Point-pushing actions and configuration-mapping spaces
aa r X i v : . [ m a t h . A T ] J u l Point-pushing actions and configuration-mapping spaces
Martin Palmer and Ulrike Tillmann nd July 2020
Abstract
Given a manifold M and a point in its interior, the point-pushing map describes a diffeo-morphism that pushes the point along a closed path. This defines a homomorphism from thefundamental group of M to the group of isotopy classes of diffeomorphisms of M that fix thebasepoint. This map is well-studied in dimension d = 2 and is part of the Birman exact se-quence. Here we describe for any d > k > k -th braid group of M on the k -punctured manifold M r z . 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 r z . Furthermore, motivated by our work in [PT], wedescribe the action of the braid group of M on the fibres of configuration-mapping spaces. Contents
1. Introduction 12. Monodromy actions 33. Point-pushing actions 44. Formulas for point-pushing actions 85. Symmetric generators 106. Loop generators 116.1 Below the maximal handle dimension. 116.2 In the maximal handle dimension. 127. Formulas for associated point-pushing actions on mapping spaces 17References 19
1. Introduction
Let M be a based, connected (smooth) manifold of dimension d > C k ( ˚ M ) theconfiguration space of k unordered distinct points in its interior. We may think of it as the modulispace of k distinct points in M . Its universal bundle is the fiber bundle U k ( M ) that associates toeach k -tuple z ∈ C k ( ˚ M ) the k -punctured manifold M r z : M r z −−−−→ U k ( M ) u y C k ( ˚ M ) . The primary goal of this paper is to describe the monodromy action (up to homotopy) of the abovefibre bundle push ( M,z ) : π ( C k ( ˚ M ) , z ) −→ π (hAut( M r z ))where hAut( M r z ) denotes the homotopy equivalences of the complement of z in M ; when M has boundary we will consider the relative homotopy equivalences. Mathematics Subject Classification : 55R80, 57N65, 55R10
Key words and phrases : Monodromy actions, point-pushing actions, Birman exact sequence, configuration-mappingspaces. X, ∗ ) be a fixed connected based space. Applying the functor Map( ; ( X, ∗ )) (or anothercontinous functor) to u defines a new fibre bundle:Map(( M r z, ∗ ) , ( X, ∗ )) −−−−→ CMap ∗ k ( M ; X ) p y C k ( ˚ M ) . Our second goal is to give explicit formulas for the monodromy action for p (up to homotopy). Thetotal space is an example of the configuration-mapping spaces studied in [EVW; PT]. Indeed, ourinterest in the monodromy actions was motivated by our study of the homology of configuration-mapping and -section spaces. In [PT] we use the results from this paper to analyse the E -pageof the Serre spectral sequence associated to p ; see also Remark 7.4.When z is just a single point the monodromy map can be defined in terms of the point pushingmap: it sends an element [ α ] ∈ π ( M, z ) to the pointed isotopy class of the diffeomorphism thatpushes the point z along the curve α and is the identity outside a tubular neighbourhood. It isnot 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 ( C k ( ˚ M ) , z ) −→ π 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 ( M,z ) . For z a single point in M , consider the fibration Diff( M ; z ) −→ Diff( M ) eval −→ M where eval denotes the map that evaluates a diffeomorphism at z . As M is path-connected, thisgives rise to the exact sequence0 −→ K −→ π ( M, z ) −→ π Diff( M ; z ) −→ π Diff( M ) −→ . By definition the kernel K of the smooth point-pushing map is a quotient of π Diff(
M, z ) andhence is abelian.For M = S a surface of negative Euler characteristic, the connected components of Diff( M ) arecontractible [EE69] [ES70] and hence the fibration gives rise to the Birman exact sequence [Bir69]0 −→ π ( S, z ) −→ π Diff( S ; z ) −→ π Diff( S ) −→ . When α has a two-sided neighbourhood in S , its image is a product of the two Dehn twists aroundthe two curves that form the boundary of a tubular neighbourhood of α . When α has a one-sidedneighbourhood (and S is not-orientable) the corresponding mapping class is known as a cross-capslide, first introduced in [Lic63]; compare also [Kor02]. On the other hand, when S = T is thetorus, Diff( T ) ≃ T ⋊ SL ( Z ) [Gra73] and eval induces an isomorphism on fundamental groups: π (Diff( T ); id T ) ∼ = K = π ( T, z ) ∼ = Z . Thus the smooth point-pushing map (and hence also the non-smooth version) is well-understoodwhen d = 2. Recently, Banks [Ban17] completely determined the kernel K also when d = 3.In particular she shows that K lies in the centre of π ( M, z ) and is trivial unless the manifold M is prime and Seifert fibered via an S action. However, little seems to be known about thepoint-pushing map nor its smooth analogue in higher dimensions. Outline and results.
The paper is organised as follows. Section 2 contains basic recollectionsabout (relative) monodromy actions associated to fibrations and Section 3 discusses equivalentdefinitions of the of point-pushing map (see Figure 3.1), and considers the induced actions forassociated fibre bundles obtained from the universal bundle u by applying a continuous functor.2n Section 4, we restrict to manifolds with boundary and dimension d >
3. Then for a k -tuple z up to homotopy M r z decomposes as a wedge of M with a k -fold wedge product of spheres S d − , M r z ≃ M ∨ W k where W k := _ k S d − . and π ( C k ( ˚ M ) , z ) is the wreath product π ( M ) k ⋊ Σ k . Thus the task of understanding the monodromy action is divided into understanding (on each ofthe terms M and W k ) the action of the symmetric group elements, which is done in Section 5, andthe more complicated action of the loop elements, considered in Section 6. The elements of thesymmetric group act (up to homotopy) by the identity on M and by permuting the k summandsin the wedge product W k ; compare Proposition 5.1. The precise action of a loop α ∈ π M is thecontent of Propositions 6.2 and 6.3. Roughly, when α is the i -th factor in the wreath product, itacts on the summand W k by taking the i th sphere S d − and mapping a neighborhood of its basepoint around α before covering itself by a degree ± α lifts to a loopin the orientation double cover of M . The other factors of W k are mapped by the inclusion. Theaction of α on M depends only on the sequence of intersections of α with the ( d −
1) cells of M ,compare formula (6.8) and Figure 6.2. So, if there are no such intersections, or M has no ( d − M is simply given by the the inclusion. Finally in Section 7 the inducedaction on the fibres of configuration mapping spaces is described.
2. Monodromy actions
We first recall the monodromy action associated to a fibration. Let f : E → B be a continuousmap and write F = f − ( 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 , f is a Hurewicz fibration, or if f is a Serre fibration and F isa CW-complex. In particular it holds whenever f is a fibre bundle and either F is a CW-complexor B is paracompact. 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 topologicalmonoid under composition, and grouplike , i.e. the discrete monoid π (hAut( F )) is a group (it isthe automorphism group of F in the homotopy category).For a pair of spaces ( F, F ), we write End( F | F ) for the topological monoid (with the compact-open topology) of self-maps of F that are the identity on F and we write hAut( F | F ) ⊆ End( F | F )for the union of those path-components of End( F | F ) corresponding to the invertible elements ofthe discrete monoid π (End( F | F )). Note that hAut( F | ∅ ) = hAut( F ). Definition 2.2 ( Monodromy actions. ) Under the above assumptions, the monodromy action as-sociated to f is the action-up-to-homotopymon f : π ( B, b ) −→ π (hAut( F )) (2.1)of π ( B, b ) on F defined as follows. For an element [ γ ] ∈ π ( B, b ) represented by a loop γ : [0 , → B , let g : F × [0 , → E be a choice of lift in the diagram: F EF × [0 ,
1] [0 , B incl γ ( − , f (2.2)and define mon f ([ γ ]) = [ g ( − , F ⊆ F be a subspace and assume that f satisfies the relative homotopy lifting property with respect to the pairs of spaces ( F, F ) and( F, F ) × [0 , f is a Hurewicz fibration, or if f is a Serre fibration and( F, F ) is a relative CW-complex. Also assume that we have a topological embedding i : F × B ֒ → E such that f ◦ i is the projection onto the second factor and i ( − , b ) is the inclusion F ⊆ F ⊆ E .(This says, essentially, that f contains the trivial fibration over B with fibre F as a sub-fibration.) Definition 2.3 ( Relative monodromy actions. ) Under these assumptions, the relative monodromyaction associated to f and F is the action-up-to-homotopymon f : π ( B, b ) −→ π (hAut( F | F )) , (2.3)where hAut( F | F ) is as in Definition 2.1, constructed as follows. For an element [ γ ] ∈ π ( B, b )represented by a loop γ : [0 , → B , let g : F × [0 , → E be a choice of lift in the diagram:( F × [0 , ∪ ( F × { } ) EF × [0 ,
1] [0 , B ( i ◦ (id F × γ )) ∪ incl γ incl f (2.4)and define mon f ([ γ ]) = [ g ( − , Lemma 2.4
The monodromy action (2.1) and relative monodromy action (2.3) are well-defined.Proof.
For the monodromy action (2.1), the proof is given in [PT, Lemma 5.3]. The proof for therelative monodromy action (2.3) is similar.
3. Point-pushing actions
This section defines the point-pushing action associated to a manifold M and a finite subset z ⊂ ˚ M of its interior. We give two definitions, one (Definition 3.1) via the monodromy action ofthe “universal” bundle (3.1), and a smooth version (Definition 3.2) via the long exact sequence ofthe bundle (3.3), as well as a simple geometric description in Lemma 3.4 for manifolds of dimensionat least 3. We then describe point-pushing actions on mapping spaces and other spaces associatedfunctorially to the complement M r z (see Definitions 3.11 and 3.12). Definition 3.1 ( The point-pushing action. ) For a manifold-with-boundary M and a finite subset z ⊆ ˚ M of cardinality k , the point-pushing action of π ( C k ( ˚ M ) , z ) on M r z is defined as follows.First, define ¯ C ,k ( M ) to be the configuration space of k unordered green points in the interiorof M and one red point in M , which may lie on the boundary. There is a fibre bundle u : U k ( M ) = ¯ C ,k ( M ) −→ C k ( ˚ M ) , (3.1)given by forgetting the red point, whose fibre is F = u − ( z ) ∼ = M r z . This is the universal bundle referred to in the introduction. Let F = ∂M ⊆ M r z and note that ( M r z, ∂M ) is a relativeCW-complex, since it is a (smooth) manifold with boundary. There is an obvious embedding i : ∂M × C k ( ˚ M ) ֒ −→ ¯ C ,k ( M )satisfying the conditions of Definition 2.3. By Definition 2.3 and Lemma 2.4, there is therefore awell-defined relative monodromy actionpush ( M,z ) : π ( C k ( ˚ M ) , z ) −→ π (hAut( M r z | ∂M )) . (3.2)This is, by definition, the point-pushing action of π ( C k ( ˚ M ) , z ) on M r z . For [ γ ] ∈ π ( C k ( ˚ M ) , z ),the homotopy class of mapspush γ = push ( M,z ) ([ γ ]) : M r z −→ M r z (fixing ∂M pointwise) is called the point-pushing map of [ γ ] on M r z .4 efinition 3.2 ( A smooth version. ) The monodromy action (3.2) may be refined to an action byisotopy classes of diffeomorphisms of M fixing ∂M . Let Diff ∂ ( M ) denote the topological groupof diffeomorphisms of M fixing ∂M pointwise, in the smooth Whitney topology. There is a fibrebundle ( cf . [Pal60; Cer61; Lim63]): Diff ∂ ( M ) −→ C k ( ˚ M ) , (3.3)defined by ϕ ϕ ( z ), whose fibre over z is the subgroup Diff ∂ ( M, z ) of diffeomorphisms fixing z as a subset. Denote by push sm ( M,z ) : π ( C k ( ˚ M )) −→ π (Diff ∂ ( M, z ))the connecting homomorphism in the long exact sequence of homotopy groups of (3.3). This is,by definition, the smooth point-pushing action of π ( C k ( ˚ M )) on M r z . For [ γ ] ∈ π ( C k ( ˚ M ) , z ),the isotopy class of diffeomorphismspush sm γ = push sm ( M,z ) ([ γ ]) : ( M, z ) −→ ( M, z )(fixing ∂M pointwise and z setwise) is the smooth point-pushing map of [ γ ] on ( M, z ).One may check that these constructions are related as follows.
Lemma 3.3
The point-pushing actions of Definitions 3.1 and 3.2 are related by the commutativediagram π ( C k ( ˚ M ) , z ) π ( C k ( ˚ M ) , z ) π (Diff ∂ ( M, z )) π (hAut( M r z | ∂M )) , push sm ( M,z ) push ( M,z ) = i (3.4) where i is induced by the inclusion Diff ∂ ( M, z ) ֒ → hAut( M r z | ∂M ) given by ϕ ϕ | M r z . 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 ( C k ( ˚ M ) , z ) determines a certain number of orientedloops γ , . . . , γ j in M , each passing through at least one point of z , such that exactly one of theloops passes through each point of z . (The number j k of such loops is the number of cycles inthe cycle decomposition of the permutation of z induced by γ .) Choose representatives of the loops γ , . . . , γ j that are smoothly embedded and have pairwise disjoint images. Also choose pairwisedisjoint closed tubular neighbourhoods T , . . . , T j of these loops, which we assume to be containedin the interior of M . Define a diffeomorphism ϕ ( T ,...,T j ) : ( M, z ) −→ ( M, z )fixing ∂M pointwise and z setwise as follows. On the complement of the tubular neighbourhoods, ϕ ( T ,...,T j ) is the identity. Suppose that the tubular neighbourhood T i contains k i of the points of z (so k + · · · + k j = k ) and identify T r ( z ∩ T ) with(( D d − × R ) r ( { } × Z )) / ∼ , where ∼ is either the equivalence relation given by ( x, t ) ∼ ( x, t + k i ) or the equivalence relationgiven by ( x, t ) ∼ ( r ( x ) , t + k i ), where r : D d − → D d − is a fixed reflection in a hyperplane passingthrough 0, depending on whether or not the loop γ i lifts to a loop in the orientation double coverof M . Choose a smooth function λ : [0 , → [0 ,
1] that takes the value 1 on [0 , ǫ ] and the value 0on [1 − ǫ,
1] for some ǫ >
0. Then the restriction of ϕ ( T ,...,T j ) to T i , under this identification, isdefined by ϕ ( T ,...,T j ) ( x, t ) = ( x, t + λ ( | x | )) . See Figure 3.1 for an illustration. We record this geometric description in the following lemma.5 d0 4 T T r non-orientable loop M Figure 3.1
An example of the point-pushing action for | z | = 6 and where the loop γ ∈ π ( C ( ˚ M ) , z )induces a permutation of z with one 4-cycle and one 2-cycle. Lemma 3.4 ( Geometric point-pushing. ) Let M be a smooth manifold-with-boundary of dimension d > and let [ γ ] ∈ π ( C k ( ˚ M ) , z ) . Choose a collection of smoothly embedded loops γ , . . . , γ j andtubular neighbourhoods T , . . . , T j as described above. Then [ ϕ ( T ,...,T j ) ] = push sm ( M,z ) ([ γ ]) ∈ π (Diff ∂ ( M, z )) . Associated point-pushing actions.
We have so far described the “universal” point-pushingaction of π ( C k ( ˚ M ) , z ) on the complement M r z , for a subset z ⊂ ˚ M with | z | = k . We nowdiscuss 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 (3.8)). Definition 3.5 ( Associated fibre bundles. ) We first recall that, if f : E → B is a fibre bundle withfibre F (and structure group Homeo( F ) in the compact-open topology), and if T : Top → Topis a continuous endofunctor (covariant or contravariant) of the topologically-enriched category ofspaces, there is an associated fibre bundle f T : T fib ( E ) −→ B (3.5)with fibre T ( F ), constructed by “applying T fibrewise” to E . More precisely, the functor T restrictsto a continuous group (anti-)homomorphismHomeo( F ) −→ Homeo( T ( F )) , (3.6)and we define (3.5) to be the Borel construction Prin( E ) × Homeo( F ) T ( F ), where Prin( E ) → B isthe principal Homeo( F )-bundle associated to f , and where Homeo( F ) acts on T ( F ) via (3.6). (See[Ste51, §§8–9] for more details.)There is an exactly analogous construction if f is equipped with a section and T : Top ∗ → Top ∗ is a continuous endofunctor of the topologically-enriched category of based spaces. In this case the6tructure group of f reduces to the based homeomorphism group Homeo ∗ ( F ) and T restricts to acontinuous group homomorphismHomeo ∗ ( F ) −→ Homeo ∗ ( T ( F )) , (3.7)so we may define (3.5) to be the Borel construction Prin ∗ ( E ) × Homeo ∗ ( F ) T ( F ), where Prin ∗ ( E ) → B is the principal Homeo ∗ ( F )-bundle associated to f , and where Homeo ∗ ( F ) acts on T ( F ) via (3.7). Definition 3.6 ( Configuration-mapping spaces. ) Let X be any space and consider the (contravari-ant) continuous functor T = Map( − , X ) : Top −→ Top . The fibre bundle associated by T to the bundle (3.1) is denoted by T fib ( ¯ C ,k ( M )) = CMap k ( M ; X ) −→ C k ( ˚ M ) , and its total space is the k -th configuration-mapping space of M and X . A point in CMap k ( M ; X )consists of a configuration z ⊂ ˚ M in the interior of M and a continuous map M r z → X .If ∂M = ∅ , the fibre bundle (3.1) admits a canonical section given by z ( z, ∗ ), where ∗ ∈ ∂M is a choice of basepoint. Thus, choosing a basepoint for X , we may also consider the fibre bundleassociated to (3.1) by the continuous functor T = Map ∗ ( − , X ) : Top ∗ → Top ∗ , which is denotedby T fib ( ¯ C ,k ( M )) = CMap ∗ k ( M ; X ) −→ C k ( ˚ M ) . A point in CMap ∗ k ( M ; X ) consists of a configuration z ⊂ ˚ M in the interior of M together with a based continuous map M r z → X . Definition 3.7 ( Associated fibre bundles, II. ) The structure group of the bundle (3.1) may bereduced further to Homeo ∂M ( M, z ), the group of self-homeomorphisms of M that fix z setwise and ∂M pointwise. Hence any continuous functor T : Homeo ∂M ( M, z ) −→ Top (3.8)(i.e., any space with a continuous action of Homeo ∂M ( M, z )) associates to (3.1) a new fibre bundle T fib ( ¯ C ,k ( M )) −→ C k ( ˚ M ) (3.9)by taking the Borel construction of the associated principal Homeo ∂M ( M, z )-bundle.
Remark 3.8
For comparison, the associated fibre bundles of Definition 3.5 above correspond tocontinuous functors (3.8) that are of the formHomeo ∂M ( M, z ) −| M r z −−−−−→ Homeo( M r z ) ⊂ Top −→ Top , in other words, that extend to an endofunctor of Top. However, there are interesting (and moresubtle) examples that do not extend in this way, as we show in the next example. Definition 3.9 ( Configuration-mapping spaces, II. ) Fix a basepoint ∗ ∈ ∂M , a based space X anda subset c ⊆ [ S d − , X ] of unbased homotopy classes of maps S d − → X . If M is non-orientablewe assume that c consists of fixed points under the involution of [ S d − , X ] given by a reflection of S d − . There is a continuous functorMap c ∗ ( − , X ) : Homeo ∂M ( M, z ) −→ Top (3.10)defined as follows. The unique object on the left-hand side is sent to the space (with the compact-open topology) of based, continuous maps f : M r z → X with “monodromy” contained in c . Thelast condition means that, if e : D d → M is an embedding such that z ∩ e ( D d ) is a single pointin the interior of e ( D d ), then the homotopy class of f ◦ e | ∂D d lies in c . (If M is orientable, we fixan orientation and require that e is orientation-preserving in the preceding sentence.) One maythen check that the natural action of ϕ ∈ Homeo ∂M ( M, z ) on the mapping space Map ∗ ( M r z, X )preserves the subspace Map c ∗ ( M r z, X ). The fibre bundle associated by (3.10) to the bundle (3.1)is denoted by CMap c, ∗ k ( M ; X ) −→ C k ( ˚ M ) , (3.11)and its total space is the k -th based configuration-mapping space of M and X with “ monodromy ”or “ charge ” c . 7 emark 3.10 Configuration-mapping spaces are discussed in more detail in [PT, §2], and maybe generalised to configuration-section spaces , which are defined in [PT, §3]. There are of coursemany other natural continuous functors T : Top → Top or T : Homeo ∂M ( M, z ) → Top that maybe used to construct interesting fibre bundles associated to the “universal” bundle (3.1).
Definition 3.11 ( Associated point-pushing action. ) For a space T with a continuous action ofHomeo ∂M ( M, z ), viewed as a continuous functor T : Homeo ∂M ( M, z ) → Top, we have from Defi-nition 3.7 a fibre bundle (3.9) T fib ( ¯ C ,k ( M )) −→ C k ( ˚ M )with fibre T . The associated point-pushing action of π ( C k ( ˚ M ) , z ) on T is then the monodromyaction of this fibre bundle, denoted bypush ( M,z,T ) : π ( C k ( ˚ M ) , z ) −→ π (hAut( T )) . (3.12) Definition 3.12 ( Point-pushing action on mapping spaces. ) In particular, if we specialise to thecase T = Map c ∗ ( M r z, X ) for a based space X and a subset c ⊆ [ S d − , X ], as in Definition 3.9,we have an associated point-pushing actionpush ( M,z,X,c ) : π ( C k ( ˚ M ) , z ) −→ π (hAut(Map c ∗ ( M r z, X ))) . which is the monodromy action of the fibre bundle (3.11). This can of course be straightforwardlygeneralised to a point-pushing action of π ( C k ( ˚ M ) , z ) on Map c (( M r z, D ) , ( X, ∗ )) for any subset D ⊆ ∂M .The following elementary lemma relates the point pushing action of π ( C k ( ˚ M ) , z ) on M r z (Def-inition 3.1) and its associated point-pushing action on the mapping space Map c (( M r z, D ) , ( X, ∗ ))(Definition 3.12). Lemma 3.13
The point-pushing action of π ( C k ( ˚ M ) , z ) on Map c (( M r z, D ) , ( X, ∗ )) is obtainedfrom its point-pushing action on M r z by pre-composition. In other words, the following diagramcommutes: π ( C k ( ˚ M ) , z ) π ( C k ( ˚ M ) , z ) π (hAut( M r z | ∂M )) π (hAut(Map c (( M r z, D ) , ( X, ∗ )))) , push ( M,z ) push ( M,z,X,c ) = ◦ (3.13) where the vertical homomorphism ◦ is defined by composition. In particular, the action up tohomotopy of π (hAut( M r z | ∂M )) on the mapping space Map(( M r z, D ) , ( X, ∗ )) preserves thesubspace Map c (( M r z, D ) , ( X, ∗ )) for each subset c ⊆ [ S d − , X ] .
4. Formulas for point-pushing actions
Let M be a connected manifold of dimension d >
3, let z ⊂ ˚ M be a k -point configuration inits interior, D ⊆ ∂M an embedded ( d − X a based space and c ⊆ [ S d − , X ] a non-empty set of unbased homotopy classes of maps S d − → X . Our goal is to giveexplicit formulas for the point-pushing action of π ( C k ( ˚ M ) , z ) on M r z (Definition 3.1). These willbe given in the following two sections; in this section we first fix notation and the identificationsthat we will use. Notation 4.1
Let W k denote a wedge W k S d − of k copies of the ( d − Construction 4.2
Let us choose an explicit homotopy equivalence of pairs( M r z, D ) ≃ ( M ∨ W k , ∗ ) , (4.1)as follows (see Figure 4.1 for an illustration). Choose a d -dimensional closed disc B in M containingthe configuration z in its interior and such that B ∩ ∂M is a ( d − ∂M ∪ = B = B ′ = M ′ ∗ = M δ Figure 4.1
An embedding of M ∨ ( W k S d − ) into M r z as a deformation retract, together with aloop δ in B ′ ∪ M ′ based at z ∩ B ′ and a tubular neighbourhood T of its intersection with M ′ . containing (but not equal to) D . (In Figure 4.1, we may assume that D = ∂M ∩ B ′ .) Note thatthe closure M ′ of M r B in M is also homeomorphic to M . Choose a basepoint ∗ of M in theintersection ∂M ∩ B ∩ M ′ . Choose also k embedded ( d − B such that each sphereintersects ∂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 bythe condition that B r 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 thereis a deformation retraction of M r z onto this subspace, supported in B r z , fixing the basepoint ∗ and sending D onto {∗} . Notation 4.3
From now on, we will write π ( C k ( ˚ M ) , z ) just as π ( 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 Definition 3.2), an element γ ∈ π ( 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 ( T ,...,T j ) given by Lemma3.4 if dim( M ) >
3. We denote its restriction to a self-diffeomorphism of M r z by π γ : M r z −→ M r z. By abuse of notation, we also denote by π γ the (homotopy class of a) homotopy self-equivalenceof M ∨ W k fixing ∗ induced via the deformation retraction (4.2): M r z M r zM ∨ W k M ∨ W k . π γ π γ ≃ incl ≃ (4.2) (4.2)Recall [Til16, Lemma 4.1] that, for dim( M ) >
3, the fundamental group π ( C k ( M )) decomposesas the semi-direct product π ( M ) k ⋊ Σ k . In the next two sections we give explicit formulas for thebottom horizontal map of (4.2) for γ = ( α , . . . , α k ; σ ) ∈ π ( M ) k ⋊ Σ k under this decomposition. Notation 4.5
We collect here some additional notation that will be used in the following twosections. • 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. 9 For a map f : A ∨ B → C we will sometimes write f as a (1 × f = (cid:0) f A f B (cid:1) , where f A = f ◦ inc A and f B = f ◦ inc B . Note that f A and f B jointly determine f , since ∨ isa coproduct. • For a map f : A ∨ B → C ∨ D we will also sometimes write f as a (2 × f = (cid:0) f A f B (cid:1) (cid:18) C f A C f BD f A D f B (cid:19) , where C f A = pr C ◦ f ◦ inc A , etc. Note that the pair of C f A and D f A does not determine f A (since ∨ is not a product), so the (2 × ” instead of “=” in this case.) • As mentioned above, we have for dim( M ) > π ( C k ( M )) ∼ = π ( M ) k ⋊ Σ k . Thus,for each σ ∈ Σ k and α ∈ π ( M ), we have elements(1 , . . . , σ ) and ( α, , . . . ,
1; id) ∈ π ( C k ( M )) , which we will denote simply by σ and α by abuse of notation. We will always use these lettersfor elements of these two subgroups of π ( C k ( M )), and we will denote a general element of π ( C k ( M )) by γ . • We take the basepoint of S d − to be the south pole, and writepinch : S d − −→ S d − ∨ S d − for the map that collapses the equator of S d − to a point. This is a based map, where wetake the convention that the basepoint of S d − ∨ S d − is contained in the left-hand summand(note that it is not the point at which the wedge sum is taken). • We write coll : S d − −→ [0 , S d − ⊂ R d onto the d -th coordinate (so the south polegoes to − x ( x + 1). Remark 4.6
Since π ( C k ( M )) is generated by elements of the form (1 , . . . , σ ) and ( α, , . . . ,
1; id)(which we henceforth denote simply by σ and α ) for σ ∈ Σ k and α ∈ π ( M ), it will suffice to giveexplicit formulas for π σ and π α : M ∨ W k −→ M ∨ W k up to basepoint-preserving homotopy, for all σ ∈ Σ k and α ∈ π ( M ). This will be done in sections5 and 6 respectively. Terminology 4.7
The elements σ = (1 , . . . , σ ) will be called symmetric generators of π ( C k ( M ))and the elements α = ( α, , . . . ,
1; id) will be called loop generators of π ( C k ( M )).
5. Symmetric generators
The action of the symmetric generators of π ( C k ( M )) on M ∨ W k is fairly easy to describe. Proposition 5.1
For any element σ ∈ Σ k we have π σ = id M ∨ σ ♯ = (cid:0) inc M inc W k ◦ σ ♯ (cid:1) (cid:18) id M ∗∗ σ ♯ (cid:19) , (5.1) where σ ♯ denotes the obvious self-map of W k = W k S d − determined by the permutation σ .Proof. In the geometric model ϕ ( T ,...,T j ) (see Lemma 3.4) for the point-pushing diffeomorphism of( M, z ) induced by γ = (1 , . . . , σ ), we may assume that the tubular neighbourhoods T , . . . , T j areall contained in the codimension-zero ball B ⊂ M (see Figure 4.1). Since ϕ ( T ,...,T j ) is the identityoutside of the tubular neighbourhoods, this implies that π σ = id M ∨ ψ , for some automorphism ψ of W k . Moreover, it is clear from this geometric model that (up to homotopy) ψ simply permutesthe k embedded ( d − . Loop generators For any α ∈ π ( M, ∗ ), the point-pushing map π α : M r z → M r z may be assumed (up tobasepoint-preserving homotopy) to be supported in a tubular neighbourhood of a 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 tubularneighbourhood T to be contained in M ′ ∪ B ′ , so the support of π α : M r z → M r z is containedin M ′ ∪ B ′ . Under the identification (4.1), this implies the following. Lemma 6.1
For any α ∈ π ( M ) , up to based homotopy, π α : M ∨ W k → M ∨ W k is of the form π α = ¯ π α ∨ id W k − , where ¯ π α is a self-map of M ∨ S d − , unique up to based homotopy. We therefore just have to describe the map ¯ π α for each α ∈ π ( M ). We first do this under anadditional assumption on the manifold M . Recall that the handle-dimension of a manifold is thesmallest i such that M may be constructed using handles of degree at most i . Using the cores ofsuch a handle decomposition, this implies that M deformation retracts onto an embedded CW-complex of dimension equal to the handle dimension of M . Since M , in our situation, is connectedand has non-empty boundary, its handle-dimension is necessarily at most dim( M ) − Proposition 6.2
Suppose that the handle dimension of M is at most dim( M ) − . Then, for anyelement α ∈ π ( M ) we have ¯ π α = (cid:0) inc M (( α ◦ coll) ∨ sgn( α )) ◦ pinch (cid:1) (cid:18) id M α ◦ coll ∗ sgn( α ) (cid:19) , (6.1) where sgn( α ) : S d − → S d − has degree +1 if α lifts to a loop in the orientation double cover of M and degree − otherwise. The other notation is explained in Notation 4.5. If the handle dimension of M is equal to dim( M ) − π α is more complicated. The following proposition gives the general formula. Proposition 6.3
For any element α ∈ π ( M ) we have ¯ π α = (cid:0) ⋔ α (( α ◦ coll) ∨ sgn( α )) ◦ pinch (cid:1) (cid:18) id M α ◦ coll ⋔ α sgn( α ) (cid:19) , (6.2) 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 statementof Proposition 6.3 (Definitions 6.5 and 6.6) and then prove Proposition 6.3. In this subsection we prove Proposition 6.2. Letus write • ¯ π Mα : M → M ∨ S d − for the restriction of ¯ π α to the M summand of M ∨ S d − ; • ¯ π Sα : S d − → M ∨ S d − for the restriction of ¯ π α to the S d − summand of M ∨ S d − .In this notation, to prove Proposition 6.2, we need to show that¯ π Mα ≃ inc M and ¯ π Sα ≃ (( α ◦ coll) ∨ sgn( α )) ◦ pinch . (6.3)We first prove the right-hand side of (6.3). This may in fact be seen purely geometrically fromFigure 4.1. We need to describe the effect of π α on the loop (representing a ( d − π α maybe 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 thetube T on the ( d − ∗ pictured in the figure, it is easier first to replace it, up to11omotopy equivalence, by a ( d − z ∩ B ′ together with a “tether”connecting this sphere to the basepoint ∗ (this corresponds to the pinch and collapse maps in theformula (6.3)). Point-pushing along T has the effect on the tether of sending it around a loophomotopic to α . On the ( d − ± T is orientable or not, in other words, whetheror not α lifts to a loop in the orientation double cover of M , which is exactly sgn( α ). Putting thisall 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: • ¯ π Mα ≃ inc M ◦ θ α for some self-map θ α : M → M ; • θ α ≃ id M .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 − π Mα to K is a map of the form K −→ M ∨ S d − . We may homotope this to be cellular , i.e., so that every r -cell of K is mapped into a cell ofdimension at most r . This implies that the image of the map must intersect S d − only in thebasepoint, so we have a factorisation up to homotopy¯ π Mα | K : K −→ M ֒ −→ M ∨ S d − , for some map K → M . Since the inclusion of K into M is a homotopy equivalence, this impliesalso that ¯ π Mα itself factorises up to homotopy as a self-map θ α of M followed by the inclusion into M ∨ S d − . This establishes the first claim above.We next have to prove that θ α is homotopic to the identity. Consider the following diagram. M MM ∨ S d − M ∨ S d − M M θ α ¯ π α id (6.4)The upper vertical inclusions are both the inclusion of the M summand into M ∨ S d − . The lowervertical inclusions are both the embedding of M ∨ S d − into M illustrated in Figure 4.1. Thebottom square commutes up to homotopy since any point pushing map becomes homotopic to theidentity once the puncture(s) have been filled in. The top square commutes up to homotopy bywhat we have just proven: that ¯ π Mα factors through θ α up to homotopy. The composition of theleft-hand vertical maps is homotopic to the identity M → M , and similarly for the right-hand side.Hence three out of the four sides of the outer square of (6.4) are homotopic to the identity, so thefourth 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 thetwo statements ¯ π Mα ≃ ⋔ α and ¯ π Sα ≃ (( α ◦ coll) ∨ sgn( α )) ◦ pinch , (6.5)and in the proof above we did not use the hypothesis on the handle-dimension of M when provingthe right-hand side of (6.3), which is the same as the right-hand side of (6.5). 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 6.3implies that they do not depend on these additional choices up to homotopy (see Remark 6.7).12 efinition 6.5 Let K ⊂ M be a CW-complex of dimension at most d − M suchthat 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 restrictionΦ τ | int( D i ) : int( D i ) −→ K ⊂ M is a smooth embedding. This exists since M is connected and has non-empty boundary, so itshandle-dimension is at most d −
1: such a CW-complex K may be constructed from the cores ofa handle decomposition of M with one 0-handle. Let α ∈ π ( M ) and choose a representative loopof α that is a smooth embedding, transverse to the interior of every cell of K and also transverseto ∂M . (For the assumption that the representative of α may be chosen to be an embedding , weare using the fact that M has dimension at least 3.)Given these choices, we define the map ⋔ α : M → S d − as follows: ⋔ α : M −→ K −→→ K/K ( d − ∼ = _ τ S d − −→ S d − , (6.6)where the map M → K is a homotopy inverse of the inclusion, the index τ runs over all ( d − K and the τ -th component of the last map is a map S d − → S d − of degree ♯ ( τ, α ), which isthe 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 − witha wedge of ( d − ♯ ( τ, α ) is well-defined.For the first point, we simply choose, arbitrarily and once and for all, an orientation of S d − and an orientation of each open ( d − τ (int( D d − )) of K . The identification of K/K ( d − with a wedge of copies of S d − is then well-defined, up to based homotopy, by taking it to be orientation-preserving on each open ( d − ♯ ( τ, α ) is well-defined, weneed an orientation of α and of each open ( d − τ , as well as a local orientation of M at eachintersection point of α with the interior of τ , i.e., each point ofΦ τ (int( D d − ) ∩ α ([0 , . (6.7)We have already chosen orientations of each open ( d − τ , and α is an oriented loop, so itremains to choose local orientations of M at each point of (6.7). We do this in several steps: • We have already chosen an orientation of S d − , which is embedded into B ′ (see Figure 4.1). • By radial expansion, this determines an orientation of ∂M ∩ B ′ . • In particular, it determines a local orientation of ∂M at the basepoint ∗ . • This, together with α , determines a local orientation of M at ∗ as follows: we take it to bethe local orientation of M at ∗ such that the algebraic intersection number of α | [1 − ǫ, with ∂M at ∗ is +1. • If M is orientable, this then determines an orientation of M , and in particular local orienta-tions of M at each point of (6.7). • If M is non-orientable, we have to be more careful. Choose ǫ > α ([ ǫ, T of α | [ ǫ, .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 − and for each open ( d − τ of K : • Suppose that we reverse the orientation of one ( d − τ . This affects the identificationof K/K ( d − with the wedge of ( d − W τ S d − that sends each sphere to itself, has degree − τ componentand has degree +1 on all other components. However, it also has the effect of reversing thesign of the algebraic intersection number ♯ ( τ , α ), so these effects cancel each other out aftercomposing all maps in (6.6). 13 Suppose that we reverse the orientation of S d − . This affects the identification of K/K ( d − with the wedge of ( d − W τ S d − that sends each sphere to itself and has degree − 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 − . Definition 6.6
Let K ⊂ M be an embedded CW-complex as in Definition 6.5. We have alreadyassumed that K has a unique 0-cell ∗ , and we now assume further that, for each i -cell τ of K , for i >
0, the image of its attaching map φ τ : ∂D i → K ( i − contains ∗ . Choose a representative loopof α ∈ π ( M ) as in Definition 6.5.We now define a map ⋔ α : M → M ∨ S d − whose composition with pr S d − : M ∨ S d − → S d − is ⋔ α . This is the map ⋔ α : M −→ K −→ M ∨ S d − (6.8)where the first map is a homotopy inverse of the inclusion and the second map is defined as follows.On the ( d − K ( d − ⊂ K ⊂ M ⊂ M ∨ S d − . We nowextend this to each ( d − K , in other words, for each ( d − τ of K , we define a map ⋔ α,τ : D d − −→ M ∨ S d − (6.9)whose restriction to ∂D d − is equal to the attaching map φ τ : ∂D d − → K ( d − of τ followed bythe inclusion K ( d − ⊂ K ⊂ M ⊂ M ∨ S d − . We define the map (6.9) in several steps: • Choose a point e ∗ ∈ ∂D d − such that φ τ ( e ∗ ) = ∗ . • Denote the intersection points of α with the interior of τ byΦ τ (int( D d − )) ∩ α ([0 , { y , . . . , y n } and write x i = Φ − τ ( y i ) ∈ int( D d − ). Let p i be the straight-line path in D d − from e ∗ to x i . • Fix orientations of D d − and S d − . Choose an embedding e n : W n S d − ֒ −→ D d − taking the basepoint to e ∗ and every other point to the interior of D d − , such that the images ofthe n copies of S d − are non-nested in D d − (see Figure 6.1). There is a unique identification D d − / im( e n ) ∼ = D d − ∨ W n S d − that is orientation-preserving away from the basepoint. Wetherefore have a map c n : D d − −→ D d − ∨ W n S d − , from which we obtain the map (see Notation 4.5 and Figure 6.1 for a picture):¯ c n = (id ∨ W n ((coll ∨ id) ◦ pinch)) ◦ c n : D d − −→ D d − ∨ W n (cid:0) [0 , ∨ S d − (cid:1) . (6.10) • Finally, we define (6.9) by ⋔ α,τ = ⋔ ⋄ α,τ ◦ ¯ c n , where the map ⋔ ⋄ α,τ : D d − ∨ W n (cid:0) [0 , ∨ S d − (cid:1) −→ M ∨ S d − is defined on each component as follows. • On the D d − component, ⋔ ⋄ α,τ is the characteristic map Φ τ : D d − → K ( d − followedby the inclusion K ( d − ⊂ K ⊂ M ⊂ M ∨ S d − . • On the i -th [0 ,
1] component, ⋔ ⋄ α,τ is the element of π ( M ) given by α | [ α − ( y i ) , · (Φ τ ◦ p i ) . • On the i -th S d − component, ⋔ ⋄ α,τ is a map S d − → S d − of degree ǫ i ∈ {± } , wherethe sign ǫ i is determined as follows. • As in Definition 6.5, the chosen orientation of S d − determines a local orientationof M at ∗ . 14 n W n S d − D d − e ∗ c n D d − W n S d − pinch and collapse¯ c n Figure 6.1
The quotient map ¯ c n : D d − −→→ D d − ∨ W n ([0 , ∨ S d − ) from Definition 6.6. • We have also chosen an orientation of D d − , and Φ τ is a smooth embedding on theinterior of D d − , so we also have an orientation of Φ τ (int( D d − )). This determinesa local orientation of M at the intersection point y i : namely the one with respectto which the intersection number of Φ τ (int( D d − )) with α ([0 , y i is +1. • If M is orientable, these two local orientations each determine an orientation of M ,and we set ǫ i to be +1 if they agree and − • If M is non-orientable, we have to be more careful, just as in Definition 6.5. Choose δ > y , . . . , y n are contained in α ([ δ, T of α | [ δ, . Since T is an orientable codimension-zero 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 ǫ i = +1 if they agree and ǫ i = − ⋔ α is independent of the choices oforientation of S d − and D d − . It is also independent of the choice of pre-image e ∗ of the basepoint ∗ ∈ K under the attaching map of τ : modifying this choice affects the map ¯ c n and the map ⋔ ⋄ α,τ oneach [0 ,
1] component, and these effects cancel out when we compose them to form ⋔ α,τ = ⋔ ⋄ α,τ ◦ ¯ c n . Remark 6.7
A priori, the maps ⋔ α : M → S d − and ⋔ α : M → M ∨ S d − described in Definitions6.5 and 6.6 depend on the choice of embedded CW-complex K and the choice of representative of α ∈ π ( 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 ( M ). This is because Proposition6.3 identifies these two maps with certain maps derived from the point-pushing map π α , whichdepends up to homotopy only on α ∈ π ( M ). Proof of Proposition 6.3.
As pointed out in Remark 6.4, we have already proven one half of Propo-sition 6.3 while proving Proposition 6.2. The remaining statement to prove is¯ π Mα ≃ ⋔ α : M −→ M ∨ S d − . (6.11)We will first prove the two (jointly weaker) statements:pr M ◦ ¯ π Mα ≃ id M and pr S d − ◦ ¯ π Mα ≃ ⋔ α , (6.12)which correspond to the (2 × π α on the right-hand side of (6.2). Consider15he following homotopy-commutative diagram. M M ∨ S d − M ∨ S d − M M ¯ π α idid (6.13)(The square is the same as the bottom square of (6.4).) The two vertical inclusions are both theembedding of M ∨ S d − into M illustrated in Figure 4.1. But this is homotopic to the projectionpr M of M ∨ S d − onto its first summand, so pr M ◦ ¯ π Mα is the composition from the top-left to thebottom-right of the diagram, and hence homotopic to the identity. This proves the left-hand sideof (6.12).Next, we prove the right-hand side of (6.12). We start by giving another description of the map w α = pr S d − ◦ ¯ π Mα : M −→ S d − using Figure 4.1. Choose a path p in B ′ from ∗ to the point z ∩ B ′ and choose a loop δ in B ′ ∪ M ′ ,intersecting ∂M ′ transversely in two points, in the homotopy class of p · γ · ¯ p . Also choose atubular neighbourhood T of δ ∩ M ′ in M ′ . Geometrically, the map w α : M → S d − is then givenby starting in M ′ , including into M , applying the point pushing map along the loop δ and thencollapsing onto the copy of S d − contained in B ′ . Clearly the complement M ′ r T of the tubularneighbourhood T is sent to the basepoint under this map. To describe how w α acts on T , we usethe following identifications. The intersection T ∩ ∂B ′ consists of two disjoint ( d − T and T , where we assume that T contains the intersection point of δ ∩ ∂B ′ where δ is pointing into M ′ and T contains the intersection point of δ ∩ ∂B ′ where δ is pointing into B ′ . We may thenidentify T with T × [0 ,
1] and describe the map w α on T by T ∼ = T × [0 , −→ T −→ T /∂T ≃ S d − , (6.14)where the two maps are the obvious projections and T /∂T ≃ S d − is the composition of thecanonical identifications T /∂T ≃ ∂B ′ ≃ S d − , given respectively by the fact that T is a closed disc in the sphere ∂B ′ and the fact that ∂B ′ deformation retracts onto the copy of S d − embedded in B ′ .We now use this geometric description of w α to show that it is homotopic to the map ⋔ α definedin Definition 6.5. Let K be a CW-complex of dimension at most d − M ′ , suchthat M ′ deformation retracts onto K . We need to show that the restriction of w α to K factors as K −→→ K/K ( d − ∼ = W τ S d − −→ S d − , (6.15)where the τ -th component of the right-hand map is a map f τ : S d − → S d − of degree ♯ ( τ, δ ). Bysmooth approximation and transversality, we may assume that each ( d − τ of K is smoothlyembedded into M ′ and that δ and T have been chosen so that (a) each r -cell of K , for r d −
2, isdisjoint from T and (b) each τ ∩ T , for τ a ( d − K , consists of finitely many ( d − δ transversely in one point.By property (a), and since M ′ r T is sent to the basepoint by w α , we see that its restrictionto K must factor through the projection K ։ K/K ( d − . So we just have to show that f τ hasdegree ♯ ( τ, δ ). By property (b) and the description (6.14) of w α | T , each component of the disjointunion of ( d − τ ∩ T contributes either +1 or − f τ ). Being careful about (local)orientations as explained in Definition 6.5, we see that the sum of these +1’s and − ♯ ( τ, δ ) of τ and δ .This completes the proof that w α | K factors as in (6.15), and hence that w α ≃ ⋔ α , in otherwords, the right-hand side of (6.12).The proof of (6.11) is similar to the proof above of the right-hand side of (6.12): looking at Figure4.1 and using a geometric model for the point-pushing map supported in a tubular neighbourhood16 ′ B y y y τTα ∗ α ∩ τ viewed in D d − τ inc ◦ Φ τ α | [ α − ( x ) , ± id M MS d − α ∩ τ viewed in M αy y y S d − Figure 6.2
Two views of the effect of point-pushing along an embedded arc α on a ( d − τ :(1) Embedded in M . — (2) Intrinsically in the disc parametrising the cell τ . of an embedded loop representing α , one must check carefully that the definition of ⋔ α given inDefinition 6.6 is a correct description of ¯ π Mα up to homotopy. Rather than go through this insymbols, we refer the reader instead to Figure 6.2, which depicts the map ¯ π Mα induced by point-pushing along α , and which one may compare to the definition of ⋔ α in Definition 6.6.
7. Formulas for associated point-pushing actions on mapping spaces
As an immediate corollary of Proposition 5.1, Lemma 6.1 Proposition 6.2 and Lemma 3.13,we obtain (under certain assumptions on M ) a formula for the associated point-pushing action(Definition 3.12) of π ( C k ( M )) on the mapping space Map c ∗ ( M r z, X ), under the identificationMap c ∗ ( M r z, X ) ≃ Map ∗ ( M, X ) × (Ω d − c X ) k (7.1)induced by the identification (4.2) of M r z with M ∨ W k S d − . On the right-hand side of (7.1),Ω d − c X denotes the union of path-components of Ω d − X corresponding to the subset c ⊆ [ S d − , X ]. Remark 7.1
There are two natural actions on the space Ω d − c X . First, there is an up-to-homotopyaction of π ( X ) on Ω d − X , which restricts to an action-up-to-homotopy on the subspace Ω d − c X (this is because the subset c ⊆ [ S d − , X ] corresponds to a union of π ( X )-orbits of π d − ( X )).Second, there is an involution of Ω d − X given by precomposition with a reflection of S d − ina hyperplane containing the basepoint; this involution commutes with the up-to-homotopy actionof π ( X ). If c ⊆ [ S d − , X ] is invariant under the corresponding involution of [ S d − , X ], then thisinvolution restricts to the subspace Ω d − c X . In our situation, the involution will only be relevantif M is non-orientable, in which case we have assumed (see Definition 3.9) that c ⊆ [ S d − , X ] is asubset of the fixed points under the involution, so in particular it is invariant under the involution. Corollary 7.2 If d = dim( M ) > and M satisfies at least one of the following conditions: • M is simply-connected, or • the handle-dimension of M is at most d − ;then the point-pushing action of γ = ( α , . . . , α k ; σ ) ∈ π ( C k ( M )) ∼ = π ( M ) k ⋊ Σ k on the mappingspace Map c ∗ ( M r z, X ) , under the identification (7.1) , is given as follows ( see also Figure 7.1 )( α , . . . , α k ; σ ) · ( f, g , . . . , g k ) = ( f, ¯ g , . . . , ¯ g k ) , (7.2)17 ....... ( ) ( ) σ α α α k f ∗ ( α ) .g σ (1) . sgn( α ) f ∗ ( α ) .g σ (2) . sgn( α ) f ∗ ( α k ) .g σ ( k ) . sgn( α k ) f α σ − (1) α σ − (2) α σ − ( k ) g g g k f push γ = ( α , . . . , α k ; σ ) = γ Figure 7.1
The action of the point-pushing map associated to γ = ( α , . . . , α k ; σ ) ∈ π ( C k ( M )) onthe mapping space Map ∗ ( M, X ) × (Ω d − c X ) k . The loop γ is represented in blue, the elements of themapping space in black and the point-pushing map is represented in green. where ¯ g i = f ∗ ( α i ) .g σ ( i ) . sgn( α i ) , and • for an element α ∈ π ( M ) we write sgn( α ) = +1 if α lifts to a loop in the orientation doublecover of M and sgn( α ) = − otherwise, • the actions of π ( X ) and of {± } on Ω d − c X are as described in Remark 7.1 above.Proof. It suffices to check this for elements of the form (1 , . . . , σ ) and ( α, , . . . ,
1; id) (symmetricand loop generators), which we denote simply by σ and α by abuse of notation.By Proposition 5.1, the action of σ on M r z ≃ M ∨ W k is the identity on the M summand andpermutes the k copies of S d − in W k = W k S d − . Lemma 3.13 tells us that the associated point-pushing action of σ on Map ∗ ( M, X ) × (Ω d − c X ) k is induced from its point-pushing action on M ∨ W k by precomposition, so we deduce that it acts by the identity on the Map ∗ ( M, X ) component and theΩ d − c X components are permuted by σ − (the inverse occurs since precomposition is contravariant).Similarly, Lemma 3.13 implies that the point-pushing action of α on Map ∗ ( M, X ) × (Ω d − c X ) k is induced from the point-pushing action of α on M ∨ W k , which is described by Lemma 6.1and Proposition 6.2, by precomposition. Putting this together, we see that α sends the tuple( f, g , . . . , g k ) to the tuple ( f, f ∗ ( α ) .g . sgn( α ) , g , . . . , g k ), as desired. Specifically, the f entry inthis tuple follows from the left-hand side of (6.3), the f ∗ ( α ) .g . sgn( α ) entry follows from the right-hand side of (6.3) and the remaining entries follow from Lemma 6.1. Remark 7.3
Part of the formula (7.2) remains valid without the additional hypothesis on M .More precisely, assuming still that dim( M ) > M is nowallowed to be non-simply-connected and to have maximal handle-dimension), the formula for theaction of γ = ( α , . . . , α k ; σ ) becomes( α , . . . , α k ; σ ) · ( f, g , . . . , g k ) = (? , ¯ g , . . . , ¯ g k ) , (7.3)where the entry ? is not in general f , but rather a based map M → X that depends in a subtle wayon f , the loop γ and the elements g i . For example, when γ = ( α, , . . . ,
1; id), the map ? : M → X is given by the compositionfold ◦ ( f ∨ g ) ◦ ⋔ α : M −→ M ∨ S d − −→ X ∨ X −→ X, where ⋔ α is the map defined in Definition 6.6. To see this, recall that the equations (6.3) describethe point-pushing action of a loop generator α under the additional assumptions on M , and theequations (6.5) describe the point-pushing action of α without these assumptions. The right-handequation of (6.3) agrees with the right-hand equation of (6.5), which is why the tuple (¯ g , . . . , ¯ g k )occurs in (7.3), just as in (7.2). However, the left-hand equation of (6.3) is simply ¯ π Mα ≃ inc M ,whereas the left-hand equation of (6.5) is ¯ π Mα ≃ ⋔ α .18 emark 7.4 Corollary 7.2 is used in [PT, §9] to prove a certain split-injectivity result for mapsbetween configuration-mapping spaces. More precisely, there is a natural map of spectral sequencesconverging to the map on homology induced by the stabilisation map
CMap c, ∗ k ( M ; X ) −→ CMap c, ∗ k +1 ( M ; X ) . Under the hypotheses on M assumed in Corollary 7.2, this map of spectral sequences is split-injective on E pages. For the precise statement, see [PT, Theorem 9.1].Corollary 7.2 may also be used to understand the path-components of configuration-mappingspaces of manifolds of dimension at least 3. As an example, we have the following. Corollary 7.5
Suppose that dim( M ) > , M is orientable and either • M is simply-connected, or • the handle-dimension of M is at most d − .Then there is a natural bijection π (CMap c, ∗ k ( M ; X )) ∼ = G f ∈h M,X i SP k ( c f ) , (7.4) where h M, X i = π (Map ∗ ( M, X )) , the notation SP k ( ) means ( ) k / Σ k and c f is the pre-imageof c ⊆ [ S d − , X ] under the quotient map π d − ( X ) /f ∗ ( π ( M )) −→ π d − ( X ) /π ( X ) = [ S d − , X ] . Proof.
By the long exact sequence associated to the bundle (3.11), the left-hand side of (7.4) isnaturally in bijection with the set of orbits of π (Map c ∗ ( M r z, X )) ∼ = h M, X i × e c k under the monodromy (i.e., point-pushing) action of π ( C k ( M )), where e c denotes the pre-imageof c ⊆ [ S d − , X ] under the quotient map π d − ( X ) → π d − ( X ) /π ( X ) = [ S d − , X ]. Corollary 7.2implies that the elements of π ( C k ( M )) act on a tuple ([ f ] , [ g ] , . . . , [ g k ]) by (i) permuting the [ g i ]’sand (ii) acting on each [ g i ] (individually) by f ∗ ( π ( M )) π ( X ). The formula (7.4) follows. References [Ban17] J. E. Banks.
The Birman exact sequence for 3-manifolds . Bull. Lond. Math. Soc. ↑ Mapping class groups and their relationship to braid groups . Comm. Pure Appl.Math.
22 (1969), pp. 213–238 ( ↑ Topologie de certains espaces de plongements . Bull. Soc. Math. France
89 (1961),pp. 227–380 ( ↑ A fibre bundle description of Teichmüller theory . J. Differential Ge-ometry ↑ Teichmüller theory for surfaces with boundary . J. Differential Ge-ometry ↑ Homological stability for Hurwitz spaces andthe Cohen-Lenstra conjecture over function fields, II . ArXiv:1212.0923v1. ( ↑ Le type d’homotopie du groupe des difféomorphismes d’une surface compacte . Ann.Sci. École Norm. Sup. (4) ↑ Algebraic topology . Cambridge University Press, Cambridge, 2002, pp. xii+544 ( ↑ Mapping class groups of nonorientable surfaces . Geom. Dedicata
89 (2002), pp. 109–133 ( ↑ Homeomorphisms of non-orientable two-manifolds . Proc. Cambridge Philos.Soc.
59 (1963), pp. 307–317 ( ↑ On the local triviality of the restriction map for embeddings . Comment. Math. Helv.
38 (1963), pp. 163–164 ( ↑ May99] J. P. May.
A concise course in algebraic topology . Chicago Lectures in Mathematics. Universityof Chicago Press, Chicago, IL, 1999, pp. x+243 ( ↑ Local triviality of the restriction map for embeddings . Comment. Math. Helv. ↑ Homology of configuration-mapping and -section spaces . In prepa-ration. ( ↑
1, 2, 4, 8, 19).[Ste51] N. Steenrod.
The Topology of Fibre Bundles . Princeton Mathematical Series, vol. 14. PrincetonUniversity Press, Princeton, N. J., 1951, pp. viii+224 ( ↑ Homology stability for symmetric diffeomorphism and mapping class groups . Math.Proc. Cambridge Philos. Soc. ↑ Institutul de Matematică Simion Stoilow al Academiei Române, 21 Calea Grivit , ei, 010702 Bucures , ti,RomâniaMathematical Institute, University of Oxford, Andrew Wiles Building, Woodstock Road, Oxford,OX2 6GG, UK [email protected]@[email protected]@maths.ox.ac.uk