d − 2, there is an embedded CW-complex K ⊂ M of dimension at most d − 2, such that M deformation retracts onto K. (Constructed, for example, using the cores of a handle decomposition of M with handles of index at most d−2.) The restriction ofπ M α to K is a map of the form Choose a CW-complex structure on M extending that of K and give S d−1 the unique CW-complex structure with a single 0-cell and a single (d − 1)-cell. With respect to these choices, we may homotope the map above to be cellular, so that every r-cell of K is mapped into a cell of dimension at most r. This implies that the image of the map must intersect S d−1 only in the basepoint, so we have a factorisation up to homotopȳ for some map K → M . Since the inclusion of K into M is a homotopy equivalence, this implies also thatπ M α itself factorises up to homotopy as a self-map θ α of M followed by the inclusion into M ∨ S d−1 . This establishes the first claim above.
We next have to prove that θ α is homotopic to the identity. Consider the following diagram. Hence three out of the four sides of the outer square of (6.4) are homotopic to the identity, so the fourth side θ α must also be homotopic to the identity.
This completes the proof of Proposition 6.2.

Remark 6.4
This also proves half of Proposition 6.3, since that proposition is equivalent to the two statementsπ and in the proof above we did not use the hypothesis on the handle-dimension of M when proving the right-hand side of (6.3), which is the same as the right-hand side of (6.5).

In the maximal handle dimension.
In this subsection, we first define the maps α and α appearing in the statement of Proposition 6.3. These depend, a priori, on some additional choices, including a CW-complex K ⊂ M onto which M deformation retracts. However, Proposition 6.3 implies that they do not depend on these additional choices up to homotopy (see Remark 6.7).
Definition 6.5 Let K ⊂ M be a CW-complex of dimension at most d − 1 embedded into M such that M deformation retracts onto K. Assume also that K has exactly one 0-cell and that, for any i-cell τ of K, if Φ τ : D i → K denotes its characteristic map, then the restriction is a smooth embedding. This exists since M is connected and has non-empty boundary, so its handle-dimension is at most d − 1: such a CW-complex K may be constructed from the cores of a handle decomposition of M with one 0-handle. Let α ∈ π 1 (M ) and choose a representative loop of α that is a smooth embedding, transverse to the interior of every cell of K and also transverse to ∂M . (For the assumption that the representative of α may be chosen to be an embedding, we are using the fact that M has dimension at least 3.) Note that the fact that α is transverse to the cells of K implies that it must be disjoint from the (d − 2)-skeleton K (d−2) of K.
Given these choices, we define the map α : M → S d−1 as follows: where the map M → K is a homotopy inverse of the inclusion, the index τ runs over all (d−1)-cells of K and the τ -th component of the last map is a map S d−1 → S d−1 of degree (τ, α), which is the algebraic intersection number of (the interior of) τ with α.
There are two subtleties in this definition: we need to choose the identification of K/K (d−2) with a wedge of (d − 1)-spheres unambiguously and we need to ensure that the algebraic intersection number (τ, α) is well-defined. Since T is an orientable codimension-zero submanifold of M containing * and each point of (6.7), we may use it to transport the local orientation of M at * to a local orientation of M at each point of (6.7).
We note that this definition does not depend on our arbitrary choices of orientations for S d−1 and for each open (d − 1)-cell τ of K: • Suppose that we reverse the orientation of one (d − 1)-cell τ 0 . This affects the identification of K/K (d−2) with the wedge of (d − 1)-spheres in a way that corresponds to inserting an automorphism of τ S d−1 that sends each sphere to itself, has degree −1 on the τ 0 component and has degree +1 on all other components. However, it also has the effect of reversing the sign of the algebraic intersection number (τ 0 , α), so these effects cancel each other out after composing all maps in (6.6). • Suppose that we reverse the orientation of S d−1 . This affects the identification of K/K (d−2) with the wedge of (d − 1)-spheres in a way that corresponds to inserting an automorphism of τ S d−1 that sends each sphere to itself and has degree −1 on each component. However, it also has the effect of reversing the local orientations of M at each intersection point (6.7) for each τ , and so it reverses the sign of each algebraic intersection number (τ, α). Again, these effects cancel each other out after composing all maps in (6.6).
This completes the definition of the map α : M → S d−1 .
For the definition of α , we again use an embedded CW-complex K ⊂ M as in Definition 6.5, and choose a representative loop of α ∈ π 1 (M ) as in Definition 6.5. Definition 6. 6 We now define a map α : M → M ∨ S d−1 whose composition with the projection pr S d−1 : M ∨ S d−1 → S d−1 is α . This is the map where the first map is a homotopy inverse of the inclusion and the second map is defined as follows.
On the (d − 2)-skeleton it is defined to be the inclusion We now extend this to each (d − 1)-cell of K, in other words, for each (d − 1)-cell τ of K, we define a map α,τ : D d−1 −→ M ∨ S d−1 (6.9) whose restriction to ∂D d−1 is equal to the attaching map φ τ : We define the map (6.9) in several steps: • Denote the intersection points of α with the interior of τ by There is a canonical homeomorphism 10) where the notation ∪ n indicates that we are taking the union along n distinct basepoints, more precisely we identify given by x i + y −→ x i + (|y|/ − 1)y (i.e. "stretching" inwards by a factor of two). Let be the quotient map D d−1 D d−1 /∼ followed by the identification (6.10). Composing this with the "pinch and collapse map" (coll ∨ id) • pinch (see Notation 4.5) on each S d−1 factor we obtain a quotient mapc (6.11) See Figure 6.1 for a visual illustration of this construction. • Finally, we define (6.9) by α,τ = α,τ •c n , where the map is defined on each piece of the domain as follows.
the sign i is determined as follows.
• As in Definition 6.5, the chosen orientation of S d−1 determines a local orientation of M at * . • We have also chosen an orientation of D d−1 , and Φ τ is a smooth embedding on the interior of D d−1 , so we also have an orientation of Φ τ (int (D d−1 )). This determines a local orientation of M at the intersection point y i : namely the one with respect to which the intersection number of Φ τ (int(D d−1 )) with α([0, 1]) at y i is +1. • If M is orientable, these two local orientations each determine an orientation of M , and we set i to be +1 if they agree and −1 if they disagree. • If M is non-orientable, we have to be more careful, just as in Definition 6.5. Choose δ > 0 such that all intersection points y 1 , . . . , y n are contained in α([δ, 1]) and choose a tubular neighbourhood T of α| [δ,1] . Since T is an orientable codimensionzero submanifold of M containing * and y i , the two local orientations of M (at * and at y i ) each determine an orientation of T . We set i = +1 if they agree and i = −1 if they disagree. One may see, as in Definition 6.5, that this construction of α is independent of the choices of orientation of S d−1 and D d−1 .
Remark 6.7 A priori, the maps α : M → S d−1 and α : M → M ∨ S d−1 described in Definitions 6.5 and 6.6 depend on the choice of embedded CW-complex K and the choice of representative of α ∈ π 1 (M ) that is a smooth embedding and transverse to ∂M and each open cell of K. However, a consequence of Proposition 6.3 is that these maps, up to basepoint-preserving homotopy, do not depend on these choices; they depend only on the element α ∈ π 1 (M ). This is because Proposition 6.3 identifies these two maps with certain maps derived from the point-pushing map π α , which depends up to homotopy only on α ∈ π 1 (M ).
Proof of Proposition 6.3. As pointed out in Remark 6.4, we have already proven one half of Proposition 6.3 while proving Proposition 6.2. The remaining statement to prove is (6.12) We will first prove the two (jointly weaker) statements: which correspond to the (2 × 2)-matrix description ofπ α on the right-hand side of (6.2). Consider the following homotopy-commutative diagram. Next, we prove the right-hand side of (6.13). We start by giving another description of the map where the middle two maps are the obvious projections and the identification on the right-hand side is induced by the projection T 1 S d−1 given by where r is a reflection in the (d−1)-sphere ∂B , π is a self-surjection of ∂B with the properties that π −1 ( * ) = ∂B int(T 1 ) and π is locally orientation-preserving on int(T 1 ), and the homeomorphism ∂B ∼ = S d−1 is given by a based isotopy in B between ∂B and the embedded copy of S d−1 in B in Figure 4.1.
We now use this geometric description of w α to show that it is homotopic to the map α defined in Definition 6.5. Let K be a CW-complex of dimension at most d − 1 embedded into M , such that M deformation retracts onto K. We need to show that the restriction of w α to K factors as where the τ -th component of the right-hand map is a map f τ : S d−1 → S d−1 of degree (τ, δ). By smooth approximation and transversality, we may assume that each (d − 1)-cell τ of K is smoothly embedded into M and that δ and T have been chosen so that (a) each r-cell of K, for r d − 2, is disjoint from T and (b) each τ ∩ T , for τ a (d − 1)-cell of K, consists of finitely many (d − 1)-discs each intersecting δ transversely in one point.
By property (a), and since M T is sent to the basepoint by w α , we see that its restriction to K must factor through the projection K K/K (d−2) . So we just have to show that f τ has (whose image under the characteristic map Φτ is the cell τ ) in blue and its intersection {y 1 , . . . , yn} with α as a configuration of red points. We also choose small disc neighbourhoods of each of these points (now depicted in green), divided into three concentric regions. Translating the depiction ofπ M α from (1) into this viewpoint, we see that the blue region (the complement of the small green disc neighbourhoods) is fixed byπ M α , in other words, it is simply mapped into M by the characteristic map Φτ of the cell. For each small green disc neighbourhood, its image underπ M α is illustrated as a light blue surface in (1); projecting this onto τ ∪ α ∪ S d−1 does not change it up to homotopy, and this may then be described in (2) as follows: the outer region of each green disc is "stretched" to cover the whole green disc (and then mapped into M via the characteristic map Φτ ); the intermediate region is collapsed to an interval and then mapped into M via a terminal segment of the loop α; the central region is collapsed to a sphere and then mapped with degree ±1, depending on local orientations, to the S d−1 summand of M ∨ S d−1 . This is precisely the map (6.9) from Definition 6.6 (see in particular Figure 6.1), which is the restriction of α to the cell τ . Thus for each (d − 1)-cell τ , the restrictions ofπ M α and of α to τ are homotopic relative to its boundary; henceπ M α α.
degree (τ, δ). By property (b) and the description (6.15) of w α | T , each component of the disjoint union of (d − 1)-discs τ ∩ T contributes either +1 or −1 to deg(f τ ). Being careful about (local) orientations as explained in Definition 6.5, we see that the sum of these +1's and −1's is precisely the algebraic intersection number (τ, δ) of τ and δ.
This completes the proof that w α | K factors as in (6.16), and hence that w α α , in other words, the right-hand side of (6.13).
The proof of (6.12) is similar to the proof above of the right-hand side of (6.13): looking at Figure  4.1 and using a geometric model for the point-pushing map supported in a tubular neighbourhood of an embedded loop representing α, one checks carefully that the definition of α from Definition 6.6 is a correct description ofπ M α up to homotopy. This is explained in Figure 6.2, which depicts the mapπ M α induced by point-pushing along α and compares it to the definition of α .

Examples
To illustrate the more complicated setting where M is non-simply-connected and has maximal handle dimension, we discuss some explicit examples, namely and more generally which all have maximal handle-dimension dim(M ) − 1 = 2 and which have fundamental groups Z and F g , the free group on g generators, respectively. Indeed, the following computations generalise to all for d 3 and g > 0.
Example 7.1 First, consider M = (S 1 × S 2 ) int(D 3 ) and let α be a generator of π 1 (M ) ∼ = Z. By Proposition 6.3, the point-pushing map has a simple explicit description when restricted to the S 2 summand, and is homotopic to the (in general complicated) map α : M → M ∨ S 2 of Definition 6.6 when restricted to the M summand.
In this example, M is homotopy equivalent to S 1 ∨S 2 (see Figure 7.1 for a picture of an embedded S 1 ∨ S 2 onto which it deformation retracts). So, under this identification, the point pushing map π α is an endomorphism of S 1 ∨ S 2 ∨ S 2 . We will label the 1-and 2-spheres with subscripts α, τ and p to indicate which of the spheres they correspond to (light or dark red spheres in Figure 7.1). Thus our aim is to describe (up to based homotopy) the map This is an element of the homotopy set X, X = π 0 (Map * (X, X)), which becomes a monoid under composition. In fact, we know of course thatπ α must be an invertible element of this monoid, i.e. an element of π 0 (hAut * (X)), but we will describe it as an element of the larger monoid X, X . In order to do this, we first describe the monoid X, X explicitly.
First, note that there is a bijection X, X ∼ = π 1 (X) × π 2 (X) × π 2 (X), and that π 1 (X) ∼ = Z{α}, the free (abelian) group generated by α. The second homotopy group of X is the same as that of its universal cover, and using Hilton's theorem [Hil55] to compute homotopy groups of wedges of spheres, we see that π 2 (X) ∼ = Z{α n p, α n τ | n ∈ Z}, the free abelian group generated by the symbols α n p and α n τ for each n ∈ Z. Moreover, the action of π 1 (X) = Z{α} is given by α.α n p = α n+1 p and α.α n τ = α n+1 τ . This means that we may write {p, τ } as a free module over the group-ring of π 1 (X). Putting these identifications together, we have as a set. To describe the monoid operation (composition) on X, X under this identification, it is useful to include it into the larger monoid X, X , where is the universal cover of X. Since ∨ is the coproduct for pointed spaces, we have the monoid of 2×2 block matrices whose entries are vertically-finite Z×Z matrices with entries in Z.
(Vertically-finite means that each column has only finitely many non-zero entries.) For example, the (i, j) entry in the bottom-left block of the matrix corresponding to f : X → X records the degree of the map Once a compatible base point in X is fixed, each based self-map of X lifts uniquely up to homotopy to a based self-map of X, so there is an injection X, X → X, X . Under the identifications (7.1) and (7.2), this is given by where each of the matrices A, B, C, D is a diagonally constant matrix of slope −k, in other words its (i, j) entry is equal to its (i − jk, 0) entry; in particular it is determined by its 0th column, and the 0th columns of A = (a ij ), B = (b ij ), C = (c ij ), D = (d ij ) are given by For example, the identity X → X corresponds to (α, p, τ ), which is sent to ( I 0 0 I ), and the map X → X that is the identity on the two S 2 factors and collapses the S 1 factor to the basepoint corresponds to (0, p, τ ), which is sent to 10 0 0 10 , where 1 0 is the matrix with 1s on the 0th row and 0s elsewhere.
Since the operation on X, X is just multiplication of matrices, one may use this inclusion of monoids to deduce a formula for the operation on X, X under the identification (7.1), which is given as follows: By considering the action on the universal cover, and using Definition 6.6 and Proposition 6.3, we may write the elementπ α ∈ X, X in terms of these explicit descriptions of X, X as follows: Similarly, we may calculate that:π As a sanity check, let us verify that these are indeed inverse elements in the monoid. After including into the larger monoid X, X , we havē where A ( ) denotes the matrix obtained by shifting A vertically upwards by steps, and these matrices are clearly inverses. This description also in particular encodes the fact thatπ α acts on π 1 (X) ∼ = Z{α} by the identity and on H 2 (X; Z) ∼ = Z{p, τ } by ( 1 1 0 1 ). (The action on H 2 is obtained by applying the operation M vf Z (Z) → Z that takes the sum of the entries in the 0th column to each entry of the 2 × 2 block matrix.) The elementπ α = (α, αp, τ + p) ∈ X, X has infinite order: this can be detected by its action on H 2 (−; Z), but one may also directly calculate: (π α ) n = (α, αp, τ + p) • · · · • (α, αp, τ + p) n = (α, α n p, τ + (1 + α + · · · + α n−1 )p), using the inclusion into X, X and the identity I ( ) I (k) = I ( +k) . Hence the point-pushing homomorphism is injective. This factors through the point-pushing homomorphism which is therefore also injective.
Generalising the discussion in the previous example, suppose that X is a wedge of a number of circles indexed by a set A and a number of two-spheres indexed by a set B. We then have where F A is the free group on the set A, Z[F A ] is its integral group-ring and Z[F A ]B is the free Z[F A ]-module on the set B. The underlying set of the monoid X, X is therefore To understand the operation of composition, it is again convenient to embed this into the larger monoid X, X , by lifting self-maps of X to self-maps of its universal cover This monoid is isomorphic to the monoid M B (M vf F A (Z)) of B × B block matrices whose entries are F A × F A integer matrices that are vertically finite (each column has only finitely many non-zero entries).
In particular, each of these matrices is determined by its 1st column. Finally, the 1st columns of each of these matrices are determined by setting (a u,1 ) ij equal to the coefficient of uτ i in f j .
We may now describe the point-pushing mapsπ α1 , . . . ,π αg and their inverses under the identifications (7.4). Namely, we havē π αi = (α 1 , . . . , α g , α i τ 0 , τ 1 , . . . , τ i−1 , τ i + τ 0 , τ i+1 , . . . , τ g ) as elements of X, X , and From this description, we deduce a formula forπ w for any word w in the generators α 1 , . . . , α g . Note that we are not assuming that the word w is reduced. Proposition 7.3 Let w be a word in the generators α 1 , . . . , α g . Then as an element of X, X , and as an element of X, X , where the non-commutative Laurent polynomials f i (w) and matrices A i (w) are defined as follows. Write where the w j are words not involving α ±1 i and j ∈ {±1} and letw j be the initial subword Proof. The description ofπ w as an element of X, X will follow from its description as an element of X, X via the embedding of monoids described earlier, so we only have to prove the latter. Multiplying out the matrices (7.5) corresponding to the letters of the word w, it is clear thatπ w is of the form  , so we just have to verify that ? i (w) = A i (w). We first note that, directly from the definition, the matrices A i (w) have the following property: if w = w w , then We also observe from (7.5) that the equality ? i (w) = A i (w) is true if w is the letter α i or its inverse.
We now prove that ? i (w) = A i (w) by induction on the number of letters of w that are equal to α i or α −1 i . If this is zero, i.e. if w does not contain α ±1 i , then A i (w) = 0 and also ? i (w) = 0, since it is the (0, i) entry in a product of matrices that each have the property that their ith rows and columns agree with the identity matrix. This establishes the base case. If there are 1 letters of w that are equal to α i or α −1 i , then we may write w = w α i w , where w does not contain α ±1 i . Applying the base case to w , the inductive hypothesis to w and using the observation above, we already know that Writing just the rows and columns indexed by 0 and i, we therefore have where we apply the identity (7.6) twice to deduce the final equality.
In particular, we note that the coefficient of the generator p = τ 0 in the middle component ofπ w is g . This implies that the point-pushing homomorphism (7.3) is injective.
The two examples above go through identically if S 1 ×S 2 is replaced with S 1 ×S d−1 for any d 3; we obtain the same formulas for the point-pushing mapsπ α and the point-pushing homomorphism α →π α is injective. Thus we have seen that, for any manifold of the form for d 3 and g 0, the point-pushing homomorphism is injective. For d = 2 this is also true: Recall the point-pushing homomorphism is part of the Birman exact sequence [Bir69a]: In the next section, we put these facts into context by discussing the kernel of the point-pushing map more generally and for any number of configuration points.

The kernel of the point-pushing map
Let M be a smooth, connected manifold of dimension d 3 and fix a ball D ⊂ M in the interior of M containing the base configuration z. This determines an identification (4.3) of π 1 (C k (M )) with the semi-direct product π 1 (M ) k Σ k . For Cat ∈ {Diff, Homeo, hAut}, recall from §3 that the point-pushing map is the monodromy of the bundle C k,1 (M ) → C k (M ), viewed either as a smooth bundle, a topological bundle or a Serre fibration. 1 Except when Cat = hAut and k 2, this may equivalently be described as a connecting homomorphism in the long exact sequence of the fibration Cat(M ) → C k (M ) taking an automorphism ϕ to its evaluation ϕ(z) at the base configuration z. (Note that such a description is impossible for Cat = hAut and k 2, since homotopy automorphisms need not be injective, so there is no well-defined map hAut(M ) → C k (M ) in this case.) Thus if Cat ∈ {Diff, Homeo}, or if Cat = hAut and k = 1, the point-pushing map fits into an exact sequence of the form 1 −→ ker(p k ) −→ π 1 (C k (M )) −→ π 0 (Cat(M, z)) −→ π 0 (Cat(M )) −→ 1.
Despite the differences between the categories of Diff and Homeo on the one hand and hAut on the other, the following results hold for all three. Note though that ker(p 1 ) and ker(p k ) may be different groups for the three different categories. i.e. the kernel of (8.1) is equal to the diagonal of ker(p 1 ) k ⊆ π 1 (M ) k ⊆ π 1 (C k (M )), where we use the identification of π 1 (C k (M )) with π 1 (M ) k Σ k fixed above. If ∂M = ∅, then (8.2) is injective.
The first proposition is an immediate consequence of the following basic lemma.

Lemma 8.3 ([Hat02, page 40])
For any space X, the image of the map π 1 (hAut(X)) → π 1 (X) induced by evaluation at some point x ∈ X has image contained in the centre Z(π 1 (X)).
On the other hand, in the somewhat degenerate special case of k = 2 and π 1 (M ) = 1, the centre of π 1 (M ) k Σ k is Σ 2 .) Next, we consider the commutative diagram where the image of ( * ) is ker(p k ) and the image of ( * * ) is ker(p 1 ) (these identifications follow, again, from the relevant long exact sequences, for general k and for k = 1 respectively). We know already that ker(p k ) is equal to ∆(G) ⊆ G k for a certain subgroup G ⊆ Z(π 1 (M )) ⊆ π 1 (M ). Since this is a diagonal subgroup of the product π 1 (M ) k , the projection onto the first factor restricts to an isomorphism of ∆(G) onto G ⊆ π 1 (M ). By commutativity of the above diagram, it follows that G = ker(p 1 ). This concludes the proof of the first statement of the proposition. For the second statement, we repeat the same arguments with Cat replaced by Cat ∂ everywhere to obtain a similar formula, and then apply Proposition 8.1.
For the proof of Proposition 8.2 in the homotopy setting, we will use the following basic lemmas. Proof. The first step is to prove that, for any space Z, the inclusion Z × A → Z × X is also a cofibration. This is most easily seen using the characterisation [Hat02, Proposition A.18] of cofibrations A → X as those inclusions for which X × [0, 1] retracts onto (

Lemma 8.4 Let
If r is a retraction witnessing that A → X is a cofibration, then id Z × r is a retraction witnessing that Z × A → Z × X is a cofibration.
Now suppose that we have a homotopy lifting problem as follows: By taking adjoints twice (since X and A are exponentiable), we may rewrite this as: This admits a lift h : is a Hurewicz fibration, so in particular a Serre fibration.
In particular, this says that the restriction map Map(X, X) → Map(A, X) is a Serre fibration. In general, whenever E → B is a Serre fibration and E 0 ⊆ E is a union of path-components, the restriction E 0 → B is also a Serre fibration. Since hAut(X) is a union of path-components of Map(X, X), this implies the second statement of the lemma.
Lemma 8.5 Let A ⊆ B ⊆ X be cofibrations of exponentiable spaces such that B admits a strong deformation retraction onto A. Then the inclusion hAut B (X) → hAut A (X) is a weak homotopy equivalence.
Here we write hAut A (X) for the space of homotopy automorphisms of X that agree with the identity on A. We will also use the notation Map A (B, X) for the space of maps B → X that agree with the inclusion on A.
We will use this lemma below when X is a manifold, B ⊂ X is an embedded interval and A is a point in this interval. (Manifolds are locally compact Hausdorff, hence exponentiable.) Proof of Lemma 8.5. Consider the following commutative diagram The From the long exact sequence of homotopy group it follows that the bottom horizontal map −| A induces isomorphisms on π * for * 1 and an injection on π 0 . This map is also clearly surjective since any map A → X may be extended to B using the retraction r, so it is a weak homotopy equivalence. It then follows from the 5-lemma (and a little extra care in degree 0) that hAut B (X) → hAut A (X) is a weak homotopy equivalence.
Proof of Proposition 8.2 in the homotopy setting. In this setting, we cannot use the long exact sequence, so we give a different argument. First, the right-hand side of diagram (8.3) implies that ker(p k ) ⊆ π 1 (M ) k . We then consider the commutative square in diagram (8.7) below, where the subscript z means that z is fixed pointwise. It follows that ker(p k ) ⊆ ker(p 1 ) k .
We next show that ker(p k ) contains the diagonal ∆(ker(p 1 )). Fix an element a 1 ∈ π 1 (M ), set a = (a 1 , . . . , a 1 ) ∈ π 1 (M ) k and consider the diagram π 0 (hAut I (M )) π 1 (M ) k π 0 (hAut z (M )) for some fixed i (say i = 1), where I ⊂ M is an embedded interval containing the configuration z and again the subscript I means that I is fixed pointwise. We observe that the element p k (a) ∈ π 0 (hAut z (M )) may be lifted to an element ϕ ∈ π 0 (hAut I (M )), defined as follows. Choose an isotopy of embeddings I → M starting at the inclusion, pulling the interval I around the loop a 1 and then ending at the inclusion again. This may be constructed similarly to the explicit description of the (smooth) point-pushing map in Lemma 3.4 and Figure 3.1, using a tubular neighbourhood of an embedded representative of the loop a 1 , which is a D d−1 -bundle over a 1 , and a choice of trivial sub-I-bundle. Extend this by the isotopy extension theorem to a path in Diff(M ) from id to ϕ. Then ϕ is a diffeomorphism (hence homotopy automorphism) of M fixing I pointwise and representing p k (a) when considered as a homotopy automorphism of M fixing z ⊂ I pointwise. Now if we assume that a 1 ∈ ker(p 1 ), it follows that ϕ = 1 ∈ π 0 (hAut I (M )), since the inclusion hAut I (M ) → hAut(M, z 1 ) is a weak homotopy equivalence by Lemma 8.5, so in particular it induces an injection on π 0 . It then also follows that a = (a 1 , . . . , a 1 ) ∈ ker(p k ).
Finally, suppose that ker(p k ) = ∆(ker(p 1 )). Then there must be an element a = (a 1 , . . . , a k ) ∈ ker(p k ) ∆(ker(p 1 )). Since a 1 ∈ ker(p 1 ), we already know that (a −1 1 , . . . , a −1 1 ) ∈ ker(p k ), so we 1 , in particular b 1 = 1. Choose embedded paths A i from z 1 to z i for each i ∈ {2, . . . , k} that are pairwise disjoint except at z 1 . Also choose embedded loops based at z i representing b i (also denoted b i by abuse of notation) for each i ∈ {2, . . . , k}. We may assume that the loops b i are pairwise disjoint, and also disjoint from the arcs A j except at z i . We also assume that the point-pushing automorphism p k (b) ∈ hAut(M, z) has support contained in a small tubular neighbourhood of the union of the loops b i . See Figure  8.1. By assumption, there is a homotopy id p k (b) of self-maps (M, z) → (M, z). Restricting this to the embedded path A i , we see that A i p k (b)(A i ) relative to endpoints. Thus, we have = (1, . . . , 1), which is a contradiction.
This finishes the proof of the first statement of the proposition. For the second statement, just as before, we repeat the same arguments with hAut replaced by hAut ∂ everywhere to obtain a similar formula, and then apply Proposition 8.1.

Remark 8.6
The kernel of (8.1) for k = 1, in the 3-dimensional topological (equivalently smooth) setting, has been understood completely by [Ban17]. By Proposition 8.2, it is therefore also understood completely for all k in the 3-dimensional topological/smooth setting.

Remark 8.7
If M does not necessarily have boundary, but it is equipped with marked points that are required to be fixed under automorphisms, then the corresponding point-pushing map p k : π 1 (C k (M P )) −→ π 0 (Cat P (M, z)) is injective when the set P ⊂ M of marked points is non-empty, just as in the ∂M = ∅ setting. For k = 1 this follows since ker(p 1 ) is the image of the map on π 1 induced by evaluation Cat P (M ) → M at a point z 1 ∈ M P , which is homotopic to evaluation at a point in P , hence nullhomotopic. For higher k, the proof above adapts to show that ker(p k ) = ∆(ker(p 1 )) also in this setting, and hence p k is also injective. of the mapping class group of S relative to D is trivial: this is because its image in π 0 (hAut * (S)) is clearly trivial -one may simply untwist T D while keeping the point * fixed -and the map π 0 (hAut D (S)) → π 0 (hAut * (S)) is injective by Lemma 8.5. In contrast, the element [T D ] ∈ π 0 (hAut ∂D (S int(D))) (8.9) Figure 8.2 A nullhomotopy of the "fake Dehn twist" in the (homotopy automorphism version of the) mapping class group of S relative to D. The blue lines indicate twisting in an annular region. The central grey disc is D. The green annulus surrounding D in some of the pictures indicates that the inner boundary of the green annulus is mapped to itself by the identity (as it must be), the outer boundary of the green annulus is sent the the midpoint of the disc, and the interior of the green annulus is "turned inside out" and mapped onto the grey disc D. The untwisting of the Dehn twist in steps 3 and 4 is well-defined exactly because the outer boundary of the green annulus is collapsed to a point. The homotopies in steps 1 and 6 are given by gradually "folding" the green annulus inwards, while keeping the grey disc fixed, until the outer boundary of the green annulus is collapsed to the midpoint of the disc. Steps 2 and 5 are not strictly necessary, since one could directly perform the homotopies of steps 1 and 6 with the larger green annulus, but they perhaps make the picture more intuitive.
is well-known to be non-trivial (and of infinite order) in the mapping class group of S int(D) relative to the boundary-component ∂D, as long as S is not the 2-sphere or the 2-disc. 2 Notice that such an apparent discrepancy cannot occur if hAut(−) is replaced with Homeo(−) or Diff(−), since in these two cases there is a canonical homeomorphism between Cat D (S) and Cat ∂D (S int(D)) for Cat ∈ {Homeo, Diff}.
The reason for this apparent discrepancy in the Cat = hAut setting is illustrated by exhibiting an explicit nullhomotopy of (8.8): see Figure 8.2. This nullhomotopy depends on the fact that points may be mapped into the disc D (hence why it does not work for (8.9)) and also the fact that homotopy equivalences may be non-injective (hence why it does not work for Cat ∈ {Homeo, Diff}).

Remark 8.9
There is a subtle difference between the space hAut A (X) involved in Lemma 8.5 and the proof of Proposition 8.2 and the space hAut(X|A) defined in §2, namely: hAut(X|A) = {f ∈ Map(X, X) | f | A = id A and f admits a homotopy inverse relative to A} hAut A (X) = {f ∈ Map(X, X) | f | A = id A and f admits a homotopy inverse}, so clearly hAut(X|A) ⊆ hAut A (X). In general, if A ⊆ X is a cofibration and f : X → X restricts to the identity on A and admits a homotopy inverse, then one may find both a left homotopy inverse for f relative to A and a right homotopy inverse for f relative to A, but these may not necessarily coincide. On the other hand, if the space Map(A, A) is simply-connected, then one may always find a two-sided homotopy inverse for f relative to A, and so in this case the two spaces hAut A (X) and hAut(X|A) are equal. In particular, this holds if A = D is a disc.
2 To see this, write γ for a curve in the interior of S int(D) parallel to ∂D, so that Tγ = T D , and choose an arc α in S int(D) with both endpoints on ∂D so that i(α, γ) = 2, where i(−, −) is the minimal geometric intersection number amongst isotopic representatives. Then i(T k γ (α), α) is strictly increasing as k → ∞. See [FM12, Proposition 3.2] for details (in the case of closed surfaces, which may easily be adapted to compact surfaces with boundary). where the entry ? is not in general f , but rather a based map M → X that depends in a subtle way on f , the loop γ and the elements g i . For example, when γ = (α, 1, . . . , 1; id), the map ? : M → X is given by the composition where α is the map defined in Definition 6.6. To see this, recall that the equations (6.3) describe the point-pushing action of a loop generator α under the additional assumptions on M , and the equations (6.5) describe the point-pushing action of α without these assumptions. The right-hand equation of (6.3) agrees with the right-hand equation of (6.5), which is why the tuple (ḡ 1 , . . . ,ḡ k ) occurs in (9.3), just as in (9.2). However, the left-hand equation of (6.3) is simplyπ M α inc M , whereas the left-hand equation of (6.5) isπ M α α .
Remark 9.4 Corollary 9.2 is used in [PT21,§8] to prove a certain split-injectivity result for maps between configuration-mapping spaces. More precisely, there is a natural map of spectral sequences converging to the map on homology induced by the stabilisation map Under the hypotheses on M assumed in Corollary 9.2, this map of spectral sequences is splitinjective on E 2 pages. For the precise statement, see [PT21,Theorem 8.12].
Corollary 9.2 may also be used to understand the path-components of configuration-mapping spaces of manifolds of dimension at least 3. As an example, we have the following. individually) by f * (π 1 (M )) π 1 (X). The formula (9.4) follows.