Spherical posets from commuting elements

Cihan Okay

doi:10.1515/jgth-2018-0008

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Spherical posets from commuting elements

Cihan Okay

Veröffentlicht/Copyright: 28. März 2018

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Aus der Zeitschrift Journal of Group Theory Band 21 Heft 4

Abstract

In this paper, we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial p-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of r-spheres where 2⁢r≥4 is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles as introduced in [2].

1 Introduction

The motivation behind this work is initiated by our interest in a certain filtration of the classifying space BG of a group G introduced in [1]. This filtration consists of subspaces B⁢(q,G)⊂B⁢G for q≥2 defined as geometric realizations of certain simplicial spaces. When q=2, the set of n-simplices is given by the set of pairwise commuting n-tuples of group elements. As shown in [13, 14] for extraspecial p-groups, B⁢(2,G) has non-trivial higher homotopy group. In this paper, we completely determine the homotopy type of the universal cover of B⁢(2,G) when G is an extraspecial p-group. The space B⁢(2,G) is a classifying space for transitionally commutative principal G-bundles [2, Theorem 2.2]. Therefore our result implies the existence of non-trivial transitionally commutative principal G-bundles over the r-sphere 𝕊r where 2⁢r is the rank of the Frattini quotient of G. Another motivation is to understand the cohomology ring of B⁢(2,G) in order to gain a better understanding of the cohomology ring of the group G. The case of extraspecial p-groups is particularly interesting. Its cohomology ring in mod p coefficients is completely determined by Quillen [15] when p=2. For odd p only partial results are known, see [4]. We believe that a description of H*⁢(B⁢(2,G),𝔽p) may help to gain more insight into the cohomology rings of extraspecial p-groups.

Let E⁢(2,G)→B⁢(2,G) denote the pull-back of the universal principal G-bundle E⁢G→B⁢G along the inclusion B⁢(2,G)⊂B⁢G. It can be shown that E⁢(2,G) is homotopy equivalent to the nerve of the poset

𝒞G⁢𝒜⁢(G)={g⁢A∣g∈G⁢ and ⁢A⊂G⁢ is an abelian subgroup}

ordered under the inclusion relation. In order to determine the homotopy type of the universal cover of B⁢(2,G) we study the coset poset 𝒞G⁢𝒜⁢(G). The latter is weakly equivalent to a simpler coset poset 𝒞V⁢ℐ⁢(V) where V is a vector space over the finite field 𝔽p with a non-degenerate alternating bilinear form 𝔟, and ℐ⁢(V) is the poset of isotopic subspaces of V. Similar posets associated to a collection of subgroups are studied extensively by various authors [19, 12, 18, 20, 6]. The techniques used in this paper span a variety of tools used in the study of topology of partially ordered sets of subgroup collections. In particular, there are two main ingredients, namely a combination of the methods used by Vogtmann in [20] and by Quillen in [18]. Our main result is the following.

Theorem 1.1.

Let E be an extraspecial p-group. There is a fibration sequence

⋁d⁢(p,r)𝕊r→B⁢(2,E)→B⁢π

where 2⁢r≥4 is the rank of the Frattini quotient of E,

(-1)r⁢d⁢(p,r)+1

=(-1)r⁢p2⁢r+1+r2+∑j=1r(-1)r-j⁢p2⁢r+1-j+(r-j)2⁢(∏t=0j-1p2⁢r-t-ptpj-pt)

and π is the kernel of the multiplication map

1→π→E×E→E/[E,E]→1.

There are two important consequences. The first is the existence of non-trivial transitionally commutative E-bundles over 𝕊r as constructed in Theorem 6.6. We also give an explicit construction of such a bundle. Another immediate application is a description of the cohomology of B⁢(2,E) in low degrees.

Corollary 1.2.

Let R denote a commutative ring, and E an extraspecial p-group. Then there is an isomorphism

Hi⁢(B⁢(2,E),R)≅Hi⁢(π,R) for ⁢i<r

where 2⁢r≥4 is the rank of the Frattini quotient of E and π denotes the fundamental group of B⁢(2,E).

We note that π is completely determined in [13, 14]. It is a central extension of E by a cyclic group of order p. For p=2 it splits as a direct product E×ℤ/2 hence its cohomology is known as a consequence of [15].

The structure of the paper is as follows. We start with preliminary results on the homotopy theory of posets and introduce coset posets in Section 2. In Section 3, we describe the decomposition theorem (Theorem 3.4) for coset posets. Next we specialize to the poset of abelian subgroups of extraspecial groups and prove some preliminary results in Section 4. The main result of the paper is proved in Section 5 in which we prove that the universal cover of 𝒞V⁢ℐ⁢(V) is spherical when G is an extraspecial p-group. Finally, in Section 6, we apply our results to B⁢(2,G).

2 Preliminaries

In this section, we recall some results on the homotopy theory of posets. We are concerned with special types of posets called coset posets. They arise naturally in the study of homotopy colimits of classifying spaces.

2.1 Homotopy theory of posets

Let 𝒳 denote a poset. We will regard a poset as a category with morphisms given by x→y whenever x≤y. Notationally we usually do not distinguish the poset 𝒳 from its nerve N⁢(𝒳) when talking about topological properties. A poset map f:𝒳→𝒴 can be regarded as a functor between the associated categories. A very useful notion associated to a map of posets is its fiber. The fiber of f over an object y∈𝒴 is defined by

f≤y={x∈𝒳∣f⁢(x)≤y},

and dually

f≥y={x∈𝒳∣f⁢(x)≥y}.

When interpreted as a category, f≤y corresponds to the comma category f↓y, and f≥y corresponds to y↓f. If f is the identity map, we simply write 𝒳≥x for f≥x. Other variations are 𝒳>x, 𝒳≤x, and 𝒳<x. Some constructions we will need are the following. The join 𝒳∗𝒴 of the posets 𝒳 and 𝒴 is defined to be the poset whose underlying set is the disjoint union 𝒳∐𝒴. The ordering agrees with the given orderings on 𝒳 and 𝒴, and x≤y for every x∈𝒳 and y∈𝒴. The (unreduced) suspension Σ⁢𝒳 is defined to be the join {0,1}∗𝒳 where {0,1} has only the reflexive relation at each object.

Next we recall some results on the homotopy theory of posets due to Quillen. Let f,g:𝒳→𝒴 be two poset maps. If the maps satisfy the condition that f⁢(x)≤g⁢(x) (or dually f⁢(x)≥g⁢(x)) for all x∈𝒳, then they are homotopic, which will be denoted by f≃g. This property implies that a poset with an initial (or terminal) object is contractible since the identity map will be homotopic to the constant map. Another fact we will use is a version of Quillen’s Theorem A for posets.

Theorem 2.1 ([18, Proposition 1.6]).

Let f:X→Y be a map of posets. If the fiber f≥y (or f≤y) is contractible for all objects y in Y, then f is a homotopy equivalence.

The dimension dim⁡(𝒳) of a poset 𝒳 is defined to be the supremum of the integers k such that there is a chain x0<x1<⋯<xk. A poset of dimension n is said to be n-spherical if it (or rather its nerve) is (n-1)-connected. In the study of spherical posets, the following theorem of Quillen is very useful. This version is the dual of the one stated in [18, Theorem 9.1].

Theorem 2.2 ([18, Theorem 9.1]).

Let f:X→Y be a map of posets. If Y is (dim⁡Y)-spherical, f≥y is dim⁡(Y≥y)-spherical and Y<y is dim⁡(Y<y)-spherical for all y, then X is (dim⁡Y)-spherical.

2.2 Coset posets

Let G be a finite group and ℱ a collection of subgroups of G. We define the coset poset associated to the collection ℱ as a set

𝒞G⁢ℱ={g⁢A∣A∈ℱ⁢ and ⁢g∈G}

and regard it as a poset ordered under inclusion. There is a morphism g⁢A→g′⁢A′ whenever gA is contained in g′⁢A′ as a set. Note that 𝒞G⁢ℱ is a poset with G-action given by left translation and its nerve is a G-space, i.e. a simplicial set with G-action. We also define a relative version. Let H⊂G be a subgroup. Given a (left) coset g⁢H⊂G we define

𝒞g⁢H⁢ℱ={g⁢h⁢A∣A∈ℱ⁢ and ⁢g⁢h∈g⁢H}.

The poset structure is similarly induced by inclusion as sets, and g⁢H⁢g-1 acts by left translation. An element x∈G induces a map x:𝒞g⁢H⁢ℱ→𝒞x⁢g⁢H⁢ℱ defined by g⁢h⁢A↦x⁢g⁢h⁢A.

We describe a basic result which usually allows us to work with smaller collections of subgroups. Let ℱ∩H denote the collection {A∩H∣A∈ℱ}. There is an H-equivariant map

iH:𝒞H⁢ℱ→𝒞H⁢ℱ∩H

defined by h⁢A↦h⁢(A∩H). Under suitable conditions this map turns out to be a weak equivalence. In fact, it becomes a weak H-equivalence. This means that the induced map on fixed points is a weak equivalence for all subgroups.

Definition 2.3.

We say ℱ is H-stable if for all A∈ℱ the intersection A∩H is also in ℱ.

For example the collection of abelian subgroups satisfies this property with respect to any subgroup.

Proposition 2.4.

If F is H-stable, then iH:CH⁢F→CH⁢F∩H is a weak H-equivalence.

Proof.

Note that by assumption we have ℱ∩H⊂ℱ. Let K⊂H and consider the restriction of iH to the fixed points iH:(𝒞H⁢ℱ)K→(𝒞H⁢ℱ∩H)K. If the latter fixed point set is not empty, then the fiber over a coset h⁢B∈𝒞H⁢ℱ∩H fixed under the action of K is given by

(iH)≥h⁢B={h′⁢A∈(𝒞H⁢ℱ)K∣h′⁢(A∩H)⊃h⁢B}

={h⁢A∈(𝒞H⁢ℱ)K∣A∩H⊃B}

={h⁢A∈𝒞H⁢ℱ∣A∩H⊃B⁢ and ⁢K⊂h⁢A⁢h-1}.

Note that if h′⁢(A∩H)⊃h⁢B, then h-1⁢h′⁢(A∩H)⊃B. In particular, we have 1∈h-1⁢h′⁢(A∩H), that is, h′=h⁢x for some x∈A∩H. Therefore we have h′⁢A=h⁢A, and the first equality above follows. The fiber is contractible since hB is initial and the result follows from Theorem 2.1. ∎

2.3 Homotopy colimit of classifying spaces

Our reference for homotopy colimits of classifying spaces is [13, §3]. Let 𝐒 denote the category of simplicial sets. The classifying space of a group can be seen as a functor B:𝐆𝐫𝐩→𝐒 from the category of groups to the category of simplicial sets. Let ℱ denote a collection of subgroups of G. We regard ℱ as a partially ordered set. Restricting the classifying space functor gives B:ℱ→𝐒. There is a fibration sequence

(2.1)hocolimℱ(G/-)→hocolimℱB→BG

induced by the inclusions B⁢A⊂B⁢G for each A∈ℱ, where (G/-):ℱ→𝐒 is the functor which sends A to the coset G/A regarded as a discrete simplicial set. The fiber can be described equivalently as follows. The homotopy colimit of the cosets G/A as A runs over the poset ℱ is isomorphic to the nerve of the coset poset

𝒞G⁢ℱ={g⁢A∣g∈G⁢ and ⁢A∈ℱ}

(see [10, Proposition 5.12]). The homotopy long exact sequence of the fiber sequence (2.1) gives an exact sequence

1→π1⁢𝒞G⁢ℱ→π1⁢hocolimℱ⁡B→G→1

of fundamental groups.

Proposition 2.5.

Assume that the collection F has an initial object. Then

π1⁢hocolimℱ⁡B≅colimA∈ℱ⁡A

where the colimit is taken in the category of groups.

Proof.

This is a consequence of [8, Corollary 5.1]. ∎

Let π denote the colimit of the groups A in ℱ. There is a commutative diagram of groups

which implies that the natural map iA is a monomorphism. Therefore we can regard subgroups in ℱ as subgroups of π, and talk about the cosets π/A. Consider the fibration sequence

hocolimℱ(π/-)→hocolimℱB→Bπ

induced by the inclusions B⁢A⊂B⁢π. Note that the fiber is a simply connected space, and it can be identified with the nerve of the coset poset

𝒞π⁢ℱ={g⁢A∣g∈π⁢ and ⁢A∈ℱ}.

We will study this object using homotopy theoretic methods for posets.

3 Decomposing coset posets

We will describe a decomposition of 𝒞G⁢ℱ as a homotopy colimit. For basic properties of homotopy colimits we refer to [9, 5, 11].

Let f:𝒳→𝒴 be a map of posets. Define a functor f≥-:𝒴op→𝐒 by sending an object y to the nerve of the fiber f≥y. Here (-)op denotes the opposite category. There is a natural map

(3.1)f~:hocolim⁡f≥-→𝒳

induced by the inclusions f≥y→𝒳. This map is a weak equivalence [11, IV §5.1]. We need some notation for the next result. Let ℒ⁢(G) denote the collection of all subgroups including the trivial subgroup and the group G. For a normal subgroup H we define ℱ∨H to be the collection {A∈ℱ∣A⁢H=G} where A⁢H⊂G denotes the subgroup generated by A and H. Let xH denote a coset in G. There is a map of posets

(3.2)θx⁢H:𝒞G⁢ℱ∨H→𝒞x⁢H⁢ℱ

defined by sending a coset yA to the coset xhA where y=x⁢h⁢a for some h∈H and a∈A.

Proposition 3.1.

Let 1→H→G→𝜋G¯→1 be an exact sequence of groups where G¯ is isomorphic to a cyclic group Cp of prime order p. Choose an element g∈G such that g¯=π⁢(g) generates G¯. Assume that F contains the trivial subgroup and a subgroup not contained in H. Then there is a weak equivalence

π~:hocolim𝒞Cp⁢ℒ⁢(Cp)op⁡π≥-→𝒞G⁢ℱ

and the functor π≥- can be identified as

π≥G¯=𝒞G⁢ℱ∨H 𝑎𝑛𝑑 π≥g¯t=𝒞gt⁢H⁢ℱ.

Proof.

The map π~ is obtained as follows. Let π⁢ℱ denote the poset

π⁢ℱ={π⁢(A)∣A∈ℱ}

of subgroups of G¯. There is an induced map of posets π:𝒞G⁢ℱ→𝒞G¯⁢(π⁢ℱ) defined by g⁢A↦π⁢(g⁢A). The decomposition (3.1) applied to π gives a weak equivalence

π~:hocolim⁡π≥-→𝒞G⁢ℱ

where the colimit is over the poset 𝒞Cp⁢ℒ⁢(Cp)op. By the assumptions on ℱ this poset can be identified with

𝒞G¯⁢(π⁢ℱ)op={g¯,g¯2,…,g¯p,G¯}

partially ordered under reverse inclusions. There are two types of fibers π≥G¯ and π≥g¯t where 1≤t≤p. Then 𝒞G⁢ℱ is the homotopy colimit of the diagram

(3.3)

where θgt⁢H for t=1,2,…,p denote the natural inclusions defined above. ∎

Example 3.2.

Let V be a vector space over 𝔽p, and let 𝒯⁢(V) denote the poset of proper subspaces. Choose a subspace W of codimension one. Since 𝒯⁢(V) is W-stable,

iW:𝒞W⁢𝒯⁢(V)→𝒞W⁢𝒯⁢(V)∩W

is a weak equivalence by Proposition 2.4. Moreover, 𝒯⁢(V)∩W is contractible since W is terminal. Therefore the decomposition in Proposition 3.1 for W=H gives

𝒞V⁢𝒯⁢(V)≃⋁p-1Σ⁢(𝒞V⁢𝒯⁢(V)∨W).

3.1 Homotopy sections

We make an assumption in addition to the set-up in Proposition 3.1. Let gt⁢ℱ∨H denote the poset of left translations {gt⁢A∣A∈ℱ∨H} regarded as a subposet of 𝒞G⁢ℱ∨H. Assume that there exists a map of posets

(3.4)s:𝒞H⁢ℱ→ℱ∨H

such that θH⁢s≃∗ and the composition

𝒞H⁢ℱ→𝑠ℱ∨H→g-1g-1⁢ℱ∨H→θH𝒞H⁢ℱ

is homotopic to the identity map. Here g-1:ℱ∨H→g-1⁢ℱ∨H is defined by A↦g-1⁢A, and θH is restricted to g-1⁢ℱ∨H. The restricted map sends g-1⁢A∈g-1⁢ℱ∨H to hA where g-1=h⁢a for some h∈H.

Lemma 3.3.

Given the map s in (3.4) there exists a homotopy section

sgk⁢H:𝒞gk⁢H⁢ℱ→𝒞G⁢ℱ∨H

of θgk⁢H such that θgk-1⁢H⁢sgk⁢H≃∗ for all 1≤k≤p .

Proof.

We define sgk⁢H to be the composite

sgk⁢H:𝒞gk⁢H⁢ℱ→g-k𝒞H⁢ℱ→𝑠ℱ∨H→gk-1gk-1⁢ℱ∨H⊂𝒞G⁢ℱ∨H

for 1≤k≤p. Then we have

θgl⁢H⁢sgk⁢H=gl⁢θH⁢gk-1-l⁢s⁢g-k≃{Idif ⁢l=k,∗if ⁢l=k-1.

where we have used θgl⁢H=gl⁢θH⁢g-l. This gives a sequence of sections

such that θgk-1⁢H⁢sgk⁢H≃0. ∎

For a poset 𝒳 we write H*⁢(𝒳) (and H~*⁢(𝒳)) for the (reduced) integral homology groups of the nerve of 𝒳. Given a map of posets f:𝒳→𝒴, the induced map in homology is denoted by f*:H*⁢(𝒳)→H*⁢(𝒴) (similarly for reduced homology groups).

Theorem 3.4.

Let 1→H→G→𝜋G¯→1 be an exact sequence of groups where G¯ is isomorphic to a cyclic group Cp of prime order p generated by g¯. Let F be a collection of subgroups containing the trivial subgroup and a subgroup not contained in H. Assume that there is a map s:CH⁢F→F∨H such that θH⁢g-1⁢s≃Id and θH⁢s≃∗. Then there is a split exact sequence

0→H~i⁢(𝒞G⁢ℱ)→H~i-1⁢(𝒞G⁢ℱ∨H)⊕(p-1)→𝜃⊕k=1pH~i-1⁢(𝒞gk⁢H⁢ℱ)→0,i≥1

where θ⁢(xt)=(θgt-1⁢H)*⁢(xt)-(θgt⁢H)*⁢(xt) for 1≤t<p.

This theorem is an immediate consequence of Lemma 3.3, the homotopy colimit decomposition (3.3), and the following result based on the Mayer–Vietoris sequence.

Proposition 3.5.

Let X0,X1,…,Xm-1 denote a sequence of subspaces of a space X such that their union gives X and any pairwise intersection Xk∩Xl with k≠l gives the same subspace denoted by Xint. Assume each inclusion ιk:Xint→Xk has a homotopy section sk such that ιk-1⁢sk≃∗ where k is written mod m. Then the associated Mayer–Vietoris sequence splits to yield short exact sequences

0→H~i⁢(X)→H~i-1⁢(Xint)⊕(m-1)→𝜃⊕k=0m-1H~i-1⁢(Xk)→0.

Proof.

There is a Mayer–Vietoris sequence

⋯→H~i-1⁢(Xint)⊕m-1→𝜃⊕kH~i-1⁢(Xk)→H~i-1⁢(X)→H~i-2⁢(Xint)⊕m-1→⋯

of reduced homology groups associated to the diagram of inclusions

One can obtain this sequence as a special case of the Bousfield–Kan spectral sequence [5, §XII.5.7]. The map θ sends an element x in the k-th summand to (ιk-1)*⁢(x)-(ιk)*⁢(x). We will show that the Mayer–Vietoris long exact sequence splits by constructing a section of θ. The sections sk can be put together to define a section of θ. Let Δ:Hi-1⁢(Xint)→Hi-1⁢(Xint)⊕m-1 denote the diagonal map. Then define

sθ:⊕k=0m-1H~i-1⁢(Xk)→H~i-1⁢(Xint)⊕m-1

by sθ⁢(αk)=(sk)*⁢(αk) if k>0 and sθ⁢(α0)=Δ⁢(s0)*⁢(α0). Then sθ is a section of θ. ∎

Corollary 3.6.

Assume that the map s exists as defined in Theorem 3.4. If CG⁢F∨H and CH⁢F are (n-1)-spherical, then CG⁢F is n-spherical.

Proof.

We set Xk=𝒞gk⁢H⁢ℱ and use the notation from Proposition 3.5. Van Kampen’s theorem gives an isomorphism

(3.5)π1(X)≅(∗kπ1(Xk))/∼

where the quotient relation is generated by (ιk)*⁢(x)∼(ιk-1)*⁢(x) for x∈π1⁢(Xint) and 0≤k≤m-1. We prove now that X is simply connected by a similar argument to that used in the proof of Proposition 3.5. Fix k and let xk be an element of π1⁢(Xk). Taking x=(sk)*⁢(xk), we see that xk=(ιk)*⁢(x)∼(ιk-1)*⁢(x)=1. Therefore the equivalence relation in (3.5) identifies every generator with the identity element, and as a result we have π1⁢(X)=1. Combining this observation with Theorem 3.4 gives the desired result. ∎

4 Coset poset of abelian subgroups of extraspecial groups

Let 𝒜⁢(G) denote the poset of abelian subgroups of G. We will study 𝒞G⁢𝒜⁢(G) when G is an extraspecial p-group. The universal cover in this case turns out to be spherical.

4.1 Extraspecial groups

An extraspecial p-group is a central extension of the form

0→ℤ/p→E→𝜈V→0

where V is an elementary abelian p-group of rank 2⁢r. The kernel of the extension is both the center and the commutator of the group [3, Chapter 8]. We will denote the center by Z⁢(E). The quotient V is also the Frattini quotient. The commutator induces a bilinear form on V: 𝔟⁢(v1,v2)=[ν-1⁢(v1),ν-1⁢(v2)]. A subspace I⊂V is called isotropic if the restriction of 𝔟 on I is zero. Let ℐ⁢(V) denote the collection of isotropic subspaces of V. There is a map of posets 𝒜⁢(E)→ℐ⁢(V) defined by A↦ν⁢(A) which induces a map of posets between the coset posets ν^:𝒞E⁢𝒜⁢(E)→𝒞V⁢ℐ⁢(V) defined by x⁢A↦ν⁢(x⁢A).

Proposition 4.1.

The induced map ν^:CE⁢A⁢(E)→CV⁢I⁢(V) is a weak equivalence.

Proof.

The fiber ν^≤v+I contains the coset ν-1⁢(v+I) as a terminal object. Hence the given map is a weak equivalence by Theorem 2.1. ∎

4.2 r=1 case

Consider the coset poset 𝒞V⁢ℐ⁢(V) when r=1. Then ℐ⁢(V) consists of all the one-dimensional subspaces and the zero subspace. The corresponding space is a one-dimensional connected space hence has the homotopy type of a wedge of circles. For example, the set of abelian subgroups for the quaternion group Q8 is given by {〈i〉,〈j〉,〈k〉,〈-1〉,1}. Under the quotient map ν:E→V the maximal abelian subgroups map to the one-dimensional subspaces, the center and the identity subgroup both map to the zero subspace. In effect the weak equivalence in Proposition 4.1 implies that to determine the homotopy type of the coset poset we can restrict to maximal abelians and their intersections.

4.3 Fundamental group

In [14, §4.2] the colimit of abelian subgroups of an extraspecial p-group is computed. Recall from Section 2.3 that this colimit is isomorphic to the fundamental group of the homotopy colimit of the classifying spaces BA of abelian subgroups A⊂E.

Proposition 4.2 ([14]).

Assume r≥2. There is an isomorphism of groups

ϕ:colimA∈𝒜⁢(E)⁡A→π

where π is the kernel of the multiplication map m⁢(e,e′)=e⁢e′⁢[E,E]:

1→π→E×E→𝑚E/[E,E]→1

and ϕ⁢(a)=(a,a-1) for a∈A.

Note that π is an extension of E by a cyclic group of order p. Using the isomorphism ϕ, we see that the homotopy long exact sequence of (2.1) gives

1→π1⁢𝒞E⁢𝒜⁢(E)→π→π1E→1

where π1 is the projection onto the first factor. Therefore we obtain the following.

Corollary 4.3.

Assume r≥2. There is an isomorphism

π1⁢𝒞E⁢𝒜⁢(E)≅ℤ/p.

4.4 Heisenberg group

Let H⁢(V) denote the Heisenberg group associated to the vector space V with the bilinear form 𝔟. This group is defined to be the set V×ℤ/p with the multiplication rule given by

(v1,t1)⁢(v2,t2)=(v1+v2,𝔟⁢(v1,v2)+t1+t2).

Let us identify Z⁢(E) with ℤ/p via an isomorphism.

Lemma 4.4.

There is a surjective homomorphism of groups defined by

φ:π→H⁢(V),(e,e′)↦(ν⁢(e),e⁢e′)

whose kernel consists of pairs (a,a-1) where a∈Z⁢(E).

Proof.

We check that φ is a group homomorphism. On one hand we have

φ⁢((e1,e1′)⁢(e2,e2′))=φ⁢(e1⁢e2,e1′⁢e2′)=(ν⁢(e1⁢e2),e1⁢e2⁢e1′⁢e2′)

and on the other hand

φ⁢((e1,e1′))⁢φ⁢((e2,e2′))=(ν⁢(e1),e1⁢e1′)⁢(ν⁢(e2),e2⁢e2′)

=(ν⁢(e1)⁢ν⁢(e2),[e1,e2]⁢e1⁢e1′⁢e2⁢e2′).

Note that both give the same result since

[e1,e2]⁢e1⁢e1′⁢e2⁢e2′=e1⁢e2⁢e1-1⁢e2-1⁢e1⁢e1′⁢e2⁢e2′=e1⁢e2⁢e1′⁢e2-1⁢e2⁢e2′=e1⁢e2⁢e1′⁢e2′

where we have used the fact that e1⁢e1′ belongs to the center Z⁢(E). To see that it is surjective observe that the images of the elements of the form (e,e-1) generate H⁢(V). The kernel is as described by definition of the map. ∎

Consider the projection pr1:H⁢(V)→V onto the first factor. An isotropic subspace I can be identified with an abelian subgroup of H⁢(V) via the map a↦(a,0). We can regard ℐ⁢(V) as a collection of subgroups of H⁢(V). First we need a preliminary result. Regard the classifying space as a functor B restricted on the posets 𝒜⁢(E) and ℐ⁢(V). We will compute the fundamental group of the homotopy colimit of B:ℐ⁢(V)→𝐒.

Proposition 4.5.

There is a natural isomorphism

π1⁢hocolimℐ⁢(V)⁡B≅H⁢(V).

Proof.

The natural map of posets 𝒜⁢(E)→ℐ⁢(V) given by A↦ν⁢(A) induces a map

hocolim𝒜⁢(E)⁡B→hocolimℐ⁢(V)⁡B

between the homotopy colimits. The homotopy fiber of this map is B⁢ℤ/p. This can be seen from the diagram

where we have used Proposition 4.1 to identify the horizontal fibers found in (2.1). Both homotopy colimits have the same universal cover and they differ by B⁢ℤ/p. Therefore all the sequences in the diagram are fibration sequences. Under the identifications of Proposition 4.2 and Corollary 4.3 consider the diagram of fundamental groups

where π¯ is the fundamental group of the homotopy colimit of B:ℐ⁢(V)→𝐒. The middle column is the extension corresponding to the homomorphism in Lemma 4.4 since the cyclic subgroup ℤ/p in π is precisely ker⁡φ. Therefore π¯ is isomorphic to H⁢(V) as desired. ∎

As an immediate consequence of this computation we have the following.

Corollary 4.6.

Assume r≥2 where 2⁢r is the dimension of V. Then the natural map

pr^1:𝒞H⁢(V)⁢ℐ⁢(V)→𝒞V⁢ℐ⁢(V)

induced by the projection pr1:H⁢(V)→V is the universal covering map.

Proof.

The projection map pr1:H⁢(V)→V induces a map of fibrations

from which we can conclude that the map between the fibers is the universal covering map. ∎

Next we summarize the relationship between the coset posets associated to 𝒜⁢(E) and ℐ⁢(V).

Proposition 4.7.

The homomorphism φ:π→H⁢(V) induces a diagram

between the universal covering maps whose fibers are given by a copy of Z/p.

Our aim is to prove the following result which will follow from the corresponding result (Theorem 5.6) for the coset poset 𝒞H⁢(V)⁢ℐ⁢(V) in view of Proposition 4.7.

Theorem 4.8.

Assume r≥2. The coset poset Cπ⁢A⁢(E) is r-spherical.

We leave the proof of this theorem to Section 5 and conclude this section with a formula of the Euler characteristic involving dimensions of Steinberg representations of symplectic groups.

4.5 Euler characteristic

The geometric realization of the poset ℐ∘⁢(V)=ℐ⁢(V)-{0} can be identified with the Tits building of the symplectic group Sp⁡(V). It is (r-1)-spherical by the Solomon–Tits theorem [19] where 2⁢r=dim⁡(V). The top dimensional homology affords the Steinberg representation of Sp⁡(V). We will describe the top dimensional homology of 𝒞H⁢(V)⁢ℐ⁢(V) as the kernel of a long exact sequence which consists of Steinberg representations for symplectic groups.

As in Section 2.3 the coset space 𝒞H⁢(V)⁢ℐ⁢(V) can be identified with the homotopy colimit of the functor HV/-:ℐ(V)→𝐒 which sends an isotropic subspace I to the coset H⁢(V)/I. There is a Bousfield–Kan spectral sequence [5, §XII.5.7] computing the integral homology of this homotopy colimit whose E2-page is given by

colimi⁡Hj⁢(H⁢V/I)⇒Hi+j⁢(𝒞H⁢(V)⁢ℐ⁢(V))

which collapses onto the j=0 axis since coset spaces are discrete. Therefore we have

Hk(𝒞H⁢(V)ℐ(V))≅colimkℤ[HV/-]

where ℤ[HV/-] is the functor ℐ⁢(V)→𝐀𝐛 which sends I to the free abelian group generated by the coset H⁢V/I. We will describe a spectral sequence to compute such derived colimits. For details of (the dual version of) this construction see [13, §4].

The derived colimits of a functor F:𝒫→𝐀𝐛 can be described as the homology H*⁢(𝒫,F) of a chain complex

Ck⁢(𝒫;F)=⊕p0<p1<⋯<pkF⁢(p0)

whose differential is induced by removing an object p0<p1<⋯<p^i<⋯<pk. We describe a nice filtration on the poset. For an object p∈𝒫 we define ht⁡(p)=-dim⁡(𝒫≥p). This yields a filtration on the chains

C*⁢(𝒫;Fr)⊂⋯⊂C*⁢(𝒫;F1)⊂C*⁢(𝒫;F0)

where the functors Fi are defined as follows:

Fi⁢(p)={0if ⁢ht⁡(p)>-i,F⁢(p)otherwise.

Associated to the filtration there is a spectral sequence in the eighth octant:

Ei,j1=⊕p∈𝒫ht⁡(p)=-iHi+j⁢((𝒫≥p,𝒫>p),F⁢(p))⇒Hi+j⁢(𝒫,F).

Now we apply this spectral sequence to the poset 𝒫=ℐ⁢(V) and the functor F=ℤ[HV/-]. The first page consists of a direct sum of homology groups

(4.1)Hi+j⁢((ℐ⁢(V)≥I,ℐ⁢(V)>I),F⁢(I))≅ℤ⁢[H⁢V/I]⊗Hi+j⁢(Σ⁢ℐ∘⁢(I⊥/I))

over isotropic subspaces I of height equal to -i. Note that ht⁡(I)=dim⁡(I)-r hence in fact the sum is over I of dimension r-i. In this case, the quotient I⊥/I is of dimension 2⁢i and ℐ∘⁢(I⊥/I) is (i-1)-spherical. Therefore the term in (4.1) is only non-zero if j=0 and the spectral sequence collapses to give a long exact sequence

0→ℤ⁢[H⁢V]⊗Hr-1⁢(ℐ∘⁢(V))→⊕I1ℤ⁢[V/I1]⊗Hr-2⁢(ℐ∘⁢(I1⊥/I1))

→⋯→⊕Ir-1ℤ⁢[V/Ir-1]⊗H0⁢(ℐ∘⁢(Ir-1⊥/Ir-1))→⊕Irℤ⁢[V/Ir]→0

whose homology computes the homology of 𝒞H⁢(V)⁢ℐ⁢(V). Here the direct sum runs over isotropic subspaces Ik of dimension k.

Corollary 4.9.

The Euler characteristic of CH⁢(V)⁢I⁢(V) is given by

(-1)r⁢d⁢(p,r)+1

=(-1)r⁢p2⁢r+1+r2+∑j=1r(-1)r-j⁢p2⁢r+1-j+(r-j)2⁢(∏t=0j-1p2⁢r-t-ptpj-pt).

Proof.

Let Nj denote the number of j-dimensional isotropic subspaces Ij in V, and Dj the dimension of the Tits building associated to the quotient Ij⊥/Ij. When j=r, the quotient is the zero space and we set Dr=1. Then the Euler characteristic is given by the alternating sum

∑j=0r(-1)r-j⁢|H⁢V/Ij|⁢Nj⁢Dj

where the number of isotropic subspaces is given by

Nj=∏t=0j-1p2⁢r-t-ptpj-pt

and we take Nj=1 if j=0. Note that the number of j linearly independent isotropic vectors is given by (p2⁢r-1)⁢(p2⁢r-1-p)⁢⋯⁢(p2⁢r-(j-1)-pj-1) since the dimension of the orthogonal complement of an isotropic subspace I is equal to 2⁢r-dim⁡(I). Dividing this product by the number of j linearly independent vectors in a j-dimensional vector space gives the formula for Nj. The dimension Dj of the Tits building of Ij⊥/Ij equals the order of the unipotent radical of the Borel subgroup of Sp⁡(Ij⊥/Ij) which is p(r-j)2 (see [22, §3.5.4]). ∎

5 Coset poset of isotropic subspaces

In this section, we focus on the coset poset 𝒞H⁢(V)⁢ℐ⁢(V) as introduced in Section 4. In Theorem 5.6 we prove that this poset is r-spherical where 2⁢r=dim⁡(V). We start with some results about subspace collections of vector spaces of arbitrary dimension.

5.1 Poset of proper subspaces

Let 𝒯⁢(V) denote the poset of proper subspaces of V, and 𝒯∘⁢(V)=T⁢(V)-{0}. The geometric realization of the latter poset is usually called the Tits building associated to the general linear group GL⁡(V). Its homotopy type is described by the Solomon–Tits theorem.

Proposition 5.1 ([19]).

Assume dim⁡V≥1. Then T∘⁢(V) is (dim⁡V-2)-spherical.

In his work on discrete series representations of the classical groups, Lusztig studied the poset 𝒞V⁢𝒯⁢(V). In [12, Theorem 1.9] he proves a homological version of the following.

Proposition 5.2.

The coset poset CV⁢T⁢(V) is (dim⁡V-1)-spherical.

This result is a special case of [6, Proposition 11] where it is extended to cosets of proper subgroups of a solvable group.

Let W be a subspace of V where codim⁡W=1. Recall that we have the following maps from Proposition 2.4 and (3.2):

iW:𝒞W⁢𝒯⁢(V)→𝒞W⁢𝒯⁢(W) and θW:𝒞V⁢𝒯⁢(V)∨W→𝒞W⁢𝒯⁢(V)

where we identified 𝒯⁢(V)∩W with 𝒯⁢(W) in the first map. Fix v∉W and define a map of posets

θv:𝒯⁢(V)∨W→𝒞W⁢𝒯⁢(W)

where θv⁢(A)=(-v+A)∩W. This map is well-defined since it can be factored as

(5.1)

where θW is restricted on the subposet -v+𝒯⁢(V)∨W⊂𝒞V⁢𝒯⁢(V)∨W of cosets of the form -v+A.

Proposition 5.3.

Assume dim⁡V≥1. The map θv:T⁢(V)∨W→CW⁢T⁢(W) is a weak equivalence. In particular, T⁢(V)∨W is (dim⁡V-2)-spherical.

Proof.

The homotopy inverse of θv is given by the map

sv:𝒞W⁢𝒯⁢(W)→𝒯⁢(V)∨W

defined by sv⁢(w+A)=〈v+w,A〉. One checks that θv⁢sv⁢(w+A)⊃w+A, and sv⁢θv⁢(B)⊂B. Therefore sv is the homotopy inverse by the basic result described in Section 2.1 and the homotopy type of the coset poset is given in Proposition 5.2. ∎

There is a variation of this result when dim⁡V≥2 whose proof is essentially the same. Let U be a subspace of W with dim⁡U=1. Let 𝒯⁢(V)∧U denote the poset {A∈𝒯⁢(V)∣A∩U=0}. We will denote the intersection 𝒯⁢(V)∨W∩𝒯⁢(V)∧U simply by 𝒯⁢(V)UW. Consider the restriction of θv to the subposet 𝒯⁢(V)UW. The image of the restricted map θv′ lies in 𝒞W⁢𝒯⁢(W)∧U. Therefore we have a diagram

where the vertical maps are natural inclusions. The map θv′ is also a weak equivalence whose homotopy inverse is given by the restriction of sv to 𝒞W⁢𝒯⁢(W)∧U. We record this result.

Proposition 5.4.

Assume dim⁡V≥2. The map θv′:T⁢(V)UW→CW⁢T⁢(W)∧U is a weak equivalence.

When dim⁡V=2, the poset 𝒯⁢(V)UW consists of one-dimensional subspaces different than W. There are p many of such subspaces. The coset poset 𝒞W⁢𝒯⁢(W)∧U consists of the cosets of the zero subspace in W. Again there are p many of such cosets. The poset of subspaces 𝒯⁢(V)UW will appear as fibers of certain maps between coset posets. Next we show that this poset is spherical.

Theorem 5.5.

Assume dim⁡V≥2. Let U⊂W be subspaces in V and codim⁡W=dim⁡U=1. Then T⁢(V)UW is (dim⁡V-2)-spherical.

Proof.

By Proposition 5.4 the poset 𝒯⁢(V)UW is weakly equivalent to the coset poset 𝒞W⁢𝒯⁢(W)∧U. The statement of the theorem holds when dim⁡V=2 since the resulting space is a disjoint union of points. For larger values of dim⁡V we will proceed by induction on the dimension. Assume that the statement of the theorem holds for vector spaces of dimension less than dim⁡V. We will decompose 𝒞W⁢𝒯⁢(W)∧U with respect to a subspace L of W of codimension one which contains U. By Corollary 3.6 it suffices to show that 𝒞L⁢𝒯⁢(W)∧U and 𝒞W⁢𝒯⁢(W)UL are (dim⁡W-2)-spherical, and construct the map s. We start with the homotopy type of the first one. Since the poset 𝒯⁢(W)∧U is L-stable, Proposition 2.4 implies that

iL:𝒞L⁢𝒯⁢(W)∧U→𝒞L⁢(𝒯⁢(W)∧U)∩L=𝒞L⁢𝒯⁢(L)∧U

is a weak equivalence and by Proposition 5.4 there is a weak equivalence

𝒞L⁢𝒯⁢(L)∧U≃𝒯⁢(W)UL

which is (dim⁡W-2)-spherical by the induction hypothesis. Next we determine the homotopy type of 𝒞W⁢𝒯⁢(W)UL by applying Theorem 2.2 to the natural map ϵ:𝒞W⁢𝒯⁢(W)UL→𝒯⁢(W)UL defined by w+A↦A. We need to consider the posets (𝒯⁢(W)UL)<A and ϵ≥A and show that both are spherical. The first can be identified as follows:

(𝒯⁢(W)UL)<A=(𝒯⁢(W)∨L)<A=𝒯⁢(A)∨L∩A

where the latter is spherical by Proposition 5.3. The fiber can be identified as

ϵ≥A=𝒞W⁢(𝒯⁢(W)UL)≥A=𝒞W⁢(𝒯⁢(W)∧U)≥A

and there is a map of posets

α:𝒞W⁢(𝒯⁢(W)∧U)≥A→𝒞W/A⁢𝒯⁢(W/A)∧U⁢A/A

defined by w+B↦w+B⁢A/A. We claim this is a weak equivalence. To see this let B¯ denote a coset in the image and consider the fiber α≤B¯. The fiber is contractible since it has a terminal object. Hence Theorem 2.1 implies that α is a weak equivalence. Finally, again using Proposition 5.4 and the induction hypothesis, we see that the fiber is spherical. Therefore both spaces satisfy the requirements in Theorem 2.2 and we conclude that 𝒞W⁢𝒯⁢(W)UL is (dim⁡W-2)-spherical. It remains to define the map s:𝒞L⁢𝒯⁢(W)∧U→𝒯⁢(W)UL. According to Theorem 3.4 this map needs to satisfy the following two properties: θL⁢s≃∗ and the composite

θL⁢(-v)⁢s:𝒞L⁢𝒯⁢(W)∧U→𝑠𝒯⁢(W)UL→-v-v+𝒯⁢(W)UL→θL𝒞L⁢𝒯⁢(W)∧U

is homotopic to the identity map for some fixed element v∈W-L. Here we restricted the map θL:𝒞W⁢𝒯⁢(W)UL→𝒞L⁢𝒯⁢(W)∧U to the subposet of cosets of the form -v+A where A∈𝒯⁢(W)UL. Our candidate for s is closely related to the homotopy inverse sv of the map

θv′:𝒯⁢(W)UL→𝒞L⁢𝒯⁢(L)∧U

considered in Proposition 5.4. Note that by diagram (5.1) we have θv′=iL⁢θL⁢(-v). Let j be a homotopy inverse to iL. We claim s=sv⁢iL is the map we need. We have θL⁢(-v)⁢s≃j⁢θv′⁢sv⁢iL≃j⁢iL≃Id. It remains to check θL⁢s=θL⁢sv⁢iL≃∗. For this recall that sv sends a coset l+A to the subspace 〈v+l,A〉. By direct verification we see that θL⁢sv⁢iL⁢(l+A) is the trivial coset 〈v+l,A∩L〉 which contains 0, yielding a contracting homotopy θL⁢s≃∗. ∎

5.2 Poset of isotropic subspaces

Let 𝔟 be an alternating bilinear form on V. This means 𝔟⁢(v,v′)=-𝔟⁢(v′,v) for all v,v′ in V. The orthogonal complement of a subspace A⊂V is defined by

A⊥={w∈V∣𝔟⁢(w,a)=0⁢ for all ⁢a∈A}.

For an element v we write v⊥ to denote the orthogonal complement of the subspace 〈v〉. We assume 𝔟 non-degenerate, i.e. V⊥=0. A subspace is called isotropic if A⊂A⊥. Let ℐ⁢(V) denote the collection of isotropic subspaces. Let H⁢(V) denote the Heisenberg group associated to the vector space V with the bilinear form 𝔟 as introduced in Section 4.4. We can regard ℐ⁢(V) as a collection of abelian subgroups for H⁢(V) under the inclusion I→H⁢(V) defined by a↦(a,0). Then the centralizer of I in H⁢(V) can be identified with the subgroup H⁢(I⊥).

Theorem 5.6.

There is a weak equivalence

𝒞H⁢(V)⁢ℐ⁢(V)≃⋁d⁢(p,r)𝕊r

where 2⁢r=dim⁡V and d⁢(p,r) is defined in Corollary 4.9.

The proof of this theorem will occupy the rest of this section. We fix a symplectic basis b={x1,x2,…,xr,x¯1,x¯2,…,x¯r} of V with 𝔟⁢(xi,x¯j)=1 when |i-j|=0 and zero otherwise. Let us fix x=xr and x¯=x¯r for the rest. We will apply Corollary 3.6 with H=H⁢(x⊥) which is a normal subgroup of H⁢(V) with a quotient isomorphic to a cyclic group of order p. The coset space 𝒞H⁢(V)⁢ℐ⁢(V) is the homotopy colimit of the diagram

where g=(x¯) and H=H⁢(x⊥). Note that gp=(p⁢x¯)=(0) is the identity element of H⁢(V). We have two main objectives:

construct the map s, and
show that 𝒞H⁢(V)⁢ℐ⁢(V)∨H is (r-1)-spherical.

5.3 Construction of s

We will construct a map of posets

s:𝒞H⁢ℐ⁢(V)→ℐ⁢(V)∨H

such that θH⁢g-1⁢s≃Id and θH⁢s≃∗ as required by Corollary 3.6. The quotient map p:x⊥→x⊥/〈x〉 induces a weak equivalence

(5.2)p:𝒞H⁢ℐ⁢(x⊥)→𝒞H⁢(x⊥/〈x〉)⁢ℐ⁢(x⊥/〈x〉)

since the fibers canonically contract to a terminal object. Here H⁢(x⊥)/〈x〉 is identified with H⁢(x⊥/〈x〉). Moreover, by Proposition 2.4 the map

(5.3)iH:𝒞H⁢ℐ⁢(V)→𝒞H⁢ℐ⁢(V)∩H=𝒞H⁢ℐ⁢(x⊥)

is a weak equivalence since ℐ⁢(V) is H-stable. Let ϕ denote the composition

(5.4)ϕ=p⁢iH:𝒞H⁢ℐ⁢(V)→𝒞H⁢(x⊥/〈x〉)⁢ℐ⁢(x⊥/〈x〉)

and let θ¯H denote the composition ϕ⁢θH where θH:ℐ⁢(V)∨H→𝒞H⁢ℐ⁢(V).

Lemma 5.7.

Suppose that the map of posets

(5.5)θ¯=θ¯H⁢g-1:ℐ⁢(V)∨H⁢(x⊥)→𝒞H⁢(x⊥/〈x〉)⁢ℐ⁢(x⊥/〈x〉)

has an inverse denoted by s¯ such that θ¯H⁢s¯≃∗. Then s=s¯⁢ϕ satisfies the required properties in Corollary 3.6.

Proof.

If φ is a homotopy inverse of ϕ, then multiplying θ¯⁢s¯=Id by φ on the left and by ϕ on the right, we get

Id≃φ⁢ϕ=φ⁢θ¯⁢s¯⁢ϕ=φ⁢ϕ⁢θH⁢g-1⁢s≃θH⁢g-1⁢s.

Moreover, θH⁢s≃∗ since ϕ⁢θH⁢s⁢ϕ≃θ¯H⁢s¯≃∗. ∎

We proceed with the construction of s¯. The quotient x⊥/〈x〉 can be identified with the subspace Z spanned by b-{x,x¯}. Let j:x⊥/〈x〉→Z⊂x⊥ denote this identification. There is a commutative diagram

where k⁢(u)=u-𝔟⁢(u,x¯)⁢x. There is a corresponding commutative diagram

(5.6)

of Heisenberg groups, where the maps are still denoted by the same letters. We define a map of posets

s¯:𝒞H⁢(x⊥/〈x〉)⁢ℐ⁢(x⊥/〈x〉)→ℐ⁢(V)∨H

which sends a coset (w,t)⁢A⊂j⁢(H⁢(x⊥/〈x〉)) to the subgroup

〈(x¯+t⁢x+w),(𝔟⁢(w,a)⁢x+a)∣a∈A〉.

Let us check that this definition gives an abelian subgroup of H. Note that the commutator of two elements in H⁢(V) is given by

[(v1),(v2)]=(v1,0)⁢(v2,0)⁢(-v1,0)⁢(-v2,0)

=(v1+v2,𝔟⁢(v1,v2))⁢(-v1-v2,𝔟⁢(v1,v2))=(0,2⁢𝔟⁢(v1,v2)).

Therefore s¯ applied to a coset gives an abelian subgroup since

𝔟⁢(x¯+t⁢x+w,𝔟⁢(w,a)⁢x+a)=𝔟⁢(w,a)⁢𝔟⁢(x¯,x)+𝔟⁢(w,a)=0.

Next we prove a key lemma.

Lemma 5.8.

The map s¯ is the inverse of θ¯ as defined in (5.5). It satisfies θ¯H⁢s¯≃∗.

Proof.

First we show that s¯⁢θ¯⁢(A)=A for all A in ℐ⁢(V)H. Such isotropic subspaces satisfy A+x⊥=V where we regard A as a subgroup of H⁢(V) via the mapping a↦(a,0). We have x¯+u∈A for some u∈x⊥ since A and x⊥ span the whole space V. Then A=〈x¯+u,A∩x⊥〉. Recall that θ¯=θ¯H⁢g-1 sends a space A to the quotient ((g-1⁢A)∩H)/〈x〉 where H=H⁢(x⊥) and g=(x¯). Although the function θ¯ is defined on subspaces, it will be convenient to think that it is defined as follows on vectors v such that 〈v〉+x⊥=V:

θ¯⁢(v)=p⁢((x¯)-1⁢(v)∩H)=(p⁢(v-x¯),𝔟⁢(v,x¯))

where p:x⊥→x⊥/〈x〉. Then we can write

j⁢θ¯⁢(v)=(k⁢(v-x¯),𝔟⁢(v,x¯))

by the commutativity of (5.6). In particular, we have j⁢θ¯⁢(x¯+u)=(k⁢(u),𝔟⁢(u,x¯)). More generally for a∈A∩x⊥ we compute

j⁢θ¯⁢(x¯+u+a)=(k⁢(u+a),𝔟⁢(u+a,x¯))

=(k⁢(u),𝔟⁢(u,x¯))⁢(k⁢(a),𝔟⁢(a,x¯)+𝔟⁢(a,u))

=(k⁢(u),𝔟⁢(u,x¯))⁢(k⁢(a),𝔟⁢(a,x¯+u))

=(k⁢(u),𝔟⁢(u,x¯))⁢(k⁢(a),0)

where in the last step we have used the fact that a and x¯+u are elements of an isotropic space A, i.e. 𝔟⁢(a,x¯+u)=0. Therefore the composition j⁢θ¯ maps A to the coset

A=〈x¯+u,A∩x⊥〉↦(k⁢(u),𝔟⁢(u,x¯))⁢k⁢(A)

in j⁢(H⁢(x⊥/〈x〉)). Applying s¯ to this coset gives a subgroup generated by

(x¯+𝔟⁢(u,x¯)⁢x+k⁢(u))=(x¯+u)

and

(𝔟⁢(k⁢(u),k⁢(a))⁢x+k⁢(a))=(𝔟⁢(u,a)⁢x+k⁢(a))=(𝔟⁢(x¯+u,a)⁢x+a)=(a)

where we have used the fact that k is defined by k⁢(u)=u-𝔟⁢(u,x¯)⁢x and respects the bilinear form. Therefore

s¯⁢θ¯⁢(A)=〈(x¯+u),(a)∣a∈A∩x⊥〉

which is exactly equal to A when regarded as a subgroup of H⁢(V).

For the converse let (w,t)⁢A be a coset in the image of j. After applying s¯ consider the element (x¯+t⁢x+w+𝔟⁢(w,a)⁢x+a,𝔟⁢(x¯,a)) obtained as the product of the two generators (x¯+t⁢x+w) and (𝔟⁢(w,a)⁢x+a). To see what θ¯ does to s¯⁢((w,t)⁢A) it suffices to check its effect on this element:

(-x¯)⁢(x¯+t⁢x+w+𝔟⁢(w,a)⁢x+a,𝔟⁢(x¯,a))

=(t⁢x+w+𝔟⁢(w,a)⁢x+a,t+𝔟⁢(w,a))

≡(w+a,t+𝔟⁢(w,a))mod(x)

=(w,t)⁢(a,0)mod(x)

which implies that θ¯⁢s¯ is the identity.

Finally, the last statement follows from θ¯H⁢s¯⁢((w,t)⁢A)⊃(0) since in effect this composite is computed by intersecting the subgroup s¯⁢((w,t)⁢A) with H⁢(x⊥) and taking the quotient by 〈x〉. Therefore this composition contracts to the constant map at the identity element. ∎

5.4 Proof of Theorem 5.6

Restricting the domains of the maps i=iH in (5.3) and p in (5.2), we obtain the following maps:

(5.7)𝒞H⁢ℐ⁢(V)∨H→𝑖𝒞H⁢ℐ⁢(x⊥)∧〈x〉→𝑝𝒞H⁢(x⊥/〈x〉)⁢ℐ⁢(x⊥/〈x〉)

where ℐ⁢(x⊥)∧〈x〉 denotes the poset of subspaces A in ℐ⁢(x⊥) such that A∩〈x〉=0. Let us consider the fibers of each of these maps. Since both maps are equivariant with respect to the left action of H, it suffices to consider the fiber over a coset of the form B. Other cosets α⁢B are isomorphic to this one under the action of α∈H.

Lemma 5.9.

Let B∈I⁢(x⊥/〈x〉). The fiber p≤B is (dim⁡B)-spherical.

Proof.

Lifting B to an isotropic subspace B~=〈j⁢(B),x〉 in x⊥, we have

p≤B=𝒞B~⁢𝒯⁢(B~)∧〈x〉≃𝒯⁢(B~⊕𝔽p)〈x〉B~

where the weak equivalence follows from Theorem 5.5 with V=B~⊕𝔽p and hence it is (dim⁡(B~⊕𝔽p)-2)-spherical as desired. ∎

Lemma 5.10.

Let B∈I⁢(x⊥)∧〈x〉. The fiber i≥B is (r-dim⁡B-1)-spherical.

Proof.

Let B⊥ denote the orthogonal complement of B in V with respect to 𝔟. Note that B⊥ contains x since B⊂x⊥. We have

i≥B=(ℐ⁢(V)∨H)≥B≅ℐ⁢(B⊥/B)∨H⁢(B⊥∩x⊥)/B

where the last poset is isomorphic to 𝒞H⁢(B⊥∩x⊥/〈B,x〉)⁢ℐ⁢(B⊥∩x⊥/〈B,x〉) by Lemma 5.8. The dimension of a maximal isotropic subspace in B⊥∩x⊥/〈B,x〉 is r-dim⁡B-1. ∎

Now we can finish the proof of Theorem 5.6 by collecting the results obtained so far. We will do so by induction on r where 2⁢r=dim⁡V. When r=2, the coset poset 𝒞H⁢(V)⁢ℐ⁢(V) is a one-dimensional connected space hence it is 1-spherical. We already constructed the section s in Lemma 5.7 and Lemma 5.8. Using the weak equivalence ϕ in (5.4) and induction, we see that 𝒞H⁢ℐ⁢(V) is (r-1)-spherical. It remains to prove that

𝒳=𝒞H⁢(V)⁢ℐ⁢(V)∨H

is (r-1)-spherical. We will use Lemma 5.9 and Lemma 5.10. First consider the map

p:𝒳′→𝒴′

in (5.7). By the induction assumption 𝒴′ is (r-1)-spherical. Let B∈ℐ⁢(x⊥/〈x〉). The fiber p≤B is (dim⁡B)-spherical by Lemma 5.9, and

𝒴>B′=𝒞H⁢(B⊥/B)⁢ℐ⁢(B⊥/B)

is (r-1-dim⁡B)-spherical by inspection. Then 𝒳′ is (r-1)-spherical by Theorem 2.2 applied to the opposite of the map 𝒳′→𝒴′ (original statement in [18, Theorem 9.1]). We turn to the other map in (5.7), and denote it simply by

i:𝒳→𝒳′.

Let B∈ℐ⁢(x⊥)∧〈x〉. In this case, 𝒳<B′=𝒞B⁢𝒯⁢(B)∧〈x〉 is (dim⁡B-1)-spherical by Proposition 5.3 and i≥B is (r-dim⁡B-1)-spherical by Lemma 5.10. Therefore 𝒳 is (r-1)-spherical by Theorem 2.2 and decomposes as a wedge of spheres where the number of spheres is equal to d⁢(p,r) by Corollary 4.9. This concludes the proof of Theorem 5.6.

6 Classifying space for commutativity

In [1] a natural filtration {B⁢(q,G)}q≥2 of the classifying space BG is introduced. These spaces can be described as homotopy colimits of classifying spaces and coset posets naturally occur in the study of such objects. In this section, we apply our main result Theorem 4.8 to B⁢(2,G) when G is an extraspecial p-group.

6.1 The space B⁢(2,G)

As a simplicial set B⁢(2,G) has n-simplices given by the set of group homomorphisms Hom⁡(ℤn,G) and the simplicial structure is induced from BG. Let E⁢(2,G)→B⁢(2,G) denote the pull-back of the universal principal G-bundle E⁢G→B⁢G along the natural inclusion B⁢(2,G)⊂B⁢G. It is known that the natural map

hocolim𝒜⁢(G)⁡B→B⁢(2,G)

is a weak equivalence [1, §4]. As a consequence of Theorem 4.8 we determine the universal cover of B⁢(2,G) when G is an extraspecial p-group.

Theorem 6.1.

Let E be an extraspecial p-group. There is a fibration sequence

⋁d⁢(p,r)𝕊r→B⁢(2,G)→B⁢π

where 2⁢r≥4 is the rank of the Frattini quotient of E, the number d⁢(p,r) is defined in Corollary 4.9, and π is the kernel of the multiplication map

1→π→E×E→E/[E,E]→1.

Proof.

By Proposition 4.2 the fundamental group of B⁢(2,G) is isomorphic to π. The universal cover of B⁢(2,G) is identified with the coset poset 𝒞π⁢𝒜⁢(G) and hence with 𝒞H⁢V⁢ℐ⁢(V) by Proposition 4.7. ∎

Let us introduce a variant of B⁢(2,G). Let V be a vector space over 𝔽p with a non-degenerate alternating bilinear form 𝔟. We define a simplicial set B⁢(𝔟,V) whose set of n-simplices is the set of n-tuples (v1,v2,…,vn) where 𝔟⁢(vi,vj)=0 for all 0≤i,j≤n. The simplicial structure is induced via the inclusion B⁢(𝔟,V)⊂B⁢V. Similar to B⁢(2,G) we can describe this space as a homotopy colimit. There is a natural weak equivalence

hocolimℐ⁢(V)⁡B→B⁢(𝔟,V)

induced by the inclusions B⁢I⊂B⁢(𝔟,V) where I is an isotropic subspace. Let E⁢(𝔟,V)→B⁢(𝔟,V) denote the pull-back of the universal bundle E⁢V→B⁢V along the inclusion B⁢(𝔟,V)⊂B⁢V. There is a weak equivalence

E⁢(𝔟,V)≃𝒞V⁢ℐ⁢(V)

since both spaces are the homotopy fibers of the map B⁢(𝔟,V)→B⁢V.

Proposition 6.2.

The projection E→V induces a diagram

where the horizontal maps are fibration sequences. Moreover, there is a fibration sequence

⋁d⁢(p,r)𝕊r→B⁢(𝔟,V)→B⁢(H⁢(V))

where H⁢(V) is the Heisenberg group.

Proof.

It is a direct verification to show that E⁢(2,E)→E⁢(𝔟,V) is a Kan fibration by checking the horn lifting property [11, p. 11]. The idea is that it is always possible to lift an isotropic subspace I to an abelian subgroup of E. We think of the zero element as the base point ∗ of E⁢(𝔟,V) and over this point the fiber is E⁢ℤ/p by definition of the map. A similar argument works for the map B⁢(2,E)→B⁢(𝔟,V). The last statement follows from Theorem 6.1 and Corollary 4.6 where we showed that the fundamental group of the homotopy colimit of BI where I∈ℐ⁢(V) is the Heisenberg group H⁢(V). ∎

6.2 Subdivision

Let X be a simplicial set. We denote its subdivision by sd⁡X. It comes with a map l:X→sd⁡X called the last vertex map. This map is a weak equivalence [11, p. 193]. For certain simplicial sets there is a nicer description of the subdivision construction. Let Ba⁡(X) denote the poset of non-degenerate simplices of X ordered under the face relation: x≤y if x is a face of y. If any non-degenerate n-simplex of X has n+1 distinct vertices, then sd⁡X is isomorphic to the nerve of the poset Ba⁡(X), called the Barratt nerve [21, Lemma 2.2.11]. Some examples of such simplicial sets are E⁢G,E⁢(2,G),E⁢(𝔟,V),… We identify the subdivision of such a simplicial set with its Barratt nerve. There is a map of posets

(6.1)θ:sd⁡E⁢(𝔟,V)→𝒞V⁢ℐ⁢(V)

defined by sending a non-degenerate simplex (v,a1,a2,…,an) to the coset v+〈a1,a2,…,an〉.

6.3 Transitionally commutative bundles

A principal G-bundle p:E→X is called transitionally commutative if there exists an open cover {Uj}j∈J of X such that the restricted bundle p|Uj is trivial for all j∈J and the transition functions commute when simultaneously defined. Two transitionally commutative bundles p0 and p1 are said to be isomorphic if there exists a transitionally commutative principal G-bundle p:E→X×[0,1] such that p|X×0=p0 and p|X×1=p1.

Theorem 6.3 ([2, Theorem 2.2]).

Let X be a finite CW-complex and G a Lie group. There is a one-to-one correspondence between the set of homotopy classes of maps [X,B⁢(2,G)] and the set of isomorphism classes of transitionally commutative principal G-bundles.

The bundle E⁢(2,G)→B⁢(2,G) is the universal example of a transitionally commutative G-bundle in the sense that any transitionally commutative G-bundle over X is isomorphic to a pull-back bundle f*⁢(E⁢(2,G))→X for some map f:X→B⁢(2,G).

Corollary 6.4.

Let E be an extraspecial p-group. Isomorphism classes of transitionally commutative principal E-bundles over Sr are given by

[𝕊r,B⁢(2,E)]=[𝕊r,⋁d𝕊r]=ℤd

where 2⁢r≥4 is the Frattini quotient of E.

Proof.

Since r≥2 any map 𝕊r→B⁢(2,E) lifts to the universal cover whose homotopy type is a wedge of spheres as determined by Theorem 6.1. The result follows from the Hurewicz theorem. ∎

This result says that although any principal E-bundle over 𝕊r is trivial for r≥2 such bundles cannot be trivialized through transitionally commutative principal E-bundles.

6.4 A non-trivial bundle

Next we construct a non-trivial transitionally commutative principal E-bundle over the r-sphere. Let {xi,x¯i}i=1r denote a symplectic basis for V. Set

x=∑i=1r(xi+x¯i) and x¯=0.

Consider the join of two point discrete posets

J={x,x¯}∗{x1,x¯1}∗⋯⁢{xr,x¯r}

whose geometric realization is homeomorphic to the r-sphere. For each object j of J let us fix a lift j~ with respect to the projection E→V. Identifying J with its nerve, we define a map of simplicial sets

τ:J→B⁢(2,E)

by sending an n-simplex j0→j1→⋯→jn to the n-simplex

((j~0)-1⁢j~1,(j~1)-1⁢j~2,…,(j~n-1)-1⁢j~n).

Our goal is to show that the class represented by τ in the homotopy group πr⁢B⁢(2,E) is non-trivial. For this we introduce another map which will turn out to be closely related to τ. Let us define a map of posets

τ~:sd⁡J→𝒞H⁢(V)⁢ℐ⁢(V)

by sending a chain j1<j2<⋯<js of objects in the poset J to the coset (j1)⁢〈j2-j1,j3-j1,…,js-j1〉 in H⁢(V).

Lemma 6.5.

The class represented by τ~ in the homotopy group πr⁢CH⁢(V)⁢I⁢(V) is non-trivial.

Proof.

We first show that τ~ factors as

sd⁡J→τ¯ℐ⁢(V′)∨H⁢(x0⊥)→θ¯𝒞H⁢(V)⁢ℐ⁢(V)

where V′=〈x0,x¯0〉⊕V is a symplectic vector space of dimension 2⁢(r+1), x0⊥ is the orthogonal complement in V′ with respect to the standard symplectic form, and θ¯ is the isomorphism in (5.5) defined with respect to the vector x0 as follows:

A↦(((x¯0)-1⁢A)∩H⁢(x0⊥))/〈x0〉.

To define τ¯ we start with the canonical chamber in the Tits building associated to Sp⁡(V′) defined by a map of posets

c:sd⁡J′→ℐ⁢(V′)∘

where J′={x0,x¯0}∗{x1,x¯1}∗⋯∗{xr,x¯r} and a chain of objects i1<i2<⋯<it maps to the subspace 〈i1,i2,…,it〉. This map is non-trivial in homology by the Solomon–Tits theorem and hence gives a non-trivial map in homotopy. Let g denote the symplectic transformation on V′ defined by

x0↦∑j=0r(xj+x¯j),x¯0↦x¯0,xi↦xi+x¯0,x¯i↦x¯i+x¯0,1≤i≤r.

Let g^ denote the map of posets ℐ⁢(V′)∘→ℐ⁢(V′)∘ defined by A↦g⁢(A). The image of the composite g^⁢c lies in the poset ℐ⁢(V′)∨H⁢(x0⊥). We define τ¯=g^⁢c⁢h where h is the isomorphism between the subdivisions sd⁡J→sd⁡J′ induced by the map of posets J→J′ defined by x↦x0, x¯↦x¯0 and the remaining xi, x¯i for 1≤i≤r are fixed. Note that by construction τ¯ is not homotopic to the constant map. It remains to check that τ~=θ¯⁢τ¯. Let j1<j2<⋯<js be an object of sd⁡J. Under τ¯ it maps to a subgroup 〈j1′,j2′,…,js′〉 where j1′=j1+x0+x¯0 if j1=x or j1′=j1+x¯0 if j1≠x, and the remaining generators are given by jk′=jk+x¯0 for 1<k≤s. Now the elements in the image of θ¯ come from elements of the form α1⁢j1′+α2⁢j2′+⋯+αs⁢js′ where ∑kαk≡1 mod p. We can write

∑k=1sαk⁢jk′=j1+∑k=2sαk⁢(jk′-j1)

≡x¯0+j1+∑k=2sαk⁢(jk-j1)mod〈x0〉

and thus the image under θ¯ is the coset (j1)⁢〈j2-j1,…,js-j1〉 in H⁢(V). ∎

Theorem 6.6.

Assume r≥2. The class represented by τ in the homotopy group πr⁢B⁢(2,E) is non-trivial. In particular, the pull-back bundle τ*⁢(E⁢(2,E))→J is a non-trivial transitionally commutative E-bundle over the r-sphere.

Proof.

Identifying J with its nerve, we define a map of simplicial sets

τ′:J→E⁢(𝔟,V)

by sending an n-simplex j0→j2→⋯→jn to the n-simplex (j0,j1-j0,j2-j1,…,jn-jn-1). Observe that there is a commutative diagram

where θ is introduced in (6.1) and the map between the coset posets is the universal covering map. To see that the right-hand square commutes we regard the maps as maps of posets. A chain j0<j1<⋯<js is sent to j0+〈j1-j0,j2-j0,…,js-j0〉 under the composition of τ~ and the covering map. On the other hand under the composition θ⁢sd⁡τ′ it is sent to j0+〈j1-j0,j2-j1,…,js-js-1〉. Note that the resulting cosets are equal. Now [τ′] is non-trivial since [τ~] is non-trivial by Lemma 6.5. Here we have used the fact that the last vertex map is a weak equivalence, and J is simply connected. Finally, from the commutative diagram

we see that τ:J→B⁢(2,E) represents a non-trivial class in the homotopy group. ∎

Other examples of non-trivial bundles can be obtained by considering the action of the automorphism group of E on the space B⁢(2,E). Equivalently, one can consider the action of Sp⁡(V) on the coset poset 𝒞H⁢(V)⁢ℐ⁢(V). The representation of Sp⁡(V) on the top dimensional homology of this coset poset could be an interesting representation of the symplectic group.

6.5 Group cohomology

We first observe a direct consequence of Theorem 6.1. Then we consider the natural map H*⁢(G,𝔽p)→H*⁢(B⁢(2,G),𝔽p) in relation to Quillen’s F-isomorphism theorem.

Corollary 6.7.

Let R denote a commutative ring, and E an extraspecial p-group. Then there is an isomorphism

Hi⁢(B⁢(2,E),R)≅Hi⁢(B⁢π,R) for ⁢i<r.

Proof.

When r=1, the universal cover of B⁢(2,E) is contractible since 𝒞V⁢ℐ⁢(V) is a wedge of spheres (Section 4.2). Hence we have an isomorphism for all i in this case. If r≥2, then the result follows from the Serre spectral sequence of the fibration in Theorem 6.1. ∎

Quillen’s F-isomorphism theorem [16, 17] implies that the kernel of the natural map

(6.2)H*⁢(B⁢G,𝔽p)→limA∈𝒜⁢(G)⁡H*⁢(B⁢A,𝔽p)

has nilpotent kernel. This map factors through the cohomology of B⁢(2,G). Therefore the kernel of the natural map

H*⁢(B⁢G,𝔽p)→H*⁢(B⁢(2,G),𝔽p)

is nilpotent [1, Proposition 3.2].

There are two types of extraspecial 2-groups E+ and E- (see [3, §23.14]). The group E+ is a central product of dihedral groups D8. The cohomology of E+ is detected on elementary abelian subgroups, that is, the map in (6.2) is injective. This forces the natural map

H*⁢(B⁢E+,𝔽2)→H*⁢(B⁢(2,E+),𝔽2)

to be injective as well. But this is not true in general. The quaternion group Q8 is of type E-, and we will show that in this case this map is not injective.

The cohomology ring of Q8 is given by

H*⁢(B⁢Q8,𝔽2)=𝔽2⁢[x,z]/(x2+x⁢z+z2,x⁢z2+x2⁢z)⊗𝔽2⁢[w4]

where x,z are the duals of the generators of the central quotient, and w4 is a 4-dimensional class. Next we compute the cohomology ring of B⁢(2,Q8). In general, the cohomology ring of B⁢(2,Q2q) is computed in [7].

Proposition 6.8.

The cohomology ring of B⁢(2,Q8) is given by

H*⁢(B⁢(2,Q8),𝔽2)=𝔽2⁢[x,y,z]/(x⁢y,x⁢z,y⁢z,x2+y2+z2)⊗𝔽2⁢[w2]

where x,y,z are of degree one and w2 is a class of degree two. As a consequence the natural map

H*⁢(B⁢Q8,𝔽2)→H*⁢(B⁢(2,Q8),𝔽2)

is not injective.

Proof.

The quadratic group Q8 has three maximal abelian subgroups isomorphic to ℤ/4 all intersecting at the center which is a cyclic group of order two. The space B⁢(2,Q8) is weakly equivalent to the classifying space

B⁢(ℤ/4∗ℤ/2ℤ/4∗ℤ/2ℤ/4)

of the amalgamated product of the maximal abelians along the center. Consider the map between the central extensions

where x↦x, y↦x+z, and z↦z. Here x, y, z are the images of the generators of the cyclic groups of order four and each generates a copy of ℤ/2. In the Serre spectral sequence of the lower extension under the differential d2 of the E2-page, w maps to the k-invariant x2+x⁢z+z2, where we denote by the same letter the dual generator in the cohomology ring. The map between the cohomology rings of the quotient groups is given by

ϵ:𝔽2⁢[x,z]→𝔽2⁢[x,y,z]/(x⁢y,x⁢z,y⁢z)

where x↦x+y and z↦z+y. Then in the spectral sequence of the upper extension under d2 the class w maps to ϵ⁢(x2+x⁢z+z2)=x2+y2+z2 by naturality and

d2⁢(Sq1⁡w)=Sq1⁡d2⁢(w)=Sq1⁡(x2+y2+z2)=0.

Therefore the spectral sequence collapses at the E3-page and we obtain the desired result. The map between the cohomology rings can be described explicitly as

𝔽2⁢[x,z](x2+x⁢z+z2,x⁢z2+x2⁢z)⊗𝔽2⁢[w4]→𝔽2⁢(x,y,z)(x⁢y,y⁢z,x⁢z,x2+y2+z2)⊗𝔽2⁢[w2]

where the map on the first factor is induced by ϵ and w4 maps to w22. Note that x2⁢z maps to zero. ∎

For odd primes p it is tempting to ask how close the map

H*⁢(B⁢E,𝔽p)→H*⁢(B⁢(2,E),𝔽p)

is to being injective? Note that cohomology rings of extraspecial p-groups are not completely known for p>2.

6.6 Almost extraspecial 2-group

There is another type of extraspecial group when p=2, also referred to as the complex type extraspecial group. It is defined to be the central product E′=E∘ℤ/4 along ℤ/2. It sits in a central extension

0→ℤ/2→E′→V′→0

where V′=V⊕V0 is isomorphic to (ℤ/2)2⁢r+1. The commutator induces a bilinear form 𝔟′ on V′. We have V0=(V′)⊥ with respect to 𝔟′. The projection map q:V′→V induces a map of posets

q^:𝒞V′⁢ℐ⁢(V′)→𝒞V⁢ℐ⁢(V)

where the fibers q≤v+I are contractible. As a consequence of Theorem 2.1 this map is a weak equivalence. Therefore the universal cover of B⁢(2,E′) is a wedge of r-spheres. Analogously we can define B⁢(𝔟′,V′) with respect to 𝔟′. There is a fibration sequence

B⁢ℤ/2→B⁢(2,E′)→B⁢(𝔟′,V′)

induced by the map E′→V′. The natural map

H*⁢(B⁢E′,𝔽2)→H*⁢(B⁢(2,E′),𝔽2)

can be shown to be injective using Quillen’s computation in [15].

Communicated by Robert Guralnick

Acknowledgements

The author would like to thank Alejandro Adem and Ergün Yalçın for helpful discussions.

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Received: 2017-02-22

Revised: 2018-02-28

Published Online: 2018-03-28

Published in Print: 2018-07-01

Artikel in diesem Heft

https://doi.org/10.1515/jgth-2018-0008