Configuration-mapping spaces and homology stability
aa r X i v : . [ m a t h . A T ] J u l Homology of configuration-mapping and -section spaces
Martin Palmer and Ulrike Tillmann nd July 2020
Abstract
For a given bundle ξ : E → M over a manifold, configuration-section spaces on ξ parametrisefinite subsets z ⊆ M equipped with a section of ξ defined on M r z , with prescribed “charge”in a neighbourhood of the points z . These spaces may be interpreted physically as spaces offields that are permitted to be singular at finitely many points, with constrained behaviournear the singularities. As a special case, they include the Hurwitz spaces , which parametrisebranched covering spaces of the 2-disc with specified deck transformation group.We prove that configuration-section spaces are homologically stable (with Z coefficients)whenever the underlying manifold M is connected and has non-empty boundary and the chargeis “small” in a certain sense, and describe a model for the stable homology. This has a partialintersection with the work on Hurwitz spaces of Ellenberg, Venkatesh and Westerland. Contents
1. Introduction 12. Configuration-mapping spaces 43. Configuration-section spaces 74. On E m -modules over E n -algebras 145. Monodromy actions 206. Braid categories 247. Stabilisation maps and extension to C ( M ) 258. Polynomiality and stability 299. Extension to B ♯ ( M ) and split-injectivity 3410. Group-completion and stable homology 36References 44
1. Introduction
Configuration spaces of points in manifolds have been intensively studied in topology and geom-etry, and may be interpreted physically as a model for particles moving in a background space. In labelled configuration spaces, each particle is equipped with an additional parameter, taking valuesin a fixed space X or, more generally, in a bundle over the underlying manifold. A more physicallyrelevant setting corresponds to equipping not the particles, but instead their complement , with amap to X or a section of a bundle over the underlying manifold (this viewpoint is suggested in[Seg14], for example). For maps to a fixed space X , these are the configuration-mapping spaces ,introduced in [EVW]. Since these spaces are intended to model particles moving in physical fields,which typically take values in a (possibly non-trivial) bundle over the underlying manifold, one isnaturally led to consider, more generally, configuration-section spaces , which we introduce in §3.Roughly, configuration-mapping spaces are defined as follows. Given a d -dimensional manifold M , a space X and a set c ⊆ [ S d − , X ] of unbased homotopy classes of maps from S d − to X , a Mathematics Subject Classification : 55R80, 57N65, 55N25
Key words and phrases : Configuration-mapping spaces, configuration-section spaces, homological stability, stablehomology, group-completion, polynomial functors, braid categories, Swiss cheese operads. k -th configuration-mapping space CMap ck ( M ; X )consists of a subset z ⊂ ˚ M of the interior of M of cardinality k and a continuous map f : M r z → X .Moreover, we require that the restriction of f to a small punctured neighbourhood of each pointof z lies in one of the homotopy classes in c . The homotopy class of the germ of f near p ∈ z maybe thought of as the “ charge ” (or “ monodromy ” or “ singularity type ”) of the particle p , and c istherefore the set of allowed charges of the particles in the system being modelled. For a subset D ⊆ ∂M (usually a point or a disc) and a basepoint ∗ ∈ X , one may also impose the boundarycondition that f ( D ) = {∗} . See §2 for precise definitions, including how to topologise this set, and§3 for the generalisation to configuration-section spaces. Some examples.
In the special case d = 2 and X = BG (so c is a subset of [ S , BG ] = Conj( G ))the space CMap ck ( M ; BG ), up to homotopy equivalence, parametrises branched coverings of thesurface M with deck transformation group G and with monodromy around the branch pointslying in c . In particular, when M = D , these are the Hurwitz spaces associated to the pair(
G, c ); see Remark 2.14 and Example 3.27 for more details. The main result of [EVW16] is arational homological stability theorem for Hurwitz spaces when G is a finite group and c is a singleconjugacy class that generates G and is “non-splitting” ( cf . Remark 1.1 below).As mentioned above, the physical motivation for studying configuration-section spaces is theirinterpretation as spaces of particles moving in fields defined on their complement. For example, wemay take X to be the Eilenberg-MacLane space K ( Z , d ) (in this case we have [ S d − , X ] = {∗} so c = {∗} ) and consider the configuration-mapping space CMap c, ∗ k ( D d ; K ( Z , d )). Its points consistof a configuration z ⊂ R d ∼ = ˚ D d together with a based map f : D d r z → K ( Z , d ), which may bethought of ( non-canonically ) as associating a “phase” in S ≃ Ω d − K ( Z , d ) to each particle p ∈ z .However, this description of f as a separate phase for each particle cannot be made consistent asthe configuration z varies, so it is really modelling some non-local data associated to the wholeconfiguration of particles. In particular, when d = 3, this could be a model for an asymptotic partof the moduli space of magnetic monopoles of total charge k in R ( cf . [AH88, Proposition 3.12]and [Seg97]). See Example 3.30 for further details.Further examples, which are configuration-section spaces but not configuration-mapping spaces(in general), include configuration spaces where the complement of the configuration is equippedwith a tangential structure (e.g. orientation, spin, etc) that need not extend to the whole of M (see Example 3.28), or with a tuple of linearly independent vector fields (see Example 3.29). Homological stability.
Our main result is that configuration-section spaces are homologicallystable , subject to a condition on the “charge” c . Theorem A
Let M be a connected manifold of dimension d > with basepoint ∗ ∈ ∂M . Let X bea based space and choose an element g ∈ π d − ( X ) that is fixed under the natural action of π ( X ) .Set c = ⊆ π d − ( X ) /π ( X ) = [ S d − , X ] . Then there are stabilisation maps CMap c, ∗ k ( M ; X ) −→ CMap c, ∗ k +1 ( M ; X ) (1.1) inducing isomorphisms on H i ( − ; Z ) in the range k > i + 4 and surjections in the range k > i + 2 .With field coefficients, these ranges may be improved to k > i + 2 and k > i respectively. See Theorem 8.5 for the precise statement, including the generalisation to configuration-sectionspaces, where the analogue of the “charge” c is slightly more subtle to define.This is an analogue of classical homological stability ([Seg73; McD75; Seg79]; see also [Ran13b])for ordinary (unordered) configuration spaces on a connected, open manifold. Remark 1.1 ( Relation with the result of [EVW16] . ) In the case of Hurwitz spaces mentionedabove, our assumption is that c = ⊆ Conj( G ) is a single conjugacy class of size 1 (correspond-ing to an element g of the centre of G ). The result of [EVW16], by contrast, allows larger conjugacy2lasses, although it is specific to the setting of Hurwitz spaces Hur cG,k ≃ CMap c, ∗ k ( D ; BG ). Thesetwo results are therefore somewhat orthogonal in terms of generality – and indeed our methodsare entirely different to those of Ellenberg, Venkatesh and Westerland.In more detail, the main result of [EVW16] says that, in a stable range, the rational homologygroups H i (Hur cG,k ; Q ) are periodic with respect to k (with a period depending on the pair ( G, c )),as long as G is finite, the conjugacy class c generates G and c is “non-splitting” in the sense that H ∩ c is either empty or a single conjugacy class for every subgroup H G . On the other hand,specialising Theorem A to the case ( M = D , X = BG ), we show that the integral homology groups H i (Hur cG,k ; Z ) are constant (1-periodic) in a stable range, as long as | c | = 1. The intersection ofour results is the case where G is finite cyclic and c = { g } for a generator g ∈ G . Split-injectivity.
For ordinary configuration spaces, and configuration spaces where the points(rather than the complement) are labelled, the analogous stabilisation maps induce split-injections on homology in all degrees. This follows from an easy argument (see [McD75, page 103]; also[MT14, §4] for a diffeomorphism equivariant stable splitting) that depends on the existence ofmulti-valued maps C k ( M ) C k − r ( M ) that “forget r points from the configuration in all (cid:0) kr (cid:1) possible ways”.However, such maps do not exist for configuration-section spaces, since in this setting one cannotsimply forget a point; one must also extend the section on the complement of the configurationto the forgotten point, which is in general impossible. Indeed, we expect that split-injectivity onhomology does not hold for configuration-section spaces in general. Nevertheless, we do have a par-tial positive result in this direction, for configuration-mapping spaces under additional hypotheseson the underlying manifold M : a certain map of spectral sequences, converging to the stabilisationmap (1.1) on homology, is split-injective on E pages. See Theorem 9.1 for the precise statement.This relies on a detailed study of the monodromy action associated to the fibration (1.2), which iscarried out in the companion paper [PT]. Stable homology.
It is well-known ([Seg73; McD75; Böd87]) that ordinary (unordered) configu-ration spaces on a connected, open manifold model function and more generally section spaces, thehomology of which coincides with the stable homology of the configuration spaces. This remainstrue also in our case.
Theorem B
Let M be a connected manifold of dimension d > with basepoint ∗ ∈ ∂M and ∗ ∈ L ( ∂M be a proper submanifold of the boundary. For any based space X and subset c ⊆ [ S d − , X ] ,there exists a bundle E d ( X, c ) over M containing the trivial bundle M × X such that CΓ c, ∗ ( M, L ; X ) −→ Γ((
M, L ); E d ( X, c )) is a weak homotopy equivalence; here Γ((
M, L ); E d ( X, c )) denotes the space of sections that take ∗ ∈ ∂M to ∗ ∈ X and on ∂M r L take values in X . This is the main result of the first part of [EVW]. In §10 below we extend their results to thesetting of configuration-section spaces. Combining Theorem B with Theorem A and an applica-tion of the group completion theorem gives the following computation of the homology of finiteconfiguration-section spaces.
Corollary C
Suppose that d > and the subset e c D ⊆ π d − ( X ) has size . Then the scanningmaps CΓ c, ∗ k ( M ; X ) −→ Γ( M ; E d ( X, c )) [ k ] induce isomorphisms on H i ( − ; Z ) in the range k > i + 4 and surjections in the range k > i + 2 .With field coefficients, these ranges may be improved to k > i + 2 and k > i respectively. Herethe subscript [ k ] on the right indicates those components that intersect non-trivially with the image. See Theorem 10.12 and Corollary 10.16 for precise and more general versions of the above results.3 utline.
The paper is organised as follows. Sections 2 and 3 contain the precise definitions ofconfiguration-mapping spaces (recalled from [EVW]) and configuration-section spaces (a naturalgeneralisation that we introduce); several different classes of examples are also discussed in section3 (see Roadmap 3.1 for a more detailed plan of section 3). The structure on configuration-sectionspaces that we need, including the stabilisation maps, is encapsulated in the statement that theyform an E -module over an E d − -algebra ( cf . Remark 4.1 for the reasoning behind this terminol-ogy). In section 4, we first explain precisely what this means (recalling along the way severaldifferent flavours of Swiss cheese operads) and then define this structure on configuration-sectionspaces in appropriate models (Proposition 4.10). In section 5 we then define the (up-to-homotopy)monodromy action of the fundamental group π ( C k ( ˚ M )) coming from the forgetful mapCΓ c, ∗ k ( M ; ξ ) −→ C k ( ˚ M ) , (1.2)where CΓ c, ∗ k ( M ; ξ ) denotes the k -th configuration-section space associated to a bundle ξ : E → M and charge c . In section 7 we show that these actions, for k ∈ N , extend to a monodromy functor C ( M ) −→ Ho(Top) , (1.3)where C ( M ) is a certain braid category on M recalled in section 6. Theorem A is proved in section8 (see Theorem 8.5 for the precise statement). First, by a spectral sequence argument, it sufficesto prove twisted homological stability for ordinary configuration spaces with coefficients in thecomposition of the monodromy functor (1.3) with homology in any fixed degree q . To apply theknown twisted homological stability results for configuration spaces, we prove (Proposition 8.6)that this composite functor is polynomial of degree q . In section 9 we discuss the extension ofthe monodromy functor to a larger braid category B ♯ ( M ) ⊇ C ( M ) and prove the partial split-injectivity result mentioned above (Theorem 9.1). Finally, in section 10, we extend (and correct)the arguments of [EVW] to the setting of configuration-section spaces to prove Theorem B andCorollary C on the stable homology of configuration-section spaces.
2. Configuration-mapping spaces
We begin by recalling the definition of configuration-mapping spaces from [EVW], which gen-eralises the classical notion of
Hurwitz spaces [Cle72; Hur91]. In fact, we slightly extend thedefinition of [EVW] by considering also non-orientable manifolds (we will generalise this furtherto configuration-section spaces in the next section).Let M be a smooth, compact, connected manifold with non-empty boundary of dimension d > X be a space. Also, let c ⊆ [ S d − , X ]be a non-empty subset of the set of (unbased) homotopy classes of maps S d − → X . There is a Z / S d − , X ] given by precomposition by a reflection of the sphere, and, in thecase when M is non-orientable , we assume that c is a collection of fixed points of this action (seeRemark 2.3 for why). If M is orientable , there is no condition on c . Remark 2.1 If X is path-connected, the set [ S d − , X ] may be identified with the set of orbits π d − ( X ) /π ( X ) for any choice of basepoint of X . When d = 2 this is the set of conjugacy classesof π ( X ). Definition 2.2 ( Configuration-mapping spaces, I. ) For a positive integer k , the underlying set ofthe configuration-mapping space CMap ck ( M ; X )is the set of pairs ( z, f ), where z is a subset of the interior ˚ M of M of cardinality k and f is acontinuous map M r z → X with the following property: for any embedding e : D d ֒ → M where e ( D d ) ∩ z consists of a single point in the interior of e ( D d ), we have[ f ◦ e ◦ i ] ∈ c, (2.1)where i denotes the inclusion S d − ֒ → D d . 4 emark 2.3 The reason for assuming that c is a collection of fixed points of the involution on[ S d − , X ], in the case when M is non-orientable, is the following. In the definition of configuration-mapping spaces above, the set c of homotopy classes of maps S d − → X is to be thought of as theset of permitted “monodromies” of the continuous map to X that is defined on the complement of aconfiguration in M . If M is non-orientable, the monodromy of any such map around a configurationpoint must automatically lie in a fixed point of [ S d − , X ] under the involution. Hence, in this case,we may as well ignore the non-fixed points of [ S d − , X ], since it is impossible for them to occur,and instead just consider subsets c of the set of fixed points of [ S d − , X ] under the involution. Thisis illuminated in more detail in Example 3.22, in the context of configuration-section spaces .To topologise the set of Definition 2.2, we will give a second definition of CMap ck ( M ; X ) that hasa natural topology, and then prove that there is a natural bijection between the two definitions.To do this, we first recall some auxiliary definitions and results. Definition 2.4 ([Pal60; Cer61]) If G is a topological group with a continuous left-action on a space X , the action admits local sections if each x ∈ X has an open neighbourhood U and a continuousmap γ : U → G such that γ ( x ′ ) .x = x ′ for each x ′ ∈ U .The utility of this definition is given by the following result. Proposition 2.5 ([Pal60, Theorem A]) If X and Y are left G -spaces such that the action of G on Y admits local sections, and f : X → Y is G -equivariant, then f is a fibre bundle. Definition 2.6
For a smooth manifold-with-boundary M , we write Diff ∂ ( M ) for the topologicalgroup of self-diffeomorphisms of M that restrict to the identity on ∂M , equipped with the subspacetopology induced by the smooth Whitney topology on C ∞ ( M, M ), the space of all smooth self-maps of M .For a smooth manifold (without boundary) M , we write Diff c ( M ) for the topological group ofself-diffeomorphisms ϕ of M such that are compactly-supported , meaning that { p ∈ M | ϕ ( p ) = p } is relatively compact in M . This is topologised as follows. For a compact subset K of M , writeDiff K ( M ) for the topological group of self-diffeomorphisms ϕ of M such that { p ∈ M | ϕ ( p ) = p } ⊆ K , equipped with the subspace topology induced by the Whitney topology on C ∞ ( M, M ). Notethat, as a set, Diff c ( M ) is the union of Diff K ( M ) over all choices of K . We define the topology ofDiff c ( M ) to be the colimit of the topologies of Diff K ( M ) over all choices of K . Lemma 2.7 If M is a smooth manifold without boundary, the continuous left-action of Diff c ( M ) on C k ( ˚ M ) admits local sections.Proof. Theorem B of [Pal60] implies, as a special case, that the action of Diff ∂ ( M ) on the ordered configuration space F k ( ˚ M ) admits local sections. Then one may use this, and the fact that thecovering map F k ( ˚ M ) → C k ( ˚ M ) is Diff ∂ ( M )-equivariant, to construct local sections for the actionof Diff ∂ ( M ) on C k ( ˚ M ). Alternatively, the statement follows directly from Proposition 4.15 of[Pal20]. Definition 2.8 ( Configuration-mapping spaces, II. ) Fix a subset ˆ z ⊆ ˚ M of cardinality k andlet Map c ∗ ( M r ˆ z, X ) denote the space of continuous maps M r ˆ z → X satisfying the condition(2.1), equipped with the compact-open topology. Write Diff ∂ ( M ) for the group of diffeomorphismsof M that fix a neighbourhood of ∂M , equipped with the smooth Whitney topology, and letDiff ∂ ( M, ˆ z ) denote the subgroup of diffeomorphisms that fix ˆ z as a subset. This acts (on the right)on Map c ( M r ˆ z, X ) by precomposition, and on Diff ∂ ( M ) by right-multiplication, and we define:CMap ck ( M ; X ) := Map c ( M r ˆ z, X ) × Diff ∂ ( M )Diff ∂ ( M, ˆ z ) . (2.2) Lemma 2.9
There is a bijection between the set defined in Definition 2.2 and the space (2.2) .Proof.
Consider the map p : CMap ck ( M ; X ) −→ C k ( ˚ M ) (2.3)5iven by [ f, ϕ ] ϕ (ˆ z ). There is a continuous left-action of Diff ∂ ( M ) on CMap ck ( M ; X ) induced byits action on itself by left-multiplication and (2.2), and on C k ( ˚ M ) given by sending a configurationto its image under a diffeomorphism. The map (2.3) is equivariant with respect to these actions.The action of Diff ∂ ( M ) on C k ( ˚ M ) admits local sections by Lemma 2.7 and hence (2.3) is a fibrebundle by Proposition 2.5.The fibre of (2.3) over a configuration z ∈ C k ( ˚ M ) is p − ( z ) = Map c ( M r ˆ z, X ) × Diff ∂ ( M, ˆ z ) · ϕ Diff ∂ ( M, ˆ z ) , (2.4)where Diff ∂ ( M, ˆ z ) · ϕ denotes the coset of ϕ ∈ Diff ∂ ( M ) under the action of Diff ∂ ( M, ˆ z ), where ϕ isany diffeomorphism taking ˆ z to z . There is a canonical identification of (2.4) with Map c ( M r z, X )via [ f, ϕ ] f ◦ ϕ − . Hence a point of CMap ck ( M ; X ) may be specified by its image under (2.3),namely an unordered configuration z ∈ C k ( ˚ M ), together with an element of the fibre p − ( z ), whichis a continuous map M r z → X satisfying condition (2.1). This gives a natural bijection betweenthe set CMap ck ( M ; X ) defined in Definition 2.2, and the formal definition (2.2). Definition 2.10 ( Configuration-mapping spaces with a boundary condition. ) If we fix a subset D ⊆ ∂M and a basepoint ∗ ∈ X , we may defineCMap c,Dk ( M ; X )to be the subspace of CMap ck ( M ; X ) consisting of pairs ( z, f ) such that f ( p ) = ∗ for all p ∈ D .Equivalently (via the proof of Lemma 2.9), we replace Map c ( M r ˆ z, X ) in (2.2) with its subspaceMap c (( M r ˆ z, D ) , ( X, ∗ )) of maps taking D to {∗} . Typically, we will take D = D d − ⊆ ∂M to bean embedded disc. Definition 2.11 ( The associated fibre bundle. ) From Definition 2.8 and the proof of Lemma 2.9,we have a fibre bundle p : CMap ck ( M ; X ) −→ C k ( ˚ M ) (2.5)whose fibre over a configuration z ∈ C k ( ˚ M ) is the space of maps M r z → X satisfying condition(2.1), and whose total space CMap ck ( M ; X ) is the configuration-mapping space . For D ⊆ ∂M and based spaces X , there are also restricted versions of the configuration-mapping space, fromDefinition 2.10, CMap c,Dk ( M ; X ) ⊂ CMap ck ( M ; X ) , corresponding to restricting each fibre of (2.5) to the space of maps of pairs ( M r z, D ) → ( X, ∗ )satisfying condition (2.1): p : CMap c,Dk ( M ; X ) −→ C k ( ˚ M ) , (2.6)with p − ( z ) = Map c (( M r z, D ) , ( X, ∗ )). Remark 2.12
Condition (2.1) depends only on the homotopy class of the map f , so the subspaceMap c ( M r z, X ) of Map( M r z, X ) is a union of path-components (a similar statement also holdsfor the version with boundary condition on D ). Remark 2.13
When X is a point (so necessarily c = [ S d − , X ] = {∗} ), we have, by definition(2.2), CMap ck ( M ; ∗ ) = Diff ∂ ( M ) / Diff ∂ ( M, ˆ z ). In this case, by (2.4), each fibre of the fibre bundle(2.3) is a single point (here we are essentially using the fact that Diff ∂ ( M ) acts transitively on C k ( ˚ M )), so (2.3) is a homeomorphism (since fibre bundles are open maps):CMap ck ( M ; ∗ ) = Diff ∂ ( M ) / Diff ∂ ( M, ˆ z ) ∼ = C k ( ˚ M ) , identifying the configuration-mapping space with the usual (unordered) configuration space in thiscase. Remark 2.14
When M = D is the 2-disc with D = I ⊆ ∂D an embedded interval in itsboundary and X = BG is the classifying space of a discrete group G (so c is a set of conjugacyclasses of G ), we have a homotopy equivalence:CMap c,Ik ( D ; BG ) ≃ Hur cG,k , cG,k is the corresponding Hurwitz space , the moduli space (topologisedappropriately) of the data (up to an appropriate notion of equivalence) consisting of (
S, s , ν, i ),where: • S is a Riemann surface with basepoint s ∈ ∂S • ν : S → D is a based covering map, branched at k point in the interior of D , • i : G ֒ → Deck( ν ) is an embedding of groups,such that G acts transitively on the generic fibres of ν and the monodromy of ν around each of itsbranch points lies in one of the conjugacy classes in c . See also Example 3.27 and in particular theweak equivalences (3.20).
3. Configuration-section spaces
We now explain how to generalise this notion to configuration-section spaces , where we considersections of a bundle over the complement of a configuration, instead of just a map to a fixed space.This includes, for example, moduli spaces of configurations whose complement is equipped witha tangential structure or with a tuple of linearly independent vector fields (which need not extendover the whole manifold).
Roadmap 3.1 ( Plan of the section. ) We define the underlying set of configuration-section spacesin Definition 3.2 and topologise it in Definition 3.6 (Lemma 3.7 gives the correspondence betweenthe two definitions). A version with a boundary condition is given in Definition 3.9. Up to thispoint, however, these are the definitions of configuration-section spaces without any restriction onthe allowed “charge” of the section near a particle of the configuration (this is the analogue ofthe subset c ⊆ [ S d − , X ] for configuration-mapping spaces in the previous section, and can also bethought of as the allowed “monodromy” of the section near a particle, or as a “singularity condition”on the section). In order to define the appropriate analogue for configuration-section spaces, wefirst construct in Definition 3.11 the covering space of local sections Σ( ξ ) → ˚ M associated to abundle over a manifold ξ : E → M . Any configuration-section of the bundle ξ determines a sectionof the associated covering space Σ( ξ ) (Construction 3.17 and Lemma 3.18), and this is then usedin Definition 3.19 to define configuration-section spaces with prescribed “monodromy” or “charge”.The forgetful fibration associated to this configuration-section space is described in Definition 3.21.After this setting up of the theory, we discuss several families of examples. Example 3.22 firstexplains precisely how the notion of configuration-section space recovers the notion of configuration-mapping space when the bundle ξ : E → M is trivial (in fact, certain subtleties in the case wherethe base manifold M is non-orientable are explained more naturally from the configuration-sectionviewpoint). There is a slight variation of configuration-section spaces where the section is givenonly up to homotopy (Definition 3.23), which includes the example of configurations equippedwith a cohomology class on the complement (Example 3.24). We then discuss the examples of Hurwitz spaces (Example 3.27), configurations with a tangential structure or tuple of linearlyindependent vector fields on their complement (Examples 3.28 and 3.29) and “ widely separatedmagnetic monopoles ” (Example 3.30).
Definition 3.2 ( Configuration-section spaces, without prescribed monodromy, I. ) Fix a smooth,connected d -manifold M (possibly with boundary) and a fibre bundle ξ : E → M . For a non-negative integer k , the (unrestricted) configuration-section space , as a set, is given byCΓ k ( M ; ξ ) = { ( z, s ) | z ⊆ ˚ M subset of cardinality k, s : M r z → E section of ξ | M r z } . (3.1)We will topologise this, and construct the associated fibre bundle over C k ( ˚ M ), in a similar wayas for configuration-mapping spaces in §2. Definition 3.3
For a smooth manifold-with-boundary M and fibre bundle ξ : E → M , the auto-morphism group Aut ∂ ( ξ ) is the subgroupAut ∂ ( ξ ) Diff ∂ ( M ) × Homeo( E )of those pairs ( ϕ, g ) such that ξ ◦ g = ϕ ◦ ξ . For a subset z ⊆ ˚ M , we write Aut ∂ ( ξ, z ) for thesubgroup of ( ϕ, g ) such that ϕ ( z ) = z . Similarly, for subsets z, z ′ ⊆ ˚ M of the same cardinality, we7rite Aut ∂ ( ξ, z z ′ ) for the subgroup of ( ϕ, g ) such that ϕ ( z ) = z ′ . If ∂M = ∅ , we write Aut c ( ξ )(resp. Aut c ( ξ, z ) and Aut c ( ξ, z z ′ )) if we replace Diff ∂ ( M ) with Diff c ( M ) in the definition. Remark 3.4
An element ( ϕ, g ) ∈ Aut ∂ ( ξ ) is determined by its second component g ∈ Homeo( E ),since a self-homeomorphism of the total space E can descend to a self-diffeomorphism of the basemanifold M in at most one way. Hence there is a continuous injection Aut ∂ ( ξ ) ֒ → Homeo( E ),although the topology on Aut ∂ ( ξ ) is generally finer than the subspace topology induced by thisinjection. Lemma 3.5 If M is a smooth manifold-with-boundary and ξ : E → M is a fibre bundle, thecontinuous left-action of Aut ∂ ( ξ ) on C k ( ˚ M ) given by ( ϕ, g ) : z ϕ ( z ) admits local sections.Proof. Let z ∈ C k ( ˚ M ) and choose an embedded codimension-zero ball B ⊆ ˚ M such that z ⊆ ˚ B . ByLemma 2.7, there is an open neighbourhood U of z in C k ( ˚ B ) and a continuous map γ : U → Diff c ( ˚ B )such that γ ( z ′ ) .z = z ′ for all z ′ ∈ U . Choose a trivialisation of ξ | ˚ B . This induces a continuous grouphomomorphism Diff c ( ˚ B ) → Aut c ( ξ | ˚ B ). Now extending both the diffeomorphism of the underlyingmanifold and the homeomorphism of the total space by the identity on M r ˚ B and on ξ − ( M r ˚ B )defines a continuous group homomorphism Aut c ( ξ | ˚ B ) → Aut ∂ ( ξ ). Composing both of these with γ completes the construction of local sections for the action of Aut ∂ ( ξ ) on C k ( ˚ M ). Definition 3.6 ( Configuration-section spaces, without prescribed monodromy, II. ) Fix a subsetˆ z ⊆ ˚ M of cardinality k , and defineCΓ k ( M ; ξ ) := Γ( M r ˆ z, ξ ) × Aut ∂ ( ξ )Aut ∂ ( ξ, ˆ z ) , (3.2)where, for a subspace S ⊆ M , we write Γ( S, ξ ) = { s ∈ Map(
S, E ) | ξ ◦ s = incl } . The right-actionof Aut ∂ ( ξ, ˆ z ) on Γ( M r ˆ z, ξ ) is given by ( ϕ, g ) : s g − ◦ s ◦ ϕ . Lemma 3.7
There is a bijection between the set defined in Definition 3.2 and the space (3.2) .Proof.
There is a continuous map p : CΓ k ( M ; ξ ) −→ C k ( ˚ M ) (3.3)given by [ s, ( ϕ, g )] ϕ ( z ). This map is equivariant with respect to the continuous left-actionsof Aut ∂ ( ξ ), and the action of Aut ∂ ( ξ ) on C k ( ˚ M ) admits local sections by Lemma 3.5. Thus, byProposition 2.5, the map (3.3) is a fibre bundle. The fibre of (3.3) over z ∈ C k ( ˚ M ) is p − ( z ) = Γ( M r ˆ z, ξ ) × Aut ∂ ( ξ, ˆ z z )Aut ∂ ( ξ, ˆ z ) . (3.4)Note that Aut ∂ ( ξ ) acts transitively on C k ( ˚ M ). (This can be seen as follows. Let z, z ′ ∈ C k ( ˚ M )and choose an embedded codimension-zero ball B ⊆ ˚ M such that z ∪ z ′ ⊆ ˚ B . Since Diff c ( ˚ B )acts transitively on C k ( ˚ B ), we may find a diffeomorphism ϕ of ˚ B such that ϕ ( z ) = z ′ , lift it to anautomorphism of ξ | ˚ B by a choice of trivialisation of ξ | ˚ B and then extend it by the identity to obtainan element of Aut ∂ ( ξ ) sending z to z ′ .) Thus the subspace Aut ∂ ( ξ, ˆ z z ) is a coset of Aut ∂ ( ξ, ˆ z )in Aut ∂ ( ξ ), and may be identified with Γ( M r z, ξ ) via [ s, ( ϕ, g )] g ◦ s ◦ ϕ − . Thus there is anatural bijection between the set defined in Definition 3.2 and the formal definition (3.2). Definition 3.8 ( The associated fibre bundle. ) From Definition 3.6 and the proof of Lemma 3.7,we have a fibre bundle p : CΓ k ( M ; ξ ) −→ C k ( ˚ M ) (3.5)whose fibre over a configuration z ∈ C k ( ˚ M ) is the space of sections Γ( M r z, ξ ). Definition 3.9 ( Boundary conditions. ) If we fix a subset D ⊆ ∂M and a section s D ∈ Γ( D, ξ ),we have a restricted version of the configuration-section space:CΓ Dk ( M ; ξ ) = { ( z, s ) ∈ CΓ k ( M ; ξ ) | s | D = s D } . (3.6)8oreover, since D is contained in the boundary of M , the subspace CΓ Dk ( M ; ξ ) ⊆ CΓ k ( M ; ξ ) isinvariant under the action of Aut ∂ ( ξ ), so Proposition 2.5 and Lemma 3.5 imply that the restriction p : CΓ Dk ( M ; ξ ) −→ C k ( ˚ M ) (3.7)of the map (3.3) is a fibre bundle. The fibre p − ( z ) over a configuration z ∈ C k ( ˚ M ) is the space ofsections { s ∈ Γ( M r z, ξ ) | s | D = s D } (generalising Definition 3.8, which corresponds to D = ∅ ).We now wish to define configuration-section spaces with prescribed monodromy CΓ ck ( M ; ξ ) ⊆ CΓ k ( M ; ξ ) . In other words, we wish to specify a certain “local behaviour” c of the section s ∈ Γ( M r z, ξ ) nearthe “singularities” z ⊆ ˚ M . To do this, we first construct a covering space over the interior ˚ M of M depending on the fibre bundle ξ : E → M . Definition 3.10 ( The covering space of local sections of ξ , informal. ) Informally, the coveringspace η ( ξ ) : Σ( ξ ) −→ ˚ M is defined by prescribing that its fibre η ( ξ ) − ( p ) over a point p in the interior of M consists of theset of all germs of sections of ξ defined on a small punctured-disc neighbourhood of p in M . Definition 3.11 ( The covering space of local sections of ξ , formal. ) Fix a point p ∈ ˚ M . A localsection near p of ξ is a pair ( B, σ ) consisting of a subset B ⊆ M homeomorphic to a d -dimensionalball, containing p in its interior, together with a section σ of ξ | ∂B . Let ∼ be the equivalence relationon such pairs generated by ( B, σ ) ∼ ( B ′ , σ ′ ) if B is contained in the interior of B ′ and the section σ ⊔ σ ′ defined on ∂B ⊔ ∂B ′ extends to a section over B ′ r int( B ). Write [ B, σ ] p for the equivalenceclass containing ( B, σ ).Let Σ( ξ ) be the set of all pairs ( p, [ B, σ ] p ) where p ∈ ˚ M and [ B, σ ] p is an equivalence class oflocal sections near p of ξ . We define a topology on Σ( ξ ) as follows. Let ( p, [ B, σ ] p ) ∈ Σ( ξ ) andchoose a representative ( B, σ ) for the equivalence class [
B, σ ] p . Define N p,B,σ = { ( q, [ B, σ ] q ) | q ∈ int( B ) } ⊆ Σ( ξ ) . Then one may check that the collection N of the sets N p,B,σ for all ( p, [ B, σ ] p ) ∈ Σ( ξ ) is a basisfor a topology T on Σ( ξ ) such that the map η ( ξ ) : Σ( ξ ) −→ ˚ M given by ( p, [ B, σ ]) p is a covering map. Remark 3.12
Although the basis N depends on a choice, for each point ( p, [ B, σ ] p ) ∈ Σ( ξ ), ofa representative ( B, σ ) of the equivalence class [
B, σ ] p , the topology T that it generates does notdepend on these choices.There is also a basepointed version of the covering space η ( ξ ). As before, let M be a d -dimensionalmanifold and ξ : E → M a fibre bundle. Choose basepoints ∗ ∈ ∂M and ∗ ∈ E such that ξ ( ∗ ) = ∗ . Definition 3.13 ( The covering space of pointed local sections. ) Fix a point p ∈ ˚ M . A pointed localsection near p of ξ is a pair ( B, σ ) consisting of a subset B ⊆ M homeomorphic to a d -dimensionalball, containing p in its interior and ∗ in its boundary, together with a section σ of ξ | ∂B such that σ ( ∗ ) = ∗ . Let ∼ be the equivalence relation on such pairs generated by ( B, σ ) ∼ ( B ′ , σ ′ ) if B iscontained in int( B ′ ) ∪ {∗} and the section σ ∨ σ ′ defined on ∂B ∨ ∂B ′ extends to a section over B ′ r int( B ). Write [ B, σ ] ∗ p for the equivalence class containing ( B, σ ).Let Σ ∗ ( ξ ) be the set of all pairs ( p, [ B, σ ] ∗ p ) where p ∈ ˚ M and [ B, σ ] ∗ p is an equivalence class ofpointed local sections near p of ξ . Analogously to Definition 3.11, we define a topology on Σ ∗ ( ξ )such that the map η ∗ ( ξ ) : Σ ∗ ( ξ ) −→ ˚ M p, [ B, σ ] ∗ p ) p is a covering map. There is a map of covering spaces over ˚ M Σ ∗ ( ξ ) −→ Σ( ξ ) (3.8)given by ( p, [ B, σ ] ∗ p ) ( p, [ B, σ ] p ). Definition 3.14 A singularity condition for ξ : E → M is a set of trivial components of thecovering η ( ξ ) : Σ( ξ ) → ˚ M . In other words, it is a subset of Γ( η ( ξ )), the set of sections of η ( ξ ).Similarly, a pointed singularity condition is a subset of Γ( η ∗ ( ξ )). Remark 3.15
The map (3.8) of covering spaces induces a function(3.8) ◦ − : Γ( η ∗ ( ξ )) −→ Γ( η ( ξ )) , so a pointed singularity condition determines an (unpointed) singularity condition. Remark 3.16
Over a point p ∈ ˚ M , the fibre of η ( ξ ) : Σ( ξ ) → ˚ M may be identified with the set of(unbased) homotopy classes of maps [ S d − , F ], where F = ξ − ( p ). This identification is canonicalonce we have chosen a local orientation of ˚ M at p . Similarly, the fibre of η ∗ ( ξ ) : Σ ∗ ( ξ ) → ˚ M maybe identified with π d − ( F ), after choosing a basepoint of F . Under these identifications, the map(3.8) | p is the quotient π d − ( F ) ։ π d − ( F ) /π ( F ) = [ S d − , F ].In order to define configuration-section spaces with prescribed singularity conditions, we needto show that any configuration-section of the bundle ξ : E → M induces a section of the covering η ( ξ ) : Σ( ξ ) → ˚ M . Construction 3.17
There is a locally-constant maploc ξ : CΓ k ( M ; ξ ) −→ Γ( η ( ξ )) (3.9)that records the local behaviour of a given configuration-section ( z, s ) in a punctured neighbour-hood of each p ∈ ˚ M . This is possible since, although s is not defined on all of ˚ M , it is definedon a punctured neighbourhood of every point of ˚ M , since it is only undefined on a discrete subset z ⊆ ˚ M .Formally, the construction is as follows. Given ( z, s ) ∈ CΓ k ( M ; ξ ) and p ∈ ˚ M , we need to choosea point in the fibre η ( ξ ) − ( p ). To do this, first modify ( z, s ) if necessary, staying within the samepath-component of CΓ k ( M ; ξ ), so that p ∈ z . We pause the construction briefly to prove that thisis always possible: Lemma 3.18
Given any ( z, s ) ∈ CΓ k ( M ; ξ ) and p ∈ ˚ M , there is a path in CΓ k ( M ; ξ ) from ( z, s ) to ( z ′ , s ′ ) such that p ∈ z ′ .Proof. By the connectivity of M , we may find an embedded D d ⊆ M containing p and a point q ∈ z in its interior, and disjoint from z r { q } . Over this embedded disc, ξ is isomorphic to a trivialbundle D d × X → D d , so we have a mapCMap ∂D d ( D d ; X ) −→ CΓ k ( M ; ξ )that extends a 1-point configuration-mapping in D d (agreeing with s | ∂D d on ∂D d ) by z r { q } and s | M r ( z ∪ D d ) . It therefore suffices to find a path in CMap ∂D d ( D d ; X ) from ( { q } , s | D d ) to ( { p } , s ′ ) forsome map s ′ : D d r { p } → X . Next, note that there is a natural left-action of Homeo( D d ; ∂D d ) onCMap ∂D d ( D d ; X ) given by pre-composition, so it suffices to find a path in Homeo( D d ; ∂D d ) fromthe identity to a homeomorphism ϕ such that ϕ ( p ) = q . But Homeo( D d ; ∂D d ) acts transitively onint( D d ) and is also path-connected (in fact contractible) by the Alexander trick.Continuing with the construction, we may assume that p ∈ z . Choose an embedded d -dimensionalball B ⊆ M containing p in its interior and disjoint from z r { p } . Then defineloc ξ ( z, s )( p ) = ( p, [ B, s | ∂B ] p ) . One may then easily check that this gives a well-defined locally-constant map of the form (3.9).10 efinition 3.19 ( Configuration-section spaces with prescribed monodromy. ) As in Definition 3.2,let M be a smooth, connected d -manifold and let ξ : E → M be a fibre bundle. Also choose a singularity condition c ⊆ Γ( η ( ξ )) ( cf . Definition 3.14). ThenCΓ ck ( M ; ξ ) := loc − ξ ( c ) ⊆ CΓ k ( M ; ξ ) . In other words, it is the subspace of CΓ k ( M ; ξ ) of configuration-sections ( z, s ) such that, if B ⊆ M is a subset that is homeomorphic to a d -dimensional ball, contains one point p of z in its interiorand is disjoint from z r { p } , then ( p, [ B, s | ∂B ] p ) ∈ Σ( ξ ) lies in the image of one of the sections in c .If we fix a subset D ⊆ ∂M and a section s D ∈ Γ( D, ξ ), we define (just as in (3.6)):CΓ c,Dk ( M ; ξ ) = { ( z, s ) ∈ CΓ ck ( M ; ξ ) | s | D = s D } . (3.10) Remark 3.20
Since (3.9) is locally-constant, the subspace CΓ ck ( M ; ξ ) of CΓ k ( M ; ξ ) is a union ofpath-components, and similarly for the restricted version (3.10) ⊆ (3.6). (Compare Remark 2.12.) Definition 3.21 ( The associated fibration. ) From Definition 3.9 we have a fibre bundle (3.7) withtotal space CΓ Dk ( M ; ξ ). By Remark 3.20, the subspace CΓ c,Dk ( M ; ξ ) ⊆ CΓ Dk ( M ; ξ ) is a union ofpath-components. Thus, the restriction p : CΓ c,Dk ( M ; ξ ) −→ C k ( ˚ M ) (3.11)of the fibre bundle (3.7) to the subspace CΓ c,Dk ( M ; ξ ) is a Hurewicz fibration. This is the associatedfibration of the configuration-section space CΓ c,Dk ( M ; ξ ). Its fibre over z ∈ C k ( ˚ M ) is denoted byΓ c,D ( M r z ; ξ ) = { s ∈ Γ( M r z ; ξ ) | s | D = s D and loc ξ ( z, s ) ∈ c } . As our first family of examples, we explain how to recover the notion of configuration-mappingspace of §2 in the case of a trivial bundle over M . Example 3.22 If ξ is the trivial bundle M × X → M for a space X , then we clearly have anidentification CΓ k ( M ; ξ ) = CMap k ( M ; X ) . (3.12)If M is orientable, then the covering space η ( ξ ) : Σ( ξ ) → ˚ M is simply the disjoint union of copiesof the trivial (identity) covering ˚ M → ˚ M , one for each element of [ S d − , X ]. In other words, η ( ξ )is isomorphic as a covering space to the projection ˚ M × [ S d − , X ] → ˚ M . Similarly, the coveringspace η ∗ ( ξ ) : Σ ∗ ( ξ ) → ˚ M is isomorphic as a covering space to the projection ˚ M × π d − ( X ) → ˚ M .In each case, the isomorphism of covering spaces depends on a choice of orientation of M .If M is non-orientable, then the covering space η ( ξ ) : Σ( ξ ) → ˚ M is a disjoint union of a numberof copies of the identity covering ˚ M → ˚ M and a number of copies of the orientation double covering˚ M or → ˚ M . More precisely, consider the involution on the set [ S d − , X ] given by precompositionby a reflection of S d − . There is one copy of the identity covering in η ( ξ ) for each fixed point ofthis action and one copy of the orientation double covering in η ( ξ ) for each orbit of size two. Inother words, writing O = [ S d − , X ] Z / for the set of orbits of size 1 and O for the set of orbitsof size 2 in [ S d − , X ], we have that η ( ξ ) is isomorphic as a covering space topr ◦ (id ⊔ (or × id)) : ( ˚ M × O ) ⊔ ( ˚ M or × O ) −→ ˚ M × ( O ⊔ O ) −→ ˚ M, (3.13)where or is the orientation double covering of ˚ M . The isomorphism is canonical up to the actionof the 2 O deck transformations of (3.13) corresponding to the deck transformations of each of thecopies of ˚ M or (acting independently). There is an analogous involution on the set π d − ( X ), andthe covering space η ∗ ( ξ ) : Σ ∗ ( ξ ) → ˚ M is a disjoint union of one copy of the identity covering foreach fixed point of π d − ( X ) and one copy of the orientation double covering for each orbit of sizetwo in π d − ( X ). This may also be written in the form (3.13), where O and O now denote theorbits of size 1 and 2 of the involution on π d − ( X ), and the isomorphism is again canonical up tothe 2 O deck transformations of (3.13) described above.A singularity condition c ⊆ Γ( η ( ξ )) therefore corresponds to (compare Remark 2.3): The base space of the fibre bundle (3.7) is paracompact, so it is a Hurewicz fibration. In general, if f : E → B is a Hurewicz fibration and E ⊆ E is a union of path-components, the composition f ◦ incl : E ֒ → E → B is alsoa Hurewicz fibration. (if M is orientable) a subset of [ S d − , X ], • (if M is non-orientable) a subset of [ S d − , X ] Z / , the set of fixed points of [ S d − , X ] underthe involution given by pre-composition with a reflection of S d − .In the orientable case, this correspondence depends on a choice of orientation of M ; in the non-orientable case, it does not depend on any choice. Analogously, a pointed singularity conditioncorresponds either to a subset of π d − ( X ) or a subset of π d − ( X ) Z / , depending on whether M isorientable or not, respectively. Again, this correspondence depends on a choice of orientation of M if M is orientable.Interpreting the singularity condition c in this way on the right-hand side, the identification(3.12) restricts to identifications: CΓ ck ( M ; ξ ) = CMap ck ( M ; X ) (3.14)for each c ⊆ Γ( η ( ξ )).There is a natural variant of configuration-section spaces where configurations are equipped justwith homotopy-classes of sections on their complements: Definition 3.23 ( Configuration-section spaces up to homotopy. ) Define h CΓ k ( M ; ξ ) to be thequotient of CΓ k ( M ; ξ ) by the equivalence relation given by ( z, s ) ∼ ( z ′ , s ′ ) if z = z ′ and thesections s, s ′ : M r z → E are homotopic through sections of ξ | M r z , in other words, lie in the samepath-component of the space Γ( M r z, ξ ). The locally-constant map (3.9) = loc ξ factors into thequotient map CΓ k ( M ; ξ ) → h CΓ k ( M ; ξ ) and a locally-constant map h loc ξ : h CΓ k ( M ; ξ ) −→ Γ( η ( ξ )) , (3.15)since homotopic sections of ξ | M r z have the same local germs in punctured neighbourhoods of everypoint in M . We then define h CΓ ck ( M ; ξ ) := h loc − ξ ( c )for any singularity condition c ⊆ Γ( η ( ξ )). We may equivalently define h CΓ ck ( M ; ξ ) to be thequotient of CΓ ck ( M ; ξ ) by the restriction of the equivalence relation above. More generally, if wefix a subset D ⊆ ∂M and a section s D ∈ Γ( D, ξ ), we may define h CΓ c,Dk ( M ; ξ ) to be the quotientof CΓ c,Dk ( M ; ξ ) by the equivalence relation given by ( z, s ) ∼ ( z ′ , s ′ ) if z = z ′ and the sections s, s ′ lie in the same path-component of the spaceΓ( M r z, ξ ; s D ) = { s ∈ Γ( M r z, ξ ) | s | D = s D } . Example 3.24 ( Configurations equipped with a cohomology class of the complement. ) As a specialcase, of course, we have configuration-mapping spaces up to homotopy h CMap ck ( M ; X ) = h CΓ ck ( M ; M × X ) . As a set, h CMap ck ( M ; X ) consists of pairs ( z, f ), where z ⊆ ˚ M has cardinality k and f is ahomotopy class of maps M r z → X whose monodromy around each point of z lies in c ⊆ [ S d − , X ].If we take X to be the Eilenberg-MacLane space K ( G, d −
1) for an abelian group G , then c is asubset of H d − ( S d − ; G ) ∼ = G and a point in h CMap ck ( M ; K ( G, d − z ⊆ ˚ M equipped with a cohomology class α ∈ H d − ( M r z ; G ) whose restrictionto each embedded sphere S d − ⊆ M r z that encloses exactly one point of z , lies in c .Before describing the next example of configuration-mapping spaces up to homotopy, we notethat, under certain conditions, configuration-mapping spaces up to homotopy have the same weakhomotopy type as the corresponding configuration-mapping spaces (not up to homotopy).12 emma 3.25 Let M be a compact, connected d -manifold with basepoint ∗ ∈ ∂M and X a based,path-connected space with a choice of subset c ⊆ [ S d − , X ] . Assume that M is homotopy equivalentto a wedge of ( d − -spheres and that X is d -coconnected, meaning that π i ( X ) = 0 for all i > d .Then, for any k > , the quotient map CMap c, ∗ k ( M ; X ) −→ h CMap c, ∗ k ( M ; X ) (3.16) is a weak homotopy equivalence. In particular, this holds if M = S is a compact, connected surface, X = BG is the classifying space of a discrete group G and c is a set of conjugacy classes of G .Proof. The quotient map fits into a map of fibration sequences:Map c, ∗ ( M r z, X ) π (Map c, ∗ ( M r z, X ))CMap c, ∗ k ( M ; X ) h CMap c, ∗ k ( M ; X ) C k ( ˚ M ) ( ⋆ )(3.16) (3.17)Since M is homotopy equivalent to a wedge of some number ℓ of ( d − M r z ≃ W | z | + ℓ S d − and thusMap c, ∗ ( M r z, X ) ≃ (Ω d − X ) ℓ × (Ω d − c X ) | z | , (3.18)where Ω d − c X ⊆ Ω d − X is the union of path-components corresponding to c ⊆ [ S d − , X ] underthe identification of [ S d − , X ] with π d − ( X ) /π ( X ). Since X is d -coconnected, the space (3.18) is1-coconnected, in other words weakly contractible, and so the map ( ⋆ ) of fibres in (3.17) is a weakhomotopy equivalence, and therefore so is the map (3.16). Remark 3.26
Lemma 3.25 generalises in an analogous way to configuration-section spaces, for abundle ξ : E → M equipped with a section over {∗} ⊆ ∂M , i.e., a point e ∈ E with ξ ( e ) = ∗ . Example 3.27 ( Hurwitz spaces. ) Following on from Example 3.24, let S be a compact, connectedsurface with boundary and now take X = K ( G,
1) = BG for a (not necessarily abelian) discretegroup G . Note that S r z , for any finite subset z ⊆ ˚ S , is aspherical, so we have a natural bijection[ S r z, BG ] ∼ = Hom( π ( S r z ) , G ) /G, where the quotient on the right-hand side is by the action of G given by post-composition by innerautomorphisms. A point in h CMap ck ( S ; BG )therefore consists of a configuration z ⊆ ˚ S equipped with a homomorphism π ( S r z ) → G moduloinner automorphisms of G . If we now take D = {∗} ⊆ ∂S to be a point, we have a natural bijection h S r z, BG i ∼ = Hom( π ( S r z ) , G ) , where h− , −i denotes based homotopy classes of based maps, and so a point in h CMap c, ∗ k ( S ; BG )consists of a configuration z ⊆ ˚ S equipped with a homomorphism π ( S r z ) → G . In particular,if S = D we have a homeomorphism h CMap c, ∗ k ( D ; BG ) ∼ = Hur cG,k , c, ∗ k ( S ; BG ) −→ h CMap c, ∗ k ( S ; BG ) (3.19)is a weak homotopy equivalence for any compact, connected surface-with-boundary S . In the case S = D , we therefore have weak homotopy equivalences (compare Remark 2.14):CMap c,Ik ( D ; BG ) ≃ CMap c, ∗ k ( D ; BG ) ≃ h CMap c, ∗ k ( D ; BG ) ∼ = Hur cG,k . (3.20) Example 3.28 ( Tangential structures on the complement. ) If θ : E → BO ( n ) is a type of tangentialstructure, and τ M denotes the homotopy class of maps M → BO ( n ) classifying the tangent bundleof M , then a tangential structure of type θ on M is a lift of τ M up to homotopy to a map M → E .Equivalently, we may pick a specific map T M in the homotopy class τ M , and a tangential structureis then a lift up to homotopy of T M . In other words, it is a section up to homotopy of the pullbackbundle ( T M ) ∗ ( θ ) : ( T M ) ∗ ( E ) → M . The configuration-section space h CΓ k ( M ; ( T M ) ∗ ( θ ))is therefore the moduli space of k -point configurations in M whose complement is equipped witha tangential structure of type θ , and its subspaces corresponding to a singularity condition c maybe interpreted as moduli spaces of configurations whose complement is equipped with a tangentialstructure of type θ , whose monodromy around the configuration points is prescribed. Example 3.29 ( Linearly independent vector fields. ) Let
T M → M denote the tangent bundleof M and write ξ r : Lin r ( T M ) → M for the associated fibre bundle whose fibre over p ∈ M isthe subspace of ( T p M ) r consisting of linearly independent r -tuples of tangent vectors at p . Theconfiguration-section space CΓ k ( M ; ξ r )is then the moduli space of k -point configurations z ⊆ ˚ M equipped with an r -tuple of linearlyindependent vector fields on M r z . In particular, when r = 1, this is the space of configurationsequipped with a non-vanishing vector field on the complement. Example 3.30 ( Magnetic monopoles. ) Going back to configuration-mapping spaces, we may con-sider CMap ∗ k ( D d ; K ( Z , d )). As remarked in the introduction, for a given configuration z ⊂ R d ∼ =˚ D d , the based map f : D d r z → K ( Z , d ) may be thought of as a phase in S ≃ Ω d − K ( Z , d )associated to each particle p ∈ z , after choosing a deformation retraction of D d r z onto a wedgeof k copies of S d − . However, this deformation retraction cannot be chosen consistently as theconfiguration z varies, so this description of an element of CMap ∗ k ( D d ; K ( Z , d )) as a configurationof particles equipped with phases in S is valid only in small neighbourhoods of the configuration-mapping space. Globally, an element of CMap ∗ k ( D d ; K ( Z , d )) models a configuration equippedwith some non-local data that does not decompose into pieces associated to each particle.When d = 3, this space could be a model for an asymptotic part of the moduli space of magneticmonopoles M k of total charge k in R , namely the part consisting of “widely separated” monopoles( cf . [AH88, Proposition 3.12], [Seg97]). We note that it is not the case that CMap ∗ k ( D ; K ( Z , whole moduli space M k , since the fundamental group of M k is known to be Z (combining[Don84] with [Seg79, Proposition 6.4]), whereas the fundamental group of CMap ∗ k ( D ; K ( Z , k via the forgetful map to C k ( R ). It is interesting to note that the full modulispace M k is also homologically stable with respect to the “magnetic charge” k (combining [Don84]with [Seg79, Proposition 1.1]).
4. On E m -modules over E n -algebras All of the structure that we will use in studying configuration-section spaces will arise from theirstructure as an E -module over an E d − -algebra , which will be defined explicitly, in appropriatemodels, in this section. Before this, we recall the notion of E m -module over an E n -algebra for any0 m n , and explain why, for n fixed, these notions coincide for all m ∈ { , , . . . , n − } .14 emark 4.1 The last sentence above means that, when d >
3, we may equally well describethe structure that we construct as an “ E -module” over an E d − -algebra. However, since we alsoconsider the case of dimension d = 2, we prefer to use the term “ E -module” throughout, forconsistency. This diverges from the terminology of [Kra19], where the name “ E -module” is used– this is (as we explain below) equivalent since that paper considers only E n -algebras where n > n > D n denote the closed unit disc in R n . For a space X and an integer k >
0, let ¯ F k ( X ) denotethe ordered configuration space of k points in X labelled by positive real numbers:¯ F k ( X ) = { (( x , r ) , . . . , ( x n , r n )) ∈ ( X × (0 , ∞ )) n | x i = x j for i = j } . Fix an integer n > Definition 4.2
The little n -discs operad D n has one colour a . For any integer k > D n ( a k ; a ) is the subspace of ¯ F k ( D n ) of configurations satisfying(i) | x i | − r i (ii) | x i − x j | > r i + r j for i = j .Interpreting such configurations as little n -discs in D n whose interiors are disjoint, with centres x i and radii r i – see Figure 4.1(a) – the operadic composition is defined by embedding D n into thesesmaller discs by translations and dilations. The symmetric action is given by the natural action ofΣ k on ¯ F k ( D n ). Definition 4.3 A D n -module operad is any operad O with two colours a and m , and whose spaceof operations O ( a k , m l ; a ), for any integers k, l >
0, is equal to D n ( a k ; a ) for l = 0 and empty for l >
1. Moreover, its operadic composition, restricted to the colour a , must be equal to that of D n .Fix two integers n > m >
0. We now define five different D n -module operads, the Swiss cheeseoperad ( SC m,n ), as well as its extended ( ESC m,n ), variant ( VSC m,n ), concentric ( CSC m,n ) and linear ( LSC n ) cousins. In each case it will suffice to specify the spaces of operations O ( a k , m l ; m )for integers k, l > D n to the colour m . Remark 4.4
The original Swiss cheese operad SC , was introduced by Voronov [Vor99], inspiredby constructions of Kontsevich [Kon94; Kon03]. The extended Swiss cheese operad ESC m,n wasintroduced by Willwacher [Wil], who cites V. Turchin for its invention in codimension 1 (when n = m + 1). The variant Swiss cheese operad VSC m,n was introduced by Idrissi [Idr]. The linearSwiss cheese operad
LSC n was introduced (under the name SC n ) by Krannich [Kra19, §2.1]. Definition 4.5 ( Extended, variant and original Swiss cheese operads. ) Let
ESC m,n ( a k , m l ; m ) bethe subspace of ¯ F k + l ( D n ) of configurations satisfying (i) and (ii) above, and(iii) x i ∈ D m for i > k + 1.The space VSC m,n ( a k , m l ; m ) is the subspace of configurations additionally satisfying the condition(iv) dist( x i , D m ) > r i for i k .The space SC m,n ( a k , m l ; m ) is the subspace of configurations satisfying conditions (i)–(iii) and(v) x i ∈ R m × ( r i , ∞ ) n − m for i k .(Note that condition (v) is stronger than condition (iv).) Interpreting such configurations again aslittle non-overlapping n -discs in D n – see Figure 4.1(b–d) – we may extend the operadic compositionof D n to each of these 2-coloured operads by embedding D n into these smaller discs by translationsand dilations. Definition 4.6 ( The concentric Swiss cheese operad. ) The space of operations
CSC m,n ( a k , m l ; m )is empty unless l = 1, in which case it is the subspace of ¯ F k +1 ( D n ) of configurations satisfying (i),(ii) and (v) above, as well as(vi) x k +1 = 0.Its operadic composition is defined as before, interpreting these configuration spaces as spaces oflittle non-overlapping discs – see Figure 4.1(e). 15 ∈ D ( a ; a )(a) ∈ ESC , ( a , m ; m )(b) ∈ VSC , ( a , m ; m )(c)
13 212 ∈ SC , ( a , m ; m )(d)
13 2 ∈ CSC , ( a , m ; m )(e)
13 2 t ∈ LSC ( a , m ; m )(f)Figure 4.1 Some operations of the little discs operad and of different flavours of Swiss cheese operadsin dimensions (1 , m colour) discs are always centred on the x -axis. Little red ( a colour)discs are unrestricted in ESC , (except that they must not overlap each other or the little blue discs,of course). In VSC , , little red discs must be disjoint from the x -axis (little red discs are called aerial and little blue discs terrestrial in [Idr]). In SC , , little red discs must lie in the upper half-disc (and wemay choose to think of the little blue discs simply as little half-discs attached to the x -axis). In CSC , the same conditions apply to little red discs, and there is now required to be exactly one little bluedisc, centred at the origin. In LSC , little red discs may lie anywhere in the rectangle. See Definitions4.2, 4.5, 4.6 and 4.7 for the precise definitions. Definition 4.7 ( The linear Swiss cheese operad. ) The space of operations
LSC n ( a k , m l ; m ) is emptyunless l = 1, in which case it is the subspace of ¯ F k ([0 , ∞ ) × [ − , n − ) × [0 , ∞ ) of configurations(( x , r ) , . . . , ( x k , r k )) and t > x i ∈ [ r i , t − r i ] × [ r i − , − r i ] n − .Its operadic composition is given by interpreting these configurations as configurations of little non-overlapping discs and embedding D n by translations and dilations, as well as placing copies of thecuboid [0 , t ] × [ − , n − end-to-end in the first coordinate direction and adding the correspondingvalues of t . See Figure 4.1(f), and also [Kra19, Definition 2.1] for precise formulas for the operadiccomposition.By definition, there are inclusions of operads CSC m,n ֒ −→ SC m,n ֒ −→ VSC m,n ֒ −→ ESC m,n (4.1)that restrict to the identity map of D n on the a colour. The natural inclusions of discs D n ֒ → D n +1 and cubes [ − , n − ֒ → [ − , n induce dimension-increasing inclusions (see Figure 4.2) LSC n −→ LSC n +1 and XSC m,n −→ XSC m,n +1 (4.2)(for X ∈ { E , V , ∅ , C } ) and similarly XSC m − ,n −→ XSC m,n (4.3)(for X ∈ { E , V , ∅ , C } ), which commute with the “flavour-changing” inclusions (4.1). The connectionbetween the four operads (4.1) and the linear Swiss cheese operads is given by the following lemma.Recall that a weak equivalence of operads is a map of operads O → P such that the maps of spaces O ( a k , m l ; a ) → P ( a k , m l ; a ) and O ( a k , m l ; m ) → P ( a k , m l ; m ) are all weak equivalences. Lemma 4.8
For any m n there is a map of operads ι m,n : LSC n −→ CSC m,n (4.4)16
Figure 4.2
The dimension-increasing inclusion
ESC , ( a , m ; m ) → ESC , ( a , m ; m ). The number-ing of the 3-discs has been omitted on the right-hand side to avoid overloading the diagram. t e − t e − t e − t ι , ι , (4.3) ∼ ∼ Figure 4.3
The map ι m,n and the homotopy-commutativity of the right-hand triangle of (4.5). commuting with (4.2) and (4.3) in the sense that the following diagrams commute up to homotopy: LSC n CSC m,n
LSC n +1 CSC m,n +1 ι m,n ι m,n +1 (4.2) (4.2) LSC n CSC m − ,n CSC m,n ι m − ,n ι m,n (4.3) (4.5) When m n − , the map ι m,n is a weak equivalence of operads, so (4.3) : CSC m − ,n → CSC m,n is also a weak equivalence of operads in this range.Proof.
The map ι m,n is defined as pictured (for the cases ( m, n ) = (1 ,
2) and ( m, n ) = (0 , , ∞ ) of theconfiguration points (interpreted as radii). The fact that ι m,n is a weak equivalence of operads for m n − m = n it is a quotient map from the rectangle onto thewhole annulus). The left-hand square of (4.5) commutes on the nose, and the right-hand triangleof (4.5) commutes up to a “stretching” homotopy that is also pictured in Figure (4.5) (in the case( m, n ) = (1 , Remark 4.9
The map ι n,n : LSC n → CSC n,n , on the other hand, is not a weak equivalence: forexample, the space
LSC n ( a , m ; m ) is contractible, whereas the space CSC n,n ( a , m ; m ) is homotopyequivalent to S n − . Algebras.
Recall that, in general, an algebra over a two-coloured operad O (with colours a and m ) consists of a pair of spaces ( X a , X m ) together with maps O ( a k , m l ; a ) × ( X a ) k × ( X m ) l −→ X a and O ( a k , m l ; m ) × ( X a ) k × ( X m ) l −→ X m Y = X a (correspondingto the red colour in Figure 4.1) and X = X m (corresponding to the blue colour in Figure 4.1). Algebras over Swiss-cheese operads.
Algebras over D n are E n -algebras , by definition. Now,the restriction of SC m,n to the a colour is D n and its restriction to the m colour is isomorphic to D m , so an algebra over SC m,n is a pair ( X, Y ) consisting of an E n -algebra Y , an E m -algebra X ,together with an additional structure intertwining them. The same remarks apply also to VSC m,n and
ESC m,n , so algebras over each of these three operads consist of an E n -algebra acting on an E m -algebra, where the precise meaning of “acting” depends on the flavour.The restriction of the concentric Swiss cheese operad
CSC m,n to the a colour is again D n , butnow its restriction to the m colour is trivial (it has no l -ary operations except when l = 1, and itsspace of 1-ary operations is homeomorphic to (0 , CSC m,n arethus E m - modules over E n -algebras, without any E m - algebra structure on the module. By Lemma4.8, we have CSC m,n ≃ CSC m ′ ,n for any m, m ′ n −
1, so the notions of “ E m -module over an E n -algebra ”, for fixed n , are equivalent for all m ∈ { , , . . . , n − } , and are equivalently encodedby the linear Swiss cheese operad
LSC n . On the other hand, the notion of “ E n -module over an E n -algebra ” is stronger, and not encoded by the linear Swiss cheese operad, as pointed out inRemark 4.9. Linear Swiss cheese structures on configuration-section spaces.
Below, we define certainhomotopy equivalent models ˙ C ( M ) ≃ C ( ˚ M ) and ˙CΓ c,D ( M ; ξ ) ≃ CΓ c,D ( M ; ξ ) for configurationspaces and configuration-section spaces (see Definitions 4.12 and 4.13, and Lemma 4.14). Definition4.13 also explains how a singularity condition c ⊆ Γ( η ( ξ )) determines a subset c D ⊆ [ S d − , X ]. Weuse a slight abuse of notation by writing CMap c,∂ ( D d ; X ) instead of CMap c D ,∂ ( D d ; X ). We alsoabbreviate ∂D d to ∂ and write ∂ for one hemisphere of ∂D d . Proposition 4.10
We have the following linear Swiss cheese structures on configuration andconfiguration-section spaces: (i) . ( ˙ C ( M ) , C ( ˚ D d )) is an algebra over LSC d , (ii) . ( ˙CΓ c,D ( M ; ξ ) , CMap c,∂ ( D d ; X )) is an algebra over LSC d , (iii) . ( ˙CΓ c,D ( M ; ξ ) , CMap c, ∂ ( D d ; X )) is an algebra over LSC d − .Moreover, the maps of pairs ( ˙CΓ c,D ( M ; ξ ) , CMap c,∂ ( D d ; X )) ֒ → ( ˙CΓ c,D ( M ; ξ ) , CMap c, ∂ ( D d ; X )) → ( ˙ C ( M ) , C ( ˚ D d )) (4.6) are maps of LSC d − -algebras, and their composition is a map of LSC d -algebras. Here, the first mapis the inclusion and the second map sends a configuration-section to its underlying configuration,forgetting the section. Point (i) is essentially [Kra19, Lemma 5.1], and part of points (ii) and (iii) – the D d -algebrastructure, but not the LSC d - and LSC d − -algebra structures – is [EVW, Propositions 2.6.1 and2.6.2]. Before proving Proposition 4.10, we first define the appropriate models for configurationand configuration-section spaces referred to above. Definition 4.11
Let M be a manifold equipped with an embedded codimension-zero disc D ⊆ ∂M in its boundary and a collar neighbourhood of ∂M , namely an embedding b : ( −∞ , × ∂M → M such that b (0 , x ) = x for all x ∈ ∂M . Define:ˆ M = M ∪ b ( R × D ) and ˆ M r = M ∪ b (( −∞ , r ] × D )for r ∈ [0 , ∞ ). Diagrammatically, ˆ M may be seen as follows, where M is green and R × D is blue(and hence ( −∞ , × D , which is identified with b (( −∞ , × D ) ⊆ M , is turquoise).18 im( b ) D∂M −∞ r ∞ ˆ M Definition 4.12
Let ˙ C k ( M ) be the subspace of C k ( ˆ M ) × (0 , ∞ ) of pairs ( z, t ) with z ⊆ int( ˆ M t ),and define ˙ C ( M ) = G k ∈ N ˙ C k ( M ) . Definition 4.13
Let M , D and b be as in Definition 4.11. Also choose a bundle ξ : E → M , asubset c ⊆ Γ( η ( ξ )) ( cf . Definition 3.14) and a section s D of ξ | D . Denote by ∗ the centre of the disc D ⊆ ∂M , let X = ξ − ( ∗ ) and choose a point x ∈ X . Choose a trivialisation ϕ : ξ | D ∼ = D × X suchthat s D corresponds to the constant section of D × X at x . Using ϕ , glue ξ along D = D × { } to the trivial X -bundle over [0 , ∞ ) × D to obtain a bundle on ˆ M , which we denote by ˆ ξ . Sincethe construction η ( − ) of Definition 3.11 commutes with restriction, we have restriction maps ofsections Γ( η ( ˆ ξ )) −| M −−−−→ Γ( η ( ξ )) −| D −−−→ Γ( η ( ξ | D )) ∼ = [ S d − , X ] , where the last bijection is induced by the trivialisation ϕ of ξ | D and the fact that η of a trivialbundle over an orientable d -manifold with fibre X is the trivial covering space with fibre [ S d − , X ](see Example 3.22). Note that the first restriction map −| M is a bijection, since any section of η ( ξ )may be extended uniquely to a section of η ( ˆ ξ ), due to the fact that η ( ˆ ξ ) is trivial over D × [0 , ∞ ).Thus the subset c ⊆ Γ( η ( ξ )) determines subsetsˆ c ⊆ Γ( η ( ˆ ξ )) and c D ⊆ [ S d − , X ] . Define ˙CΓ c,Dk ( M ; ξ ) to be the subspace of CΓ ˆ ck ( ˆ M ; ˆ ξ ) × (0 , ∞ ) of elements ( z, s, t ) consisting of areal number t > z, s ) such that • z ⊆ int( ˆ M t ), • s is the constant section at x on the subspace [ t, ∞ ) × D ⊆ ˆ M ,and let ˙CΓ c,D ( M ; ξ ) = G k ∈ N ˙CΓ c,Dk ( M ; ξ ) . We will also use the following slight abuse of notation:CMap c,∂ ( D d ; X ) = G k ∈ N CMap c D ,∂D d k ( D d ; X ) , CMap c, ∂ ( D d ; X ) = G k ∈ N CMap c D ,∂ D d k ( D d ; X ) , where ∂ D d = ∂D d ∩ { x d } is the southern hemisphere of ∂D d . Lemma 4.14
For each k , there are natural embeddings C k ( M ) ֒ −→ ˙ C k ( M ) and CΓ c,Dk ( M ; ξ ) ֒ −→ ˙CΓ c,Dk ( M ; ξ ) admitting deformation retractions. In particular, C ( M ) ≃ ˙ C ( M ) and CΓ c,D ( M ; ξ ) ≃ ˙CΓ c,D ( M ; ξ ) .Proof. We will prove this just for the second embedding (for configuration-section spaces), sincethe proof for the first embedding (for configuration spaces) is identical, forgetting the sections andconsidering just configurations of points. The embedding is defined by sending a configuration-section ( z, s ) of ξ to the element ( z, ˆ s,
1) of ˙CΓ c,Dk ( M ; ξ ), where ˆ s is the section of ˆ ξ | ˆ M r z given byextending s by the constant section at x on D × [0 , ∞ ).We now construct a deformation retraction of ˙CΓ c,Dk ( M ; ξ ) onto the image of this embedding.First we choose a family of diffeomorphisms ϕ : [0 , × [0 , ∞ ) → Diff( ˆ M ) with the properties that19
13 2 t t ′ t ′ t ′ + t Figure 4.4
The maps (4.7) defining the
LSC d action on ( ˙CΓ c,D ( M ; ξ ) , CMap c,∂ ( D d ; X )), in dimension d = 2 and for k = 3. The light green, yellow, blue and orange colours represent sections defined on thecomplement of each configuration. Light grey indicates regions where the section is constant at thebasepoint of X (note that ξ is trivial with fibre X over the grey regions, so this makes sense). (i) ϕ (0 , t ) = id and (ii) ϕ (1 , t )( ˆ M t ) ⊆ M . Namely, we define ϕ ( u, t ) to be the identity outside of N = ( ∂M × ( −∞ , ∪ ( D × [0 , ∞ )), and for a point ( x, v ) of N , define ϕ ( u, t )( x, v ) = ( x, v − ut ).This may be lifted to a family of automorphisms e ϕ : [0 , × [0 , ∞ ) → Aut( ˆ ξ ) such that e ϕ ( u, t ) covers ϕ ( u, t ), using the observation that the obvious projection p : N → ∂M is a homotopy equivalence,so we may identify ˆ ξ | N with p ∗ ( ˆ ξ | ∂M ). The deformation retraction is defined by sending ( z, s, t ),at time u ∈ [0 , (cid:0) ϕ ( u, t )( z ) , e ϕ ( u, t ) ◦ s ◦ ϕ ( u, t ) − , (1 − u ) t + u (cid:1) . Proof of Proposition 4.10.
The
LSC d action on ( ˙CΓ c,D ( M ; ξ ) , CMap c,∂ ( D d ; X )) is determined bymaps LSC d ( a k ; a ) × (CMap c,∂ ( D d ; X )) k −→ CMap c,∂ ( D d ; X ) LSC d ( a k , m ; m ) × (CMap c,∂ ( D d ; X )) k × ˙CΓ c,D ( M ; ξ ) −→ ˙CΓ c,D ( M ; ξ ) , (4.7)which are defined by picture in Figure 4.4. The LSC d − action on ( ˙CΓ c,D ( M ; ξ ) , CMap c, ∂ ( D d ; X ))is determined by maps LSC d − ( a k ; a ) × (CMap c, ∂ ( D d ; X )) k −→ CMap c, ∂ ( D d ; X ) LSC d − ( a k , m ; m ) × (CMap c, ∂ ( D d ; X )) k × ˙CΓ c,D ( M ; ξ ) −→ ˙CΓ c,D ( M ; ξ ) , (4.8)which are defined by picture in Figure 4.5. The fact that these are well-defined actions of linearSwiss cheese operads may be verified easily from the construction. This gives the action of LSC d forcase (ii) and of LSC d − for case (iii). The action of LSC d for case (i) is identical to that for case (ii),forgetting all sections and remembering just the configurations (compare the bottom line of Figure4.4 with [Kra19, Figure 4]). The statements that the maps (4.6) respect the LSC d − structures(and that their composition respects the LSC d structures) is also clear from the construction.
5. Monodromy actions
Definition 5.1
Let f : E → B be a Serre fibration and F = f − ( b ) for a point b ∈ B . Assumeeither that F is a CW-complex or that f is a Hurewicz fibration. Then the monodromy action of f is the action-up-to-homotopy mon f : π ( B, b ) −→ π (hAut( F )) (5.1)20 t t ′ t ′ t ′ + t Figure 4.5
The maps (4.8) defining the
LSC d − action on ( ˙CΓ c,D ( M ; ξ ) , CMap c, ∂ ( D d ; X )), also indimension d = 2 and for k = 3. The light green, yellow, blue and orange colours represent sectionsdefined on the complement of each configuration. Light grey indicates regions where the section isconstant at the basepoint of X (note that ξ is trivial with fibre X over the grey regions, so this makessense). Dotted regions indicate that the map to X is extended into this region by defining it to beindependent of the vertical direction in this region. ( Cf . Figure 7.1 for a 3-dimensional picture.) of π ( B, b ) on the fibre F defined as follows. Given an element [ γ ] ∈ π ( B, b ) and a representativeloop 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 (5.2)and define mon f ([ γ ]) = [ g ( − , . Remark 5.2
In fact, all one needs for Definition 5.1 is a continuous map f : E → B and a point b ∈ B such that f satisfies the homotopy lifting property with respect to F and F × [0 ,
1] ( cf . theproof of Lemma 5.3 below). Lemma 5.3
The construction of Definition 5.1 using the lifting diagram (5.2) gives a well-definedgroup homomorphism (5.1) .Proof.
Suppose that γ ′ is another representative of [ γ ] and that g ′ is a lift of γ ′ in the diagram(5.2) (with γ replaced by γ ′ ). Let k : F × [0 , → E be a choice of lift in the diagram: F × (cid:0) (cid:1) EF × [0 , [0 , B g ∪ const incl ∪ g h incl f (5.3)where h is a homotopy γ ≃ γ ′ relative to endpoints. Then k | F ×{ }× [0 , is a homotopy g ( − , ≃ g ′ ( − ,
1) of self-maps of F . This implies that the construction of Definition 5.1 gives a well-definedfunction mon f : π ( B, b ) → π (Map( F, F )). It remains to prove that mon f is a homomorphism ofmonoids, since it will then follow that it has image contained in the underlying group π (hAut( F ))of π (Map( F, F )). It is clear that mon f takes the constant loop to the identity map of F , since inthis case we may take the lift in (5.2) to be the projection F × [0 , ։ F followed by the inclusion F ֒ → E . We therefore just have to prove thatmon f ([ γ .γ ]) = mon f ([ γ ]) ◦ mon f ([ γ ])21or elements [ γ ] , [ γ ] ∈ π ( B, b ). Choose lifts g and g in the diagrams: F EF × [0 ,
1] [0 , B incl γ ( − , fg F EF × [1 ,
2] [1 , B incl γ ( − , fg (5.4)so we have mon f ([ γ ]) ◦ mon f ([ γ ]) = [ g ( − , ◦ g ( − , F × [0 , → E of the diagram: F EF × [0 ,
2] [0 , B incl γ .γ ( − , f (5.5)by: ( x, t ) ( g ( x, t ) t ∈ [0 , g ( g ( x, , t ) t ∈ [1 , . By definition, it follows that mon f ([ γ .γ ]) = [ g ( g ( − , , g ( − , ◦ g ( − , Notation 5.4
From now on, we fix, once and for all, choices of the objects of Definitions 4.11 and4.13, namely: • a manifold M equipped with an embedded codimension-zero disc D ⊆ ∂M with centre ∗ , • a collar neighbourhood of ∂M , namely an embedding b : ( −∞ , × ∂M ֒ → M so that b (0 , − )is the inclusion ∂M ⊂ M , ⊲ This determines the manifold ˆ M and its submanifolds ˆ M r ( r ∈ [0 , ∞ )) as in Definition 4.11. • a fibre bundle ξ : E → M , with basepoint x ∈ X := ξ − ( ∗ ), • a subset c ⊆ Γ( η ( ξ )) ( cf . Definition 3.14), • a trivialisation θ : ξ | D ∼ = D × X . ⊲ We write s D for the section of ξ | D corresponding to the constant section of D × X at x . ⊲ Using the trivialisation θ , we extend ξ by a trivial X -bundle to obtain a bundle ˆ ξ over ˆ M .Recall the homotopy-equivalent models ˙ C k ( M ) ≃ C k ( ˚ M ) and ˙CΓ c,Dk ( M ; ξ ) ≃ CΓ c,Dk ( M ; ξ ) forconfiguration spaces and configuration-section spaces from §4 (Definitions 4.12 and 4.13). Lemma 5.5
There is a Hurewicz fibration ˙CΓ c,Dk ( M ; ξ ) −→ ˙ C k ( M ) (5.6) given by forgetting the section data of a configuration-section.Proof. The forgetful map (5.6) is defined by ( z, s, t ) ( z, t ), where t > z is aconfiguration in the interior of ˆ M t and s is a section of ˆ ξ over ˆ M t r z . There are homeomorphisms˙ C k ( M ) ∼ = C k ( ˚ M ) × (0 , ∞ ) and ˙CΓ c,Dk ( M ; ξ ) ∼ = CΓ c,Dk ( M ; ξ ) × (0 , ∞ ) (5.7)under which (5.6) corresponds to the map p × id (0 , ∞ ) , where p is the Hurewicz fibration (3.11).Hence (5.6) is also a Hurewicz fibration. The homeomorphisms above may be defined as follows. Let D ′ be an open codimension-zero disc in ∂M containing D in its interior. Choose an identificationof b (( −∞ , × D ′ ) ⊆ M with ( −∞ , × D so that b ( { } × D ) corresponds to { } × D (Figure 5.1).This induces an embedding D × R ֒ → ˆ M , and we obtain a homeomorphism ψ r : ˆ M → ˆ M for each r ∈ R by defining ψ r ( x, t ) = ( x, t + r ) for ( x, t ) ∈ D × R and ψ r ( y ) = y for y ∈ ˆ M r ( D × R ). Theleft-hand homeomorphism of (5.7) may then be defined by( z, t ) ( ψ − t ( z ) , t ) . im( b ) D −∞ ∞ Figure 5.1
The turquoise region is b (( −∞ , × D ′ ) and is identified with ( −∞ , × D so that b ( { }× D )corresponds to { } × D . For illustration, four other slices { t } × D under this identification are drawn. Choosing a trivialisation of ˆ ξ | D × R (extending the identity trivialisation of ˆ ξ | D × [0 , ∞ ) , which is trivialby construction), we may lift ψ r : ˆ M → ˆ M to a bundle-homeomorphism e ψ r : ˆ ξ → ˆ ξ . The right-handhomeomorphism of (5.7) may then be defined by( z, s, t ) ( ψ − t ( z ) , e ψ t ◦ s ◦ ψ − t , t ) . Remark 5.6
The homeomorphisms (5.7) constructed in the proof of Lemma 5.5 give an alternativeproof of Lemma 4.14, although the construction of (5.7) is a little more ad hoc.
Definition 5.7
For a real number r >
1, let p r = ( ∗ , r − ) ∈ D × [0 , ∞ ) ⊆ ˆ M , where ∗ denotesthe centre of the disc D . For an integer k >
1, the k -th “standard configuration” in ˆ M is definedto be z k = { p , p , . . . , p k } , and the basepoint of ˙ C k ( M ) is defined to be ( z k , k ). Remark 5.8
Stabilisation maps for configuration spaces and configuration-section spaces will bedefined in §7, using the E -module structure of §4. However, at the level of configuration spaces,it is already clear that the stabilisation map ˙ C k ( M ) → ˙ C k +1 ( M ) should be defined by( z, t ) ( z ⊔ { p t +1 } , t + 1) . Lemma 5.9
The fibre of (5.6) over ( z k , k ) ∈ ˙ C k ( M ) is the space Γ c,Dk ( M ; ξ ) := Γ c D ,D ×{ k } (cid:0) ˆ M k r z k ; ˆ ξ (cid:1) (5.8) of sections s of ˆ ξ defined over ˆ M k r z k ⊆ ˆ M such that • the restriction of s to a small punctured neighbourhood of p i lies in c D ⊆ [ S d − , X ] for each i ∈ { , . . . , k } , where c D is as in Definition 4.13, • the restriction of s to D × { k } ⊆ ∂ ˆ M k is constant at the basepoint x of X . Corollary 5.10
We have a Hurewicz fibration sequence of the form Γ c,Dk ( M ; ξ ) −→ ˙CΓ c,Dk ( M ; ξ ) −→ ˙ C k ( M ) . (5.9) Proof of Corollary 5.10.
This follows immediately from Lemmas 5.5 and 5.9.
Proof of Lemma 5.9.
Directly from the definitions, the fibre of (5.6) over ( z k , k ) may be describedas written, except that the first condition says that s must satisfy the singularity condition ˆ c ⊆ Γ( η ( ˆ ξ )), where ˆ c is determined by c ⊆ Γ( η ( ξ )) as explained in Definition 4.13. But all of the pointsof z k lie in D × [0 , ∞ ) ⊆ ˆ M , over which the bundle ˆ ξ is trivial with fibre X , so the singularityconditions around these points are equivalent to the conditions written in the lemma. Definition 5.11
Let Br( M ) = (cid:2) π ( ˙ C k ( M )) (cid:3) k ∈ N denote the groupoid whose objects are N , whose automorphism group of k ∈ N is π ( ˙ C k ( M )) andwhich has no morphisms between distinct objects.23 efinition 5.12 Fix (
M, D, ∗ , b, ξ, x , c, θ ) as in Notation 5.4. The associated monodromy functor Mon c,D ( M, ξ ) : Br( M ) −→ Ho(Top) (5.10)takes the object k ∈ N to the fibre Γ c,Dk ( M ; ξ ) of (5.6). On automorphisms of k , it is defined bythe monodromy action (5.1), with f = (5.6).
6. Braid categories
Definition 6.1 ([Gra76, p. 219]) The
Quillen bracket construction hD , Ci of a category C equippedwith a left-action of a monoidal category D is the category with the same objects as C , and withmorphisms given by hD , Ci ( c, c ′ ) = colim D ( C ( − ⊕ c, c ′ )), where ⊕ is the action of D on C . In otherwords, a morphism c → c ′ in hD , Ci is an equivalence class of morphisms ϕ : d ⊕ c → c ′ in C with d in D , and where ( d , ϕ ) ∼ ( d , ϕ ) if there exists θ : d → d in D such that ϕ = ϕ ◦ ( θ ⊕ id c ).This comes equipped with a canonical functor C −→ hD , Ci given by the identity on objects and by ϕ ⊕ ϕ on morphisms, where 0 is the unit object of themonoidal structure on D . Lemma 6.2
The groupoid
Br( D d ) has a monoidal structure given by taking the boundary connectedsum of two discs. If d > it is braided and if d > it is symmetric. Now let M be a connected d -manifold with non-empty boundary, and let D ⊆ ∂M be an embedded ( d − -dimensional disc.There is a well-defined action of Br( D d ) on Br( M ) given by boundary connected sum along D .Proof. Let us write Br( M ) = π ( ˙ C ( M ) , { z k } k ∈ N ), the fundamental groupoid of ˙ C ( M ) with respectto the set of basepoints { z k ∈ ˙ C k ( M ) | k ∈ N } . This is a skeleton for, hence equivalent to, thefull fundamental groupoid π ( ˙ C ( M )), with objects all points of ˙ C ( M ). When M = D d we have π ( ˙ C ( D d )) ≃ π ( C ( ˚ D d )), since ˙ C ( D d ) and C ( ˚ D d ) are homotopy equivalent. By Proposition 4.10(i), C ( ˚ D d ) is an E d -algebra and ˙ C ( M ) is an E -module over it. Passing to fundamental groupoids andpulling back along the equivalences, this gives rise to the structure on Br( D d ) and Br( M ) claimedin the lemma. Remark and Notation 6.3
Note that Br( D ) is the free monoidal category on one object, Br( D )is the free braided monoidal category on one object and Br( D ) is the free symmetric monoidalcategory on one object. We will therefore abbreviate these groupoids by M , B and S respectively.The standard inclusions D ֒ → D ֒ → D induce monoidal functors M ֒ → B ֒ → S (the second oneis also braided monoidal), and for d > D ֒ → D d induces an isomorphism S ∼ = Br( D d ). Definition 6.4 ([Kra19, §5.2]) Let M be a connected d -manifold with non-empty boundary, with d >
2, and let D ⊆ ∂M be an embedded ( d − D d ) on Br( M ). The standard inclusion D ֒ → D d induces a braided monoidal functor B →
Br( D d ), and hence an action of B on Br( M ).Using Definition 6.1, we may therefore define the category C ( M ) = hB , Br( M ) i , which is equippedwith a canonical functor Br( M ) −→ C ( M ) . (6.1) Lemma 6.5
The functor (6.1) is the inclusion of the underlying groupoid of C ( M ) .Proof. This is an immediate generalisation of the proof of [RW17, Proposition 1.7].
Remark 6.6
Our notation is the reverse of that of [Kra19], since we are using the oppositeconvention of considering left -actions of monoidal categories.24 efinition 6.7 ([Pal18, §2.3 and §3.1]) The categories B ( M ) and B ♯ ( M ) both have N as their setof objects. A morphism k → ℓ in B ♯ ( M ) is a path γ , up to endpoint-preserving homotopy, in thespace C r ( ˚ M ) for some r , satisfying γ (0) ⊆ { p , . . . , p k } and γ (1) ⊆ { p , . . . , p ℓ } . Composition isdefined by analogy with composition of partially-defined functions: given morphisms γ : k → ℓ and δ : ℓ → m , let r = | γ (1) ∩ δ (0) | and define δ ◦ γ to be the path in C r ( ˚ M ) obtained by concatenatingthe corresponding restrictions of γ and δ . A morphism k → ℓ in the subcategory B ( M ) ⊆ B ♯ ( M )is a path γ as above, with r = k . Remark 6.8
The categories B ( M ) and B ♯ ( M ) were denoted by B f ( M ) and B ( M ) respectively in[Pal18]. We have modified that notation to fit more naturally with the standard notation FI andFI ♯ for the categories of finite sets and injections , and of finite sets and partially-defined injections ,respectively, as well as with the notation of [Kra19]. We note ( cf . [Kra19, Remark 5.10]) that, ifdim( M ) > M is simply-connected, then B ( M ) ∼ = FI and B ♯ ( M ) ∼ = FI ♯ . Lemma 6.9
Under the conditions of Definition 6.4, there is a canonical functor C ( M ) −→ B ( M ) , which is the identity on objects and is also: • full, for any M , • faithful (and hence an isomorphism) if and only if dim( M ) > and M is simply-connected.Proof. The set of morphisms m → n of C ( M ) is empty if m > n , and if m n it is naturallyidentified with the orbit set B n ( M ) /B m , where B n ( M ) = π ( C n ( M ) , { p , . . . , p n } ) and B m actson B n ( M ) via the homomorphism v mn − m : B m → B n ( M ) of Definition 7.1 below followed by right-multiplication of B n ( M ) on itself. On the other hand, the set of morphisms m → n of B ( M ) isalso empty if m > n , and if m n it is a homotopy class of paths in C m ( M ) from the basepointconfiguration { p , . . . , p m } to a subconfiguration of { p , . . . , p n } .The functor C ( M ) → B ( M ) is defined on morphisms as follows. Given a morphism m → n of C ( M ), represented by a loop of configurations γ in C n ( M ) based at { p , . . . , p n } , forget all strandsof γ that start at p i for m +1 i n . The result is a path of configurations in C m ( M ) representinga morphism m → n of B ( M ).Given any path of configurations δ in C m ( M ) from { p , . . . , p m } to a subconfiguration of { p , . . . , p n } ,it is always possible to “adjoin strands” to δ to extend it to a loop of configurations γ in C n ( M ).This implies that the functor C ( M ) → B ( M ) is full.The fact that the functor C ( M ) → B ( M ) is faithful if dim( M ) > M is simply-connectedis stated in Remark 5.10 of [Kra19]. It also follows from Lemma 4.1 of [Til16], which implies thatthe functor is given by Σ n / Σ m → Inj( m, n ) on morphism-sets (from m to n ); we already knowthat this is surjective, so injectivity follows from a simple counting argument.If M is 2-dimensional or π ( M ) is non-trivial, it is easy to construct pairs of distinct morphisms1 → C ( M ) that become equal in B ( M ), i.e., after forgetting the second strand of a 2-strandbraid on M . Thus the conditions that dim( M ) > M is simply-connected are necessary for C ( M ) → B ( M ) to be a faithful functor. Summary 6.10
Let M be a connected d -manifold with non-empty boundary, with d >
2, and let D ⊆ ∂M be an embedded ( d − M ) ֒ −→ C ( M ) −→→ B ( M ) ֒ −→ B ♯ ( M ) (6.2)that all act by the identity on their common set of objects (which is N ). The first and third functorsare faithful, and the second functor is full. The second functor is also faithful (and therefore anisomorphism) if and only if M is simply-connected and has dimension at least 3.
7. Stabilisation maps and extension to C ( M ) As in §5, we fix the data (
M, D, ∗ , b, ξ, x , c, θ ) of Notation 5.4, namely a bundle ξ : E → M overa d -manifold, a disc D ⊆ ∂M , etc. 25 (a) Choices of elements of
LSC d − ( a , m ; m ) and of CMap c, ∂ ( D d ; X ).The light green region represents a map D d r { } → X in one of the homotopyclasses c D ⊆ [ S d − , X ], where c D is determined by c as in Definition 4.13,sending the southern hemisphere of ∂D d to { x } ⊆ X. ˆ M t t t + 1 (b) The stabilisation maps of (7.1). Given a (blue) configuration-section( z, s ) lying in ˆ M t with s | D ×{ t } = const x , add a new point at p t +1 = ( ∗ , t + )and extend the section as illustrated: grey indicates the constant map to x ,green indicates the configuration-section in D d chosen above and dotted-greenindicates that the green region is extended, as in Figure 4.5, by defining it tobe independent of the vertical direction in this region. Figure 7.1
Stabilisation maps for configuration-section spaces.
Stabilisation maps.
Let us choose, once and for all, an element of
LSC d − ( a , m ; m ) consisting ofone ( d − d − c, ∂ ( D d ; X ) wherethe configuration has exactly one point. See Figure 7.1(a).By Proposition 4.10, the pair ( ˙CΓ c,D ( M ; ξ ) , CMap c, ∂ ( D d ; X )) is an algebra over the linear Swisscheese operad LSC d − ( i.e. ˙CΓ c,D ( M ; ξ ) is an E -module over the E d − -algebra CMap c, ∂ ( D d ; X )).Moreover, the pair ( ˙ C ( M ) , C ( ˚ D d )) is an algebra over LSC d , and hence also over LSC d − by restric-tion, and the maps ( ˙CΓ c,D ( M ; ξ ) , CMap c, ∂ ( D d ; X )) −→ ( ˙ C ( M ) , C ( ˚ D d ))that send a configuration-section to its underlying configuration (forgetting the section) are mapsof LSC d − -algebras. This structure induces (horizontal) stabilisation maps ˙CΓ c,Dk ( M ; ξ ) ˙CΓ c,Dk +1 ( M ; ξ )˙ C k ( M ) ˙ C k +1 ( M ) (7.1)commuting with the (vertical) forgetful maps, for all k ∈ N . Concretely, the top horizontal map isgiven by the second line of (4.8), plugging in our choices of elements above. The bottom horizontalmap is defined similarly, ignoring sections and considering just configurations. See Figure 7.1(b).Note that the bottom horizontal map of (7.1) is exactly as already described in Remark 5.8. Definition 7.1
The map π ( ˙ C k ( M )) −→ π ( ˙ C k +1 ( M ))of fundamental groups induced by the stabilisation map (7.1) will be denoted by σ k . Identifyingthe interior of the ( d − D ⊆ ∂M with ( − , d − , we also have inclusions( − , × { } d − × ( k, k + ℓ ) ֒ −→ int( ˆ M k + ℓ )26 −→ k − k − k k + ℓp k − p k − p k Figure 7.2
The map (7.2) inducing v ℓk : B ℓ → π ( ˙ C k + ℓ ( M )). for integers k > ℓ >
1, which induce maps C ℓ (( − , × { } d − × ( k, k + ℓ )) −→ ˙ C k + ℓ ( M ) (7.2)given by S ( S ⊔ { p , . . . , p k } , k + ℓ ) as illustrated in Figure 7.2. The induced map B ℓ ∼ = π ( C ℓ (( − , × { } d − × ( k, k + ℓ ))) −→ π ( ˙ C k + ℓ ( M )) (7.3)of fundamental groups is denoted by v ℓk .The maps σ k : π ( ˙ C k ( M )) → π ( ˙ C k +1 ( M )) and v ℓk : B ℓ → π ( ˙ C k + ℓ ( M )) of Definition 7.1 maybe used to characterise extensions of functors along the inclusion Br( M ) ⊂ C ( M ): Proposition 7.2 ([Kra19, §5.2])
Let M be a connected d -manifold, for d > , and let D ⊆ ∂M bean embedded ( d − -dimensional disc. Choose a collar neighbourhood b as in Notation 5.4 and takethe basepoint ∗ ∈ ∂M to be the centre of D . Let F : Br( M ) → D be a functor. An extension of F to the larger category C ( M ) ⊇ Br( M ) is equivalent to a choice of morphism s k : F ( k ) → F ( k + 1) of D for each k ∈ N such that, for any k > , ℓ > , α ∈ π ( ˙ C k ( M )) and β ∈ B ℓ , the followingtwo diagrams commute, where s ℓk = s k + ℓ − ◦ · · · ◦ s k +1 ◦ s k . F ( k ) F ( k + 1) F ( k ) F ( k + 1) s k s k F ( α ) F ( σ k ( α )) F ( k ) F ( k + ℓ ) F ( k + ℓ ) s ℓk s ℓk F ( v ℓk ( β )) (7.4) Proposition 7.3
The stabilisation maps (7.1) determine an extension of the monodromy functor (5.10) = Mon c,D ( M, ξ ) : Br( M ) → Ho(Top) to a functor g Mon c,D ( M, ξ ) : C ( M ) −→ Ho(Top) . (7.5) Proof.
We will apply Proposition 7.2 with D = Ho(Top) and F = (5.10) = Mon c,D ( M, ξ ). Recallthat Mon c,D ( M, ξ ) sends k to the fibre Γ c,Dk ( M, ξ ) of the Hurewicz fibration (5.6) over the basepoint( z k , k ) ∈ ˙ C k ( M ). The bottom horizontal map of the map of Hurewicz fibrations (7.1) preservesbasepoints, so its top horizontal map restricts to a map of fibres F ( k ) = Γ c,Dk ( M, ξ ) −→ Γ c,Dk +1 ( M, ξ ) = F ( k + 1) , which we define to be s k . It remains to check the two conditions (7.4) of Proposition 7.2.The element α ∈ π ( ˙ C k ( M )) acts on F ( k ) = Γ c,Dk ( M, ξ ) through a “point-pushing” diffeomor-phism θ α of ˆ M k r z k . (Only the isotopy class of θ α is important, since the diagrams (7.4) live inthe homotopy category.) The element σ k ( α ) ∈ π ( ˙ C k +1 ( M )) is the extension of α to a loop of k + 1points in ˆ M k +1 given by leaving the point p k +1 fixed. Hence we may choose the point-pushingdiffeomorphism θ σ k ( α ) of ˆ M k +1 r z k +1 (through which σ k ( α ) acts on F ( k + 1) = Γ c,Dk +1 ( M, ξ )) tobe the extension of θ α by the identity on ( D × [ k, k + 1]) r { p k +1 } . This implies that the left-handsquare of (7.4) commutes up to homotopy.Now consider any element s ∈ F ( k ), a section of ˆ ξ defined on ˆ M k r z k . Extend it in a standardway ℓ times, as shown in Figure 7.1(b), to obtain a section ¯ s of ˆ ξ defined on ˆ M k + ℓ r z k + ℓ . The27estriction of ¯ s to P = ( D × [ k, k + ℓ ]) r { p k +1 , . . . , p k + ℓ } may be thought of as a map P → X ,since ˆ ξ is trivial over P . Note that this map P → X does not in fact depend on s : it just consistsof ℓ concatenated copies of the standard map to X illustrated on the right-hand side of Figure7.1(b). We denote it by f ℓ : P → X . Any element β ∈ B ℓ determines a self-diffeomorphism (up toisotopy) β : P → P by point-pushing. Claim: f ℓ ◦ β is homotopic to f ℓ through maps sending the two ends D × { k } and D × { k + ℓ } to the basepoint x ∈ X .This claim implies that ¯ s · β is homotopic to ¯ s through maps sending D × { k + ℓ } to x , in otherwords, there is a path ¯ s · β ∼ ¯ s in Γ c,Dk + ℓ ( M, ξ ) = F ( k + ℓ ), and moreover these paths may be chosencontinuously in s . In other words, the right-hand triangle of (7.4) commutes up to homotopy.It therefore remains to prove the claim above, and it will suffice to prove it when β is the standardgenerator of B ℓ that interchanges the two punctures p i and p i +1 for k + 1 i ℓ −
1. We will dothis diagrammatically for dimensions d > d = 2.First assume that d >
3. In this case, Figure 7.3 illustrates a “bird’s eye” view of the map f ℓ : P → X , where P = ( D × [ k, k + ℓ ]) r { p k +1 , . . . , p k + ℓ } , by collapsing the “vertical” direction ofFigures 7.1(b) and 7.2 (corresponding to the last copy of [ − ,
1] in our identification of D ⊆ ∂M with [ − , d − ). Since d > d − > β : P → P corresponding to the generator of the braid group that interchanges the punctures p i and p i +1 . From Figure 7.3, and due to the fact that (the bird’s eye view of) f ℓ is the same in asmall neighbourhood of each puncture p k +1 , . . . , p k + ℓ , it is clear that the homotopy class of f ℓ ◦ β isthe same as that of f ℓ . Moreover, a homotopy connecting them may be chosen to be supported ina small neighbourhood of an arc connecting p i and p i +1 , so it does not affect the ends D × { k } and D × { k + ℓ } of P , which are therefore still mapped to the basepoint x ∈ X during the homotopy.This proves the claim when d > d = 2 we do not collapse the vertical direction, and in this case Figure 7.4(a) illustratesthe map f ℓ : P → X without any dimension reduction, and Figure 7.4(b) illustrates the effect ofthe point-pushing diffeomorphism β : P → P that interchanges the punctures p i and p i +1 . We willargue that 7 . ≃ . P → X sending the left, bottom and right edges of the rectangle to the basepoint x ∈ X corresponds, up to relative homotopy, to an ordered ℓ -tuple of elements of π ( X, x ). Recall that,when defining the stabilisation maps, we chose an element of CMap c, ∂ ( D ; X ) (see Figure 7.1(a)),which is, up to homotopy, a choice of element κ ∈ π ( X, x ) lying in one of the conjugacy classes c D ⊆ [ S , X ] = Conj( π ( X, x )), where c D is determined by c as explained in Definition 4.13.The map 7 . P → X therefore corresponds to the ordered ℓ -tuple ( κ, κ, . . . , κ ). Now the effectof the point-pushing diffeomorphism β : P → P , under this correspondence, is to interchange the i -th and ( i + 1)-st elements of an ordered tuple while conjugating one by the other; in symbols:( . . . , λ i , λ i +1 , . . . ) ( . . . , λ i λ i +1 λ − i , λ i , . . . ). But the ordered ℓ -tuple ( κ, κ, . . . , κ ) correspondingto 7.4(a) is clearly fixed under this action, so 7 . . ◦ β also corresponds to ( κ, κ, . . . , κ ),and therefore 7 . ≃ . Remark 7.4
The general setting of [Kra19] is for E -modules over E -algebras; the theoremstated in the next section (Theorem 8.7) in terms of the category C ( M ) is a rephrasing of this ina special case. If d >
3, the fact that the configuration-section spaces on M form an E -moduleover an E d − -algebra, hence in particular over an E -algebra, automatically implies that we havean extension to C ( M ). On the other hand, when d = 2, it is not tautological that we have anextension to C ( M ), although it is still true, by Proposition 7.3.28 k + ℓp k +1 p k +2 p i p i +1 p k + ℓ · · · · · · Figure 7.3
A bird’s eye view of the map f ℓ : P → X , where P = ( D × [ k, k + ℓ ]) r { p k +1 , . . . , p k + ℓ } .Grey indicates the constant map to the basepoint x ∈ X and the green regions are mapped to X asin Figure 7.1(b), depending in particular on our choices in Figure 7.1(a). The red arrows illustrate theeffect of the point-pushing diffeomorphism β : P → P . (a) k k + ℓp k +1 p k +2 p i p i +1 p k + ℓ · · · · · · (b) k k + ℓp k +1 p k +2 p i p i +1 p k + ℓ · · · · · · Figure 7.4 (a)
The map f ℓ : P → X when d = 2, with colour-coding as in Figure 7.3. (b) The map f ℓ ◦ β : P → X , where β is the point-pushing diffeomorphism interchanging the punctures p i and p i +1 .
8. Polynomiality and stability
In this section we complete the proof of our main homological stability result for configuration-section spaces. First, we show that the composition of (7.5) with the functor H i ( − ; K ) : Ho(Top) → Vect K → Ab is “of degree i ” for all i > K . Via a result of [Kra19] (recalled belowas Theorem 8.7), this implies twisted homological stability for configuration spaces with coefficientsin H i ((7.5); K ), which then implies the desired result by a spectral sequence comparison argument. Polynomial functors.
The category C ( M ) has a canonical endofunctor s : C ( M ) −→ C ( M )defined as follows. The maps σ k : π ( ˙ C k ( M )) → π ( ˙ C k +1 ( M )) induced by the stabilisation maps( cf . Definition 7.1) induce an endofunctor of Br( M ) given by k k + 1 on objects. Similarly, thestandard inclusions of braid groups B k → B k +1 induce an endofunctor of B given by k k + 1on objects. The endofunctor B → B is braided monoidal and the two endofunctors
B → B andBr( M ) → Br( M ) are compatible with the left-action of B on Br( M ), so they induce an endofunctorof C ( M ) = hB , Br( M ) i , which we denote by s . There is moreover a natural transformation ι : id C ( M ) −→ s given by the morphisms ι k = (1 , id k +1 ) : k → k + 1 of C ( M ) for k ∈ N . (Recall that a morphism a → b in C ( M ) is determined by an object c of B and a morphism a + c → b of Br( M ).) We notethat s ( ι k ) = (1 , v k ( τ )) = v k ( τ ) ◦ ι k +1 , where τ ∈ B is the standard generator and v k : B → π ( ˙ C k +2 ( M )) is the homomorphism defined in Definition 7.1.29 efinition 8.1 For a category C equipped with an endofunctor s : C → C and natural transfor-mation ι : id C → s , and an abelian category A , the degree of a functor T : C → A takes values in {− } ∪ N ∪ {∞} and is defined recursively as follows. The only functor T of degree − T = 0. Iffunctors of degree d have been defined, we say that T has degree d + 1 if and only if the naturaltransformation T ι : T → T s is split injective in the functor category Fun( C , A ) and the functor∆ T = coker( T ι : T → T s ) :
C → A has degree d . Once all functors of finite degree have been defined, all remaining functors are saidto have degree ∞ . Remark 8.2
For C = C ( M ), equipped with the endofunctor and natural transformation describedabove, this corresponds to the notion of “ split degree at
0” of [Kra19, Definition 4.6]. There arean analogous endofunctor s and natural transformation ι on the category B ♯ ( M ) (which commutewith the functor C ( M ) → B ♯ ( M ) from Summary 6.10), and when C = B ♯ ( M ) equipped with these,the definition above corresponds to the degree of [Pal18, Definition 3.1]. In this setting, the degreeof T has an alternative characterisation in terms of cross-effects of T (Definition 3.15 and Lemma3.16 of [Pal18]). See also [Pal] for a more general overview of notions of degree of a functor viarecursion (as above) or via cross-effects. Lemma 8.3
Let C be a category as in Definition 8.1, A an abelian category, T and T : C → A functors and A ∈ ob( A ) . Then we have • deg( T ⊕ T ) = max { deg( T ) , deg( T ) } , • deg( T ⊗ A ) deg( T ) , and, more generally, • deg( T ⊗ T ) deg( T ) + deg( T ) whenever deg( T ) and deg( T ) are non-negative,where we assume that ( A , ⊗ ) is an abelian monoidal category for the second and third points.Proof. This is a direct generalisation of [Pal18, Lemma 3.18], whose proof also generalises directly.
Definition 8.4
Let C be as in Definition 8.1. We say that a functor T : C −→
Ho(Top)has slope λ , for λ ∈ (0 , ∞ ), if for every field K and for each integer i >
0, the composite functor H i ( − ; K ) ◦ T : C −→
Vect K has degree λi in the sense of Definition 8.1.Our main homological stability result is the following. Suppose that we have chosen a bundleover a manifold ξ : E → M , a disc D ⊆ ∂M , a singularity condition c ⊆ Γ( η ( ξ )), etc, as describedin Notation 5.4. In particular, the singularity condition c determines a subset c D ⊆ [ S d − , X ],where X is the fibre of ξ over the basepoint of M , as explained in Definition 4.13, and thus asubset e c D ⊆ π d − ( X ), defined to be the preimage of c D under the canonical projection π d − ( X ) −→ π d − ( X ) /π ( X ) ∼ = [ S d − , X ] . Theorem 8.5
Suppose that the subset e c D ⊆ π d − ( X ) has size . Then the stabilisation maps CΓ c,Dk ( M ; ξ ) −→ CΓ c,Dk +1 ( M ; ξ ) induce isomorphisms on integral homology up to degree k − and surjections up to degree k − .With coefficients in a field, both of these ranges may be improved by one. In particular, this implies Theorem A of the introduction. The main technical input for this isthe following.
Proposition 8.6 If | e c D | = 1 , the functor (7.5) = g Mon c,D ( M, ξ ) has slope .
30e will also use part of Theorem D of [Kra19], which we recall in the following:
Theorem 8.7 ([Kra19, part of Theorem D])
Let ( M, D, b, ∗ ) be as in Proposition 7.2 and let G : C ( M ) −→ Ab be a functor to the category of abelian groups. If deg( G ) r , then the maps H i ( ˙ C k ( M ); G ( k )) −→ H i ( ˙ C k +1 ( M ); G ( k + 1)) , induced by the stabilisation maps (7.1) together with the functor G , are isomorphisms in the rangeof degrees i k − r − and surjections in the range of degrees i k − r .Proof of Theorem 8.5 (homological stability for configuration-section spaces). From the fibration se-quences (5.9) and the stabilisation maps (7.1), we have a map of fibration sequences of the form:Γ c,Dk ( M, ξ ) Γ c,Dk +1 ( M, ξ )˙CΓ c,Dk ( M ; ξ ) ˙CΓ c,Dk +1 ( M ; ξ )˙ C k ( M ) ˙ C k +1 ( M ) , (8.1)which has an associated map of Serre spectral sequences H p ( ˙ C k ( M ); H q (Γ c,Dk ( M, ξ ); K )) H p ( ˙ C k +1 ( M ); H q (Γ c,Dk +1 ( M, ξ ); K )) H p + q ( ˙CΓ c,Dk ( M ; ξ ); K ) H p + q ( ˙CΓ c,Dk +1 ( M ; ξ ); K ) ⇒ ⇒ (8.2)for any field K . By Proposition 8.6, the functor H q ( − ; K ) ◦ g Mon c,D ( M, ξ ) has degree q for each q >
0, and hence Theorem 8.7 implies that (8.2) is an isomorphism on E pages in the range ofbidegrees 2 p k − q −
2, and a surjection for 2 p k − q . In particular, it is an isomorphism fortotal degree p + q k − p + q k . By a spectral sequence comparisonargument (see [Zee57, Theorem 1] or [CDG13, Remarque 2.10]), the same statements hold alsoin the limit. Composing with the homotopy equivalences of Lemma 4.14, we conclude that thestabilisation mapsCΓ c,Dk ( M ; ξ ) ≃ ˙CΓ c,Dk ( M ; ξ ) −→ ˙CΓ c,Dk +1 ( M ; ξ ) ≃ CΓ c,Dk +1 ( M ; ξ ) (8.3)induce isomorphisms on K -homology up to degree k − k . Applyingthis for K = F p and using the maps of long exact sequences induced by the short exact sequencesof coefficients 0 → Z /p r → Z /p r +1 → Z /p →
0, we deduce the same statements for homology withcoefficients in Z ( p ∞ ), where Z ( p ∞ ) is the direct limit of Z /p → Z /p → Z /p → · · · . Then usingthe short exact sequence of coefficients0 → Z −→ Q −→ Q / Z = M p Z ( p ∞ ) → , we conclude that the stabilisation maps (8.3) induce isomorphisms on integral homology up todegree k − k − Proof of Proposition 8.6.
Let K be a field and i > F i = H i ( − ; K ) ◦ g Mon c,D ( M, ξ ) : C ( M ) −→ Vect K .
31n this notation, we need to show that deg( F i ) i , where deg( − ) is as in Definition 8.1.Recall that we have assumed that the subset e c D ⊆ π d − ( X ) has cardinality 1, and denote by Z the corresponding path-component of Ω d − X . Also write Y = Γ D ( M, ξ ) for the space of sectionsof ξ : E → M that restrict to the fixed section s D ( cf . Notation 5.4) on D ⊆ ∂M . There are naturalmaps e k : Y × Z k −→ Γ c,Dk ( M, ξ ) = g Mon c,D ( M, ξ )( k ) , (8.4)defined in Figure 8.1, such that the square Y × Z k Y × Z k +1 Γ c,Dk ( M, ξ ) Γ c,Dk +1 ( M, ξ ) e k e k +1 commutes up to homotopy, where the bottom horizontal map is the stabilisation map (namely thetop horizontal map of (8.1), which is also g Mon c,D ( M, ξ )( ι k )) and the top horizontal map is theobvious inclusion ( s, f , . . . , f k ) ( s, f , . . . , f k , ∗ ), where ∗ ∈ Z is any basepoint (exactly whichbasepoint does not matter since Z is path-connected). Moreover, the map (8.4) is a topologicalembedding, and it is not hard to define a deformation retraction of Γ c,Dk ( M, ξ ) onto its image –hence (8.4) is a homotopy equivalence.Now consider an automorphism α ∈ Aut C ( M ) ( k ) = π ( ˙ C k ( M )), which, via the endofunctor s of C ( M ), induces an automorphism s ( α ) = σ k ( α ) ∈ π ( ˙ C k +1 ( M )). (Recall that the notation σ k ( − )was introduced in Definition 7.1.) One may check that, under the identifications (8.4), we have anequality g Mon c,D ( M, ξ )( s ( α )) = g Mon c,D ( M, ξ )( α ) × id Z (8.5)in the group of homotopy automorphisms up to homotopy π (hAut( Y × Z k +1 )).Next, consider the morphism ι k : k → k + 1 of C ( M ). As observed earlier in this section, we havethe identity s ( ι k ) = v k ( τ ) ◦ ι k +1 , where τ ∈ B is the standard generator and the homomorphism v k : B → π ( ˙ C k +2 ( M )) is as in Definition 7.1. We also noted above that, under the identifications(8.4), the map g Mon c,D ( M, ξ )( ι k ) corresponds to the obvious inclusion of Y × Z k into Y × Z k +1 using the fixed basepoint ∗ of Z . Using this, one may check that, under the identifications (8.4),we have an equality g Mon c,D ( M, ξ )( s ( ι k )) = g Mon c,D ( M, ξ )( ι k ) × c Z (8.6)in the homotopy set π (Map( Y × Z k +1 , Y × Z k +2 )) = [ Y × Z k +1 , Y × Z k +2 ], where c Z : Z → Z is the identity map if d >
3, and if d = 2 it is the homotopy automorphism of Z ⊆ Ω X given byconjugating a given loop in the path-component Z of Ω X by the fixed loop ∗ ∈ Z .The morphisms of C ( M ) are generated by its automorphisms together with the morphisms ι k for k ∈ N , so the identifications (8.4), (8.5) and (8.6) imply that we have a natural isomorphism g Mon c,D ( M, ξ ) ◦ s ∼ = g Mon c,D ( M, ξ ) × C( c Z ) (8.7)of functors C ( M ) → Ho(Top), where, for an endomorphism f : A → A in Ho(Top), the functorC( f ) : C ( M ) → Ho(Top) sends each object to A , each automorphism to id A and each morphism ι k to f . Applying H i ( − ; K ) and using the Künneth theorem (including the naturality of the Künnethisomorphism), the decomposition (8.7) induces an isomorphism of functors F i ◦ s ∼ = i M j =0 F i − j ⊗ H j (C( c Z ); K ) , (8.8)such that the natural transformation F i ι : F i → F i ◦ s corresponds, under (8.8), to the inclusion ofthe j = 0 summand, where we are using the fact that Z is path-connected to identify H (C( c Z ); K )with the constant functor at K . 32 k M · · · D Figure 8.1
The map e k : Y × Z k −→ Γ c,Dk ( M, ξ ) from the proof of Proposition 8.6, where Z is a givenpath-component of Ω d − X = Map(( D d − , ∂D d − ) , ( X, { x } )) and Y = Γ D ( M, ξ ).Given inputs ( s, f , . . . , f k ), the section e k ( s, f , . . . , f k ) of ˆ ξ over ˆ M k r z k is given by s in the yellowregion (namely M ) and by the maps f , . . . , f k on the red arcs (representing embedded ( d − D × [0 , k ]). Recall that, over ( D × [0 , k ]) r z k , the bundleis trivial with fibre X , so we may think of sections as maps ( D × [0 , k ]) r z k → X . The bottom face of D × [0 , k ] is sent to the basepoint x of X . We then extend the map in the red regions by defining it tobe constant along radii centred at the punctures z k , and we extend it in the green regions by definingit to be constant in the vertical direction. Using (8.8) and the fact that F i ι corresponds to the inclusion of the j = 0 summand under thisidentification, we deduce (i) that F i ι is split-injective in the functor category Fun( C ( M ) , Vect K )and (ii) that we have an isomorphism of functors∆ F i ∼ = i M j =1 F i − j ⊗ H j (C( c Z ); K ) . (8.9)The fact that c Z is a homotopy automorphism, i.e., invertible in Ho(Top), implies that the functor H j (C( c Z ); K ) sends each ι k to an isomorphism, which implies, by definition, thatdeg( H j (C( c Z ); K )) . (8.10)We now prove that deg( F i ) i by induction on i >
0. For i = 0, the identification (8.9) saysthat ∆ F = 0. Together with the fact that F ι is split-injective, this implies that deg( F ) i >
1, using the identification (8.9), Lemma 8.3, the fact (8.10) and the inductive hypothesis,we see that deg(∆ F i ) = deg (cid:16) i M j =1 F i − j ⊗ H j (C( c Z ); K ) (cid:17) max ij =1 { deg( F i − j ) } i − . Together with the fact that F i ι is split-injective, this implies by definition that deg( F i ) i . Remark 8.8 ( The hypothesis of Proposition 8.6. ) The assumption | e c D | = 1 (where e c D ⊆ π d − ( X )is the subset induced by the “singularity condition” c ⊆ Γ( η ( ξ ))) of Proposition 8.6 means thatthe corresponding subspace Z ⊆ Ω d − X is a single path-component, rather than a union of severalpath-components. The path-connectedness of Z is used in a key way for the identification (8.9)of ∆ F i in terms of F i − , F i − , . . . . If Z were disconnected, we would have H (C( c Z ); K ) = K π ( Z ) and, denoting by Z the path-component of Z containing its basepoint, the identification (8.9)would become ∆ F i ∼ = F i ⊗ K π ( Z ) r { Z } ⊕ i M j =1 F i − j ⊗ H j (C( c Z ); K ) , which would break the inductive argument, since this decomposition involves F i itself. Remark 8.9 ( Naturality of the Künneth theorem. ) In order to transform the identification (8.7)of functors C ( M ) → Ho(Top) into the identification (8.8) of functors C ( M ) → Vect K , we used theKünneth theorem for field coefficients, which does not involve any Tor terms. If we had worked33nstead with Z coefficients, we would have obtained a decomposition similar to (8.8) – at the levelof objects – including also some Tor terms. The appearance of the Tor terms themselves is noproblem, since one may prove an analogue of the second and third points of Lemma 8.3 for Tor( − )instead of ⊗ , so they behave as desired with respect to degree. The problem instead is that theKünneth short exact sequences are split, but not naturally split (unless – of course – the Tor termsvanish). Thus, we would not have been able to obtain a decomposition of functors analogous to(8.8), since the naturality of the Künneth theorem, in the case where the Tor terms vanish, waskey to upgrading (8.8) from an isomorphism at the level of objects to an isomorphism of functors.See also Remark 4.4 of [Pal18] for a similar comment about the (non-)naturality of the splitting ofthe Künneth short exact sequence. Remark 8.10 ( Improving the range of stability. ) Our main homological stability result states that,on homology with field coefficients, the stabilisation maps (8.3) induce isomorphisms up to degree k − k . This then implies the analogous statements for homologywith integral coefficients – but only in a range of degrees that is smaller by one, i.e., isomorphismsup to degree k − k − H ∗ (Ω d − X ; Z ) and H ∗ (Γ D ( M, ξ ); Z ) aretorsion-free in all degrees. Under this assumption, one may run the proof of Proposition 8.6 using Z in place of K , since the torsion-freeness assumption implies that one may apply the Künneththeorem for Z coefficients without the appearance of any Tor terms (see Remark 8.9 above for theimportance of the vanishing of the Tor terms, and see Remark 4.4 of [Pal18] for the analogousremark in a similar setting). We could then also run the proof of Theorem A directly with Z coefficients, without any need to pass from field coefficients to Z coefficients at the end, which iswhere we lose 1 from the range of stability.Note that, in the case where M = D and X = BG , this hypothesis amounts to requiring that H ∗ ( G ; Z ) is torsion-free in all degrees. (In particular, this means that the abelianisation of G mustbe torsion-free, which is not the case for G finite cyclic or G = A ⋊ Z /
2, where A is abelian and Z /
9. Extension to B ♯ ( M ) and split-injectivity In this brief section we prove a split-injectivity result for the homology of configuration-mappingspaces, under certain conditions on the underlying manifold M . Let us fix a connected manifold M , an embedded disc D ⊆ ∂M , a based space X and a subset c ⊆ [ S d − , X ]. Recall that thestabilisation maps CMap c,Dk ( M ; X ) → CMap c,Dk +1 ( M ; X ) fit into a map of fibre sequences of theform (8.1), inducing a map of their associated Serre spectral sequences: H p ( ˙ C k ( M ); H q (Map c,D ( ˆ M k r z k , X ); Z )) H p ( ˙ C k +1 ( M ); H q (Map c,D ( ˆ M k +1 r z k +1 , X ); Z )) H p + q (CMap c,Dk ( M ; X ); Z ) H p + q (CMap c,Dk +1 ( M ; X ); Z ) ⇒ ⇒ (9.1)(Note that above we take coefficients in Z rather than in a field, as was the case in (8.2).)The main result of this section is the following. Theorem 9.1
Let M have dimension at least and assume that either π ( M ) = 0 or the handle-dimension of M is at most dim( M ) − . Then the map of Serre spectral sequences (9.1) above issplit-injective on each entry of the E pages. Remark 9.2
Note that, since M is connected and has non-empty boundary, its handle-dimensionis at most dim( M ) −
1. Thus the only manifolds excluded by the additional hypotheses of Theorem9.1 are (i) surfaces and (ii) non-simply-connected manifolds of maximal handle dimension.Recall from §7 that we have a monodromy functor C ( M ) −→ Ho(Top) (9.2)34ncoding both the stabilisation maps and the monodromy action of π ( ˙ C k ( M ) , z k ) on the fibreMap c,D ( ˆ M k r z k , X ) of the fibration CMap c,Dk ( M ; X ) → ˙ C k ( M ) for each k (see Proposition 7.3).Recall also from §6 that we have a functor of braid categories C ( M ) −→ B ♯ ( M ) (9.3)(see Summary 6.10). Proposition 9.3
Let dim( M ) > and assume that either π ( M ) = 0 or the handle-dimension of M is at most dim( M ) − . Then the monodromy functor (9.2) extends to B ♯ ( M ) .Proof. The fibre Map c,D ( ˆ M k r z k , X ) of the fibration CMap c,Dk ( M ; X ) → ˙ C k ( M ) decomposes upto homotopy as Map ∗ ( M, X ) × (Ω d − c X ) k , where Ω d − c X denotes the union of path-componentsof the loopspace Ω d − X corresponding to the subset c ⊆ [ S d − , X ]. Since M has dimension atleast 3, the fundamental group π ( ˙ C k ( M )) decomposes as π ( M ) k ⋊ Σ k [Til16, Lemma 4.1]. Underthese identifications, and under the given assumptions on M , [PT, Corollary 7.2] implies that themonodromy action is given, for α i ∈ π ( M ), σ ∈ Σ k , f ∈ Map ∗ ( M, X ) and g i ∈ Ω d − c X , by thefollowing formula: ( α , . . . , α k ; σ ) · ( f, g , . . . , g k ) = ( f, ¯ g , . . . , ¯ g k ) , (9.4)where ¯ g i = f ∗ ( α i ) .g σ ( i ) . sgn( α i ). The element f ∗ ( α i ) ∈ π ( X ) acts on the point g σ ( i ) ∈ Ω d − c X viathe usual action-up-to-homotopy of π ( X ) on the iterated loopspaces of X . The sign sgn( α i ) ∈{± } depends on whether or not the loop α i lifts to a loop in the orientation double cover of M ,and − g σ ( i ) ∈ Ω d − c X via a reflection of S d − .The formula (9.4) may be visualised as follows ( cf . Figure 7.1 of [PT]). The element ( α , . . . , α k ; σ )of π ( M ) k ⋊ Σ k is a k -strand braid, where the strands may pass through each other, and where eachstrand is labelled by an element of π ( M ). The element ( f, g , . . . , g k ) is acted upon by pushingthe i -th element g i backwards along the i -th strand of the braid while acting on it by f ∗ ( ) of thelabel of that strand as well as the involution sgn( ) of the label of the strand.This description of the monodromy action immediately suggests how to extend the monodromyfunctor (9.2) to B ♯ ( M ), since the morphisms of B ♯ ( M ) have a similar combinatorial descriptionwhen M has dimension at least 3. Namely, a morphism m → n in B ♯ ( M ) may be viewed as a braid(whose strands may cross each other) from a subset of { , . . . , m } to a subset of { , . . . , n } , whereeach strand is labelled by an element of π ( M ) ( cf . [Kra19, Remark 5.10]). In fact, we will definean extension of the monodromy functor to B ♯ ( M ) op , but this will finish the proof since B ♯ ( M ) iscanonically isomorphic to its opposite category.To define the extension of the monodromy functor to B ♯ ( M ) op , we describe how a morphism m → n of B ♯ ( M ) acts on ( f, g , . . . , g n ) for f ∈ Map ∗ ( M, X ) and g i ∈ Ω d − c X . First, fix a basepoint ∗ ∈ Ω d − c X . If there is a strand ending at position i , we push the i -th element g i backwards alongthis strand, acting on it by f ∗ ( ) and sgn( ) of the strand’s label. We then fill in any blanks inthe resulting partial m -tuple of elements of Ω d − c X with the basepoint ∗ .In formulas, this is written as follows. A morphism m → n of B ♯ ( M ) is given by a partially-defined injective function σ from a subset of { , . . . , m } to a subset of { , . . . , n } and an element α i ∈ π ( M ) for each i ∈ dom( σ ). This morphism acts by( f, g , . . . , g n ) ( f, ¯ g , . . . , ¯ g m ) , where ¯ g i = f ∗ ( α i ) .g σ ( i ) . sgn( α i ) if i ∈ dom( σ ) and ¯ g i = ∗ otherwise. Remark 9.4
The assumption that either π ( M ) = 0 or the handle-dimension of M is at mostdim( M ) − cf . [PT, Remark 7.3]). Inparticular, there is a non-trivial action on the component f ∈ Map ∗ ( M, X ) of the tuple.
Proof of Theorem 9.1.
By [Pal18, Theorem A and Remark 1.3], for any abelian group-valued func-tor T : B ♯ ( M ) → Ab, the induced stabilisation maps on T -twisted homology H ∗ ( C k ( M ); T ( k )) −→ H ∗ ( C k +1 ( M ); T ( k + 1))35re split-injective in all degrees. This implies the statement of Theorem 9.1 by applying it to theextended monodromy functor B ♯ ( M ) → Ho(Top) of Proposition 9.3 composed with the homologyfunctor H q ( − ; Z ) : Ho(Top) → Ab.
Remark 9.5
Split-injectivity of a map of Serre spectral sequences on each entry of the E pagesdoes not, however, automatically imply split-injectivity on any of the further pages, since we donot know whether the splittings commute with the differentials.
10. Group-completion and stable homology
The identification of the stable homology follows a relatively well-established path. For con-figuration spaces of manifolds with labels in a fixed space this is semi-classical [May72] [McD75],[Seg73], [Böd87], [Sal01]. In the case of configuration-mapping spaces this has been done in [EVW].Here we also generalise the arguments to configuration-section spaces. In the classical setting an important role is played by the scanning map from the configurationspaces to certain mapping or section spaces. The part that differs from one situation to the other,is the identification of the ‘local data’, i.e. what is seen in a small disk modulo its boundary. Theremaining arguments are nearly formal and – somewhat surprisingly – remain so also in our casewhere the data associated to a configuration depends on global information rather than the localinformation in the neighborhood of the configuration points themselves.In outline, we will first consider configuration-section spaces of any number of particles wherethe particles can ‘vanish’ at the boundary or in some other subspace of the underlying manifold M and establish a quasi-fibration sequence for them. Then we identify the ‘local data’. Usingan induction on the handlebody decomposition of M we can then ‘integrate’ the local data tothe whole manifold. Finally, using the group-completion theorem we relate this to the finiteconfiguration-section spaces. Let M be a compact,connected manifold of dimension d and N ⊂ M be a co-dimension zero closed submanifold suchthat M r N is open. As before, let ξ : E → M be a fibre bundle and let S be a subspace of M .We assume that ξ has a canonical section s defined over all of M and the sections that we willconsider agree with s on S . (In previous sections, we imposed “boundary conditions” on subsets D ⊆ ∂M ; the definition generalises straightforwardly to arbitrary subsets S ⊆ M .) Let c ⊆ Γ( η ( ξ ))be a singularity condition ( cf . Definition 3.14). Definition 10.1
Let CΓ c,S ( M, N ; ξ ) be the quotient of CΓ c,S ( M ; ξ ) = ` k CΓ c,Sk ( M ; ξ ) by theequivalence relation ( z, s ) ∼ ( z ′ , s ′ ) whenever z ∩ M r N = z ′ ∩ M r N and s | M r N = s ′ | M r N . So ( z, s ) and ( z ′ , s ′ ) agree outside the interior of N .Thus points can disappear as they move into the submanifold N . It will be convenient to alsoallow points to disappear on the boundary of M . In that case we will writeCΓ c,S ( M, ∂M ; ξ )were we interpret this to mean that ∂M is thickened to a small collar b : [ − ǫ, × ∂M ⊂ M to fitthe above definition. Proposition 10.2
Let M ′ be a compact co-dimension 0 submanifold of M , such that ( M ′ , N ∩ M ′ ) is connected. Then the canonical quotient map CΓ c,S ( M, N ; ξ ) π −→ CΓ c,S ( M, M ′ ∪ N ; ξ ) We also correct the arguments in [EVW] in several places: this includes the statement of the quasi-fibrationsequence in Proposition 10.2; the proof of Theorem 10.12 and in particular the correction of diagram (10.2). s a quasi-fibration with fibre CΓ c ′ ,S ′ ∪ ∂ ′ ( M ′ , N ∩ M ′ ; ξ ′ ); here S ′ = S ∩ M ′ ; ∂ ′ = ∂M ′ ∩ ∂K where K = ( M r M ′ ) is the closure of the complement of M ′ in M ; ξ ′ := ξ | M ′ and c ′ ⊆ Γ( η ( ξ ′ )) are the restrictions of ξ and c to M ′ .Proof. We need to identify the fibre π − (( z, s )) where ( z, s ) ∈ CΓ c,S ( M, N ; ξ ) is a representative of( z, s ) ∈ CΓ c,S ( M, M ′ ∪ N ; ξ ); here s ∈ Γ( M r z, ξ ) is a section satisfying the restrictions imposed by c and S . By definition, this fibre consists of all those ( z ′ , s ′ ) that agree with ( z, s ) when restrictedto K . Thus z ′ r ( K ∩ z ′ ) can be an arbitrary subset of the interior of M ′ and s ′ is a section of ξ ′ which agrees with s on the interface ∂ ′ = ∂M ′ ∩ ∂K . Taking the relation introduced by N intoaccount, we see that π − (( z, s )) = { ( z ′ , s ′ ) ∈ CΓ c ′ ,S ∩ M ′ ( M ′ , N ∩ M ′ ; ξ ′ ) | s ′ | ∂ ′ = s | ∂ ′ } . Thus the fibre is independent of z and in case that s | ∂ ′ = s | ∂ ′ it is precisely given by the fibre ofthe proposition.To show π is a quasi-fibration in the sense of Dold-Thom [DT58] we use the natural filtrationsby number of points in the configuration of the base space: B k := { ( z, s ) ∈ CΓ c,S ( M, M ′ ∪ N ; ξ ) with | z | k } . Step 1:
We will prove that π restricted to B k r B k − is a fibration. For this note that the map B k r B k − → CΓ c ′′ ,S ∩ Kk ( K ; ξ ′′ ) sending ( z, s ) to ( z ∩ ( M r M ′ ) , s | K ) is a homeomorphism onto itsimage of points ( z ′′ , s ′′ ) where s ′′ extends to a section over all of M while restricting to s over S .Let Γ( E | ∂ ′ ) denote the space of sections of E restricted to ∂ ′ and consider the homeomorphism φ : ( B k r B k − ) × Γ( E | ∂M ′ ) CΓ c ′ ,S ∩ M ′ ( M ′ , N ∩ M ′ ; ξ ) −→ π − ( B k r B k − )defined by φ (( z ′′ , s ′′ ) , ( z ′ , s ′ )) = ( z ′′ ∪ z ′ , s ′′ ∪ ∂ ′ s ′ ) . Here the fibre product is taken using the natural maps that take (a configuration and) a sectionto its restriction over ∂ ′ . As these are fibrations, the projection of the source of φ to B k r B k − isa fibration, thus proving (1). Step 2:
Let U be a tubular neighbourhood of M ′ in M and let J t be an isotopy of M such that J = id M and J ( U ) ⊂ M ′ while for all t ∈ [0 , J t ( S ) ⊂ S and J t ( N ∩ U ) ⊂ N. We can now define U k ⊂ B k with at most k − M r U . So in particular B k − ⊂ U k .Define a homotopy H t by the formula H t ( z, s ) = ( J t ( z ) , s ◦ J − t ) . By definition this commutes with π and hence is a fiberwise homotopy. We want to show that H : π − ( z, s ) → π − ( H ( z, s )) induces a homotopy equivalence on fibres over points ( z, s ) ∈ U k .But this follows as ( M ′ , N ∩ M ′ ) is connected and both fibres can be identified withCΓ c ′ ,S ′ ( M ′ , N ∩ M ′ ; ξ ′ ) . Together
Step 1 and
Step 2 prove that π satisfies the Dold-Thom criteria for a quasi-fibration. Up to homeomorphism we may assume that none of the k points are in [ − ǫ, × ∂M ′ ⊂ K . We will now identify the ‘local data’ for the scanning process. By definition,‘local data’ is the restriction of the configuration section spaces to a disk relative to its boundary.In other words, it is the configuration section space on the disk where points can disappear at theboundary. As the fibre bundle ξ is locally trivial, the ‘local data’ in the case of configuration-sectionspaces does not see the global twisting of ξ . We can thus assume we have a trivial bundle withfixed, based fibre X and a fixed subset c ⊆ π d − ( X ) of monodromies, and thus reduce to the caseof configuration-mapping spaces.Let ( X, ∗ ) be a pointed space and let c ⊆ π d − ( X ) be a subset of monodromies. Definition 10.3
Let A d ( X, c ) be the pushout of the diagram D d × Map c ( S d − , X ) i ←− S d − × Map c ( S d − , X ) ev −→ X, where the left arrow is the inclusion and the right arrow the evaluation map ev ( t, f ) := f ( t ). Let A ′ d ( X, c ) be the homotopy fibre of the natural inclusion X → A d ( X, c ). When c = π d − ( X ) wedrop c from the notation and write A d ( X ) and A ′ d ( X ). Lemma 10.4 A d ( X, c ) ≃ CMap c ( D d , ∂D d ; X ) . Proof.
We expand on the arguments given in [EVW].Let C denote the subspace of CMap c ( D d , ∂D d ; X ) of pairs ( z, f ) where z ∩ int( D d ) has sizeeither zero or one. By radial expansion, we have a deformation retractionCMap c ( D d , ∂D d ; X ) ≃ C . The radial expansion varies continuously with the configuration z : If z ∈ z is the closest pointto zero, and z ∈ z the second most close (it could be as close as z ) then the radial expansionproceeds at rate 1 / | z | .Note that C = C ∪ C , where • C is the subspace of C of pairs ( z, f ) where z ⊆ N ( ∂ ), • C is the subspace of C of pairs ( z, f ) where z ⊆ int( D d ) has cardinality one,and hence • C ∩ C is the subspace of C of pairs ( z, f ) where z ⊆ N ( ∂ ) ∩ int( D d ) has cardinality one.Note that the inclusion C ∩ C ֒ → C is a (closed) cofibration, since we may exhibit ( C , C ∩ C )as an NDR-pair by using a closed 2 ǫ -neighbourhood of ∂D d . Hence we haveCMap c ( D d , ∂D d ; X ) ≃ C = C ∪ C = Pushout( C ← ֓ C ∩ C ֒ → C ) ≃ hPushout( C ← ֓ C ∩ C ֒ → C ) . It therefore suffices to identify the two inclusion maps C ∩ C ֒ → C and C ∩ C ֒ → C up tohomotopy with the two maps in the diagram of Definition 10.3, since A d ( X, c ) is the pushout ofthis diagram, and thus also the homotopy pushout, since the map i of the diagram is a cofibration.There are homotopy equivalences C g −−→ D d × Map c ( S d − , X ) C ∩ C g −−−→ S d − × Map c ( S d − , X ) C g −−→ X given, respectively, by( z, f ) ( z, f | S d − ) ( z, f ) (cid:16) − z | z | , f | S d − (cid:17) ( z, f ) f (0) . Strictly speaking, we extend both D d and ∂D d by a small collar [1 , ǫ ] × ∂D d . g ◦ inclusion ≃ inclusion ◦ g : C ∩ C −→ D d × Map c ( S d − , X ) , using the straight line in D d between a given point z ∈ N ( ∂ ) ∩ int( D d ) and − z | z | . This identifiesthe inclusion C ∩ C ֒ → C with the inclusion map i of Definition 10.3.Similarly, we may define a homotopy g ◦ inclusion ≃ ev ◦ g : C ∩ C −→ X, by ( z, f ) f ( − tz | z | ) for t ∈ [0 , z ∈ N ( ∂ ) ∩ int( D d ), the linesegment in D d between 0 and − z | z | does not pass through z , and hence f is defined on all of thisline segment. (This is the reason for defining the homotopy equivalence g with the negative signin the formula above; without the negative sign, we would be forced to consider the line segmentbetween 0 and z | z | instead, which passes through z , meaning that f is not defined on the whole linesegment.) This identifies the inclusion C ∩ C ֒ → C with the map ev of Definition 10.3.We will now identify multiple deloopings of configuration-mapping spaces associated to D d . Forthis it is easier to work with cubes rather than with disks. Thus we fix an identification of the d -disk with the d -cube via a homeomorphism that takes the southern hemisphere of the boundaryof the cube less one of its faces: D d ≡ I d and 12 ∂ ≡ ∂I d r ( I d − × { } ); (10.1)here I = [0 , c,Sk ( D d ; X ) , k >
0, is an E d -algebra when S = ∂ and an E d − -algebra when S = ∂ . Thus these two spaces have d and respectively d − X . As in the case of configuration spaces with labelsand other similar cases, it turns out that taking the classifying space with respect to multiplicationcorresponding to one such pair of opposite faces is equivalent to allowing the configurations tovanish on those faces which we will now make precise. We introduce the notation B k (CMap c,S ( D d ; X )) := CMap c,S ( I d , ∂I k × I d − k ; X ) . Lemma 10.5
There are homotopy equivalences(i) s : B (CMap c,∂ ( D d ; X )) ≃ −→ B (CMap c,∂ ( D d ; X )) ;(ii) s : B (CMap c, ∂ ( D d ; X )) ≃ −→ B (CMap c, ∂ ( D d ; X )) .Proof. For c = π d − ( X ), this is Lemma 3.3.1 of [EVW]. The proof follows standard arguments,compare [May72], [McD75], [Sal01], and automatically extends to an arbitrary set c of mon-odromies. Lemma 10.6
There are homotopy equivalences(i) s k : B k − (CMap c,∂ ( D d ; X )) ≃ −→ Ω B k (CMap c,∂ ( D d ; X )) for < k d ;(ii) s k : B k − (CMap c, ∂ ( D d ; X )) ≃ −→ Ω B k (CMap c, ∂ ( D d ; X )) for < k < d .Proof. For c = π d − ( X ), this is Lemma 3.5.1 of [EVW]. The proof follows standard arguments,compare [May72], [McD75], [Sal01], and automatically extends to an arbitrary set c of mon-odromies. 39 emma 10.7 There are homotopy equivalences A d ( X, c ) ≃ B d (CMap c,∂ ( D d ; X )) and A ′ d ( X, c ) ≃ B d − (CMap c, ∂ ( D d ; X )) . Proof.
For c = π d − ( X ), this is Lemma 3.5.2 of [EVW] and the proof generalises for an arbitraryset c of monodromies. Indeed, the first homotopy equivalence holds by Lemma 10.4, the definitionof B k , and because CMap c,∂ ( I d , ∂I d ; X ) ≃ CMap c ( I d , ∂I d ; X ) as the restriction on the maps tobe constant on the boundary imposes no additional restriction in the quotient configuration spacewhere all sections are identified that agree on the complement of a collar of the boundary. Corollary 10.8
There are homotopy equivalences(i) Ω B CMap c,∂ ( D d ; X ) ≃ Ω d A ( X, d ) for d > ;(ii) Ω B CMap c, ∂ ( D d ; X ) ≃ Ω d − A ′ d ( X, c ) ≃ Map ∗ (( D d , S d − ); ( A d ( X, c ); X )) for d > .Proof. To prove part ( ii ), note that there is a string of homotopy equivalencesΩ B CMap c, ∂ ( D d ; X ) ≃ Ω B (CMap c, ∂ ( D d ; X )) ≃ Ω d − B d − (CMap c, ∂ ( D d ; X )) ≃ Ω d − A ′ d ( X, c ) . The first and second homotopy equivalence follow from Lemma 10.5 and repeated application ofLemma 10.6. The last homotopy equivalence follows from Lemma 10.7. The second homotopyequivalence in part ( ii ) follows as by definition A ′ ( X, c ) is the homotopy fibre of the canonicalinclusion of X into A ( X, c ). An entirely analogous argument proves part ( i ). As in the classical case for configurationspaces with labels [McD75], the configuration-section spaces are describing section spaces of certainbundles that depend on the tangent bundle of the underlying manifold. Thus we need the followingfiberwise generalisation of A d ( X, c ).As before, let M be compact and connected, and fix a metric on M . Denote by D d M and S d − M the associated disk and sphere bundles of the tangent bundle T M . Definition 10.9
Let E d ( ξ, c ) be the fibre-wise pushout of the diagram D d M × M Map cM ( S d − M, ξ ) i ←− S d − M × M Map cM ( S d − M, ξ ) ev −→ ξ, where Map M denotes the space of fiberwise maps between bundles over M and ev is the fiberwiseevaluation. Thus E d ( ξ, c ) is a fibre bundle over M with fibres A d ( X, c ) for X a typical fibre of ξ .Write E ′ d ( ξ, c ) for the fibrewise homotopy fibre of the map of fibre bundles ξ → E d ( ξ, c ) over M . Example 10.10 If ξ is the trivial fibration with fibre X and M is parallelisable then E d ( ξ, c ) isthe trivial fibration with fibre A d ( X, c ), and E ′ d ( ξ, c ) is the trivial fibration with fibre A ′ d ( X, c ), thehomotopy fibre of the natural inclusion X → A d ( X, c ). Example 10.11
One reason to consider non-trivial bundles ξ over M is because it allows us to‘untwist’ the tangent bundle. Thus if ξ is the sphere bundle of the cotangent bundle T ∗ M then E d ( ξ, c ) is the trivial fibration with fibre A d ( S d − , c ).Let M be pointed and have non-empty boundary and let L ( ∂M be a ( d − ∂M . WriteΓ S (( M, L ); E d ( ξ, c ))for the sections of E d ( ξ, c ) → M which restrict on {∗} ∪ S to s and on ∂M r L take values in ξ | ∂M ⊂ E d ( ξ, c ) | ∂M . 40 heorem 10.12 There is a weak homotopy equivalence CΓ c, ∗ ( M, L ; ξ ) −→ Γ((
M, L ); E d ( ξ, c )) . Proof.
One proceeds by induction on a handle decomposition of M using the quasi-fibration se-quence of Proposition 10.2.As M is of dimension d and has boundary, we can choose a handle decomposition of M thatonly contains handles of index k < d . In the initial stage of our induction, the case when M = D d ,both spaces are contractible: Since M retracts via isotopies into (a collar neighborhood of) L , allconfigurations can be contracted to the empty configuration and the maps in the fibre are pointedmaps from the disk D d to X . Similarly, the section space consist of pointed maps to X .Now assume that M is obtained from M ′ by attaching a k -handle D k × D d − k for 0 < k < d along S k − × D d − k to the boundary of M ′ . As before in (10.1) we identify the d -disk with the d -cube. Then M = M ′ ⊔ ∂ ′ I d where ∂ ′ = ∂I k × I d − k . We may assume that L ( ∂M ′ does not intersect ∂ ′ . Consider the following diagram:CΓ c ′ , ∗∪ ∂ ′ ( M ′ , L ; ξ ′ ) −−−−→ CΓ c, ∗ ( M, L ; ξ ) −−−−→ CΓ c, ∗ ( M, L ∪ M ′ ; ξ ′′ ) y y y Γ ∂ ′ (( M ′ , L ); E d ( ξ ′ , c ′ )) −−−−→ Γ((
M, L ); E d ( ξ, c )) −−−−→ Γ(( I d , ∂ ′ ); E d ( ξ ′′ , c ′′ )) [ ξ ] . (10.2)Here ξ ′ and ξ ′′ denote the bundle ξ restricted to appropriate submanifolds of M , and similarly c ′ and c ′′ are restrictions of c . The upper row is a quasi-fibration by Proposition 10.2. Note that( M ′ , L ) is indeed connected.In the lower row of diagram (10.2), the subscript [ ξ ] on the right denotes the subspace of those s ′′ ∈ Γ(( I d , ∂ ′ ); E d ( ξ ′′ , c ′′ ))that are restrictions of sections in Γ(( M, L ); E d ( ξ, c )). This selects entire connected components ofthe section space determined by the homotopy class in π (Γ( ∂ ′ ; E d ( ξ ′′ , c ′′ )) defined by the restrictionof s ′′ to ∂ ′ ≃ S k − . In this case however, since s ′′ | ∂ ′ also has to extend over I d , it must benullhomotopic in the first place as a map to A ( X, d ) and can thus be extended to a section over M . Thus Γ(( I d , ∂ ′ ); E d ( ξ ′′ , c )) [ ξ ] = Γ(( I d , ∂ ′ ); E d ( ξ ′′ , c )) . If s | I d denotes the base point in Γ(( I d , ∂ ′ ); E d ( ξ ′′ , c )), then the fibre of the bottom right restrictionmap is simply given by those sections defined on M ′ that agree with s on ∂ ′ . This describes thesection space on the left if we also remember the restriction imposed by L . Thus, also the lowerrow of diagram (10.2) is a fibration sequence.The vertical maps of diagram (10.2) are the scanning maps and the diagram commutes. Considerthe right down arrow. By Lemmas 10.6 and 10.7 we haveΓ(( I d , ∂ ′ ); E d ( ξ ′′ , c ′′ )) ≃ Map ∗ (( I d , I k × ∂I d − k ); ( A d ( X, c ) , X )) ≃ Ω d − k − A ′ ( X, c ) ≃ B k (CMap c, ∗ ( I d , ∂I k × I d − k ; X )) ≃ CΓ c ′′ , ∗ ( I d , ∂ ′ ; ξ ′′ ) . Restricting both sets to those components with sections s ′′ that can be extended to all of M andnoting CΓ c ′′ , ∗ ( I d , ∂ ′ ; ξ ′′ ) [ ξ ] ≃ CΓ c, ∗ ( M, L ∪ M ′ ; ξ )shows that the right down arrow of diagram (10.2) is a weak homotopy equivalence.41ow consider the left down arrow of diagram (10.2). It is also the left down arrow of the followingcommutative diagram:CΓ c ′ , ∗∪ ∂ ′ ( M ′ , L ; ξ ′ ) −−−−→ CΓ c ′ , ∗ ( M ′ , L ∪ ∂ ′ ; ξ ′ ) | ∂ ′ −−−−→ Γ( ∂ ′ ; ξ ∂ ′ ) [ ξ ′ ] y ≃ y = y Γ ∂ ′ (( M ′ , L ); E d ( ξ ′ , c ′ )) −−−−→ Γ(( M ′ , L ∪ ∂ ′ ); E d ( ξ ′ , c ′ )) | ∂ ′ −−−−→ Γ( ∂ ′ ; ξ ∂ ′ ) [ ξ ′ ] . The space on the right is the space of based sections of ξ | ∂ ′ = ξ ′ | ∂ ′ that can be extended to sectionsof ξ ′ on M ′ r z for some configuration z ′ in M ′ or, equivalently, to sections of E d ( ξ ′ , c ′ ). Thisis exactly the image of the maps | ∂ ′ that restricts the sections s ′ to ∂ ′ and have as fibers (overthe constant map) the spaces on the left. The middle arrow is a weak homotopy equivalence byinduction hypothesis, and hence so is the left down arrow of this diagram and of diagram (10.2).Finally, invoking the Five Lemma, we see that also the middle arrow of diagram (10.2) is a weakhomotopy equivalence. We now relate the homology of thespaces in Theorem 10.12 to the homology of the configuration section spaces for finite configura-tions.We first recall the group completion theorem from [Qui94] [MS76].
Theorem 10.13
Let A be a well-pointed topological monoid. The canonical map of a monoid intoits (derived) group completion induces an isomorphism H ∗ ( A )[ π A − ] ≃ H ∗ (Ω B A ) whenever the localisation can be constructed by left fractions, and in particular whenever π ( A ) isin the centre of H ∗ ( A ) . Let π ( A ) be generated by s , . . . , s n and define s := s . . . s n . Then H ∗ ( A )[ π ( A ) − ] ≃ H ∗ ( A )[ s − ] ≃ H ∗ ( A ∞ )where A ∞ := Tel( A s −→ A s −→ . . . ) is the telescope (or homotopy colimit) on left multiplication bya representative of s in A . One can construct a comparison map α : A ∞ → Ω B A following [MS76]:Consider the map p : A ∞ × A E A → B A . The source space is the telescope of A × A E A ≃ ∗ , andhence is contractible itself. Under the conditions of the theorem, McDuff and Segal show that themap p is a homology fibration and the canonical map from the fibre to the homotopy fibre α : A ∞ −→ Ω B A is hence an H ∗ -isomorphism. If A is homotopy commutative [Ran13a] [MP15], or satisfies a some-what weaker condition [Gri], then the map p is a homology fibration for all abelian coefficients andthe map α is therefore acyclic. In particular, the fundamental groups of all components of A ∞ areperfect, and α induces a weak homotopy equivalence on the plus construction. Example 10.14
Consider the monoids A (cid:3) := CMap c,∂ ( D d ; X ) and A ⊔ := CMap c, ∂ ( D d ; X ) . For d > A (cid:3) , and for d > A ⊔ are homotopy commutative by Proposition4.10 and hence satisfy the conditions of the group completion theorem. Assuming π is finitelygenerated, we have that α is acyclic and, by Corollary 10.8, for any (abelian) local coefficientsystem L the following are isomorphisms(i) H ∗ ( A (cid:3) , ∞ ; L ) ≃ H ∗ (Ω d A ( X, d ); L ) for d > H ∗ ( A ⊔ , ∞ ; L ) ≃ H ∗ (Ω d − A ′ ( X, d ); L ) for d > A to modules M over A , or in ourmain example from the disk D d to more general manifolds M . Let A be homotopy commutative, s be a product of generators of A and let M be an A -module. Using the A -module structure definethe telescope M ∞ := Tel( M s −→ M s −→ . . . ) . Then arguing as before, the map p : M ∞ × A E A → B A is a homology-fibration. Thus the canoncialmap M ∞ −→ hofib( p : M ∞ × A E A → B A ) (10.3)from the fibre to the homotopy fibre of p is a homology isomorphism, compare [MS76] [MP15].Returning to our main example of configuration-section spaces, let ξ : E → M be a fibre bundleover a connected manifold M of dimension d > X . Let D denote a( d − M . Define M := CΓ c,D ( M ; ξ ) = G k > CΓ c,Dk ( M ; ξ ) . Theorem 10.15
For d > , there are isomorphisms H ∗ ( M ∞ ) ≃ H ∗ (Γ( M ; E d ( ξ, c ))) . By Proposition 4.10, M is a module over both A (cid:3) and A ⊔ . To identify the homotopy fibre of p we will consider the A ⊔ module structure, and hence have to assume d > A ⊔ ishomotopy commutative. Proof.
Using diagram (10.4) below we will identify the homotopy fiber of p as the space in (10.5)below. This holds for d >
2. When d > p is a homology fibration and the homology equivalence(10.3) implies the theorem.Let M = M ∪ D I d be the manifold from Definition 4.11 and write D for the copy { } × I d − of D = { } × I d − in the boundary of M . Using similar notation as in diagram (10.2), we havethe following commutative diagram M ∞ × A ⊔ E A ⊔ ≃ −−−−→ CΓ c , ∗ (( M , D ); ξ ) ≃ −−−−→ Γ(( M , D ); E d ( ξ , c )) p y π y res y B A ⊔ ≃ −−−−→ CΓ c ′ , ∗ (( I d , D ∪ D ); ξ ′ ) ≃ −−−−→ Γ(( I d , D ∪ D ); E d ( ξ ′ , c ′ )) . (10.4)The vertical map π is the quotient map of Proposition 10.2 which lets configuration points in M disappear. The right vertical map restricts sections on M to I d . Reasoning as for diagram (10.2), res is surjective and has fibreΓ D ( M, D ; E ( ξ, c )) ≃ Γ( M, D ; E ( ξ, c )) (as ∗ ∈ D ≃ ∗ ) . (10.5)By Theorem 10.12 the two horizontal maps on the right are weak homotopy equivalences andthe right square commutes up to homotopy. Indeed, let S = ∂ r D with ∗ ∈ S and replacethe configuration-section spaces and the section spaces by those decorated by S . With thesereplacements the square commutes.By by part (ii) of Lemma 10.5 we have homotopy equivalences B A ⊔ ≃ CΓ c ′ , ∂ (( I d , D ∪ D ); ξ ′ ) ≃ CΓ c ′ , ∗ (( I d , D ∪ D ); ξ ′ ) . The proof of part (ii) of Lemma 10.5 generalises to give also a homotopy equivalence M × A ⊔ E A ⊔ ≃ CΓ c, ∂ ( M + , D + ; ξ ) ≃ CΓ c, ∗ ( M + , D + ; ξ ) . The stabilisation maps induce a homotopy equivalence on these spaces, and hence also the top lefthorizontal map is a homotopy equivalence. The left square clearly commutes.43ote that we did not need to make any restriction on c to apply the group completion theorem.We also note that diagram (10.4) holds for all d > A ⊔ in order to deduce that p is a homology fibration. However, if we know independently that p isa homology fibration, as for example when we know that the components of M satisfy homologystability, the conclusion of the theorem still holds. Thus combining the above with Theorem 8.5 wecan identify the stable homology of the finite configuration-section spaces in the following cases. Corollary 10.16
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