Homology stability for asymptotic monopole moduli spaces

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

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 $M$.

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 ${\tilde{\mathcal{T}}}_{\lambda }$. The Gibbons–Manton configuration space is the quotient space ${\mathcal{T}}_{\lambda }={\tilde{\mathcal{T}}}_{\lambda }/{\Sigma }_{\lambda }$. 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 ${\tilde{\mathcal{I}}}_{\lambda }$ to be the total space of the fibre bundle associated with the principal $T^r$-bundle ${\tilde{\mathcal{T}}}_{\lambda }$ 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 ${\tilde{\mathcal{I}}}_{\lambda }$ 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 ${\mathcal{I}}_{\lambda }={\tilde{\mathcal{I}}}_{\lambda }/{\Sigma }_{\lambda }$. We have therefore made this replacement in definition 2.6. (The difference between ${\mathcal{I}}_{\lambda }$ and its finite covering space ${\tilde{\mathcal{I}}}_{\lambda }$ 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 ${\tilde{\mathcal{T}}}_{\lambda }$ 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 ${\mathcal{T}}_{\lambda }={\tilde{\mathcal{T}}}_{\lambda }/{\Sigma }_{\lambda }$ (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 ${\tilde{\mathcal{I}}}_{(1,\dots ,1)}={\tilde{\mathcal{T}}}_{(1,\dots ,1)}$. 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 ${\tilde{\mathcal{T}}}_{(1,\dots ,1)}$ 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.

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.

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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.

The stabilization map (4.2) lifts to (4.3).

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 ${\tilde{\mathcal{T}}}_{\mu }=\underset{j=1}{\overset{r+n+1}{⨁}}{S}_{\mu ,j}\to F_{r+n+1}({\mathbb{R}}^3)$ along the stabilization map $F_{r+n}({\mathbb{R}}^3)\to F_{r+n+1}({\mathbb{R}}^3)$ is

4.6

Lemma 4.6.

Let $p:E\to B$ be a fibration over a based, path-connected space $B$ admitting a universal covering $\pi :\tilde{B}\to B$. Write $\tilde{p}:\tilde{E}\to \tilde{B}$ 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 $\tilde{p}$ over each point in ${\pi }^{-1}(b_0)\subset \tilde{B}$ 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 ${\tilde{\mathcal{T}}}_{\lambda {[n]}_c}\to F_n({\mathbb{R}}^3)$. 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