HOMOTOPY EQUIVALENCE OF SIMPLICIAL SETS WITH A GROUP ACTION

RAFAEL VILLARROEL-FLORES

Abstract. We describe several theorems that allow us to establish the equivariant homotopy equivalence of two simplicial sets with a group action. They are also applied via the nerve functor to the context of categories, especially to some of those defined by Dwyer in the context of homology decompositions. These theorems generalize results by Quillen, Thévenaz and Webb, and others.

1. Introduction

Our purpose in this paper is to generalize known theorems about homeomorphism and homotopy equivalence of simplicial sets and categories to the case when such objects have the action of a group G and we are interested in equivariant homotopy type.

Since many of the G-simplicial sets that we find in practice come from small G-categories via the nerve functor, in section 4 we collect some theorems about this special case. We obtain a generalization of Quillen’s Theorem A ([13]). In section 5 we study the important concept of the homotopy colimit and show the equivariant version of its fundamental property, namely, that if a map of diagrams of simplicial sets induces a G-homotopy equivalence between corresponding values, then the homotopy colimits of the diagrams are G-homotopy equivalent.

In section 6 we consider preorders, that is, categories in which there is at most one morphism between any two objects. When the category is small this concept is the same as having a set P with a reflexive and transitive binary relation ≤, and for those categories we will use such viewpoint. If x and y belong to P we define x ∼ y if x ≤ y and y ≤ x, and x ≺ y if x ≤ y and y ≰ x. Let X_{≻x} = { y | y ≻ x }. Then, we prove the following, which generalizes Proposition 1.7 from [14]. Recall that we attach topological concepts to categories via the nerve functor. Proposition. Let X be a G-preordered set that has an upper bound for the length of chains of the form x_{0} ≺ x_{1} ≺ ⋯ ≺ x_{n}. Let Y be a G-invariant subset of X that preserves the relation ∼. Assume that for all x ∈ X − Y, we have that X_{≻x} is G_{x}-contractible. Then the inclusion Y → X is a G-homotopy equivalence.

In section 7 we consider the categories X_{C}^{α}, X_{C}^{β} defined by Dwyer in [3, 5]. Here C denotes a collection of subgroups of the group G. These categories can be seen to be G-preordered sets, which we will denote by the same symbols. The elements of X_{C}^{α} are the pairs (H, i) such that H ∈ C and i: H → G is a monomorphism. The G-action is given by g(H, i) = (H, c_{g}i) and the order relation is given by (H, i) ≤ (K, j) if i(H) ⊆ j(K). On the other hand, the elements of X_{C}^{β} are all the cosets of the form xH where H ∈ C and we have xH ≤ yK whenever ^{x}H ⊆ ^{y}K. The G-action is just given by g(xH) = gxH. Let S_{p}(G) be the collection of nontrivial p-subgroups of G where p is prime, A_{p}(G) be the subset of S_{p}(G) that consists of the elementary abelian subgroups, and B_{p}(G) be the subset of S_{p}(G) that consists the subgroups H that are equal to the maximal normal p-subgroup of N_{G}(H). In section 8 we prove the following facts, which are analogous to known properties of the complexes involved. Proposition. For any finite group G and any collection C such that A_{p}(G) ⊆ C ⊆ S_{p}(G), we have that

Date: 30th March 2000.

1991 Mathematics Subject Classification. Primary: 55N91; Secondary: 18B35,20J06,55U10.

Key words and phrases. simplicial sets, preordered sets, homotopy colimit.

X_{C}^{α} ≃_{G} X_{S_{p}(G)}^{α}

Proposition. For any finite group G and any collection C such that B_{p}(G) ⊆ C ⊆ S_{p}(G), we have that

X_{C}^{β} ≃_{G} X_{S_{p}(G)}^{β}

In section 2 we state the basic terminology that we will use. In section 3 we prove theorems that deal with our more general case: that of simplicial sets. We include here two facts about diagonals of bisimplicial G-sets X. First, in Theorem 3.3 we prove that the two simplicial G-topological spaces that are obtained from X by fixing one of the variables and realizing with respect to the other, are G-homeomorphic to the diagonal, then in Theorem 3.5 we prove that if a map of bisimplicial G-sets is a G-homotopy equivalence in each dimension, then the diagonals of the given bisimplicial sets are G-homotopy equivalent.

This paper includes material from my Ph. D. thesis, written at the University of Minnesota under the direction of Peter Webb.

2. Preliminaries

We fix some notation. Let G be a finite group. We denote by G the category with only one object ∗ and hom_{G}(∗, ∗) = G, where the composition is given by group multiplication. Let SCat be the category of small categories. We will denote by G-SCat the category of small G-categories, which is defined as the category of functors SCat^{G} ([11, page 40]). A G-category can be thought of as a category C on which the elements of G act as functors.

We will use Δ to denote the category with obj Δ = { [n] = {0, 1, …, n} | n ∈ ℕ } and hom_{Δ}([n], [m]) the set of monotone maps. Hence, for a category D, the category of simplicial objects (see [12]) in D, denoted sD, is equal to D^{Δ^{op}}. We will denote by Δ[n] the simplicial set given by the contravariant functor hom_{Δ}(−, [n]). We will consider Δ[n] as a G-simplicial set with the trivial action.

We define the category G-sSet of G-simplicial sets as sSet^{G}. This category is isomorphic to the category of simplicial objects in the category of G-sets (the former is equal to (Set^{Δ^{op}})^{G}, the latter is equal to (Set^{G})^{Δ^{op}}) and we will use either viewpoint as needed. We have the nerve functor N: G-SCat → G-sSet that assigns the simplicial set given by the contravariant functor hom_{SCat}(−, C) to the category C. In particular, the n-simplices of N(C) are diagrams of objects and maps in C of the form

(1)  X_{0} ⟶^{ϕ_{1}} X_{1} ⟶^{ϕ_{2}} ⋯ ⟶^{ϕ_{n}} X_{n}

The group G acts on N(C) in the obvious way. If F: C_{1} → C_{2} is a functor between small G-categories, then

(2) (X_{0} ⟶^{ϕ_{1}} X_{1} ⟶^{ϕ_{2}} X_{2} ⋯ X_{n−1} ⟶^{ϕ_{n}} X_{n}) ↦

(F X_{0} ⟶^{F ϕ_{1}} F X_{1} ⟶^{F ϕ_{2}} F X_{2} ⋯ F X_{n−1} ⟶^{F ϕ_{n}} F X_{n})

is a map N(F): N(C_{1}) → N(C_{2}) of G-simplicial sets.

We also can consider the geometric realization ([12, Chapter III]) as a functor | − |: G-sSet → G-Top, where G-Top is the category of G-topological spaces. For a G-category C, we will denote |N(C)| just as |C|. Definition 2.1. If X and Y are G-topological spaces, a G-homotopy from X to Y is a continuous map H: X × [0, 1] → Y such that H(gx, t) = gH(x, t) for all g ∈ G, x ∈ X and t ∈ [0, 1]. Two G-maps f_{1}, f_{2} : X → Y are G-homotopic if there is a G-homotopy H from X to Y such that H(x, 0) = f_{1}(x) and H(x, 1) = f_{2}(x). In this case we write f_{1} ≃_{G} f_{2}. The G-topological spaces X and Y are G-homotopy equivalent if there are G-maps f: X → Y and f′: Y → X such that f′f ≃_{G} 1_{X} and ff′ ≃_{G} 1_{Y}.

The following theorem is a fundamental tool for reducing questions about equivariant homotopies to ordinary ones. Theorem 2.2 (Bredon, [2]). Let X and Y be G-CW-complexes and ϕ: X → Y a G-equivariant cellular map. Then ϕ is a G-homotopy equivalence if and only if ϕ^{H}: X^{H} → Y^{H} is a homotopy equivalence for each subgroup H of G.

We now remind the reader the concept of homotopy between maps of simplicial sets. We follow ([10]) and we distinguish between the concepts of weak and strong homotopies. Definition 2.3. If K and L are G-simplicial sets and ϕ, ψ: K → L are G-maps, we say that ϕ is weakly G-homotopic to ψ if the maps |ϕ|, |ψ|: |K| → |L| are G-homotopic. In this case, we write ϕ ≃_{G} ψ.

In the same way, we say that two G-functors between G-categories F_{1}, F_{2} : C → D are weakly G-homotopic if |F_{1}| ≃_{G} |F_{2}|. Also we say that two G-categories C and D are weakly G-homotopy equivalent if |C| and |D| are G-homotopy equivalent. In particular, the G-homotopy equivalence need not be induced by a functor.

Let us denote by 0 the subsimplicial set of Δ[1] = hom_{Δ}(−, [1]) of maps that have images contained in {0} and by 1 the subsimplicial set of maps that have images contained in {1}. Hence 0 and 1 are both simplicial sets with only one simplex on each dimension. Definition 2.4. If K and L are G-simplicial sets and ϕ, ψ: K → L are G-maps, we say that ϕ is strongly (or simplicially) G-homotopic to ψ if there is a G-simplicial set map H: K × Δ[1] → L such that H restricted to K × 0 can be identified with ϕ and H restricted to K × 1 can be identified with ψ.

Similarly, we say that two G-functors between G-categories F_{1}, F_{2} : C → D are strongly G-homotopic if N(F_{1}) and N(F_{2}) are strongly G-homotopic.

3. Theorems About G-Homotopy Type of Simplicial Sets

This section contains routine generalizations to the equivariant case of wellknown theorems about homeomorphisms and homotopy equivalences of simplicial sets. We do not know of any reference in which they are explicitly stated.

Theorem 3.1. If K and L are G-simplicial sets, we have a G-homeomorphism

(3)  |K × L| ≅_{G} |K| × |L|

if the right side has a CW-structure.

Proof. See [12, Chapter III]. In there, the author defines maps |K × L| → |K| × |L| and |K| × |L| → |K × L| which are bijective and continuous whenever |K| × |L| is a CW-complex. It is not difficult to check they are also equivariant. □

Corollary 3.2. If K and L are G-simplicial sets and ϕ, ψ: K → L are strongly G-homotopic G-maps, they are also weakly G-homotopic.

Proof. The realization of the strong homotopy gives a homotopy between the realizations, since |Δ[1]| is homeomorphic to the unit interval, and given that [0, 1] is locally compact, we have |K × Δ[1]| ≅_{G} |K| × [0, 1]. □

A bisimplicial G-set X is a simplicial object in the category of simplicial G-sets. It can also be seen as a collection of G-sets {X_{mn}}_{m,n≥0}, where X_{mn} = (X_{m})_{n}, with vertical and horizontal faces and degeneracies. The diagonal diag(X) of a bisimplicial G set X is the G-simplicial set with diag(X)_{n} = X_{nn}, and an i-th degeneracy the composition of the horizontal i-th degeneracy with the corresponding vertical i-th degeneracy. The faces are defined similarly. Thus, we get a functor diag from the category of bisimplicial G-sets to the category of G-simplicial sets. Theorem 3.3. Let X be a bisimplicial G-set. Construct a simplicial G-topological space X^{I} by sending [m] to the realization of the simplicial set [n] ↦ X_{mn}. Similarly, construct another simplicial G-topological space X^{II} by sending [n] to the realization of the simplicial set [m] ↦ X_{mn}. Then we have a homeomorphism of G-topological spaces

(4)  | diag(X)| ≅_{G} |X^{I}| ≅_{G} |X^{II}|

Proof. We refer the reader to [7, I §3]. There the authors give the definition of the geometric realization of a simplicial topological space and prove the non-equivariant version of the theorem. The method used is to give a set Z and three equivalence relations on it so that by applying them in different orders we get | diag(X)|, |X^{I}| and |X^{II}| and observing that the resulting spaces have to be homeomorphic. It is immediate to check that G preserves such equivalence relations, hence the isomorphisms given are equivariant. □

Lemma 3.4. For a G-simplicial set X, we have that |X|^{G} is naturally homeomorphic to |X^{G}|.

Proof. This follows from [8, Chapter I, Proposition 2.4.], which says that the functor | − | preserves finite limits, and that X^{G} is the limit of the functor G → sSet that gives X its action of G. □

Since limits in a category of simplicial objects can be calculated dimension-wise, we also deduce from the proof that X^{G} is the subsimplicial G-set of X such that (X^{G})_{n} = (X_{n})^{G}, and the faces and degeneracies restrict to fixed points. Theorem 3.5. Let ϕ: X → Y a map of bisimplicial G-sets. Suppose that for all n we have that ϕ_{n}: X_{n} → Y_{n} is a weak G-homotopy equivalence. Then diag ϕ is a weak G-homotopy equivalence.

Proof. The non-equivariant version of this theorem is [8, Chapter IV, Proposition 1.9.]. We will use Lemma 2.2. To prove that | diag(ϕ)| is a G-homotopy equivalence we need to check that | diag(ϕ)|^{H} is an ordinary homotopy equivalence. By the previous lemma, we just need to verify that diag(ϕ^{H}): diag(X^{H}) → diag(Y^{H}) is a weak homotopy equivalence since clearly diag(Y)^{H} can be identified with diag(Y^{H}). By the non-equivariant version of Theorem 3.5, it is enough to prove that |(ϕ^{H})_{n}|: |(X^{H})_{n}| → |(Y^{H})_{n}| is a homotopy equivalence. But this follows from the fact that |(X^{H})_{n}| = |X_{n}|^{H} (similarly for Y) and that the equivariant homotopy equivalence |ϕ_{n}|: |X_{n}| → |Y_{n}| restricts to fixed points. □

4. Theorems About G-Homotopy Type Of Categories

In this section we extend to the equivariant case some results known for categories without a group action or for G-posets.

For the rest of this section, C and D will denote G-categories. We will say that C is G-contractible if C is weakly G-homotopy equivalent to a point.

Lemma 4.1. (Compare with [1, page 292]) A natural transformation η: F → F′ between the G-functors F, F′: C → D induces a strong G-homotopy between F and F′.

Proof. The natural transformation allows us to define a G-functor τ: C×{0 < 1} → D by (X, 0) ↦ F (X), (X, 1) ↦ F′(X). Since N(C × {0 < 1}) ≅_{G} N(C) × Δ[1], we have that N(τ) is a G-strong homotopy, □

Note that in the previous lemma we do not ask any kind of equivariant behavior of the natural transformation η.

Corollary 4.2. Suppose that the G-functor F: C → D is left adjoint to the Gfunctor F′: D → C, then F and F′ are strong G-homotopy equivalences.

Proof. We apply Lemma 4.1 to the unit 1_{C} → F′F and the counit F F′ → 1_{D} of the adjunction. □

Corollary 4.3. If C is a G-category with an initial object X fixed by G, then C is strongly G-contractible.

Proof. Let X be the category with only one object X and only one morphism. Define a functor S: C → X by sending every object to X, and all maps to the identity. Let T: X → C include the object X in C. Given that X is a fixed object, both S and T are G-functors. Then ST is the identity in X, and we prove that T S is homotopic to the identity because for any object C in C we can take the unique map C → X as defining the component of a natural transformation η_{C} between T S and the identity. Hence, by Corollary 4.2, we have that C is strongly G-homotopy equivalent to X, which is a point. □

Definition 4.4. Following Quillen ([13, page 93]), we make the following definition: if F: C → D is a functor and D ∈ obj D, let F/D denote the category with objects (C, v) with C ∈ obj C and v: F C → D. A morphism from (C, v) to (C′, v′) is a map w: C → C′ such that v′F (w) = v. We similarly define the category D\F with objects the pairs (C, u) with u: D → F C.

We remark that if F is a G-functor, then both F/D and D\F have an action of the stabilizer G_{D} induced by the action of G on C and D, namely g(C, v) = (gC, gv). We now prove the equivariant form of Quillen’s Theorem, which is probably the most powerful technique available to prove that a functor induces a weak equivariant homotopy equivalence of categories. It is at the same time a generalization of Quillen’s Theorem A from [13] and of (1.4) from [14]. First we need a lemma. This is Proposition 5.12 from [4]. Lemma 4.5. For C a small G-category, we denote by C^{G} the small category with obj C^{G} = (obj C)^{G}, (objects fixed under the action of G) and with hom_{C^{G}}(X, Y) = hom_{C}(X, Y)^{G}. If F: C → D is a functor between G-categories, let F^{G} be the restriction of F to the fixed categories C^{G} → D^{G}. Then we have that −^{G} is a functor G-SCat → SCat that commutes with the nerve functor.

Proof. Immediate from the definitions of G-category and the action of G on the nerve. □

Theorem 4.6. Let F: C → D be a G-functor. If the category D\F is weakly G_{D}- contractible for every object D of D, then F is a weak G-homotopy equivalence.

Proof. We follow the argument of (1.4) from [14]. We apply Lemma 2.2 to the map |F |: |C| → |D|. Let H ≤ G and consider |F |^{H}: |C|^{H} → |D|^{H}. From Lemmas 3.4 and 4.5, we have that |C|^{H} = |C^{H}|. Hence it is enough to prove that F^{H}: C^{H} → D^{H} is an ordinary homotopy equivalence. We therefore apply the non-equivariant form of Quillen’s theorem to F^{H}. Let D be an object in D^{H} so that H ≤ G_{D}, and consider D\(F^{H}). It is clear that this can be identified with (D\F)^{H}, and the G_{D}-contraction of D\F restricts to give a contraction of the fixed points, □

5. Homotopy Colimits

Remember that if F: C → D is a functor with C a small category, we call F a diagram on D. The homotopy colimit of a diagram of simplicial sets is an important tool in the study of homotopy types, given its fundamental property of homotopy invariance. In this section we show that we can define a natural G-action on the homotopy colimit of a diagram of G-simplicial sets, and then generalize the fundamental property to obtain a theorem about equivariant homotopy invariance of homotopy colimits. We also state and prove the generalization of a theorem of Thomason ([15, Theorem 1.2]) to the equivariant case. Definition 5.1. Let F: C → G-sSet be a diagram of G-simplicial sets. We define the simplicial replacement ⨆_{∗} F of F as the simplicial object in the category of simplicial G-sets with

(5)  (⨆_{∗} F)_{n} = ⨆_{\overline{X}} F X_{0}

where the coproduct is taken over all the n-simplices \overline{X} = X_{0} ⟶^{ϕ_{1}} X_{1} → ⋯ → X_{n−1} → X_{n} of N(C). Faces and degeneracies are induced by the maps

(6)  d_{0} : F X_{0} ⟶^{F ϕ_{1}} F X_{1}
(7)  d_{j}: F X_{0} ⟶^{1} F X_{0}, j > 0
(8)  s_{j}: F X_{0} ⟶^{1} F X_{0}

where in (6) the domain is indexed by \overline{X} and the range by d_{0}^{N(C)}\overline{X}, in (7) the domain is indexed by \overline{X} and the range by d_{j}^{N(C)}\overline{X} and in (8), the domain is indexed by \overline{X} and the range by s_{j}^{N(C)}\overline{X}

The action of g ∈ G on (⨆_{∗} F)_{n} is given by

(9)  ⨆_{\overline{X}} F X_{0} \underrightarrow{⨆ g} ⨆_{\overline{X}} F X_{0}

Our definition of simplicial replacement differs slightly from that in [1]. We follow [6, page 153] and [4, page 20]. Definition 5.2. Let F: C → G-sSet be a diagram of G-simplicial sets, We define the homotopy colimit of F, denoted hocolim F, by

(10)  hocolim F = diag(⨆_{∗} F)

Hence hocolim F is a G-simplicial set with n-simplices of the form (\overline{X}, y), with

\overline{X} = (X_{0} ⟶^{ϕ_{1}} X_{1} → ⋯ → X_{n−1} → X_{n}) ∈ N(C)_{n}, and y ∈ F (X_{0})_{n}. The G-action

is given by g(\overline{X}, y) = (\overline{X}, gy). This is the action considered in the work of Dwyer, for example [4, page 29].

We will prove now the equivariant homotopy invariance of the homotopy colimit. In this theorem, we prove that if we have two functors that have equivariant homotopy equivalent values by a natural transformation, then their homotopy colimits are equivariant homotopy equivalent. The nonequivariant form of this theorem can be found in [1, page 335]. Theorem 5.3 (Homotopy Invariance of the Homotopy Colimit). Let

(11)  F, F′: C → G-sSet

be two diagrams of G-simplicial sets. Let ε: F → F′ be a natural transformation such that each ε_{X}: F X → F′X is a weak G-homotopy equivalence Then there is a weak G-homotopy equivalence

(12)  hocolim F → hocolim F′

Proof. We have a map

(13)  ⨆_{∗} F = ⨆_{\overline{X}} F X_{0} \underrightarrow{⨆_{X} ε_{X_{0}}} ⨆_{\overline{X}} F′X_{0} = ⨆_{∗} F′

Taking diagonals, we get a map hocolim F → hocolim F′ given by (\overline{X}, y) ↦ (\overline{X}, ε_{X_{0}}y). It is immediate to check that it is equivariant. Since

(14)  (⨆_{∗} F)_{n} = ⨆_{\overline{X}∈N(C)_{n}} F X_{0} \underrightarrow{⨆_{X} ε_{X_{0}}} ⨆_{\overline{X}∈N(C)_{n}} F′X_{0} = (⨆_{∗} F′)_{n}

is a coproduct of weak G-homotopy equivalences, it is a weak G-homotopy equivalence. The result then follows from Theorem 3.5. □

We will now state an equivariant version of Thomason’s theorem ([15, Theorem 1.2]), which expresses the homotopy colimit of a diagram of small categories as weak homotopy equivalent to a certain category.

Definition 5.4. Let F: C → G-SCat be a functor. We define a G-category C ∫ F with objects the pairs (X, a) with X ∈ obj C, a ∈ obj F (X). A map (X, a) → (Y, b) is given by a pair (f, u) such that f: X → Y is a map in C and u: F (f)(a) → b is a map in the category F (Y). The composition is given by (f′, u′)(f, u) = (f′f, u′ ∘ F (f′)(u)). The action of g ∈ G is defined on objects as g(X, a) = (X, ga) and on maps as g(f, u) = (f, gu). (This is analogous to the action given by Dwyer in [5, 3.3]). We can check that this defines a G-category. The category C ∫ F is called the Grothendieck construction on F. Remark 5.5. It is not difficult to check that (C ∫ F)^{G} has the same objects and maps as C ∫ F^{G}. Theorem 5.6. Let C be a category and F: C → G-SCat a functor. Then there is a weak G-homotopy equivalence

(15)  hocolim N(F) → N(C ∫ F)

Proof. There is a simplicial map, given by Thomason except for a change in notation:

(16)  λ: hocolim N(F) → N(C ∫ F)

defined by sending

(17)  (X_{0} ⟶^{ϕ_{1}} ⋯ ⟶^{ϕ_{n}} X_{n}, a_{0} ⟶^{α_{1}} ⋯ ⟶^{α_{n}} a_{n})

to

(18) (X_{0}, a_{0}) ⟶^{(ϕ_{1},F (ϕ_{1})(α_{1}))} (X_{1}, F (ϕ_{1})(a_{1})) ⟶^{(ϕ_{2},F (Φ_{2})(α_{2}))} (X_{2}, F (Φ_{2})(a_{2}))

→ ⋯ → (X_{n−1}, F (Φ_{n−1})(a_{n−1})) ⟶^{(ϕ_{n},F (Φ_{n})(α_{n}))} (X_{n}, F (Φ_{n})(a_{n}))

where we have used Φ_{i} to denote ϕ_{i} ⋯ ϕ_{1}. This map is equivariant since the action of G occurs in the second variable of the Grothendieck construction, and the F (Φ_{i}) are G-functors. We have then to prove that

(19)  λ^{H}: (hocolim N(F))^{H} → N(C ∫ F)^{H}

is a homotopy equivalence. But this follows from the nonequivariant version of Theorem 5.6, since (hocolim N(F))^{H} = hocolim(N(F))^{H} = hocolim N(F^{H}), and N(C ∫ F)^{H} = N((C ∫ F)^{H}) = N(C ∫ F^{H}). □

6. Preordered Sets

We start by stating what the theorems of the previous section mean in the special context of preordered sets. A preordered set is a small category in which there is at most one morphism between any pair of objects. This is the same thing as a set P with a binary relation ≤ which is reflexive and transitive. If G is a group, we say that P is a G-preordered set if G acts on the set P and for every g ∈ G and x ≤ y in P we have gx ≤ gy. Lemma 6.1. (Compare with Lemma 4.1) Let P and Q be two G-preordered sets. If F, F′: P → Q are equivariant order-preserving maps such that F (x) ≤ F′(x) for all x ∈ P, then F is G-homotopic to F′.

An initial object x in a preordered set P seen as a category has the property that x ≤ y for all y ∈ P. In this situation we will refer to x as an initial element of P.

Corollary 6.2. (Compare with Corollary 4.3) If P is a G-preordered set with an initial element fixed by G, then P is G-contractible. Remark 6.3. (Compare with Definition 4.4) If ϕ: P → Q is a map of preordered sets and y ∈ Q, then y\ϕ can be identified with the subpreordered set { x ∈ P | y ≤ ϕ(x) } of P. We denote it as ϕ^{−1}(Y_{≥y}) Theorem 6.4. (Compare with Theorem 4.6) Let ϕ: P → Q a map of preordered sets. If ϕ^{−1}(Q_{≥y}) is G_{y}-contractible for all y ∈ Q, then ϕ is a G-homotopy equivalence.

We now generalize (1.7) of [14] to the case of a preordered set (X, ≤) with a group action. We need some notation. For x, y ∈ X, define x ∼ y if x ≤ y and y ≤ x. This gives an equivalence relation in X. Define x ≺ y if x ≤ y and x ≁ y (equivalently, if x ≤ y and y ≰ x). A chain

(20)  x_{0} ≺ x_{1} ≺ ⋯ ≺ x_{N}

is said to have length N. If x ∈ X, denote by X_{≻x} the set { y ∈ X | y ≻ x }. We say that an element x_{0} in X is minimal if w ≤ x_{0} implies w ∼ x_{0}. Note that the set of minimal elements of X is invariant under G and preserves the relation ∼, that is, if x is minimal and x ∼ z, then z is minimal. Proposition 6.5. Let X be a G-preordered set that has an upper bound for the length of the chains of the form (20). Let Y be a G-invariant subset of X that preserves the relation ∼. Assume that for all x ∈ X − Y, we have that X_{≻x} is G_{x}-contractible. Then the inclusion Y → X is a G-homotopy equivalence.

For clarity, we insert a Lemma Lemma 6.6. Let X be a G-preordered set and Y be a G-invariant subset of X that preserves the relation ∼. Then

(1) The set of minimal elements in the subpreordered set X − Y preserves the relation ∼ in X.

(2) The element x is minimal in the subpreordered set X − Y if and only if either: a) x is minimal in X or b) x has the property that w ≺ x implies w ∈ Y.

Proof of Lemma 6.6. We prove 1: Let x be minimal in X − Y and z ∈ X such that z ∼ x. Since Y preserves ∼, its complement also does, and so z ∈ X − Y. So, z is also minimal in X − Y. Now, for the proof of 2, let x minimal in X − Y and not minimal in X. Take w ≺ x. We must then have w ∈ Y because otherwise w ∈ X−Y contradicting that x is minimal in X−Y. Now for the reverse implication, if condition a) is true then clearly x is minimal in X − Y. Suppose then that x has the property stated in b). Let z ∈ X − Y such that z ≤ x. We need to prove z ≥ x. If this were not true, we would have z ≺ x. But this would imply that z ∈ Y, contrary to the choice of z. □

Proof of Proposition 6.5. We construct a chain of subpreordered sets of X

(21)  X = X_{0} ⊃ X_{1} ⊃ ⋯ ⊃ X_{n} = Y

where X_{i} is obtained from X_{i−1} by removing the minimal elements in X_{i−1} − Y. We reach Y in a finite number of steps because the length of the chains of the form (20) is bounded. Let i be arbitrary between 1 and n. We prove X_{i−1} ≃_{G} X_{i} using Theorem 6.4.

By 1 of Lemma 6.6 and induction, we know that each X_{i} preserves the equivalence relation ∼.

Let ı: X_{i} → X_{i−1} be the inclusion. Let x ∈ X_{i−1} and consider x\ı = { z ∈ X_{i} | x ≤ z }. We have two cases:

If x ∈ X_{i}, then x\ı has x as an initial element fixed by G_{x}, so x\ı is G_{x}- contractible, by Corollary 6.2.

If x ∉ X_{i}, we now prove that

(22)  { z ∈ X_{i} | x ≤ z } = X_{≻x}

Let z ∈ X_{i} with x ≤ z. We cannot then have z ∼ x, since X_{i} preserves ∼. Hence z ≻ x, and this proves one containment.

Let now z ∈ X such that z ≻ x. Then z ≥ x but we also need to prove z ∈ X_{i}. Note that since x ∈ X_{i−1} − X_{i} and Y ⊆ X_{i} we know that x ∉ Y. Now, since z is not minimal in X and z ≻ x with x not in Y, we know that z was not removed in the first step and so z ∈ X_{1}. Then, provided i > 1, using 2 of Lemma 6.6 with X = X_{1} we get that z is not minimal in X_{1} − Y and so z ∈ X_{2}. We continue in this way until we reach that z ∈ X_{i}. This proves equation (22).

Since by hypothesis X_{≻x} is G_{x}-contractible, we are done since Theorem 6.4 applies. □

Of course we can similarly prove the dual proposition, namely Proposition 6.7. Let X be a G-preordered set that has an upper bound for the length of the chains of the form (20). Let Y be a G-invariant subset of X that preserves the relation ∼. Assume that for all x ∈ X − Y, we have that X_{≺x} is G_{x}-contractible. Then the inclusion Y → X is a G-homotopy equivalence.

7. Dwyer’s spaces

The categories X_{C}^{α} and X_{C}^{β} that are described in this section were first defined by Dwyer ([3]). Definition 7.1. A collection C of subgroups of G is a set of subgroups of G that is closed under conjugation. The orbit category O_{C} is defined as the category with set of objects equal to {G/H}_{H∈C} and morphisms the G-maps. We denote by GSet the category of G-sets. Then β̃_{C} is the functor O_{C} → GSet that sends G/H to G/H itself. The G-simplicial set X_{C}^{β} is then defined as hocolim β̃_{C}.

By Theorem 5.6, this space is weakly G-homotopy equivalent to the nerve of the Grothendieck construction associated to β̃_{C}. This is a G-category which Dwyer denotes by X_{C}^{β} that has as objects the pairs (xH, G/H), where H ∈ C, xH is a coset in the G-set G/H, and a map from (xH, G/H) to (yK, G/K) is a Gmap G/H → G/K that sends xH to yK. The G-action then corresponds to g(xH, G/H) = (gxH, G/H). Note that the stabilizer of (xH, G/H) is ^{x}H.

Since the G-sets G/H are transitive, a G-map is determined by a value on a point. Hence between any two objects (xH, G/H) and (yK, G/K) there can be at most one map. It is not difficult to prove that this happens precisely when xH ⊆ ^{y}K. This says that X_{C}^{β} can be seen as a preordered set, whose elements are the cosets xH, with H ∈ C and xH ≤ yK if ^{x}H ⊆ ^{y}K. We will use preordered set notation from now.

We can consider as well the preordered set X_{C}^{α}, that has as elements the pairs (H, i) such that i: H → G is a monomorphism with i(H) ∈ C. The G-action is given by g(H, i) = (H, c_{g}i), where c_{g} is conjugation by g and the order relation is given by (H, i) ≤ (K, j) if i(H) ⊆ j(K). This preordered set has been observed to be equivariantly homotopy equivalent to its G-subpreordered set with elements of the form (H, c_{g}), which is denoted EA_{C} in [9]. For the proof of this fact, we use the dual of Theorem 6.4. Consider the inclusion ı: EA_{C} → X_{C}^{α} and take (H, i) ∈ X_{C}^{α}. The stabilizer of (H, i) is C_{G}(i(H)). We have to prove that ı/(H, i) = { (K, c_{g}) | ^{g}K ⊆ i(H) } is C_{G}(i(H))-contractible. But this set has (i(H), ı_{i(H)}) as a final element fixed by C_{G}(i(H)), where ı_{i(H)} is the inclusion of i(H) on G, so it is C_{G}(i(H))-contractible. From now on, we will use X_{C}^{α} to denote this preordered set.

We will prove theorems that allow us to simplify the preordered sets X_{C}^{α} and X_{C}^{β} up to weak equivariant homotopy equivalence.

Theorem 7.2. Let C be a collection of subgroups of G. Consider C as a G-poset ordered by inclusion and with G acting by conjugation. Let k, l: C → C be G-poset maps such that H ⊇ k(H), H ⊆ l(H) for all H ∈ C. Then k and l induce Gpreordered set maps k̃: X_{C}^{α} → X_{C}^{α} and l̃: X_{C}^{β} → X_{C}^{β} by k̃(H, i) = (k(H), i|_{k(H)}) and l̃(xH) = xl(H), and we have that X_{C}^{α} ≃_{G} im k̃ ≃_{G} X_{im k̃}^{α} and X_{C}^{β} ≃_{G} im l̃≃_{G} X_{im l̃}^{β}.

Proof. Clearly k̃ is equivariant. We show that k̃ is order preserving: Suppose that (H, c_{r}) ≤ (Q, c_{s}) in X_{C}^{α}. Then ^{r}H ⊆ ^{s}Q and given that C is closed under conjugation, we have that H, Q ∈ C. Since k is a G-poset map, we have that rk(H) ⊆ ^{s}k(Q), that is k̃(H, c_{r}) = (k(H), c_{r}) ≤ (k(Q), c_{s}) = k̃(Q, c_{s}). This proves that k̃ is a G-preordered set map. We show now that l̃ is well-defined: if xH = yH then y^{−1}x ∈ H ⊆ l(H), hence xl(H) = yl(H). It preserves the order of X_{C}^{β}, because if xH ≤ yK, then ^{x}H ⊆ ^{y}K, hence l(^{x}H) ⊆ l(^{y}K), hence ^{x}l(H) ⊆ ^{y}l(K), so xl(H) ≤ yl(K), but this is precisely l̃(xH) ≤ l̃(yK). It is also immediate that l̃ is equivariant. To prove the final statements, just note that (k(H), c_{r}) ≤ (H, c_{r}) for any (H, c_{r}) ∈ X_{C}^{α} and that xH ≤ l̃(xH) for any xH ∈ X_{C}^{β}. The theorem then follows from Lemma 6.1. □

We show some examples of maps k, l satisfying the conditions of the previous theorem.

(1) Let C be a collection of subgroups (closed under intersection) and l(K) = intersection of maximal elements of C containing K. This proves X_{C}^{β} to be G-homotopy equivalent to X_{C′}^{β}, where C′ consists of the subgroups that are intersection of maximal elements of C.

(2) C = S_{p}(G), the poset of nontrivial p-subgroups of G, Q a normal p-subgroup and l(K) = KQ. This proves X_{C}^{β} to be G-homotopy equivalent to X_{C′}^{β}, where C′ consists of the subgroups in C containing Q.

(3) Again, let C = S_{p}(G) and k(P) = the subgroup of P generated by the elements of order p in P. This proves X_{C}^{α} to be G-homotopy equivalent to X_{C′}^{α}, where C′ consists of the subgroups in S_{p}(G) which are generated by its elements of order p.

For the case of C = S_{p}(G), Propositions 8.2 and 8.5 will give subcollections C′ of C such that X_{C′}^{α} ≃_{G} X_{C}^{α} and X_{C′}^{β} ≃_{G} X_{C}^{β} and that are in general smaller than the collections given in these examples.

The set of subgroups C_{0} that appears in the following proposition is no longer a collection of subgroups of G (since it is not closed under conjugation), nevertheless, the definitions of X_{C_{0}}^{α} and X_{C_{0}}^{β} as G-preordered sets make sense. Proposition 7.3. Let C_{0} be a set of representatives of the conjugacy classes of C. Then X_{C_{0}}^{α} is G-homotopy equivalent to X_{C}^{α} and X_{C_{0}}^{β} is G-homotopy equivalent to X_{C}^{β}.

Proof. For each K ∈ C, denote by K̂ ∈ C_{0} the chosen representative and fix m(K) ∈ G such that K = ^{m(K)}K̂. If K ∈ C_{0}, choose m(K) = 1. Define a map f: X_{C}^{α} → X_{C_{0}}^{α} by f(H, i) = (Ĥ, ic_{m(H)}). It is immediate to prove that f is equivariant and order preserving. If j: X_{C_{0}}^{α} → X_{C}^{α} denotes the inclusion, then we get that jf(H, i) ≤ (H, i) for all (H, i) ∈ X_{C}^{α} and that fj = 1_{X_{C_{0}}^{α}}. This proves X_{C}^{α} ≃_{G} X_{C_{0}}^{α}. On

the other hand, we have an isomorphism of G-sets c̄_{m(K)} : G/K → G/K̂ given by xK ↦ xm(K)K̂. Define then a map f′: X_{C}^{β} → X_{C_{0}}^{β} by sending xK → c̄_{m(K)}(xK). Since we have c̄_{m(K)}(xK) ≤ xK ≤ yL ≤ c̄_{m(L)}(yL) for xK ≤ yL, we have that f′ is order preserving. We have also that gf(xK) = g(xm(K)K̂) = gxm(K)K̂ = f(gxK) and so f′ is equivariant. On the other hand, the inclusion i: X_{C_{0}}^{β} → X_{C}^{β} is of course order preserving and equivariant. Then we check: Let xK ∈ X_{C}^{β}, then if(xK) = i(xm(K)K̂) = xm(K)K̂ ≥ xK, and so if ≃ 1_{X_{C}^{β}} by Lemma 4.1. Since

fi = 1_{X_{C_{0}}^{β}}, we get X_{C_{0}}^{β} ≃ X_{C}^{β} □

8. Further applications

The preordered sets X_{C}^{α} and X_{C}^{β} are particularly interesting when C = S_{p}(G), because of the following proposition of Dwyer, whose proof we include for the sake of completeness. Proposition 8.1 ([5], 7.3, 7.4). Let C = S_{p}(G). Then for any nontrivial p-subgroup H of G, we have that (X_{C}^{α})^{H} and (X_{C}^{β})^{H} are contractible.

Proof. We have that (X_{C}^{α})^{H} = { (Q, i) | H ⊆ C_{G}(i(Q)) }. If we denote by ı the inclusion of the corresponding subgroup, we have the following contraction of (X_{C}^{α})^{H}:

(Q, i) ≤ (i(Q)Z(H), ı) ≥ (Z(H), ı)

where Z(H) is the center of H. On the other hand, we have that (X_{C}^{β})^{H} = { xK | H ⊆ G_{xK} = ^{x}K }, which is equal to (X_{C}^{β})_{≥H}, where we consider H as a coset. Since H is an initial element there, the result follows from the nonequivariant version of Corollary 6.2. □

This shows that the spaces X_{C}^{α} and X_{C}^{β} are good candidates to apply Theorem (4.1) from [17]. That result applies in the following setting: Let G be a finite group, M a Mackey functor for G, X and Y sets of subgroups of G closed under taking subgroups and conjugation and Z a G-simplicial set. Let us denote by [G\Z_{i}] a set of representatives for the orbits of the action of G on Z_{i}, we then have that if

(1) M is projective relative to X

(2) for every Y ∈ Y, M(Y) = 0

(3) for every subgroup H ∈ X − Y, the subsimplicial set of fixed points Z^{H} is contractible then there is an acyclic and split sequence of the form

(23)  ⋯ → ⨁_{σ∈[G\Z_{1}]} M(G_{σ}) → ⨁_{τ∈[G\Z_{0}]} M(G_{τ}) → M(G) → 0

This sequence has been used to calculate group homology and cohomology. For example, we can take M(G) = H_{n}(G, U)_{p}, that is the p-Sylow subgroup of the homology of G with coefficient module U, X the set of all p-subgroups of G, Y that contains only the identity subgroup, and Z equal to X_{C}^{α} or X_{C}^{β}. This is studied in another paper ([16]). For now, let us note that following Dwyer’s terminology ([5]), the collection C is centralizer-sharp (respectively subgroup-sharp), if the sequence 23 is acyclic for M(G) = H_{n}(G, 𝔽_{p}) for all n ≥ 0 and Z = N(X_{C}^{α}) (respectively Z = N(X_{C}^{β})). By proving 8.1, Dwyer shows that C is both centralizersharp and subgroup-sharp, but by Theorem (4.1) from [17] we obtain also that the sequences (23) are split.

We now prove propositions analogous to the well-known equivalences B_{p}(G) ≃_{G} S_{p}(G), A_{p}(G) ≃_{G} S_{p}(G) (see for example [14]). For a group G, let O_{p}(G) be the maximal normal p-subgroup of G. Then B_{p}(G) is defined to be the set of psubgroups H of G such that H = O_{p}(N_{G}(H)). Subgroups in B_{p}(G) are called p-radical. The collection A_{p}(G) is the collection of p-elementary abelian subgroups of G. Proposition 8.2. Let C be a collection of subgroups of G such that A_{p}(G) ⊆ C ⊆ S_{p}(G). Then

(24)  X_{C}^{α} ≃_{G} X_{S_{p}(G)}^{α}

Proof. Let Y = X_{C}^{α} and X = X_{S_{p}(G)}^{α}. We apply Proposition 6.7, so we need to prove that for any (P, i) ∈ X − Y, the preordered set

(25)  (X_{A_{p}(G)}^{α})_{≺(P,i)} = { (K, j) ∈ X_{A_{p}(G)}^{α} | j(K) ⊂ i(P) }

is G_{(P,i)}-contractible. But G_{(P,i)} = C_{G}(i(P)), and this acts trivially on (25). We continue analogously to the proof of Lemma 2.1 (i) from [14]. Clearly X satisfies the finite length condition since a chain in X under the relation ≺ induces a totally ordered chain of subgroups of G. Let Φ(i(P)) be the Frattini subgroup of i(P). We have that Φ(i(P)) is not trivial since P is not elementary abelian. We have the following, for an arbitrary (K, j) ∈ (X_{S_{p}(G)}^{α})_{≺(P,i)}, where ı denotes the inclusion

(26)  (K, j) ≤ (j(K)Φ(i(P)), ı) ≥ (Φ(i(P)), ı)

and so (X_{S_{p}(G)}^{α})_{≺(P,i)} is contractible, as we wanted to prove. □

Corollary 8.3. Let C be a collection of subgroups of G such that A_{p}(G) ⊆ C ⊆ S_{p}(G). Then C is centralizer-sharp, and the sequence (23) with Z = X_{C}^{α} is acyclic and split. Remark 8.4. Dwyer proved that A_{p}(G) is centralizer-sharp in [5, 8.1] by showing that the set of fixed points of X_{C}^{α} by a non-trivial p-subgroup is contractible. Proposition 8.5. Let C be a collection of subgroups of G such that B_{p}(G) ⊆ C ⊆ S_{p}(G). Then

(27)  X_{C}^{β} ≃_{G} X_{S_{p}(G)}^{β}

Proof. We apply Proposition 6.5 with X = X_{S_{p}(G)}^{β}, Y = X_{C}^{β} and proceed analogously to the proof of Lemma 2.1 (ii) from [14]. Clearly X satisfies the finite length condition since a chain in X under the relation ≺ gives a totally ordered chain of subgroups of G. Let xH ∈ X − Y. We want to prove that Z = X_{≻xH} is G_{xH}-contractible. Now, since G_{xH} = ^{x}H, and this acts trivially on Z, we just have to prove it is contractible. Since H ∉ B_{p}(G), then H̄ = O_{p}(N_{G}(H)) is strictly greater than H. Also for yK ∈ Z, since ^{y}K ⊃ ^{x}H, we have that N_{(^{y}K)}(^{x}H) = ^{y}K ∩ N_{G}(^{x}H) ⊃ ^{x}H. Hence we have the following, for an arbitrary yK ∈ Z:

(28)  yK ≥ x(^{x^{−1}y}K ∩ N_{G}(H)) ≤ x((^{x^{−1}y}K ∩ N_{G}(H))H̄) ≥ xH̄

and so X_{≻xH} is contractible, as we needed to prove. □

Corollary 8.6. Let C be a collection of subgroups of G such that B_{p}(G) ⊆ C ⊆ S_{p}(G). Then C is subgroup-sharp, and the sequence (23) with Z = X_{C}^{β} is acyclic and split.

Let A_{p}(G) be the collection of elementary abelian p-subgroups of G. It is also well known than A_{p}(G) ≃_{G} S_{p}(G). However, as Dwyer points out in [5, 1.10], it is not possible in general that X_{A_{p}(G)}^{β} ≃_{G} X_{S_{p}(G)}^{β}.

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Instituto de Matemáticas, UNAM, Circuito Exterior, Ciudad Universitaria, México, D.F., México

E-mail address: rafael@matem.unam.mx