Combinatorics of orbit configuration spaces
From a group action on a space, define a variant of the configuration space by insisting that no two points inhabit the same orbit. When the action is almost free, this “orbit configuration space” is the complement of an arrangement of subvarieties inside the cartesian product, and we use this structure to study its topology.
We give an abstract combinatorial description of its poset of layers (connected components of intersections from the arrangement) which turns out to be of much independent interest as a generalization of partition and Dowling lattices. The close relationship to these classical posets is then exploited to give explicit cohomological calculations akin to those of (Totaro ’96).
Key words and phrases:Dowling lattice, hyperplane arrangement, orbit configuration space
2010 Mathematics Subject Classification:Primary 05E18; Secondary 06A11, 52C35
1.1. Orbit configuration spaces
A fundamental topological object attached to a topological space is its ordered configuration space of distinct points in . Analogously, given a group acting freely on one defines the orbit configuration space by
These spaces were first defined in [XM97] and come up in many natural topological contexts, including:
Universal covers of when is a manifold with [XM97].
Classifying spaces of well studied groups, such as normal subgroups of surface braid groups with quotient [XM97].
Equivariant loop spaces of and [Xic02].
The current literature typically requires the action to be free, with main results relying on this assumption. For an action that is not free, one could simply throw out the set of singular points for the action and consider , where
the set of points fixed by a nontrivial group element. However, the excision can create more harm than good: e.g. when is a smooth projective variety, removing destroys the projective structure and causes mixing of Hodge weights in cohomology. In particular, having a projective structure makes a spectral sequence calculation more manageable (see Theorem D and §3.6). Furthermore, one is often interested in allowing orbit configurations to inhabit , e.g. in arrangements arising from type B and D root systems (see §3.4).
We propose an alternative approach: observe that inside , the orbit configuration space is the complement of an arrangement of subspaces. The cohomology can then be computed from the combinatorics of this arrangement and from . Furthermore, the wreath product of with the symmetric group , which we denote by , acts naturally on the space , and this induces an –action on the set and its complement . The induced action on can also be traced through the combinatorial computation.
1.2. Running assumptions and notation
For our study, we from hereon assume that and are finite sets, so that the arrangement is finite. Moreover, by a “space” we mean either CW-complex or an algebraic variety over an algebraically closed field.
When discussing cohomology below, we will always suppress the coefficients, as they never have a significant effect on the results. For a CW-complex, one may take the cohomology to mean singular cohomology with coefficients in any ring ; and for an algebraic variety, may be taken to mean -adic cohomology with coefficients in either or .
The combinatorics at play is the poset of layers: connected components of intersections from , ordered by reverse inclusion. This poset admits an abstract combinatorial description, that does not in fact depend on (only depending on the –set ) and it is of much independent interest. For example,
In the case of classical configuration spaces ( trivial), the poset is the lattice of set partitions of .
In the case that is a cyclic group acting on via multiplication by roots of unity, the poset is an instance of the Dowling lattice, described in [Dow73] as an analogue of the partition lattice which consists of partial –partitions of .
In §2.1, we define the poset which specializes to these classical examples and discuss the natural action of the wreath product group .
Even though is not in general a lattice, it supports a myriad of properties that have been fundamental in the modern study of posets, since it is essentially built out of partition and Dowling lattices as indicated in the following theorem (Theorem 2.4.2):
Theorem A (Local structure of ).
For any with , the interval is isomorphic to a product
where denotes a partition lattice and denotes a Dowling lattice for some subgroup . In particular, every interval is a geometric lattice and has the homology of a wedge of spheres.
In the remainder of Section §2 we study the structure of these posets: In §2.3 we discuss their functoriality in the various inputs; in §2.4 we discuss local structure and prove Theorem A. Of particular interest is §2.5, where we discuss the characteristic polynomial: a fundamental invariant of a ranked poset, which is a common generalization of the chromatic polynomial of a graph, and the Poincaré polynomial of the complement of a hyperplane arrangement. We give a factorization of the characteristic polynomial into linear factors, generalizing a long list of special cases stretching back to Arnol’d and Stanley’s work on the pure braid group and the partition lattice.
Theorem B (Characteristic polynomial).
Let be a -set. Then
where is the Möbius function of the poset and is the minimum element. An analogous factorization for the case appears in Theorem 2.5.2 below.
Lastly, in §2.6, we consider the action of on the poset , and describe its orbits. In §2.7, we study their Whitney homology as a representation of : this invariant has proved important both for topology and to the abstract theory of posets, and will be later used when discussing orbit configuration spaces in §3.
As mentioned above, the poset arises naturally in the study of orbit configuration spaces, when we take to be the set of singular points for the action of on . Section §3 is devoted to studying the topology of these spaces, and relating it with the combinatorics of Section §2.
In §3.1 we define an arrangement in , whose complement is the orbit configuration space . Recall that the poset of layers of an arrangement is the collection of connected components of intersections from , ordered by reverse inclusion. This poset encodes subtle aspects of the topology of , as we shall see here (Theorem 3.2.5):
Theorem C (Poset of layers).
The poset of layers of the arrangement is naturally and –equivariantly isomorphic to the poset .
This description opens the door to cohomology calculations: considering a spectral sequence for complements of arrangements (see [Pet17] and also [Tot96, Bib16, Dup15]), one obtains a description of the –page in terms of the poset’s Whitney homology. Furthermore, when is a smooth projective algebraic variety, a weight argument guarantees that there could be at most one nonzero differential. Thus, in this case one is closer to getting a hand on the cohomology.
We summarize the explicit description of the spectral sequence machinery, following the simplifications that arise from our combinatorial analysis, in the following (Theorem 3.6.1):
Theorem D (Simplified spectral sequence).
There is a spectral sequence with
Here, the summands are indexed by poset elements of rank , and denotes the corresponding layer in . The term denotes the reduced cohomology of the order complex for the interval , and is therefore described explicitly by Theorem 2.4.2.
When is a smooth projective variety, the sequence degenerates at the –page, i.e. all differentials vanish past the first page.
Recall that certain invariants of can be computed already from any page of a spectral sequence converging to . These are the generalized Euler characteristics, or cut-paste invariants, discussed briefly in §3.7. Universal among those is the motive, i.e. the class in the Grothendieck ring of varieties. Our combinatorial calculations then give:
Theorem E (Motivic factorization).
Let act on an algebraic variety over and algebraically closed field as above, with singular set . Then in the Grothendieck ring ,
An analogous factorization for a free action is given in Theorem 3.7.1 below.
In particular, this gives a formula for the number of -points in for every divisible by . Alternatively, when is smooth, one gets a formula for the classical Euler number of .
In §3.5 we analyze the local structure of the arrangement , i.e. its germ at every point in . A surprising conclusion is that, to first order, the arrangement is isomorphic to a product of orbit configuration spaces for groups possibly different from . As a byproduct of our analysis we get a new proof of the following result.
Corollary F (Stabilizers on curves).
Suppose acts faithfully on a algebraic curve over some algebraically closed field . Then the stabilizer in of any smooth point is cyclic.
Lastly, our handle on the combinatorics of these arrangements can be exploited to understand what happens when one removes from a set other than the set of singular points . We consider this more general case in §3.4, but note now that all of our theorems hold true for these spaces as well.
For example, when is a –invariant subset of , the group now acts on with nontrivial stabilizers. The resulting orbit configuration space space is the complement in of a subarrangement of . The new poset of layers is a subposet of , which inherits many properties from to which our study applies. These types of arrangements arise naturally, e.g. from roots systems in , and elliptic curves (see §3.4).
An extended abstract of this work will appear in the FPSAC (2018) proceedings volume of Séminaire Lotharingien Combinatoire. A follow-up paper by the same authors will continue this work in the realm of representation stability.
The authors would like to thank Emanuele Delucchi, Graham Denham, and John Stembridge for many useful conversations and insights that helped shape this paper.
2. A generalization of Dowling lattices
In [Dow73], Dowling defined a family of lattices dependent on a positive integer and a finite group . We recall his construction here.
Let . A partial -partition of is a set consisting of a partition of the subset along with projectivized -colorings: functions , defined up to the following equivalence: and are equivalent if there is some for which . The reader might benefit from thinking of such an equivalence class of colorings as a point in projective space
The zero block of a partial -partition of is the set .
We take the convention of using an uppercase letter for a set, the corresponding lowercase letter for the function , and for the equivalence class of .
The Dowling lattice is the set of partial –partitions of . We will consider the elements of as ordered pairs where is a partial –partition and is its zero block. This set is a lattice with partial order determined by the following covering relations:
where with for some , and
The lattice has rank function given by , where is the number of blocks in the partition .
While it is not necessary to record the zero block in an element of , we do so because it is useful in understanding our generalization which involves adding a coloring to the zero block by some finite -set.
2.1. Introducing the posets
Let be a finite group acting on a finite set .
Definition 2.1.1 (The –Dowling poset).
Let be the set of ordered pairs where is a partial –partition of and is an –coloring of its zero block, i.e. a function .
To denote an element we will extend the standard notation of set partitions, as illustrated by the following example:
denotes the partial set partition with projectivized colorings and respectively, and zero block colored by the function .
The set is partially ordered with similar covering relations, given by either merging two blocks or coloring one by .
where with for some , and
where is the extension of to given on by a composition
for some –equivariant function .
Just as with the Dowling lattice, the poset is ranked with the rank of given by .
When coloring a block of , the –equivariant function is determined by a choice of , where is the identity in . Then one can extend to by setting for .
Recall the wreath product , sometimes denoted by , is the semidirect product of with the symmetric group . It acts on the –Dowling poset as follows. Let and . Then we have where
with zero block ,
is given by , and
is given by .
We leave it as an exercise to the reader to verify that the action preserves the order.
While it is convenient to consider (partial) partitions of the set , it will sometimes prove to be more convenient to consider partitions of any finite set , for example in Theorem 2.4.2. That is, one could define as the set of partial –partitions of whose zero block is colored by . In the case that , we have , and in general when we have . Note that the latter isomorphism depends on the choice of bijection .
Here we introduce the primary examples, which will be carried throughout this paper. The first describes the partition and Dowling lattices as specializations of –Dowling posets.
The Dowling lattice is equal to whenever consists of a single point. The lattice of set partitions of can be realized with the trivial group and no zero block, . As in [Dow73, Thm. 1(e)], we also have .
Example 2.2.2 (Type C Dowling poset).
Let act trivially on a finite set . In the case that is 2 or 4, the poset was studied in [Bib17]. Here, the poset describes the combinatorial structure of an arrangement arising naturally from the type C root system, which we will revisit in Example 3.2.3.
The Hasse diagram for when is depicted in Figure 1.
Example 2.2.4 (Hexagonal Dowling poset).
Let , which we identify with the group of th roots of unity . Let us consider acting on the set so that the action of the generator is by the permutation given by cycle notation . This poset also arises from an arrangement, which we will revisit in Example 3.3.1 below.
Example 2.2.5 (Square Dowling poset).
Let act on the set , where the action of on is by the permutation . As with the previous examples, this is associated to an arrangement which we revisit in Example 3.3.2.
The first property we state for the posets is that it behaves naturally with respect to changes in the inputs of , , and . This proposition is straightforward to verify.
Proposition 2.3.1 (Functoriality).
The –Dowling posets are functorial in the following ways:
The two inclusions induce a –equivariant injective map of ranked posets defined by
Here the sets and could be replaced by any two finite sets.
In particular, multiplying by gives a canonical equivariant injection and exhibits as a functor, as stated next.
If is an injective map of sets, then there is an injective map of ranked posets defined by
where is defined via for each .
If is a –equivariant map of sets, then there is a –equivariant map of ranked posets defined by
Furthermore, if is surjective (resp. injective) then is also surjective (resp. injective).
If is a group homomorphism and acts on a set (hence also inducing an action of on ), then there is a –equivariant map of ranked posets defined by
where is defined via for each . Furthermore, if is surjective (resp. injective) then is also surjective (resp. injective).
2.4. Local structure of the poset
In this section, we give an explicit description of the intervals inside of the –Dowling posets. The beauty of the local structure (the intervals) is that we can view our posets as being built out of partition and Dowling lattices, which we denote by and .
While the local structure of the poset is familiar, it is important to note that the global structure is more complicated. When , the poset is not even a (semi)lattice: least upper bounds and greatest lower bounds need not exist. While has a (unique) minimum element given by the partition of into singleton blocks, it may have several maximal elements corresponding to different –colorings of .
The following theorem describes the intervals in . It is a generalization of Dowling’s [Dow73, Theorem 2 and Corollary 2.1], which are the case with . The surprising consequence of Theorem 2.4.2 below, is that when the local picture is sensitive to the orbits and stabilizers of . As mentioned in Remark 2.1.3, this theorem is most naturally stated by considering for a general finite set (not just ); we discuss this further in Remark 2.4.3 below.
Theorem 2.4.2 (Local structure).
Let be a finite set with an action of a finite group , and let denote its set of –orbits. For each orbit , pick a representative and let be the stabilizer of in .
For , we have
Furthermore, if , then
where is the set of blocks for which and is the set of blocks for which .
In particular, every closed interval is a geometric lattice.
We will prove parts (1) and (2) of Theorem 2.4.2 below; part (3) follows by combining these two. To qualitatively explain part (1), recall that the covering relations state that an element lies under if the partition is a refinement of the partition , possibly excising blocks away from the zero block . In particular, defines a partition of each block in , and furthermore includes a partial partition of the zero block. To qualitatively explain part (2), recall that the covering relation allows one to merge existing blocks, or throw entire blocks into the zero block. Thus an element above is determined by specifying blocks to be merged and the –ratios between their –colorings, and by coloring the remaining blocks by .
The isomorphism is not canonical; we make the following choices. For each , fix a representative with stabilizer subgroup . Then for each element pick a ‘transporter’ so that .
We define a map from to the product by an assignment
The first tuple is simple: for a block , define a partition of by
For the second tuple start with defining for every a partial partition of by
For each , the coloring relation in the definition of shows that we may pick a representative such that for each . Since , we can define a –coloring with .
To describe the inverse map, consider a pair of tuples . We recover a partial partition of by unioning all of these (partial) partitions,
and obtain -colorings as follows. If , then inherit from . Otherwise is a block of a partial partition. Represent its -coloring by and recolor by . Lastly, inherit the -coloring of the zero block from that of . These constructions are clearly inverses. ∎
This isomorphism is also not canonical: Let us write and pick a representative for each .
Then for we will construct an element of . For , let
Now, pick a representative for . By the covering relations, we have that for each there is such that . This defines a function ; a different representative map for would define a function in the same equivalence class of . The collection is a partial -partition of the set .
We also have that where is a union of blocks from and . In fact, the zero block of is . Since for some (unique) -equivariant , let us color so that . Then , and one can recover from this data. ∎
Example 2.4.4 (Hexagonal Dowling Poset).
Recall from Example 2.2.4 the Hexagonal Dowling poset , where . The orbits of are , , , so that we have with stabilizers , , and .
One already finds intervals that factor as Dowling lattices of different ranks and of different groups when : consider given by
Next, the element given by is mapped under the isomorphism to
Moreover, and the element is mapped under this isomorphism to .
2.5. Characteristic polynomial
A fundamental invariant attached to a ranked poset is its characteristic polynomial. Recall (and see [OS80]) that when specialized to intersection lattices of hyperplane arrangements this polynomial gives a close relative of the Poincaré polynomial of the complement, and that in the further special case of graphical arrangements it computes the chromatic polynomial of the graph. The roots of this polynomial carry subtle information, e.g. for reflection arrangements it encodes the exponents of the Coxeter group (see [Bri73]). In Theorem 2.5.2 below we factorize the polynomial associated with .
The factorization formula in Theorem 2.5.2 specializes to that computed by Dowling [Dow73, Thm. 5] when and by Ardila, Castillo, and Henley [ACH15, Thm. 1.18] when and . The same formula for the partition lattice is well-known and goes back to Arnol’d [Arn69] and Stanley [Sta72]. This is the special case when is trivial and is empty.
Recall that the characteristic polynomial of a ranked poset with minimum element is defined by
where is the Möbius function. In [Sta72] Stanley defined the notion of a supersolvable lattice, encompassing the cases of partition and Dowling lattices. There he showed that for such lattices, a partition of the atoms gives a factorization of their characteristic polynomial. We thus proceed by describing the atoms, generalizing Corollaries 1.1 and 1.2 of [Dow73].
The rank-one elements (or atoms) of are
where and , corresponding to the partial -partition whose only non-singleton block is with -coloring ; and
, where and , corresponding to the partial -partition with zero block colored by and the rest are singleton blocks.
Moreover, if has rank , then it is covered by
elements of rank .
Theorem 2.5.2 (Characteristic polynomial).
If is a nonempty finite set, then
If , then
The authors thank Emanuele Delucchi for suggesting this method of proof.
First assume that is nonempty. For each , define