Abstract
We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles. The former are Gibbons–Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monopole moduli space by the Borel construction. They form boundary hypersurfaces in a partial compactification of the classical monopole moduli spaces. Our results follow from a general homological stability result for configuration spaces equipped with non-local data.
1. Introduction
The topology of the moduli spaces of magnetic monopoles ${\mathcal{M}}_{k}$ has been the subject of intensive study for many decades. By a theorem of Donaldson [1], they have a model as spaces of rational functions on $\mathbb{C}{P}^{1}$. Via this model, their homotopy and homology groups are known to stabilize as $k\to \mathrm{\infty}$ by a theorem of Segal [2] and their homology (both stable and unstable) was completely computed by Cohen et al. [3] in terms of the homology of the braid groups, which is completely known [4].
The moduli spaces ${\mathcal{M}}_{k}$ are non-compact manifolds. Recently, a partial compactification of ${\mathcal{M}}_{k}$ has been constructed by Kottke & Singer [5] by adding certain boundary hypersurfaces ${\mathcal{I}}_{\lambda}$ to ${\mathcal{M}}_{k}$ indexed by partitions $\lambda =({k}_{1},\dots ,{k}_{r})$ of $k$.
Points in these boundary hypersurfaces are thought of as ‘ideal’ monopoles of total charge $k$, with $r$ ‘clusters’ centred at different points in ${\mathbb{R}}^{3}$, with charges ${k}_{1},\dots ,{k}_{r}$, which are ‘widely separated’ but nevertheless interact.
Our main theorem proves a homology stability result for these ideal monopole moduli spaces as the number of clusters of a fixed charge $c\ge 1$ goes to infinity:
Theorem A.
Fix a positive integer $c$ and a tuple $\lambda =({k}_{1},\dots ,{k}_{r})$, of fixed length $r$, of positive integers ${k}_{i}\ne c$. Write $\lambda {[n]}_{c}=({k}_{1},\dots ,{k}_{r},c,\dots ,c)$, where $c$ appears $n$ times. There are natural stabilization maps
We also prove an analogous result for moduli spaces of framed Dirac monopoles (in other words Gibbons–Manton torus bundles; see §2b for the definitions) and, more generally, Gibbons–Manton $\mathbf{\text{Z}}$-bundles for any sequence $\mathbf{\text{Z}}$ of path-connected ${S}^{1}$-spaces; see theorems 4.1 and 4.9.
These results follow from a general homology stability result (proposition 3.3) for unordered configuration spaces with non-local parameters. Homology stability for configuration spaces whose points are labelled by elements of a fixed space $X$ is well known; these are configuration spaces with local parameters. However, the ideal monopole moduli spaces ${\mathcal{I}}_{\lambda}$ are non-local. The key observation in §3 is that homology stability only requires the parameters associated with a configuration to satisfy much weaker properties, which allows us to consider interesting non-local parameters. In [6], we recently proved a different homology stability result for non-local configuration spaces, namely for configuration-section spaces; this encouraged us to try to prove homology stability also in the context of ideal monopole moduli spaces. Proposition 3.3 is the abstract general result that applies in our situation in the present paper. Though similar in nature, it neither is implied by nor implies the homology stability result in [6].
(a) Outline
We first recall some background on moduli spaces of magnetic monopoles in §2: first on the moduli spaces themselves in §2a and then on their partial compactifications introduced by Kottke & Singer [5] in §2b, whose boundary hypersurfaces are the ideal monopole moduli spaces. In §3, we then prove a general homology stability result for configuration spaces equipped with ‘non-local’ data, deducing it from twisted homological stability for configuration spaces [7] (see also [8]). In §4, we apply it to prove our main theorem, homology stability for ideal monopole moduli spaces, as well as an extension (theorem 4.9) to Gibbons–Manton $\mathbf{\text{Z}}$-bundles more generally.
2. Monopole moduli space and boundary hypersurfaces
(a) Monopole moduli space
We briefly recall from [9] some different monopole moduli spaces and the relations between them.
A magnetic monopole on ${\mathbb{R}}^{3}$ is a pair consisting of a connection $A$ on the trivial principal $SU(2)$-bundle on ${\mathbb{R}}^{3}$ together with a field $\varphi $ taking values in the associated Lie algebra $\mathfrak{s}\mathfrak{u}(2)$. Fixing a framing, these may be viewed, respectively, as a smooth 1-form and a smooth function on ${\mathbb{R}}^{3}$ taking values in $\mathfrak{s}\mathfrak{u}(2)$, which we may identify topologically as $\mathfrak{s}\mathfrak{u}(2)\cong {\mathbb{R}}^{3}$. These data $A$ and $\varphi $ must satisfy the Bogomolny equations and a certain finiteness condition; see [9, pp. 14–15] for details. This finiteness condition implies that $\varphi (x)\ne 0$ for $|x|$ sufficiently large, so the restriction of $\varphi $ to ${\mathbb{R}}^{3}\setminus {B}_{R}(0)$ takes values in $\mathfrak{s}\mathfrak{u}(2)\setminus \{0\}$ for $R\gg 0$. The degree of this map is the charge of the monopole and is always positive. The set of all magnetic monopoles of charge $k\ge 1$, up to gauge equivalence (automorphisms of the trivial bundle ${\mathbb{R}}^{3}\times \mathfrak{s}\mathfrak{u}(2)\to {\mathbb{R}}^{3}$), suitably topologized, is the monopole moduli space ${\mathcal{N}}_{k}$. A slight variation of the construction, quotienting by a smaller gauge group, yields a different space ${\mathcal{M}}_{k}$ related to ${\mathcal{N}}_{k}$ by a principal ${S}^{1}$-bundle
The spaces ${\mathcal{M}}_{k}$ and ${\mathcal{M}}_{k}^{0}$ admit the structure of hyperKähler manifolds of dimensions $4k$ and $4k-4$, respectively. For charge $k=1$, we have ${\mathcal{M}}_{1}^{0}=pt$ (and ${\mathcal{M}}_{1}\cong {S}^{1}\times {\mathbb{R}}^{3}$) and for $k=2$, the 4-manifold ${\mathcal{M}}_{2}^{0}$ is known as the Atiyah–Hitchin manifold and has been studied in detail in [9].
By [1], ${\mathcal{M}}_{k}$ is homeomorphic to the space ${R}_{k}$ of degree-$k$ rational self-maps of $\mathbb{C}{P}^{1}$ that send $\mathrm{\infty}$ to 0. Thus, it is also homeomorphic to the space ${R}_{k}^{\prime}$ of degree-$k$ rational self-maps of $\mathbb{C}{P}^{1}$ that send $\mathrm{\infty}$ to 1. The points of the space ${R}_{k}^{\prime}$ may conveniently be described as pairs $(p,q)$ of coprime monic polynomials with coefficients in $\mathbb{C}$, both of degree $k$. Identifying these polynomials with their sets of roots, we obtain a natural embedding
The fundamental group of ${\mathcal{M}}_{k}$ is $\mathbb{Z}$, by [2, proposition 6.4]. Also, by [9, ch. 2], the fundamental group of ${\mathcal{N}}_{k}$ is $\mathbb{Z}/k$ and the projection map (2.1) induces the reduction-mod-$k$ map $\mathbb{Z}\twoheadrightarrow \mathbb{Z}/k$. It follows from the long exact sequence that (2.1) induces isomorphisms on ${\pi}_{i}$ for all $i\ge 2$, so ${\mathcal{M}}_{k}$ and ${\mathcal{N}}_{k}$ have the same universal cover, up to homotopy equivalence, which is denoted by ${\mathcal{X}}_{k}$.
There are stabilization maps ${\mathcal{M}}_{k}\to {\mathcal{M}}_{k+1}$, which may be defined under the isomorphism ${\mathcal{M}}_{k}\cong {R}_{k}$ by adding to a given rational self-map a new zero and a new pole ‘far away’ from the origin. (This is not invariant under the circle action, so it does not descend to a stabilization map on the moduli spaces ${\mathcal{N}}_{k}$.) The stabilization maps ${\mathcal{M}}_{k}\to {\mathcal{M}}_{k+1}$ induce isomorphisms on homotopy groups (and hence also homology groups) in a stable range, by Segal [2]. Lifting to universal covers, it follows that there are also stabilization maps ${\mathcal{X}}_{k}\to {\mathcal{X}}_{k+1}$ that induce isomorphisms on homotopy (and homology) groups in a stable range.
By the main theorem of Segal [2], the homotopy colimit of the stabilization maps ${\mathcal{M}}_{k}\to {\mathcal{M}}_{k+1}\to \cdots $ is weakly equivalent to ${\Omega}_{0}^{2}{S}^{2}$. Thus, the stable homology of ${\mathcal{M}}_{k}$ is the homology of ${\Omega}_{0}^{2}{S}^{2}$ and the stable homology of ${\mathcal{X}}_{k}$ is the homology of the universal cover of ${\Omega}_{0}^{2}{S}^{2}$. Moreover, the unstable homology of ${\mathcal{M}}_{k}$ (i.e. its homology outside of the stable range) is also known: by the main result of [3,10], the homology of ${\mathcal{M}}_{k}$ is isomorphic to the group homology of the braid group ${B}_{2k}$, which is completely computed [4]. The rational unstable homology of ${\mathcal{X}}_{k}$ has also been computed by Segal & Selby [11], and is significantly more complicated than the rational unstable homology of ${\mathcal{M}}_{k}$: the rational homology ${H}_{\ast}({\mathcal{M}}_{k};\mathbb{Q})$ is the same as that of the circle, so it has total dimension 2, whereas Segal & Selby [11] show that the rational homology ${H}_{\ast}({\mathcal{X}}_{k};\mathbb{Q})$ has total dimension $k$, concentrated in degrees of the form $2(k-d)$, where $d$ is a divisor of $k$.
Notation 2.1.
The principal bundles (2.1) and (2.2) arise from a principal (in particular free) action of the product ${S}^{1}\times {\mathbb{R}}^{3}$ on ${\mathcal{M}}_{k}$. If we first quotient by ${\mathbb{R}}^{3}$ (Euclidean translations), we obtain a principal ${\mathbb{R}}^{3}$-bundle
(b) Boundary hypersurfaces
Kottke & Singer [5] have constructed a partial compactification of ${\mathcal{M}}_{k}^{c}\simeq {\mathcal{M}}_{k}$ of the form
We will not recall here the construction of ${\mathcal{I}}_{\lambda}^{c}$ in [5]; instead we will take an alternative characterization of ${\mathcal{I}}_{\lambda}^{c}$ to be its definition (see definitions 2.6 and 2.12 and remark 2.13). To begin with, we recall the definitions of ordered and unordered configuration spaces.
Definition 2.2.
For any space $M$, let us write ${F}_{r}(M)=\{({v}_{1},\dots ,{v}_{r})\in {M}^{r}\mid {v}_{i}\ne {v}_{j}\text{for}i\ne j\}$ for the ordered configuration space of $r$ points in $M$, topologized as a subspace of the product ${M}^{r}$. We also write ${C}_{r}(M)={F}_{r}(M)/{\Sigma}_{r}$ for the unordered configuration space of $r$ points in $M$.
Recall (see, for example [12, theorem V.1.1]) that the degree-$(d-1)$ cohomology of ${F}_{r}({\mathbb{R}}^{d})$ is given by:
Definition 2.3.
[5, Definition 4.6 and the paragraph preceding it] For a sequence of integers $\lambda =({k}_{1},\dots ,{k}_{r})$, the corresponding Gibbons–Manton circle factors are the principal ${S}^{1}$-bundles
Definition 2.4.
The symmetric group ${\Sigma}_{r}$ acts on ${F}_{r}({\mathbb{R}}^{3})$ by permuting the particles. Let ${\Sigma}_{\lambda}\le {\Sigma}_{r}$ be the stabilizer of $\lambda =({k}_{1},\dots ,{k}_{r})\in {\mathbb{Z}}^{r}$ under the obvious permutation action of ${\Sigma}_{r}$ on ${\mathbb{Z}}^{r}$. Then the action of ${\Sigma}_{\lambda}$ on ${F}_{r}({\mathbb{R}}^{3})$ lifts to a well-defined action on ${\stackrel{~}{\mathcal{T}}}_{\lambda}$. The Gibbons–Manton configuration space is the quotient space ${\mathcal{T}}_{\lambda}={\stackrel{~}{\mathcal{T}}}_{\lambda}/{\Sigma}_{\lambda}$. Note that there is a principal ${T}^{r}$-bundle
Remark 2.5.
One may make analogous definitions for Euclidean spaces ${\mathbb{R}}^{d}$ in general, replacing ${S}^{1}=K(\mathbb{Z},1)$ with $K(\mathbb{Z},d-2)$, so that ${\mathcal{T}}_{\lambda}$ is a principal $K{(\mathbb{Z},d-2)}^{r}$-bundle over ${F}_{r}({\mathbb{R}}^{d})$. For example, when $d=2$, it is a regular covering space with deck transformation group isomorphic to ${\mathbb{Z}}^{r}$. In particular, for $d=2$ and $\lambda =(1,1,\dots ,1)$, it is the regular covering space corresponding to the homomorphism
Definition 2.6.
The moduli space of ideal monopoles of weight $\lambda $ is defined as follows. Recall that the monopole moduli space ${\mathcal{M}}_{k}$ is equipped with a circle action. The product ${\mathcal{M}}_{{k}_{1}}\times \cdots \times {\mathcal{M}}_{{k}_{r}}$ is therefore equipped with an action of the torus ${T}^{r}$. We define ${\stackrel{~}{\mathcal{I}}}_{\lambda}$ to be the total space of the fibre bundle associated with the principal ${T}^{r}$-bundle ${\stackrel{~}{\mathcal{T}}}_{\lambda}$ by changing the fibre to ${\mathcal{M}}_{{k}_{1}}\times \cdots \times {\mathcal{M}}_{{k}_{r}}$. In other words, it is the Borel construction
Remark 2.7.
This is not yet the boundary stratum ${\mathcal{I}}_{\lambda}^{c}$ constructed by [5] in their partial compactification of ${\mathcal{M}}_{k}^{c}$, since it has the wrong dimension. Recall that the dimension of ${\mathcal{M}}_{k}^{c}$ is $4k-3$, so its boundary strata must have dimension $4k-4$, whereas the dimension of ${\mathcal{I}}_{\lambda}$ is $4k+3r$. The definition of ${\mathcal{I}}_{\lambda}^{c}$ is similar to that of ${\mathcal{I}}_{\lambda}$ (and these two spaces are homotopy equivalent; see remark 2.10), using the centred moduli spaces ${\mathcal{M}}_{{k}_{i}}^{c}$ instead of ${\mathcal{M}}_{{k}_{i}}$ and using a centred version of the configuration space, which we define next.
Definition 2.8.
The ordered centred configuration space ${F}_{r}^{c}({\mathbb{R}}^{3})\subseteq {F}_{r}({\mathbb{R}}^{3})$ is defined to be the space of all ordered configurations $({x}_{1},\dots ,{x}_{r})$ in ${F}_{r}({\mathbb{R}}^{3})$ such that
Definition 2.9.
The moduli space of centred ideal monopoles of weight $\lambda $ is defined as follows. Analogously to definition 2.6, consider the Borel construction
Remark 2.10.
Since the inclusion ${F}_{r}^{c}({\mathbb{R}}^{3})\subseteq {F}_{r}({\mathbb{R}}^{3})$ and the projection (2.3) are homotopy equivalences, we also have
However, since we focus in this paper on the homological properties of ${\mathcal{I}}_{\lambda}$, the difference between ${\mathcal{I}}_{\lambda}$ and ${\mathcal{I}}_{\lambda}^{c}$ will not be relevant to us.
Terminology 2.11.
When $\lambda =(1,1,\dots ,1)$, the moduli space ${\mathcal{I}}_{\lambda}$ is called the moduli space of widely separated magnetic monopoles. This terminology follows the intuition that points $x\in {\mathcal{I}}_{\lambda}$ should be thought of as monopoles of total charge $k$, with $r$ different ‘clusters’ centred at the points $\pi (x)$, with charges ${k}_{i}$, which are ‘widely separated’ but nevertheless interact: these interactions are encoded in the structure group ${T}^{r}$ of the bundle (2.9).
Definition 2.12.
The moduli space of framed Dirac monopoles of weight $\lambda $ is the Gibbons–Manton configuration space ${\mathcal{T}}_{\lambda}$ of definition 2.4, which has the total space of the Gibbons–Manton torus bundle (2.7) as a finite covering.
Remark 2.13. (On definitions)
Definitions 2.6 and 2.12 are not precisely the definitions given in [5]. By [5, theorem 4.9], the moduli space of ideal monopoles of weight $\lambda $—according to their definition—is equivalent to the space denoted by ${\stackrel{~}{\mathcal{I}}}_{\lambda}$ in definition 2.6. However, as pointed out in [5] (see the Remark on page 53), this is not the correct space to form the boundary hypersurfaces of the compactification ${\overline{\mathcal{M}}}_{k}$ of ${\mathcal{M}}_{k}$, and one should instead pass to the quotient space ${\mathcal{I}}_{\lambda}={\stackrel{~}{\mathcal{I}}}_{\lambda}/{\Sigma}_{\lambda}$. We have therefore made this replacement in definition 2.6. (The difference between ${\mathcal{I}}_{\lambda}$ and its finite covering space ${\stackrel{~}{\mathcal{I}}}_{\lambda}$ is not significant in [5] since they are interested primarily in studying the geometry of these spaces locally.) Similarly, by [5, proposition 4.8], the moduli space of framed Dirac monopoles of weight $\lambda $—according to their definition—is equivalent to the total space ${\stackrel{~}{\mathcal{T}}}_{\lambda}$ of the Gibbons–Manton torus bundle (2.7). For the same reasons as above, we instead consider the moduli space of framed Dirac monopoles to be the quotient space ${\mathcal{T}}_{\lambda}={\stackrel{~}{\mathcal{T}}}_{\lambda}/{\Sigma}_{\lambda}$ (definition 2.12). Henceforth, we treat definitions 2.6 and 2.12 as the definitions of the ideal and framed Dirac monopole moduli spaces, respectively.
Remark 2.14.
Another small difference between our definition and that of [5] concerns the action of the symmetric group ${\Sigma}_{\lambda}$. In [5], the ordered centred configuration spaces (cf. definition 2.8) are defined in a slightly asymmetric way, which does not allow for taking a quotient by ${\Sigma}_{\lambda}$ (as we do above), since they single out one point of the configuration to lie at $0\in {\mathbb{R}}^{3}$. We have modified the definition to be more symmetric by instead requiring the centre of mass to lie at 0. This does not change the homeomorphism type of the centred ordered configuration space and it has the advantage of having a natural action of the full symmetric group ${\Sigma}_{r}$, not just ${\Sigma}_{r-1}$.
Remark 2.15.
When $k=1$, the monopole moduli space ${\mathcal{M}}_{1}^{c}$, as an ${S}^{1}$-space, is simply ${S}^{1}$ itself. Thus, according to definition 2.6, we have ${\stackrel{~}{\mathcal{I}}}_{(1,\dots ,1)}={\stackrel{~}{\mathcal{T}}}_{(1,\dots ,1)}$. The moduli space of widely separated magnetic monopoles ${\mathcal{I}}_{(1,\dots ,1)}$ (cf. terminology 2.11) is therefore the quotient of the total space of the Gibbons–Manton torus bundle ${\stackrel{~}{\mathcal{T}}}_{(1,\dots ,1)}$ by the symmetric group ${\Sigma}_{r}$.
Remark 2.16. (Higher codimension boundary strata)
The space (2.5) is only a partial compactification of ${\mathcal{M}}_{k}$: it is a manifold with boundary whose interior is ${\mathcal{M}}_{k}$, but it is still non-compact. In a recent preprint [13], a full compactification of ${\mathcal{M}}_{k}$ is proposed,^{1} which is a smooth manifold with corners that recovers the partial compactification ${\overline{\mathcal{M}}}_{k}$ if one discards corners of codimension greater than 1. It would be interesting to extend our study of the homology of ${\mathcal{I}}_{\lambda}$ to the deeper boundary strata of this full compactification.
3. Homology stability for configurations with non-local data
The goal of this section is to prove proposition 3.3, which gives sufficient conditions that imply homology stability for configuration spaces equipped with additional (possibly ‘non-local’) parameters.
Labelled configuration spaces, where each separate point of a configuration is equipped with a label taking values in a fixed space, are the most obvious examples of this setting—we refer to these as configuration spaces with local data, since the labels are associated with individual points of the configuration. However, the key observation of this section is that the proof of homology stability requires only weaker properties of the parameters, which are satisfied also in other interesting, non-local settings.
In particular, in §4 we will apply this to our key motivating example of non-local configuration spaces, Gibbons–Manton torus bundles and moduli spaces of ideal monopoles, where the parameters are genuinely non-local, encoding the pairwise interactions of the points of the configuration.
For the general setting of non-local configuration spaces, let us consider a connected manifold $\overline{M}$ with non-empty boundary and denote its interior by $M$. We first recall the definition of the stabilization maps between the ordered and unordered configuration spaces ${F}_{n}(M)$ and ${C}_{n}(M)$ (see definition 2.2).
Definition 3.1.
Choose a collar neighbourhood of $\overline{M}$, in order words an open neighbourhood $U$ of $\mathrm{\partial}\overline{M}$ and an identification $\phi :U\cong \mathrm{\partial}\overline{M}\times [0,1)$ that restricts to $\phi (p)=(p,0)$ for $p\in \mathrm{\partial}\overline{M}\subset U$. (This exists by Brown [14].) Let $\hat{M}$ be the result of thickening the collar neighbourhood, i.e. the union of $\overline{M}$ and $\mathrm{\partial}\overline{M}\times (-1,1)$ along the identification $\phi $. Also, choose a diffeomorphism $(-1,1)\cong (0,1)$ that restricts to the identity on $(1-\u03f5,1)$ for some $\u03f5>0$. Taking the product with the identity on $\mathrm{\partial}\overline{M}$ and extending by the identity on $M\setminus U$, this determines a diffeomorphism $\theta :\hat{M}\cong M$. Finally, choose a basepoint $\ast \in \mathrm{\partial}\overline{M}$. These choices determine a stabilization map
Remark 3.2.
Up to homotopy, the stabilization maps (3.1) and (3.2) depend only on the choice of boundary-component of $\overline{M}$ containing the basepoint $\ast $. These maps (or maps homotopic to them) were introduced in [15, §4] and [2, appendix]; see also [16, §4] or [7, §2.2].
Let us now consider the sequence
Proposition 3.3.
Fix path-connected spaces $Y$ and $Z$ and suppose that ${f}_{n}^{-1}({c}_{n})={Z}^{n}\times Y$ for all $n$. Fix a basepoint $\ast \in Z$. Moreover, we assume also that
— | the monodromy ${\pi}_{1}({C}_{n}(M))\to \text{hAut}({Z}^{n}\times Y)$ of (3.6) is the projection onto the symmetric group followed by the obvious permutation action on the factors of the product ${Z}^{n}$; | ||||
— | the restriction ${Z}^{n}\times Y\to {Z}^{n+1}\times Y$ of the lifted stabilization map (3.5) to fibres over basepoints is the natural inclusion $({z}_{1},\dots ,{z}_{n},y)\mapsto (\ast ,{z}_{1},\dots ,{z}_{n},y)$. |
Example 3.4.
One source of examples of fibrations (3.6) over configuration spaces ${C}_{n}(M)$ equipped with lifted stabilization maps (3.5) that satisfy the two conditions of proposition 3.3 is configuration spaces with local data. This means that we choose a fibration $f:E\to \overline{M}$ with path-connected fibres, where $M=\text{int}(\overline{M})$, trivialized over a disc $D\subset \mathrm{\partial}\overline{M}$. Then we set
However, there also exist labelling data (3.6) and (3.5), satisfying the two conditions of proposition 3.3, that do not arise in this way. We will call these ‘non-local’ data:
Definition 3.5.
A system of configuration spaces equipped with non-local data is a choice of (3.6) and (3.5) that do not arise as described in example 3.4 above.
Remark 3.6.
Proposition 3.3, in the setting of configuration spaces with local data, is well known: see [17, appendix A] or [18, appendix B]. The point of this section is to observe that it also holds in a more general setting, requiring just the two assumptions of proposition 3.3, which includes also configuration spaces with non-local data. We will see in §4 that asymptotic monopole moduli spaces are examples of configuration spaces with non-local data: this is our key motivating example. We note, on the other hand, that configuration-mapping spaces, considered in [6], are in general not examples of configuration spaces with non-local data in the sense of proposition 3.3, as the associated monodromy action does not in general factor through the symmetric group. See [19, §9] for a detailed study of the monodromy action for configuration-mapping spaces.
In order to prove proposition 3.3, we first need to recapitulate some definitions and results from [7]. Recall that we are considering a connected manifold $\overline{M}$ with non-empty boundary whose interior we denote by $M$. Associated with this manifold, there is a certain category $\mathcal{B}(M)$, the partial braid category on $M$, whose objects are non-negative integers $\{0,1,2,\dots \}$ and whose morphisms are ‘partial braids’ in $M\times [0,1]$; the precise definition is given in [7, §2.3].^{2} This category comes equipped with an endofunctor $S$ that acts by $+1$ on objects as well as a natural transformation $\iota :\text{id}\Rightarrow S$.
Definition 3.7.
[7, Definitions 2.2 and 3.1] A twisted coefficient system for the sequence (3.4) of unordered configuration spaces on $M$, defined over a ring $R$, is a functor $T:\mathcal{B}(M)\to R\text{-}\text{Mod}$. The degree of a twisted coefficient system $T$, taking values in $\{-1,0,1,2,3,\dots \}\cup \{\mathrm{\infty}\}$, is defined recursively by setting $\text{deg}(0)=-1$ and declaring that $\text{deg}(T)\le d$ if and only if $\text{deg}(\mathrm{\Delta}T)\le d-1$, where $\mathrm{\Delta}T$ is the cokernel of the natural transformation $T\iota :T\Rightarrow TS$.
Remark 3.8.
In [7], the ground ring $R$ is always assumed to be $\mathbb{Z}$, but everything generalizes directly to an arbitrary ground ring $R$.
For any twisted coefficient system $T$, the morphisms ${\iota}_{n}:n\to Sn=n+1$, which between them constitute the natural transformation $\iota $, induce homomorphisms $T(n)\to T(n+1)$. Together with the stabilization maps (3.4), these induce homomorphisms
Theorem 3.9.
[7, Theorem A] If $T$ is a twisted coefficient system for (3.4) of degree $d$, then the map of twisted homology groups (3.7) is an isomorphism in the range of degrees $\ast \le {\textstyle \frac{1}{2}}(n-d)$.
An important family of examples of finite-degree twisted coefficient systems are defined on the category $\text{FI}\mathrm{\u266f}$,^{3} which is the category whose objects are non-negative integers and whose morphisms from $m$ to $n$ are the partially defined injections from $\{1,\dots ,m\}$ to $\{1,\dots ,n\}$. For any manifold $M$, there is a canonical functor ${f}_{M}:\mathcal{B}(M)\to \text{FI}\mathrm{\u266f}$, so any functor $\text{FI}\mathrm{\u266f}\to R\text{-}\text{Mod}$ determines a twisted coefficient system for any manifold $M$.
Construction 3.10.
[A generalization of [7, example 4.1]] Choose path-connected spaces $Y,Z$ and a basepoint $\ast \in Z$. Also choose an integer $q\ge 0$ and a field $K$. There is a functor
Lemma 3.11.
For any manifold $M$, the twisted coefficient system
Proof.
When $Y$ is the one-point space, this is [7, lemma 4.2]. The extra factor of $Y$ in the product does not affect the proof at all (as long as $Y$ is path-connected), so the proof of the general case is identical to that of [7, lemma 4.2].
This completes our recapitulation of the necessary definitions and results of [7], and we may now complete the proof of proposition 3.3.
Proof of proposition 3.3.
We will take field coefficients and prove homological stability up to degree $n/2$. This will automatically imply homological stability up to degree $n/2-1$ with integral coefficients (and hence any untwisted coefficients), via the short exact sequences of coefficients
We therefore consider the Serre spectral sequence, with coefficients in a field $K$, associated to the fibration (3.6) and the map of Serre spectral sequences induced by the stabilization maps downstairs (3.4) and upstairs (3.5). The map of ${E}^{2}$ pages is of the form
Remark 3.12.
One may prove proposition 3.3 using the twisted homological stability result [8, theorem D] instead of the twisted homological stability result [7, theorem A], although this results in a range of degrees one smaller, namely $n/2-1$ for field coefficients and $n/2-2$ for integral coefficients.
Remark 3.13.
The map (3.10) of ${E}^{2}$ pages of Serre spectral sequences is split-injective in all degrees by [7, theorem A]. However, this does not in general imply split-injectivity in the limit, so we cannot deduce from this that ${E}_{n}\to {E}_{n+1}$ induces split-injections on homology. Anticipating remark 4.7, there are obstructions to proving split-injectivity on homology for configurations with non-local data, in contrast to the case of ordinary configurations and twisted homology.
4. Homology stability for asymptotic monopole moduli spaces
Fix a positive integer $c$ and a tuple $\lambda =({k}_{1},\dots ,{k}_{r})$ of positive integers that sum to $k$. Denote by $\lambda {[n]}_{c}$ the tuple $({k}_{1},\dots ,{k}_{r},c,\dots ,c)$, where there are $n$ appearances of $c$. For simplicity, we will assume that ${k}_{i}\ne c$ for each $i$ (if this is not the case we may simply remove these entries from $\lambda $ and increase $n$ appropriately). Our main theorem is the following.
Theorem 4.1.
There are natural stabilization maps
We first prove theorem 4.1 for the Gibbons–Manton configuration spaces ${\mathcal{T}}_{\lambda {[n]}_{c}}$ in §4a. We then show in §4b that homological stability is preserved in general when replacing each circle factor in the torus fibre of ${\mathcal{T}}_{\lambda {[n]}_{c}}$ with another space that is equipped with a circle action. In particular, we deduce the second part of theorem 4.1, since moduli spaces of ideal monopoles ${\mathcal{I}}_{\lambda {[n]}_{c}}$ are special cases of this construction.
(a) Gibbons–Manton torus bundles
Recall that the Gibbons–Manton torus bundle ${\mathcal{T}}_{\lambda {[n]}_{c}}$ has base space ${F}_{r+n}({\mathbb{R}}^{3})/{\Sigma}_{\lambda {[n]}_{c}}$, where ${\Sigma}_{\lambda {[n]}_{c}}={\Sigma}_{\lambda}\times {\Sigma}_{n}$. By abuse of notation, we will write
Our first goal in this section is to lift the classical stabilization maps of configuration spaces
Our second goal is to show that these lifted stabilization maps satisfy the two hypotheses of proposition 3.3. This will imply homological stability for Gibbons–Manton torus bundles, i.e. the first part of theorem 4.1.
We begin with a lemma about pullbacks of Gibbons–Manton circle factors. To prepare for this, we first choose an explicit concrete model for the stabilization maps (4.2); i.e. we make explicit some of the choices involved in definition 3.1 in the case $M={\mathbb{R}}^{3}$. Up to homotopy, this does not make any difference, but it will be convenient for the proof of lemma 4.3 to choose a specific representative of this homotopy class of maps.
Definition 4.2.
We will in fact replace ${\mathbb{R}}^{3}$ with the open upper half-space $M={\mathbb{R}}^{2}\times (0,\mathrm{\infty})$. We may then take $\overline{M}={\mathbb{R}}^{2}\times [0,\mathrm{\infty})$ with the obvious collar neighbourhood, so $\hat{M}={\mathbb{R}}^{2}\times (-1,\mathrm{\infty})$. Take $\ast =(0,0,0)\in \mathrm{\partial}\overline{M}={\mathbb{R}}^{2}\times \{0\}$ as basepoint. With these choices (and identification of ${\mathbb{R}}^{3}$ with ${\mathbb{R}}^{2}\times (0,\mathrm{\infty})$), the stabilization map
Lemma 4.3.
Let $\lambda =({k}_{1},\dots ,{k}_{r})$ for positive integers ${k}_{i}$ and write ${\lambda}^{\prime}=({k}_{1},\dots ,{k}_{r-1})$. Then the pullback of the circle bundle ${S}_{\lambda ,j}\to {F}_{r}({\mathbb{R}}^{3})$ along the stabilization map (4.4) is ${S}_{{\lambda}^{\prime},j}\to {F}_{r-1}({\mathbb{R}}^{3})$ if $j\le r-1$ and a trivial bundle if $j=r$.
Proof.
Recall that the bundle ${S}_{\lambda ,j}\to {F}_{r}({\mathbb{R}}^{3})$ is the pullback of the universal ${S}^{1}$-bundle on $\mathbb{C}{P}^{\mathrm{\infty}}$ along the map ${F}_{r}({\mathbb{R}}^{3})\to \mathbb{C}{P}^{\mathrm{\infty}}$ given by the sum $\sum _{i=1,i\ne j}^{r}{k}_{i}.{\iota}_{ij}$ where ${\iota}_{ij}:{F}_{r}({\mathbb{R}}^{3})\to {S}^{2}\subset \mathbb{C}{P}^{\mathrm{\infty}}$ is given by
The key observation is the following. When $i=r$ and we restrict ${\iota}_{rj}$ to ${F}_{r-1}({\mathbb{R}}^{3})$ along (4.4), the vertical (third) coordinate of the point ${x}_{r}$ will always be smaller than the vertical coordinate of the point ${x}_{j}$, due to the choices made in the construction of (4.4) in definition 4.2; see figure 1 for a detailed explanation. Thus, the right-hand side of (4.5) always takes values in the bottom hemisphere of ${S}^{2}\subset \mathbb{C}{P}^{\mathrm{\infty}}$, and hence ${\iota}_{rj}$ restricted along (4.4) is nullhomotopic. By exactly analogous reasoning, when $j=r$ the map ${\iota}_{ir}$ restricted along (4.4) takes values in the top hemisphere of ${S}^{2}\subset \mathbb{C}{P}^{\mathrm{\infty}}$ and hence is also nullhomotopic.
Putting this all together, we deduce that the map ${F}_{r-1}({\mathbb{R}}^{3})\to \mathbb{C}{P}^{\mathrm{\infty}}$ classifying the pullback of ${S}_{\lambda ,r}$ is nullhomotopic, so this pullback is trivial. It also implies that the map ${F}_{r-1}({\mathbb{R}}^{3})\to \mathbb{C}{P}^{\mathrm{\infty}}$ classifying the pullback of ${S}_{\lambda ,j}$, for $j\le r-1$, is the sum $\sum _{i=1,i\ne j}^{r-1}{k}_{i}.{\iota}_{ij}$, which is by definition the map that classifies ${S}_{{\lambda}^{\prime},j}$.
Remark 4.4.
Recalling that we denote by ${\alpha}_{ij}$ the pullback of a fixed generator of ${H}^{2}({S}^{2};\mathbb{Z})$ along the map ${\iota}_{ij}:{F}_{r}({\mathbb{R}}^{3})\to {S}^{2}$, the discussion in the proof above implies that the stabilization map ${F}_{r-1}({\mathbb{R}}^{3})\to {F}_{r}({\mathbb{R}}^{3})$ acts on ${H}^{2}(-;\mathbb{Z})$, in the basis (2.6), by ${\alpha}_{ij}\mapsto {\alpha}_{ij}$ if $j\le r-1$ and ${\alpha}_{ir}\mapsto 0$. It is also easy to see that the automorphism ${\sigma}_{\ast}:{F}_{r}({\mathbb{R}}^{3})\to {F}_{r}({\mathbb{R}}^{3})$ induced by a permutation $\sigma \in {\Sigma}_{r}$ acts on generators of ${H}^{2}({F}_{r}({\mathbb{R}}^{3});\mathbb{Z})$ by ${\alpha}_{ij}\mapsto {\alpha}_{{\sigma}^{-1}(i),{\sigma}^{-1}(j)}$. It follows from this that the pullback of the circle bundle ${S}_{\lambda ,j}$ along ${\sigma}_{\ast}$ is the circle bundle ${S}_{{\sigma}^{-1}(\lambda ),{\sigma}^{-1}(j)}$.
Corollary 4.5.
Proof.
Let us write $\mu =\lambda {[n+1]}_{c}$ and ${\mu}^{\prime}=\lambda {[n]}_{c}$. Lemma 4.3 then implies that the pullback of the Gibbons–Manton torus bundle ${\stackrel{~}{\mathcal{T}}}_{\mu}=\underset{j=1}{\overset{r+n+1}{\u2a01}}{S}_{\mu ,j}\to {F}_{r+n+1}({\mathbb{R}}^{3})$ along the stabilization map ${F}_{r+n}({\mathbb{R}}^{3})\to {F}_{r+n+1}({\mathbb{R}}^{3})$ is
In order to apply proposition 3.3 to prove the first part of theorem 4.1, we recall the following general fact about mondromy actions of fibrations.
Lemma 4.6.
Let $p:E\to B$ be a fibration over a based, path-connected space $B$ admitting a universal covering $\pi :\stackrel{~}{B}\to B$. Write $\stackrel{~}{p}:\stackrel{~}{E}\to \stackrel{~}{B}$ for the pullback of $p$ along $\pi $. Let $F$ denote the fibre of $p$ over the basepoint ${b}_{0}\in B$ and note that the fibre of $\stackrel{~}{p}$ over each point in ${\pi}^{-1}({b}_{0})\subset \stackrel{~}{B}$ is also canonically identified with $F$. Then the monodromy action ${\pi}_{1}(B)\to \text{hAut}(F)$ of $p$ is equal to
Proof of theorem 4.1 for ${\mathcal{T}}_{\lambda {[n]}_{c}}$.
We first assume that $\lambda =()$ and $r=0$, so that $\lambda {[n]}_{c}$ is the tuple $(c,c,\dots ,c)$ of $n$ copies of $c\ge 1$. We are now in the setting of proposition 3.3 with (3.4) = (4.2), (3.5) = (4.3), (3.6) = (2.8) and $Z={S}^{1}$.^{4}
To complete the proof under this assumption, it suffices to check the two hypotheses of proposition 3.3. The first hypothesis says that the monodromy ${\pi}_{1}({C}_{n}({\mathbb{R}}^{3}))\to \text{hAut}({T}^{n})$ of the Gibbons–Manton torus bundle (2.8) is the obvious permutation action on the circle factors of the torus ${T}^{n}$. To check this property, we use lemma 4.6. In our setting, the universal covering of ${C}_{n}({\mathbb{R}}^{3})$ is ${F}_{n}({\mathbb{R}}^{3})$ and the pullback of ${\mathcal{T}}_{\lambda {[n]}_{c}}\to {C}_{n}({\mathbb{R}}^{3})$ is ${\stackrel{~}{\mathcal{T}}}_{\lambda {[n]}_{c}}\to {F}_{n}({\mathbb{R}}^{3})$. The deck transformation action of ${\pi}_{1}({C}_{n}({\mathbb{R}}^{3}))\cong {\Sigma}_{n}$ sends a loop (permutation) $\sigma $ to the obvious automorphism ${\sigma}_{\ast}$ of the ordered configuration space ${F}_{n}({\mathbb{R}}^{3})$. By remark 4.4, the action of ${\sigma}_{\ast}$ by pullback on Gibbons–Manton circle factors sends ${S}_{\lambda {[n]}_{c},j}$ to ${S}_{\lambda {[n]}_{c},{\sigma}^{-1}(j)}$ (here we use the fact that $\lambda {[n]}_{c}=(c,c,\dots ,c)$, so ${\sigma}^{-1}(\lambda {[n]}_{c})=\lambda {[n]}_{c}$). Hence ${\sigma}_{\ast}$ simply permutes the different circle factors in the Gibbons–Manton torus bundle; in particular its action on the torus fibre simply permutes the different copies of ${S}^{1}$, as required.
The second hypothesis of proposition 3.3 says that the restriction of the lifted stabilization map (4.3) to the fibres over the basepoints is the natural inclusion ${T}^{n}\to {T}^{n+1}$. This is immediate by construction of the lifted stabilization map: it is given (before quotienting by symmetric groups and therefore also afterwards) by including into a direct sum with a (trivial) circle bundle and then a pullback of bundles.
Proposition 3.3 therefore implies that the stabilization map ${\mathcal{T}}_{\lambda {[n]}_{c}}\to {\mathcal{T}}_{\lambda {[n+1]}_{c}}$ induces isomorphisms on homology in all degrees $\le n/2-1$ with integral coefficients and in all degrees $\le n/2$ with field coefficients, under our assumption that $\lambda =()$.
To complete the proof of theorem 4.1 for ${\mathcal{T}}_{\lambda {[n]}_{c}}$ we deduce the general case from the special case $\lambda =()$ that we have just proven. To do this, we first observe that the constructions and results so far generalize directly to Gibbons–Manton torus bundles with fixed points. In this setting, we consider the subspace of the configuration space ${C}_{\lambda ,n}({\mathbb{R}}^{3})$ where the $\lambda $-partitioned $r$-point configuration $\mathbf{\text{x}}$ is fixed and the unordered $n$-point configuration is free to move in the complement of $\mathbf{\text{x}}$; in other words, we consider the fibre of the projection ${C}_{\lambda ,n}({\mathbb{R}}^{3})\to {C}_{\lambda}({\mathbb{R}}^{3})$ over $\mathbf{\text{x}}\in {C}_{\lambda}({\mathbb{R}}^{3})$. Let us denote this subspace by ${C}_{\lambda ,n}({\mathbb{R}}^{3};\mathbf{\text{x}})$ and consider the restriction of ${\mathcal{T}}_{\lambda {[n]}_{c}}\to {C}_{\lambda ,n}({\mathbb{R}}^{3})$ to ${C}_{\lambda ,n}({\mathbb{R}}^{3};\mathbf{\text{x}})$, which we denote by ${\mathcal{T}}_{\lambda {[n]}_{c}}{|}_{\mathbf{\text{x}}}$. The difference between this setting and the $\lambda =()$ setting considered above is that (1) the unordered $n$-point configuration now lies in ${\mathbb{R}}^{3}\setminus \mathbf{\text{x}}$, (2) there are $r$ additional Gibbons–Manton circle factors encoding the pairwise interactions of the fixed points $\mathbf{\text{x}}$ with the free points, and (3) the $n$ Gibbons–Manton circle factors that encode the pairwise interactions of the $n$ free points with each other are now modified to also take into account their interactions with the fixed points $\mathbf{\text{x}}$. The arguments above generalize directly to this setting and prove that restricted stabilization maps
Remark 4.7.
For unordered configuration spaces, the stabilization maps ${C}_{n}({\mathbb{R}}^{3})\to {C}_{n+1}({\mathbb{R}}^{3})$ have the additional property that they are split-injective on homology. This is essentially a consequence of the existence of forgetful maps ${F}_{n}({\mathbb{R}}^{3})\to {F}_{r}({\mathbb{R}}^{3})$ at the level of ordered configuration spaces that forget the last $n-r$ points of a configuration. Using these maps, standard techniques using transfer maps (see [15] or [20]) imply split-injectivity on homology for stabilization maps of unordered configuration spaces. We record here the observation that the forgetful maps
More informally, one could say that the reason why we cannot naturally lift forgetful maps to Gibbons–Manton torus bundles is because of the non-local nature of the additional circle parameters: each circle parameter is associated with all configuration points simultaneously, since it encodes the pairwise interactions of one of the points with all of the others. Thus, there is no well-defined way of forgetting a subset of the configuration points in the presence of these non-local parameters.
(b) Changing the fibre
For a sequence of spaces $\mathbf{\text{Z}}=\{{Z}_{1},{Z}_{2},\dots \}$, we will consider the family of finite products of the form ${Z}_{\lambda}={Z}_{{k}_{1}}\times \cdots \times {Z}_{{k}_{r}}$ for tuples $\lambda =({k}_{1},\dots ,{k}_{r})$ of positive integers. If each ${Z}_{i}$ is a $G$-space for some topological group $G$, we consider each ${Z}_{\lambda}$ as a $G$-space via the diagonal action.
Definition 4.8.
Let $\mathbf{\text{Z}}$ be a sequence of ${S}^{1}$-spaces and let $\lambda =({k}_{1},\dots ,{k}_{r})$. Let ${\stackrel{~}{\mathcal{T}}}_{\lambda}(\mathbf{\text{Z}})$ be the total space of the fibre bundle obtained from the principal ${T}^{r}$-bundle ${\stackrel{~}{\mathcal{T}}}_{\lambda}$ by the Borel construction:
In particular, we have ${\mathcal{I}}_{\lambda}={\mathcal{T}}_{\lambda}(\mathbf{\text{Z}})$ for $\mathbf{\text{Z}}=\{{\mathcal{M}}_{1},{\mathcal{M}}_{2},{\mathcal{M}}_{3},\dots \}$. We now prove:
Theorem 4.9.
For any sequence $\mathbf{\text{Z}}=\{{Z}_{1},{Z}_{2},\dots \}$ of path-connected ${S}^{1}$-spaces, there are natural stabilization maps
Theorem 4.1 corresponds to two special cases of theorem 4.9, namely the sequences $\{{S}^{1},{S}^{1},\dots \}$ and $\{{\mathcal{M}}_{1},{\mathcal{M}}_{2},\dots \}$ of ${S}^{1}$-spaces. It therefore remains only to prove theorem 4.9.
Proof of theorem 4.9.
The proof is a direct generalization of the proof of theorem 4.1 for ${\mathcal{T}}_{\lambda {[n]}_{c}}$, so we just explain the differences. First of all, the lifts of the stabilization maps exist by the proof of corollary 4.5, where we additionally apply the (functorial) Borel construction to the outer square of (4.6) before quotienting by the symmetric group actions.
We begin by assuming that $\lambda =()$ and $r=0$, so that $\lambda {[n]}_{c}=(c,c,\dots ,c)$ where there are $n$ copies of $c\ge 1$. We are therefore in the setting of proposition 3.3 with $Z={Z}_{c}$. The two hypotheses of that proposition are satisfied by the same argument as in the proof of theorem 4.1 for ${\mathcal{T}}_{\lambda {[n]}_{c}}$, together with the evident observation that applying the Borel construction that replaces each circle factor in the fibre with the ${S}^{1}$-space ${Z}_{c}$ has the effect, on fibres, that permutation maps ${T}^{n}\to {T}^{n}$ and natural inclusions ${T}^{n}\to {T}^{n+1}$ are sent to the corresponding permutation maps ${({Z}_{c})}^{n}\to {({Z}_{c})}^{n}$ and natural inclusions ${({Z}_{c})}^{n}\to {({Z}_{c})}^{n+1}$. Thus proposition 3.3 completes the proof in the case $\lambda =()$.
This generalizes to Gibbons–Manton $\mathbf{\text{Z}}$-bundles with fixed points exactly as for Gibbons–Manton torus bundles with fixed points, and one may then deduce the general case of the theorem from this by a spectral sequence comparison argument applied to the analogue of the diagram (4.8).
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Authors' contributions
M.P. and U.T.: conceptualization, investigation, methodology, writing—original draft, writing—review and editing.
Both authors gave final approval for publication and agreed to be held accountable for the work performed therein.
Conflict of interest declaration
We declare we have no competing interests.
Funding
The first author was partially supported by a grant of the Romanian Ministry of Education and Research, CNCS—UEFISCDI, project no. PN-III-P4-ID-PCE-2020-2798, within PNCDI III.
Acknowledgements
The first author is grateful to Michael Singer for introducing him to asymptotic monopole moduli spaces, and for asking the question of whether they are homologically stable. The authors are grateful to the anonymous referee for their careful reading of and helpful comments on an earlier version of this paper.
Footnotes
1 Although full details of its (recursive) construction are deferred to forthcoming work of the same authors.
2 In [7], the theory is developed more generally for configuration spaces with (local) labels in a space $X$. We will not need this level of generality here, so we will suppress it (equivalently, we take $X$ to be the one-point space).
3 This is denoted $\Sigma $ in [7], but we use the more common notation $\text{FI}\mathrm{\u266f}$.
4 Proposition 3.3 requires us to fix a basepoint on $Z={S}^{1}$. This may initially appear problematic, since the circle fibres of the Gibbons–Manton circle factors (definition 2.3) cannot be given consistent basepoints, since the Gibbons–Manton circle factors do not admit global sections. However, proposition 3.3 only requires a choice of basepoint on a single fibre, namely the fibre over the base configuration, so this issue does not arise.
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