Combinatorial Topology of Quotients of Posets

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Combinatorial Topology of

Quotients of Posets

Dissertation

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

(Dr. rer. nat.)

Beim Fachbereich Mathematik und Informatik der Universit¨at Bremen

eingereicht von

Dipl.-Math. Ralf Donau

Gutachter:

Prof. Dr. Dmitry N. Feichtner-Kozlov Prof. Dr. Jens Gamst

Eingereicht am: 16. M¨arz 2020

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Abstract

In this thesis we study the topology of quotients of posets. By the topology of a poset we mean the topology of its order complex called nerve in this thesis. An action of some group on a poset induces an action on its nerve.

The posets we consider are partition lattices of finite sets. It is well-known that the nerve of a partition lattice is homotopy equivalent to a wedge of spheres of equal dimension. The symmetric group acts on a partition lattice in a natural way. We consider quotients of such a nerve by subgroups of the symmetric group. Especially we consider subgroups which fix at least one element. It turns out that quotients by such subgroups are also homotopy equivalent to wedges of spheres of equal dimension. Furthermore we consider sublattices of the partition lattice where certain block sizes are forbidden.

For the proofs we use Discrete Morse Theory as well as Equivariant Discrete Morse Theory. We use the notion of an acyclic matching. We also develop new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory. There exists an adaption of Discrete Morse Theory to free chain complexes. In this thesis we develop an adaption for the equivariant case.

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Introduction

For a poset P , we denote its nerve or order complex by ∆(P ) which is a regular trisp. ∆(P ) consists of so called simplices which are chains of comparable elements in P . For a group G acting on P we obtain an induced action of G on its nerve ∆(P ) in a natural way. The quotient ∆(P )/G is again a regular trisp. In this thesis we study the homotopy types of such trisp using Discrete Morse Theory for the case where P is a partition lattice.

Let n ≥ 3 be a natural number. A partition of the set [n] := {1, . . . , n} is a collection of pairwise disjoint subsets of [n] such that the union of all these subsets equals [n]. Let Πn denote the set of all partitions of [n]. Since each

subset of [n] can be partitioned again, we may define a partial order relation on Πnby refinement such that the finer partition is the smaller partition. Together

with this partial order, Πn becomes a lattice. We call it the partition lattice.

Let Πn denote the poset obtained from Πn by removing both the smallest

and greatest element, which are {{1}, . . . , {n}} and {[n]}, respectively. The action of the symmetric group Sn on [n] induces an action of Sn on Πn. As

mentioned above this induces an action of Snon ∆(Πn). Here the simplices are

chains of partition refinements.

1.1 Solved problems

It is well-known that ∆(Πn) is homotopy equivalent to a wedge of spheres of

dimension n − 3. This result has applications in arrangement theory, since the intersection lattice of a linear braid arrangement in isomorphic to a partition lattice Πn, see [15], [23]. The following theorem is the first result concerning

the topology of the quotient ∆(Πn)/G, where G is a non-trivial subgroup of Sn.

This and related facts have implications in singularity theory, see [20], [21], [25]. Theorem 1.1.1 (Kozlov [18, 17]). For any n ≥ 3, the space ∆(Πn)/Sn is

contractible.

This led to a general question of determining the homotopy type of ∆(Πn)/G

for an arbitrary subgroup G ⊂ Sn. The subgroups which would be especially

interesting to consider are the Young subgroups, written as Sk1× Sk2× · · · × Skr

with r ≥ 2 and k1+k2+· · ·+kr= n. The partition {{1, . . . , k1}, {k1+1, . . . , n}}

is fixed by such a subgroup. Hence we observe that ∆(Πn)/G is simply connected

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for n ≥ 5 and a Young subgroup G ⊂ Snby Proposition 3.2.4. First we consider

the Young subgroup S1× Sn−1:= {σ ∈ Sn| σ(1) = 1}.

Theorem 1.1.2 (Donau [10]). Let n ≥ 3 and G ⊂ S1×Sn−1be a subgroup, then

the topological space ∆(Πn)/G is homotopy equivalent to a wedge of k spheres

of dimension n − 3, where k is the index of G in S1× Sn−1.

Proof. See Proposition 3.1.4 and Section 3.1.3.

The smallest example is the case where G equals S1× Sn−1. In this case

we obtain the following result: ∆(Πn)/S1× Sn−1 is homotopy equivalent to a

sphere of dimension n − 3.

One might conjecture that ∆(Πn)/G is homotopy equivalent to a wedge of

spheres for any n ≥ 3 and any subgroup G ⊂ Sn. But this statement is not

true as the following example will show: Let p ≥ 5 be a prime number and let Cp denote the subgroup of Sp that is generated by the cycle (1, 2, . . . , p).

Then the fundamental group of ∆(Πp)/Cpis isomorphic toZ/pZ. In particular

∆(Πp)/Cp cannot be homotopy equivalent to a wedge of spheres. A proof,

which uses facts about covering spaces1_{, can be found in [11] and in Section}

3.2. Another example is the Young subgroup S2× S6 for n = 8. Here we have

H4(∆(Π8)/S2× S6) = Z/2Z. Hence it cannot be homotopy equivalent to a

wedge of spheres.

Instead of considering the homotopy type of such a quotient directly, we may find a homotopy that preserves the group action first.

Proposition 1.1.3 (Donau [6]). Let n ≥ 3, then ∆(Πn) is (S1 × Sn−1

)-homotopy equivalent to a wedge of (n − 1)! spheres of dimension n − 3, where S1× Sn−1 permutes the spheres freely.

Ragnar Freij [14] determined the (S2× Sn−2)-homotopy type of ∆(Πn) by

determining the (S2× Sn−2)-homotopy type of the complex of disconnected

graphs on n vertices, see Theorem 2.5.12.

These results were generalized further by Gregory Arone [1] for arbitrary Young subgroups.

Theorem 1.1.4 (Arone [1]). Let n ≥ 3. We consider the Young subgroup Sk1× Sk2× · · · × Skr ⊂ Sn

with k1+ k2+ · · · + kr = n. Assume gcd(k1, k2, . . . , kr) = 1, then ∆(Πn) is

(Sk1 × Sk2× · · · × Skr)-homotopy equivalent to a wedge of (n − 1)! spheres of

dimension n−3. Sk1×Sk2×· · ·×Skracts freely on these spheres by permutation.

Notice that the choice of the homotopy equivalence depends on the choice of the Young subgroup. For any subgroup G ⊂ Sk1 × Sk2 × · · · × Skr, the

underlying (Sk1 × Sk2× · · · × Skr)-homotopy equivalence is in particular a

G-homotopy equivalence. Hence ∆(Πn)/G is homotopy equivalent to a wedge of

spheres of dimension n − 3. Theorem 1.1.2 can be deduced from this fact. Furthermore Arone gives an answer for arbitrary Young subgroups. In gen-eral we do not obtain a free action on the spheres. Rather we also obtain nontrivial actions on single spheres which may lead to torsion in the homology of the corresponding quotient space.

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1.2. METHODS 9 Theorem 1.1.5 (Arone [1]). ∆(Πn)/Sk× Sn−k is homotopy equivalent to a

wedge of spheres of dimension n − 3 if and only if either 1. n = 2k and k is a prime number

2. or n = 3k and k or n − k is a prime number 3. or gcd(k, n − k) = 1.

Let Liendenote the 1n homogeneous component of the free Lie algebra on n

letters. Sn acts on Lien by permutation of the letters. Additionally the action

of Sn on ∆(Πn) induces an action on its cohomology H∗(∆(Πn);C). It is

well-known2 _{that Lie}

n and the top cohomology Hn−3(∆(Πn);C) with a sign twist

are isomorphic as Sn-modules.

Lien∼= Hn−3(∆(Πn);C) ⊗ sign

We express the sign twist by tensoring with the sign representation. By Propo-sition 4.3 in [26] there exists an isomorphism of Sn-modules

Lien−→ Hn−3(∆(Πn);C) ⊗ sign

that is strongly related to the proof of Proposition 1.1.3, see Chapter 4. Now we consider a special class of subposets of Πn. Let Πn,k denote the

poset obtained from Πn by removing all partitions where some block has more

than 1 and less than k elements. Since Πn,k is a geometric lattice, ∆(Πn,k) is

shellable, hence homotopy equivalent to a wedge of spheres, see [4].

We consider quotients ∆(Πn,k)/G for subgroups G ⊂ Sn. Notice that Πn=

Πn,2, hence the following two theorems may be considered as generalizations of

Theorem 1.1.1 and Theorem 1.1.2.

Theorem 1.1.6 (Kozlov [18]). Let n > k ≥ 2, then ∆(Πn,k)/Sn is collapsible.

Theorem 1.1.7 (Donau [9]). Let n ≥ 3 and 2 ≤ k < n, then ∆(Πn,k)/S1×Sn−1

is

1. collapsible for n ̸≡ 0, 1 mod k.

2. homotopy equivalent to a sphere of dimension 2n

k − 3 for n ≡ 0 mod k.

3. homotopy equivalent to a sphere of dimension 2(n−1)_k −2 for n ≡ 1 mod k.

1.2 Methods

For the proofs of my results I use Discrete Morse Theory as well as Equivariant Discrete Morse Theory. We use the notion of an acyclic matching. I also developed new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory.

For a finite regular trisp ∆ with a group G acting on it, we may construct some additional structure on F (∆), called an G-equivariant acyclic match-ing, such that some simplices σ ∈ F (∆) are special. These special simplices are called critical simplices. This G-equivariant acyclic matching leads to a G-homotopy equivalence by the following Theorem.

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Theorem 1.2.1 (Freij [14]). Let G be a finite group. Let ∆ be a finite regular G-trisp and let M be a G-equivariant acyclic matching on the poset F (∆) \ {0ˆ}. Let ci denote the number of critical i-dimensional simplices of ∆. Then ∆ is

G-homotopy equivalent to a G-CW complex where the cells correspond to the critical simplices of M and the action of G is the same as the action on ∆ restricted to the critical simplices of M .

If the critical simplices form a subtrisp ∆c, then ∆ collapses onto ∆c. That

means ∆c is obtained from ∆ by just deleting orbits of simplices one by one

without changing its G-homotopy type.

If we set G to the trivial group, then we obtain the original Main Theorem of Discrete Morse Theory.

We obtain an (S1× Sn−1)-equivariant acyclic matching on F (∆(Πn))

by the following proposition.

Proposition 1.2.2 (Donau [6]). There exists an (S1×Sn−1)-equivariant acyclic

matching on F (∆(Πn)), where the set of critical simplices consists of one critical

simplex of dimension 0 and (n − 1)! critical simplices of dimension n − 3. Now Proposition 1.1.3 follows by applying Theorem 1.2.1. Furthermore we obtain an explicit description of the set of critical simplices, which can be used to count the number of spheres for a quotient or obtain a basis of the cohomology. We will prove Proposition 1.2.2 by induction using a method I call Equivariant Patchwork Theorem.

In general there does not exist an (Sk× Sn−k)-equivariant acyclic matching

on F (∆(Πn)/Sk× Sn−k) such that the set of critical simplices contains exactly

one simplex of dimension 0. There are even examples where the quotient is homotopy equivalent to a wedge of spheres. One example is S2×S2, see Example

4.4.1 in Chapter 4.

For any subgroup G ⊂ S1× Sn−1we obtain an induced acyclic matching

on F (∆(Πn/G)) which allows us to determine the homotopy type of ∆(Πn/G)

directly by applying the Main Theorem of Discrete Morse Theory. In 2010 I constructed an acyclic matching on such a quotient directly.

For the proof of Theorem 1.1.7, I create a chain of homotopy equivalences using trisp closure maps to collapse ∆(Πn,k)/S1× Sn−1 to some

intermedi-ate subtrisp for further comparison. In the case where ∆(Πn,k)/S1× Sn−1 is

homotopy equivalent to a sphere, we obtain a homotopy equivalence ∆(Πn,k)/S1× Sn−1−→ ∆(Πl)/S1× Sl−1

for some natural number l using two collapses in total. Notice ∆(Πl,2) = ∆(Πl).

There exists an adaption of Discrete Morse Theory to free chain complexes, see Theorem 11.24 in [19, Chapter 11.3]. In this thesis I develop an adaption for the equivariant case3_{. This means we have a free chain complex together with a}

group acting on it. We use the same notion of an equivariant acyclic matching as in Equivariant Discrete Morse Theory. The relation between acyclic matchings and poset maps with small fibers can also be adapted to the equivariant case, which is proven in this thesis and can also be found in [7]. For the proof of the adaption in the equivariant case, it turns out that using poset maps with small fibers is a nice replacement for linear extensions of partial orders.

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Chapter 2

Preliminaries

In this section we describe the notions which are necessary to define the nerve of the partition lattice ∆(Πn), which is a regular trisp, as well as the canonical

group action of Sn on ∆(Πn) and its quotients by subgroups of Sn. We also

describe Discrete Morse Theory, a tool used to determine the homotopy types of trisps. Most of the examples given in this section are related to ∆(Πn) and

Discrete Morse Theory.

2.1 Homotopy type

In this thesis we determine the homotopy types of different triangulated spaces, called trisps. We will define the notion of a trisp later.

Definition 2.1.1. Let X and Y be topological spaces and let f, g : X −→ Y be two continuous maps. Assume we have a continuous map H : X × [0, 1] −→ Y such that H(x, 0) = f (x) and H(x, 1) = g(x) for all x ∈ X. Then we call H a homotopy between f and g. We call f and g homotopic to each other or of same homotopy type.

Definition 2.1.2. Let X and Y be topological spaces. Assume we have contin-uous maps f : X −→ Y and g : Y −→ f such that f ◦ g is homotopic to idX

and g ◦ f is homotopic to idY. Then we call X homotopy equivalent to Y .

Definition 2.1.3. We call a topological space X contractible if X is homotopy equivalent to a topological space that only consists of a single point.

Definition 2.1.4. A subspace A ⊂ X is a strong deformation retract of X if there exists a continuous map R : X × [0, 1] −→ X such that

1. R(·, 0) = idX.

2. R(a, t) = a for all a ∈ A, t ∈ [0, 1]. 3. R(x, 1) ∈ A for all x ∈ X.

We call R a strong deformation retraction.

If A is a strong deformation retract of X, then A is homotopy equivalent to X. We will use this later to simplify the construction of homotopy equivalences.

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Examples 2.1.5.

1. Rn _{is contractible for any n ≥ 0.}

2. Rn_{\ {0} is homotopy equivalent to a sphere of dimension n − 1 for n > 0.}

3. By removing an edge from a triangle and its interior we obtain a strong deformation retract.

4. The disc {(x, y) ∈R2_{| x}2_{+ y}2_{≤ 1} is contractible.}

5. A wedge of a sphere and a contractible space is homotopy equivalent to the sphere.

6. The disc is homotopy equivalent to a 2-simplex. There even exists a homeomorphism which preserves the boundaries.

In general there exists a homeomorphism fn : ∆n −→ Bn such that the

restriction ∂fn: ∂∆n−→ ∂Bn is a homeomorphism for any n > 0.

2.2 Trisps

A trisp1_{is a topological space which is glued together from standard simplices}

which are subsets ofRn_{. The gluing procedure is described by the gluing data.}

Therefore we will first define the gluing data which is described in a purely combinatorial way. Furthermore for each simplex, a trisp has an order on the set of its vertices. In [16], these spaces are called ∆-complexes.

For a trisp ∆, we will use Discrete Morse Theory to find a homotopy equiv-alence to some hopeful simpler CW complex2_{. We will also need the notion}

of a cellular map later.

1_{See also [19]} 2_{See also [16]}

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2.2. TRISPS 13 Definition 2.2.1. For two given CW complexes X, Y we denote their n-skeletons by Xn, Yn respectively. We call a continuous map f : X −→ Y a cellular map if f (Xn−1) ⊂ Yn−1 for all n ≥ 0.

Definition 2.2.2. The gluing data of a trisp ∆ consists of the following parts: 1. A sequence (Si)i≥0of sets, where the elements of Siare called i-simplices

or simplices of dimension i. The elements of S0 are called vertices.

2. For m ≤ n and each order-preserving injection f : [m + 1] −→ [n + 1] we have a map Bf : Sn−→ Sm, such that:

(a) For two composable order-preserving injections f, g we have Bf ◦g= Bg◦ Bf

(b) For each i ≥ 0 we have

Bid[i+1]= idSi

Let σ ∈ Si be a simplex, then we say that v ∈ S0is a vertex of σ if and only

if there exists some f : [1] −→ [i + 1] with Bf(σ) = v. The order on the set

of vertices of σ is obtained as follows. Let f1, f2 : [1] −→ [i + 1] be two

order-preserving injections. Then we set Bf1(σ) < Bf2(σ) if and only if f1(1) < f2(1).

We denote the set of all its vertices by V (σ).

We call a trisp a regular trisp if the vertices for any simplex are distinct, i.e. any simplex σ ∈ Sihas i + 1 pairwise distinct vertices. We call a trisp finite

if it has only finitely many simplices.

We say that τ is a subsimplex of σ if there exists an order-preserving injection f : [m + 1] −→ [n + 1], where m is the dimension of τ and n is the dimension of σ, such that Bf(σ) = τ . In this case we have V (τ ) ⊂ V (σ) which

can be easily verified. We often consider the case where m = n + 1. In this case, by the i-th boundary we mean the subsimplex that corresponds to the order-preserving injection that omits the i-th element of [n + 1]. We call a simplex a maximal simplex if it is not a proper subsimplex of any other simplex.

Assume we have a regular trisp. Let σ ∈ Snbe a simplex and let v ∈ V (σ) be

a vertex. v is obtained by an unique order-preserving injection fv: [1] −→ [n+1]

with v = Bfv(σ). There exists a unique subsimplex τ of σ of codimension 1,

such that V (τ ) = V (σ) \ {v}: We define an order-preserving injection f : [n] −→ [n + 1] such that fv(1) ̸∈ imf and set τ := Bf(σ). Then Bg(τ ) = v for some

g : [1] −→ [n] implies v = Bg(Bf(σ)) = Bf ◦g(σ) implies f ◦ g(1) = fv(1) which

is impossible by the construction of f . We say that τ is obtained from σ by deleting the vertex v.

The topological space, we call trisp, obtained by the gluing data, is obtained as follows. For each simplex σ ∈ Siwe have a copy of the standard i-simplex

∆i _{which is the convex hull of e}

1, . . . , ei+1 in Ri+1:

∆i = {λ1e1+ · · · + λi+1ei+1| λ1+ · · · + λi+1 = 1}

Each order-preserving injection f : [m + 1] −→ [n + 1] induces an R-linear inclusion

f∗:Rm+1 −→ Rn+1

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which maps the standard m-simplex into the standard n-simplex. The order-preserving injection f corresponds to one of the boundaries of the standard n-simplex. We glue all copies of standard simplices according to the order-preserving injections in the natural way.

The topological space obtained by this procedure has the following prop-erties: For each simplex σ ∈ Sn we have a continuous map fσ : ∆k −→ ∆

such that the restriction fσ|int(∆n) is a homeomorphism. Let τ be a proper

subsimplex of σ of dimension m < n. Then fτ = fσ◦ f∗, where f∗ is induced

by an order-preserving injection f : [m + 1] −→ [n + 1] such that τ = Bf(σ).

We have imfτ ⊂ ∂imfσ. Furthermore ∂imfσ is covered by proper subsimplices

of σ, hence σ is glued to simplices of dimension less than n. ∆ is equipped with the following topology: A subset X ⊂ ∆ is open if f−1

σ (X) is open for all

σ ∈⋃︁

i≥0Si.

Since the standard n-simplex ∆n is homeomorphic to the closed ball Bn, see Examples 2.1.5, we have constructed a CW complex.

v

2

v

1

e

13

e

12

v

3

e

23

F

Figure 2.1: The standard 2-simplex.

Example 2.2.3. Examples for trisps can be found in Figures 2.2, 2.3, 2.4. We give a brief explanation of the subsimplices of the 2-simplices in the drawings, see Figure 2.1. For an order-preserving injection f : [2] −→ [3], we set Bf(F ) =

ef (1)f (2). For eij, i < j and an order-preserving injection f : [1] −→ [2], we set

Bf(eij) = vi for f (1) = 1, Bf(eij) = vj for f (1) = 2. For an order-preserving

injection f : [1] −→ [3], we set Bf(F ) = vf (1). Bf ◦ Bg = Bg◦f can be easily

verified. Since all these examples have only one vertex, these trisps are not regular.

v

a

b

c

A

B

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2.2. TRISPS 15

v

a

b

c

A

B

Figure 2.3: The projective plane as a trisp.

v

a

b

c

A

B

Figure 2.4: The Klein bottle as a trisp.

Definition 2.2.4. Let ∆1 and ∆2 be two trisps. A trisp morphism F :

∆1 −→ ∆2 consists of the following data: For every n ≥ 0 and every simplex

σ ∈ Sn(∆1) we have a triple (n(σ), p(σ), F (σ)), where

1. 0 ≤ n(σ) ≤ n,

2. p(σ) : [n + 1] −→ [n(σ) + 1] is an order-preserving surjection,

3. F (σ) ∈ Sn(σ)(∆2) such that for any order-preserving injection f : [m +

1] −→ [n + 1] there exists an order-preserving injection g : [n(Bf(σ)) +

1] −→ [n(σ) + 1] such that (a) g ◦ p(Bf(σ)) = p(σ) ◦ f ,

(b) Bg(F (σ)) = F (Bf(σ)).

The composition of trisp morphisms is defined by composing the data in the natural way.

Remark 2.2.5. Let ∆1, ∆2 be regular trisps and assume we have a trisp

mor-phism F : ∆1 −→ ∆2 such that for every n and every simplex σ ∈ Sn(∆1) we

have n(σ) = n. Then the definition of a trisp morphism simplifies as follows: F is a sequence of maps Fi: Si(∆1) −→ Si(∆2), such that for all m ≤ n, we have

Bf(Fn(σ)) = Fm(Bf(σ))

for all order-preserving injections f : [m + 1] −→ [n + 1] and all σ ∈ Sn(∆1).

Remark 2.2.5 can be applied to trisp isomorphisms and quotient maps we construct later. We will only deal with this kind of trisp morphisms in this thesis.

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Figure 2.5: The torus embedded intoR3.

Such a trisp morphism F : ∆1 −→ ∆2 induces a continuous map between

the underlying topological spaces as follows. Let x ∈ ∆1, then there exists a

simplex σ ∈ Sk(∆1), such that x ∈ fσ(int ∆k). Since the restriction of fσ to

int(∆k) is a homeomorphism, we may set F (x) := fF (σ)◦ fσ−1(x). If we consider

∆1, ∆2as CW complexes, then F is also a cellular map.

2.3 Quotients

Definition 2.3.1. Let G be a group. Let C be a category and X ∈ ob(C) be an object. We consider G as a category. A group action of G on X is a functor A : G −→ C such that the unique object of G is mapped to X.

Since any morphism of the category G is an automorphism, each group element g ∈ G corresponds to an automorphism A(g) : X −→ X in the category C. We simply write g : X −→ X if the group action is clear from the context. For a functor F : C −→ D to some other category D, we obtain an induced action on F (X) via F ◦ A : G −→ D.

Definition 2.3.2. Let G be a group acting on X1, X2 ∈ ob(C) via functors

A1, A2 : G −→ C. Let f : X1 −→ X2 be a morphism. Then f is a

G-morphism if A2(g) ◦ f = f ◦ A1(g) for all g ∈ G. If f can be considered as

a map between sets via a faithful forgetful functor, then this is equivalent to f (g(x)) = g(f (x)) for all x ∈ X1, g ∈ G.

Before we are able to define the notion of a quotient we have to define the notion of a colimit first.

Definition 2.3.3. Let C be a category. Let D : I −→ C be a functor from a category I to C. We call a pair (S, (si : D(i) −→ S)i∈ob(I)), where S ∈ ob(C),

a natural sink of D, if for any choice of objects i, j of I and any morphism f : i −→ j the following diagram commutes.

D(i) si →→ D(f ) ↓↓ S D(j) sj →→

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2.3. QUOTIENTS 17 Definition 2.3.4. We call a natural sink a colimit if it satisfies the following universal property. For any natural sink (T, (ti : D(i) −→ T )i∈ob(I)), there

exists a unique morphism S −→ T such that the following diagram commutes for any i ∈ I. D(i) ti →→ si →→ S ↓↓ T

Because of the universal property, any two colimits of a functor D are iso-morphic.

Definition 2.3.5. Let G be a group acting on an object X of a category C via an action A. A colimit of the functor A : G −→ C whenever it exists is called a quotient. We denote this colimit by q : X −→ X/G. Since G has exactly one object, we may denote the natural sink by (X/G, q : X −→ X/G). The following diagram commutes for all g ∈ G:

X q →→ g ↓↓ X/G X q →→

Example 2.3.6. Let G be a group acting on a set X. Let A : G −→ C denote the corresponding functor. Then the set theoretic quotient X/G is a colimit of A : G −→ C, hence a quotient in our sense.

Now we consider group actions on trisps. Notice that we can apply Remark 2.2.5 to trisp automorphisms. Let G be a group acting on a trisp ∆. We obtain the quotient trisp ∆/G as follows. We set Si(∆/G) := Si(∆)/G and denote

the corresponding quotient map by qi : Si(∆) −→ Si(∆/G) for all i ≥ 0. For

an order-preserving injection f we set Bf([σ]) := [Bf(σ)]. This is well-defined

since gi(σ1) = σ2 implies gj(Bf(σ1)) = Bf(gi(σ1)) = Bf(σ2). The sequence

q := (qi)n≥0 is indeed a trisp morphism. q : ∆ −→ ∆/G is a colimit of the

group action. Furthermore we can apply Remark 2.2.5 to the quotient map q. Proposition 2.3.7. Let G be a group acting on a finite regular trisp ∆. Assume for all g ∈ G and any simplex σ we have that g fixes V (σ) ∩ g(V (σ)), then the quotient trisp ∆/G is again regular.

The proof can also be found in [19, Chapter 14.1].

Proof. Let [σ] be a simplex of ∆/G of dimension n represented by some simplex σ of ∆. Let f1, f2 : [1] −→ [n + 1] be two distinct order-preserving

injec-tions. Assume Bf1([σ]) = Bf2([σ]). This implies [Bf1(σ)] = [Bf2(σ)], hence

Bf1(σ) = gBf2(σ) for some g ∈ G. In particular gBf2(σ) ∈ V (σ) ∩ g(V (σ)),

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2.4 Acyclic categories

The purpose of this section is to define the notion of a nerve of a poset. Since any poset can be considered as a category, we may define the nerve of a so called acyclic category first.

Definition 2.4.1. A small category is called acyclic if only identity morphisms have inverses, and any morphism from an object to itself is an identity mor-phism.

Any poset can be considered as an acyclic category, where the objects cor-respond to the elements of the poset and for x, y ∈ P . We set hom(x, y) = {∗} for x ≤ y. Else we set hom(x, y) = ∅.

2|134 123|4 12|34 3|124 14|23 1|3|24 13|2|4 13|24 1|234 1|23|4 14|2|3 1|2|34 12|3|4

Figure 2.6: The subposet Π4 of the partition lattice Π4

Definition 2.4.2. To each acyclic category C we can associate a trisp ∆(C), called the nerve of C. The vertices of ∆(C) are the objects of C, the i-simplices are chains of i + 1 composable non-identity morphisms, e.g.

σ = a1 m1 −→ a2 m2 −→ a3−→ . . . mi −→ ai+1

The boundary simplices of σ of codimension-1 are defined as follows. ∂1σ = a2 m2 −→ a3−→ . . . mi −→ ai+1 ∂jσ = a1 m1 −→ . . .m−→ aj−2 j−1 mj◦mj−1 −→ aj+1−→ . . . mi −→ ai+1 ∂i+1σ = a0 m1 −→ a1 m2 −→ a2−→ . . . mi−1 −→ ai

Listing the boundaries for codimension-1 pairs is already sufficient to describe a trisp. The nerve of any acyclic category is a regular trisp.

Let C, D be two acyclic categories. A functor F : C −→ D induces a trisp morphism ∆(F ) : ∆(C) −→ ∆(D) in a natural way by applying F to all objects and morphisms of a chain. Hence the nerve construction is a functor. It is easy to see that Remark 2.2.5 can be applied.

Let G be a group acting on an acyclic category C, that means each g ∈ G is considered as a functor g : C −→ C. Then we obtain an induced action on its nerve ∆(C) since, in particular, any g ∈ G induces a functor.

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2.4. ACYCLIC CATEGORIES 19 1. The action of Sn on the poset3 Πn induces an action on its nerve ∆(Πn)

as described above.

2. ∆(Π3) consists of 3 points, hence it is homotopy equivalent to a wedge of

2 spheres of dimension 0.

3. Consider Π4, see Figure 2.6, then ∆(Π4) is homotopy equivalent to a wedge

of 6 spheres of dimension 1.

4. ∆(Π4)/S4, see Figure 2.7, is contractible.

2+2

1+3

1+2+1

Figure 2.7: The space ∆(Π4)/S4. The vertices are written as partitions of the

number 4.

5. ∆(Π4)/S1× S3, see Figure 2.8, is homotopy equivalent to a sphere of

dimension 1.

2⊕2

1⊕3

2⊕1+1

1⊕2+1

3⊕1

Figure 2.8: The space ∆(Π4)/S1× S3. The vertices are written as partitions of

the number 4 which distinguish the number on the left side of ⊕.

6. ∆(Π4)/S2× S2, see Figure 2.9, is homotopy equivalent to a sphere of

dimension 1.

Figure 2.9: The space ∆(Π4)/S2× S2.

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7. ∆(Π5)/S1× S4, see Figure 2.10. It contains ∆(Π4)/S1× S3by adding one

to the number on the left hand side of ⊕ in each vertex and appending the vertex v7= 2 ⊕ 1 + 1 + 1 to each simplex. This observation will later

be used in our proofs for the induction step.

0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11

v

5

v

6

v

7

v

3

v

4

v

2

v

1

v

3

v

4

v

2

v

1

Figure 2.10: The space ∆(Π5)/S1× S4 as a sphere of dimension 2 with some

vertices removed without changing its homotopy type. For example 2 ⊕ 3 is missing. Both pieces are glued together along the dashed line, which is the contained ∆(Π4)/S1× S3. v1= 3 ⊕ 2, v2= 3 ⊕ 1 + 1, v3= 4 ⊕ 1, v4= 2 ⊕ 2 + 1,

v5= 1 ⊕ 2 + 2, v6= 1 ⊕ 2 + 1 + 1, v7= 2 ⊕ 1 + 1 + 1.

Proposition 2.4.4. Let G be a finite group acting on an acyclic category C. Then ∆(C)/G is a regular trisp.

The proof can also be found in [19, Chapter 14.1].

Proof. Let σ be a simplex of ∆(C), g ∈ G and let v ∈ V (σ) ∩ g(V (σ)) be a vertex. Then v = gw for some w ∈ V (σ). σ is a chain of composable morphisms. Without loss of generality let w −→ gw be the corresponding morphism in this chain. Since G is finite we obtain a chain of morphisms

w −→ gw −→ g2w −→ · · · −→ w

In particular, by composition, we obtain a morphism gw −→ w which contra-dicts the fact that C is acyclic. Hence gw = w which implies gv = ggw = gw = v. Now we can apply Proposition 2.3.7.

Remark 2.4.5. Let P be a Poset. Then the vertices of ∆(P ) are the elements of P . The i-simplices are chains σ = (x1, . . . , xi+1) of comparable elements such

that xj ∈ P for 1 ≤ j ≤ i + 1 and xj < xj+1 for 1 ≤ j ≤ i. In particular for

any choice of elements of P there is at most one simplex that contains them as vertices.

The boundaries of a simplex σ ∈ Si are obtained by omitting elements from

the chain according to an order-preserving injection as follows: Let f : [k+1] −→ [i + 1] be an order-preserving injection. Then Bf(σ) = (xf (1), . . . , xf (k+1)) ∈ Sk.

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2.5. DISCRETE MORSE THEORY 21 Example 2.4.6. Let G be a group acting on a poset P , then in general we have ∆(P )/G ̸= ∆(P/G). For example in ∆(Π5)/S1× S4 we have two 1-simplices

that contain the vertices 3 ⊕ 2 and 1 ⊕ 2 + 1 + 1, see Figure 2.10 as opposed to ∆(Π5/S1× S4) where any pair of vertices has at most one 1-simplex that

contains them, see Remark 2.4.5.

Definition 2.4.7. Let ∆ be a trisp. The face poset F (∆) := ⋃︁

i≥0Si(∆)

consists of all simplices of ∆. For two simplices x, y ∈ F (∆) we set x ≤ y if x is a subsimplex of y.

Definition 2.4.8. Let X be a regular trisp. The barycentric subdivision of X is defined as

Bd(X) := ∆(F (X))

Remark 2.4.9. An action of a group G on a regular trisp X induces an action of G on Bd(X). Furthermore X is G-homeomorphic to Bd(X).

2.5 Discrete Morse Theory

This is a brief summary of Discrete Morse Theory. We use the notion of an acyclic matching. Further details as well as foundations of Discrete Morse Theory can be found in [13], [17], [19].

Definition 2.5.1. Let P be a poset. A partial matching on P is a subset M ⊂ P × P with the following properties:

1. (a, b) ∈ M implies b ≻ a.

2. each p ∈ P belongs to at most one element of M .

≺ denotes the cover-relation in P , i.e. for a, b ∈ P we set a ≺ b if a < b and a < x < b implies a = x or b = x.

Definition 2.5.2. A partial matching M on P is acyclic, if and only if there does not exist n ≥ 2 and (a1, b1), . . . , (an, bn) ∈ M pairwise distinct with

b1≻ a1≺ b2≻ a2≺ . . . ≺ bn≻ an ≺ b1

An element p ∈ P is called critical if it is not contained in any matching pair (a, b) ∈ M .

The following Theorem is a tool to construct an acyclic matching on a poset by gluing together smaller acyclic matchings which can be constructed much easier in some cases.

Theorem 2.5.3 (Patchwork Theorem). Let φ : P −→ Q be an order-preserving map and assume we have acyclic matchings on the subposets φ−1(q) for all q ∈ Q. Then the union of these matchings is an acyclic matching on P . Theorem 2.5.4 (Main Theorem of Discrete Morse Theory). Let ∆ be a finite regular trisp and let M be an acyclic matching on the poset F (∆) \ {0ˆ}. Let ci

denote the number of critical i-dimensional simplices of ∆. Then:

1. If the critical simplices form a subcomplex ∆c of ∆, then ∆ collapses onto

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2. ∆ is homotopy equivalent to a CW complex with ci cells of dimension i.

The proofs of Theorems 2.5.3 and 2.5.4 as well as further facts on Discrete Morse Theory can be found in [19, Chapter 11]. In Chapter 4 we have Theorem 4.2.4, a version of Theorem 2.5.4 which respects an action of a group on a trisp. Remark 2.5.5. If a trisp ∆ collapses onto a subtrisp ∆′_{, then ∆ is homotopy}

equivalent to ∆′. If ∆′ consists of one vertex, we call ∆ collapsible. Example 2.5.6.

In Figure 2.11 we have an acyclic matching on F (∆(Π4)/S1× S3) with one

critical vertex and one critical 1-simplex. Hence ∆(Π4)/S1× S3 is homotopy

equivalent to a CW complex which consists of one 0-cell and one 1-cell by Theorem 2.5.4. One feature of this acyclic matching is that it is a quotient of an equivariant acyclic matching on F (∆(Π4)) which we will describe later.

On F (∆(Π4)/S2× S2) it is also possible to construct an acyclic matching

with one critical vertex and one critical 1-simplex. But in this case it is not a quotient of an equivariant acyclic matching, see Example 4.4.1.

1⊕2+1

3⊕1

1⊕3

2⊕1+1

2⊕2

Figure 2.11: An acyclic matching on F (∆(Π4)/S1× S3) with four matching

pairs which are highlighted by the thick lines.

Closure maps

Let ∆ be a regular trisp whose set of vertices is divided into two disjoint parts B ∪ R. We call the vertices of B the blue vertices and the vertices of R the red vertices. Let ∆R be the subtrisp of ∆ that contains all simplices σ ∈ F (∆R) with V (σ) ⊂ R. For σ ∈ F (∆)\F (∆R) let m(σ) denote the smallest blue vertex of σ.

Definition 2.5.7. A map φ : B −→ R is called a closure map if for any σ ∈ F (∆) \ F (∆R_{) either φ(m(σ)) ∈ V (σ) or φ(m(σ)) can be uniquely inserted}

into σ, i.e. there exists a unique simplex τ ∈ F (∆) with φ(m(σ)) ∈ V (τ ) and σ is a subsimplex of τ with dim τ = dim σ + 1.

Theorem 2.5.8 (Kozlov [17]). Let ∆ be a regular trisp whose set of vertices is divided into two disjoint parts B ∪ R. Assume we have a closure map φ : B −→ R, then there exists an acyclic matching on F (∆) such that a simplex σ ∈ F (∆) is critical if and only if σ ∈ F (∆R). In particular ∆ collapses onto its subtrisp ∆R.

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2.5. DISCRETE MORSE THEORY 23 Proof. We define a map p : F (∆) \ F (∆R_{) −→ F (∆) \ F (∆}R_{) as follows. Let}

σ ∈ F (∆) \ F (∆R). First assume φ(m(σ)) ∈ V (σ), then we set p(σ) to the simplex obtained from σ by deleting φ(m(σ)) as described earlier. Now assume φ(m(σ)) ̸∈ V (σ), then we set p(σ) to the simplex obtained from σ by inserting φ(m(σ)). We obtain a partial matching on F (∆) by matching each σ with p(σ). Notice that V (σ) ∩ B = V (p(σ)) ∩ B. We define an order-preserving map:

f : F (∆) −→ N

σ ↦−→ |B ∩ V (σ)|

We have to show that the induced partial matchings on the fibers f−1(n) are acyclic, then we can apply Theorem 2.5.3. f−1(0) = ∅.

Assume we have b1 ≻ a1 ≺ b2, where (a1, b1), (a2, b2) are matching pairs.

Then we have B ∩ V (b1) = B ∩ V (a1) = B ∩ V (b2), i.e. all have the same set

of blue vertices. Let m be the smallest blue vertex. b2 is obtained from a1 by

inserting one vertex. We have φ(m) ∈ V (b2) and φ(m) ̸∈ V (a1). Since b1 is

obtained from a1by inserting φ(m), we get b1= b2since the insertion is unique

by Remark 2.4.5.

∆ collapses onto ∆R by Theorem 2.5.4.

Corollary 2.5.9 (Kozlov [17]). Let P be a finite poset and let φ : P −→ P be an order-preserving map such that φ2 _{= φ and φ(x) ≥ x for all x ∈ P . Then}

the order complex ∆(P ) collapses onto the subcomplex ∆(φ(P )).

We call φ a closure operator. We may replace φ(x) ≥ x by φ(x) ≤ x. Proof. We set R := φ(P ) and B := P \ φ(P ). It is easy to see that φ : B −→ R is a closure map. Now we apply Theorem 2.5.8.

We can use trisp closure maps and closure operators to prove the following results.

Theorem 2.5.10 (Kozlov [18, 17]). For any n ≥ 3, the space ∆(Πn)/Sn is

contractible.

Proof. The vertices of ∆(Πn)/Sn can be written as partitions of the number n.

A partition of the number n is a formal sum k1+ · · · + kt= n, where ki > 0 for

1 ≤ i ≤ t. Let R be the set of all vertices of ∆(Πn)/Snwhich are represented by

number partition of the form 2 + · · · + 2 + 1 + · · · + 1, i.e. only the numbers 1 and 2 are allowed. Let B be the set of all other vertices. We define a closure map φ : B −→ R which replaces each number kiby a sum ki= 2 + · · · + 2 + 1 + · · · + 1

such that the number of 2s is maximized. The subtrisp ∆R_{⊂ ∆(Π}

n)/Snis just

a simplex, hence contractible. By Theorem 2.5.8, ∆(Πn)/Sn collapses onto the

subtrisp ∆R_.

Definition 2.5.11. Let n ≥ 3. The abstract simplicial complex of disconnected graphs DGn on n vertices is constructed as follows. Let Kn denote the complete

graph on n vertices and let En denote its set of edges. A simplex of DGn is a

subset σ ⊂ En such that the subgraph of Kn with edges σ and the set of vertices

unchanged has at least two connected components.

The action of Sn on the set of vertices of Kninduces an action on its edges.

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Theorem 2.5.12. DGn is Sn-homotopy equivalent to ∆(Πn).

The proof can also be found in [19, Chapter 13]. For the part of the proof that is related to Sn we have to take a look at Chapter 4. Alternatively you can

simply ignore the group action.

Proof. We consider Bd(DGn), see Definition 2.4.8. ∆(Πn) can be considered as

a subcomplex of Bd(DGn) by sending each block b of a partition to the complete

subgraph of Kn where b is the set of vertices.

We define a map φ : F (DGn) −→ F (DGn) which takes each graph to its

transitive closure. φ is a closure operator. Furthermore φ is a Sn-map. Hence we

can apply Corollary 2.5.9 to obtain a Sn-homotopy equivalence Bd(DGn) −→

∆(Πn). On the other hand DGn is Sn-homeomorphic to Bd(DGn) by Remark

2.4.9.

2.6 Homology and Chain complexes

Homology is a tool to assign algebraic invariants to topological spaces, which are sequences of abelian groups. It turns out that homotopy equivalent topological spaces have isomorphic homology. We will consider the special case where our topological spaces are cell complexes. We will also consider chain complexes with a chosen basis and a group action.

Chain complexes

Let R be a commutative ring.

Definition 2.6.1. A chain complex consists of a sequence of R-modules (Cn)n≥0 and a sequence of R-linear maps (∂n : Cn −→ Cn−1)n>0 such that

∂n−1◦ ∂n = 0 for all n > 1. · · · −→ Cn ∂n −→ Cn−1 ∂n−1 −→ Cn−2−→ . . .

We denote such a chain complex by (C∗, ∂∗) or just C∗.

Definition 2.6.2. Let (C1

∗, ∂∗1) and (C∗2, ∂∗2) be two chain complexes. A chain

morphism f : (C1

∗, ∂∗1) −→ (C∗2, ∂∗2) is a sequence of R-linear maps (fn :

C1

n−→ Cn2)n≥0 such that

∂2_n◦ fn= fn−1◦ ∂n1

for all n > 0, i.e. the following diagram commutes for all n > 0.

. . . →→ C1 n ∂1_n _→→ fn ↓↓ C1 n−1 →→ fn−1 ↓↓ . . . . . . →→ C2 n ∂2_n →→ C2 n−1 →→ . . .

Definition 2.6.3. Let (C_∗1, ∂_∗1), (C_∗2, ∂_∗2) be two chain complexes and let f, g : (C_∗1, ∂_∗1) −→ (C_∗2, ∂2_∗) be two chain morphisms. f is homotop to g if there

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2.6. HOMOLOGY AND CHAIN COMPLEXES 25 exists a sequence of linear maps (Dn : Cn1 −→ Cn+12 )n≥0 such that fn− gn =

Dn−1∂1n+ ∂ 2 n+1Dn for all n > 0. . . . →→ C1 n+1 →→ ↙↙ f,g ↓↓ C1 n →→ Dn ↙↙ f,g↓↓ C1 n−1 →→ Dn−1 ↙↙ f,g↓↓ . . . ↙↙ . . . →→ C2 n+1 →→ Cn2 →→ Cn−12 →→ . . . Definition 2.6.4. Let C1

∗, C∗2 be two chain complexes. We call C∗1 homotopy

equivalent to C2

∗, if there exist chain morphisms f : C∗1−→ C∗2, g : C∗2−→ C∗1

such that f ◦ g is homotop to idC2

∗ and g ◦ f is homotop to idC∗1.

Definition 2.6.5. A basis of C∗ is a sequence of sets Ω = (Ωn)n≥0 such that

Ωn is a basis of Cn for all n ≥ 0.

Definition 2.6.6. Assume we have a group G acting on C∗. Then we call a

basis Ω = (Ωn)n≥0 of C∗ a G-basis if g(Ωn) = Ωn for all n ≥ 0, g ∈ G.

Homology Groups

Let R be a ring. Let ∆ be a trisp with the gluing data (Si(∆), Bf). For any

n ≥ 0 we define the n-th chain group Cn(∆; R) of ∆ with coefficients in R to

the free R-module over the set Sn(∆).

Cn(∆; R) := ⎧ ⎨ ⎩ ∑︂ σ∈Sn(∆) cσσ | cσ ∈ R ⎫ ⎬ ⎭ For any n ≥ 1 we define the boundary operator

∂n : Cn(∆; R) ⊃ Sn(∆) −→ Cn−1(∆; R) σ ↦−→ ∂n(σ) := n+1 ∑︂ i=1 (−1)i+1Bfi(σ)

where fi: [n] −→ [n + 1] denote the order-preserving injection that omits i. We

have ∂n−1◦ ∂n= 0 for all n > 1, hence (C∗(∆; R), ∂∗) is a chain complex.

We define the n-th homology group of ∆ with coefficients in R as follows:

Hn(∆; R) := ker ∂n/im∂n+1

By H(∆; R) we denote the whole sequence of homology groups.

Theorem 2.6.7. Let ∆1 and ∆2 trisps such that ∆1is homotopy equivalent to

∆2. Then H(∆1, R) and H(∆2, R) are isomorphic.

Proof. Theorem 2.10 in [16]. Examples 2.6.8.

1. Torus (T ), see Figure 2.2:

Hi(T ;Z) ∼= ⎧ ⎪ ⎨ ⎪ ⎩ Z for i = 0, 2 Z2 _{for i = 1} 0 for i ≥ 3

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H0 H1 H2 H3 H4 H5 H6 H≥7 |F | ∆(Π3)/S1× S2 Z2 0 0 0 0 0 0 0 2 ∆(Π4)/S1× S3 Z Z 0 0 0 0 0 0 10 ∆(Π4)/S2× S2 Z Z 0 0 0 0 0 0 14 ∆(Π5)/S1× S4 Z 0 Z 0 0 0 0 0 50 ∆(Π5)/S2× S3 Z 0 Z2 0 0 0 0 0 79 ∆(Π6)/S1× S5 Z 0 0 Z 0 0 0 0 276 ∆(Π6)/S2× S4 Z 0 0 Z3 0 0 0 0 520 ∆(Π6)/S3× S3 Z 0 0 Z3 0 0 0 0 620 ∆(Π7)/S1× S6 Z 0 0 0 Z 0 0 0 1760 ∆(Π7)/S2× S5 Z 0 0 0 Z3 0 0 0 3796 ∆(Π7)/S3× S4 Z 0 0 0 Z5 0 0 0 5382 ∆(Π8)/S1× S7 Z 0 0 0 0 Z 0 0 12740 ∆(Π8)/S2× S6 Z 0 0 0 Z/2Z Z3 0 0 31322 ∆(Π8)/S3× S5 Z 0 0 0 0 Z7 0 0 50972 ∆(Π8)/S4× S4 Z 0 0 0 Z/2iZ∗ Z8 0 0 59777 ∆(Π9)/S1× S8 Z 0 0 0 0 0 Z 0 103782 ∆(Π9)/S2× S7 Z 0 0 0 0 0 Z4 0 286155 ∆(Π9)/S3× S6 Z 0 ? ? ? ? Z9 0 527122 ∆(Π9)/S4× S5 Z 0 ? ? ? ? Z14 0 705697 * i ∈ {1, 2}

Table 2.1: The homology of ∆(Πn)/Sk× Sn−k with coefficients inZ for n ≤ 9,

and the cardinality of its face poset. The results are obtained by computer calculations.

2. Klein bottle (K), see Figure 2.3:

Hi(K;Z) ∼= ⎧ ⎪ ⎨ ⎪ ⎩ Z for i = 0 Z × (Z/2Z) for i = 1 0 for i ≥ 2 3. Projective plane (P2 R), see Figure 2.4: Hi(P_R2;Z) ∼= ⎧ ⎪ ⎨ ⎪ ⎩ Z for i = 0 Z/2Z for i = 1 0 for i ≥ 2

4. ∆(Πn)/Sk × Sn−k, see Table 2.1. The results are obtained by computer

calculations. Arone mentions this in [1]. I computed the Betti Numbers over different fields of finite characteristics. I used Theorem 2.6.12 to determine the characteristics to test.

Definition 2.6.9. Let F be a field. The n-th Betti number over the field F is defined as

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2.6. HOMOLOGY AND CHAIN COMPLEXES 27 Definition 2.6.10. The Euler Characteristic χ(∆) of ∆ is defined as

χ(∆) :=∑︂

n≥0

(−1)n|Sn(∆)|

Sn(∆) denotes the set of simplices of dimension n.

The Euler Characteristic is a topological invariant which only depends on the homotopy type, which can be deduced from the following theorem.

Theorem 2.6.11. Let ∆ be a trisp which has finitely many simplices. Then we have

χ(∆) =∑︂

n≥0

(−1)nβ_nF(∆) for any field F .

The proof of Theorem 2.6.11 can be found in [19, Chapter 3]. A Theorem on Transfer

Let R be a commutative ring. Let G be a finite group that acts on a trisp ∆. We have an induced action of G on the chain complex C(∆; R). We define the following R-linear chain morphism:

σ : C(∆; R) −→ C(∆; R) c ↦−→ ∑︂

g∈G

gc

We notice gσ = σ for all g ∈ G. Let π : C(∆; R) −→ C(∆/G; R) denote the chain morphism induced by the canonical projection ∆ −→ ∆/G. We define the chain morphism

µ : C(∆/G; R) −→ C(∆; R) πc ↦−→ σc

where c ∈ C(∆; R). The chain morphism µ induces a morphism µ∗: H(∆/G; R) −→ H(∆; R) in homology called the transfer. Clearly we have imµ∗⊂ H(∆; R)G_.

Theorem 2.6.12. If F is a field of characteristic zero or prime to |G|, then µ∗: H(∆/G; F ) −→ H(∆; F )G

is an isomorphism.

The proof of Theorem 2.6.12 as well as a detailed description of the Transfer can be found in [5].

Cohomology

We define the n-th cochain group Cn(∆; R) of ∆ with coefficients in R to the dual space of Cn(∆; R), i.e. Cn(∆) := homR(Cn(∆; R), R). We set the

coboundary operator to the dual map of the boundary operator. ∂n:= homR(∂n, R) : Cn−1(∆; R) −→ Cn(∆; R)

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Since taking the dual is a contravariant functor, we have ∂n+1_{◦ ∂}n _{= 0. We}

define the n-th cohomology group of ∆ with coefficients in R as follows: Hn(∆; R) := ker ∂n+1/im∂n

The set {σ∗| σ ∈ Sn(∆)} where

σ∗: Cn(∆; R) ⊃ Sn(∆) −→ R

τ ↦−→ {︄

1 for τ = σ 0 else is a basis of Cn(∆; R). It is the dual basis of Sn(∆).

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Chapter 3

Quotients of the nerve of

the partition lattice

3.1 Subgroups of S

1

× S

n−1

In this section we prove that the space ∆(Πn)/G is homotopy equivalent to

a wedge of spheres of dimension n − 3 for all natural numbers n ≥ 3 and all subgroups G ⊂ S1× Sn−1 using Discrete Morse Theory. Furthermore we find

a simple formula to compute the number of spheres. For example ∆(Πn)/S1×

Sn−1is homotopy equivalent to a sphere.

3.1.1 The collapsible subtrisp

Let n ≥ 3 be a fixed natural number throughout this section. In this section we consider subgroups of the Young subgroup

S1× Sn−1:= {σ ∈ Sn| σ(1) = 1}

We now consider the regular trisp ∆ := ∆(Πn)/G, where G ⊂ S1× Sn−1 is a

subgroup. Let ∆∗be that subtrisp of ∆ that we obtain by removing all vertices that are represented by some partition where the block containing 1 has exactly two elements and any other block is singleton. This condition is independent of the choice of the representative. Such a representative can be written as

vk := {{1, k}, {2}, . . . , ˆ︃{k}, . . . , {n}}

with k ∈ {2, . . . , n}. The element with the hat above is omitted.

Let R be the set of all vertices of ∆∗where the block containing 1 is singleton. Lemma 3.1.1. There exists an acyclic matching on the poset F (∆∗) such that a simplex σ ∈ F (∆∗) is critical if and only if V (σ) ⊂ R.

Proof. Let B be the set of all vertices that are not contained in R. Now we define a closure map φ : B −→ R. Let b ∈ B and let b˜ ∈ Πnbe a representative

of b. We modify the blocks in b˜ as follows. We replace the block A containing 1 by the two blocks {1} and A \ {1} and take the class of this new partition as φ(b). By applying Corollary 2.5 in [17] and Proposition 7 in [22] it is easy to

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see that this map is well-defined and a closure map1_{. By Theorem 2.2 in [17]}

we get an acyclic matching on F (∆∗) where a simplex σ is critical if and only if V (σ) ⊂ R.

Lemma 3.1.2. There exists an acyclic matching on F (∆∗) that has only one critical simplex which has dimension 0. In particular ∆∗ is collapsible.

Proof. We define an order-preserving map φ : F (∆∗) −→ [0, 1]

σ ↦−→ {︄

0 if V (σ) ⊂ R 1 else

By Lemma 3.1.1 we already have an acyclic matching on φ−1(1) with no critical simplices. Now we define an acyclic matching on φ−1(0). Let α be the vertex that is represented by the partition {{1}, {2, . . . , n}}. Let σ be a simplex of φ−1_{(0) with V (σ) ̸= {α}. It is clear that α can be either deleted from σ or}

uniquely inserted into σ. The matching rule therefore is the following: add α to a simplex if it is not already there, otherwise remove it. This matching is acyclic, since this rule can be formulated by a closure map that maps each vertex that does not equal α to α. We apply Theorem 2.5.3 and obtain an acyclic matching on F (∆∗) that has only one critical simplex; the simplex that has only the vertex α. Theorem 2.5.4 tells us that ∆∗ is collapsible.

What about the remaining simplices that are represented by a chain that contains a vkwith k ∈ {2, . . . , n}? Let VGbe the set of all these special vertices.

We define a poset PG := VG ∪ {0} such that 0 is the smallest element of PG

and the only element that is comparable with some other element. That means x, y ∈ PG, x < y implies x = 0. We define an order-preserving map φG :

F (∆) −→ PG as follows. Let σ ∈ F (∆), then we map σ to 0 if σ ∈ F (∆∗).

Otherwise we map σ to the special vertex of VG that belongs to σ, which is

unique. By Lemma 3.1.2 we already have an acyclic matching on φ−1_G (0) with one critical cell which has dimension 0. The next step would be to find acyclic matchings on φ−1_G (v) for all v ∈ VG and apply Theorem 2.5.3.

3.1.2 The main result

Let n ≥ 4, for each 2 ≤ k ≤ n we define the embedding ik : S1× Sn−2 −→

S1× Sn−1, where ik(σ) : [n] −→ [n] x ↦−→ {︄ k if x = k fk◦ σ ◦ fk−1(x) else

and fk is the order-preserving bijection fk : [n − 1] −→ [n] \ {k}. Notice that

σ(1) = 1 and hence ik(σ)(1) = 1. The image of ik consists of all permutations

σ ∈ S1× Sn−1with σ(k) = k. It also easy to see that ik is injective.

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3.1. SUBGROUPS OF S1× SN −1 31

Lemma 3.1.3. Let n ≥ 4 and G ⊂ S1× Sn−1 be an arbitrary subgroup. Let

2 ≤ k ≤ n and assume we have an acyclic matching on F (∆(Πn−1)/i−1k (G)),

where the set of critical simplices consists of one critical simplex of dimension 0 and l critical simplices of dimension d, where l, d ≥ 0. Let v ∈ VG be the vertex

that is represented by vk. Then there exists an acyclic matching on φ−1G (v) such

that the set of critical simplices consists of l critical simplices of dimension d+1. Proof. We define a map

ψ : F (∆(Πn−1)/i−1k (G)) −→ φ −1

G (v) \ {v}

as follows. Let σ ∈ F (∆(Πn−1)/i−1k (G)) be a simplex, then we choose a

rep-resentative σ˜ ∈ F (∆(Πn−1)) and describe what happens to the vertices of σ˜,

which are partitions of the set [n − 1]. Let p = {B1, B2, . . . , Br}, where B1

is the block that contains 1, be a partition in the chain σ˜. For p we define p∗ _{:= {f}

k(B1) ∪ {k}, fk(B2), . . . , fk(Br)}, which is a partition of the set [n]. σ˜

can be written as

σ˜ = (p1< . . . < pt)

We define ψ(σ) as the simplex that is represented by the chain (vk< p∗1< . . . <

p∗_t).

This map is well-defined, for: Let (p1< . . . < pt) and (p′1< . . . < p′t) be two

representatives. That means there exists g ∈ i−1_k (G) with p′_i= gpifor all i ∈ [t].

Assume i ∈ [t], pican be written as pi= {B1, B2, . . . , Br} with 1 ∈ B1, then we

also have p′_i= {g(B1), g(B2), . . . , g(Br)} with 1 ∈ g(B1). For 1 < j ≤ r we have

fk(g(Bj)) = ik(g)(fk(Bj)). It also follows fk(g(B1))∪{k} = ik(g)(fk(B1)∪{k}),

since ik(g)(k) = k. Therefore ik(g) gives us our relation. It is easy to see that

ψ is order-preserving. ψ has an inverse

ψ−1: φ−1_G (v) \ {v} −→ F (∆(Πn−1)/i−1k (G))

that maps a σ ∈ φ−1_G (v) \ {v} as follows. The smallest vertex of σ is v, we choose a representative σ˜ = (vk < p1 < . . . < pt) ∈ F (∆(Πn)) such that

vk is the smallest partition in the chain σ˜. Let p = {B1, B2, . . . , Br}, where

B1 is the block that contains 1, be a partition in σ˜. In particular we have

k ∈ B1, since vk refines any partition in σ˜. For p we define the partition

p∗:= {f_k−1(B1\ {k}), fk−1(B2), . . . , fk−1(Br)}. We define ψ−1(σ) as the simplex

that is represented by (p∗1< . . . < p∗t).

ψ−1 _{is well-defined, for: Let (v}

k < p1 < . . . < pt) and (vk < p′1 < . . . < p′t)

be two representatives. That means there exists g ∈ G with p′

i = gpi for all

i ∈ [t]. Clearly we have g(1) = 1 and g(k) = k, and therefore g lies in the image of ik. i−1k (g) gives us the relation we are looking for. It is easy to see that ψ

−1

is order-preserving and the inverse of ψ.

Via ψ we get an acyclic matching on φ−1_G (v) that has l critical simplices of dimension d + 1, one critical simplex τ of dimension 1, and additionally we have the critical simplex that has only the vertex v. Finally we match v with τ and this matching is acyclic since v is the smallest element of φ−1_G (v).

Proposition 3.1.4. Let n ≥ 3 and G ⊂ S1× Sn−1 be a subgroup, then the

topological space ∆(Πn)/G is homotopy equivalent to a wedge of spheres of

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Proof. We show there exists an acyclic matching on the poset F (∆(Πn)/G),

where the set of critical simplices consists of one critical simplex of dimension 0 and the other critical simplices have dimension n − 3. Then we can apply Theorem 2.5.4.

For n = 3 we have the cases G = S1× S2 and G = {id[3]}, hence ∆(Πn)/G

consists of two or three points. We do not match anything and get an acyclic matching with one critical simplex of dimension 0 and another one or two critical simplices of dimension 0. Now we assume n > 3 and proceed by induction.

By the discussion at the end of Section 3.1.1 it remains to show that there exists an acyclic matching on φ−1_G (v) for each v ∈ VG such that all critical

simplices have dimension n − 3. Let v ∈ VG, then we choose a 2 ≤ k ≤ n such

that vk is a representative of v. By induction hypothesis we have an acyclic

matching on ∆(Πn−1)/i−1_k (G) which has one critical simplex of dimension 0

and critical simplices of dimension n − 4. We apply Lemma 3.1.3 and get our acyclic matching on φ−1_G (v).

3.1.3 Counting spheres

We are now interested in the number of spheres we have. This can be computed either by looking into the proof of Proposition 3.1.4 or by determining the Betti Number βn−3(∆(Πn)/G), which is the dimension of Hn−3(∆(Πn)/G;C).

The action of S1×Sn−1on ∆(Πn) induces an action on Hn−3(∆(Πn);C). By

applying Corollary 3.3 in [12] and Theorem 1.5 in [24] we know that Hn−3(∆(Πn);C)

and

C · (S1× Sn−1) =

⨁︂

σ∈S1×Sn−1

C · σ

are isomorphic as (S1× Sn−1)-modules. By applying Theorem 2.6.12 we obtain

Hn−3(∆(Πn)/G;C) ∼= Hn−3(∆(Πn);C)G∼= (C · (S1× Sn−1))G

The dimension of (C · (S1× Sn−1))Gis the index of G in S1× Sn−1. Hence we

obtain the following result: ∆(Πn)/G is homotopy equivalent to a wedge of k

spheres of dimension n − 3, where k is the index of G in S1× Sn−1.

3.2 Torsion in homology

One might conjecture that ∆(Πn)/G is homotopy equivalent to a wedge of

spheres for any n ≥ 3 and any subgroup G ⊂ Sn. We disprove this by showing

that ∆(Πp)/Cp is not homotopy equivalent to a wedge of spheres for any prime

number p ≥ 5. Notice that Cp is not a subgroup of a proper Young subgroup.

3.2.1 Fundamental Group

In this section we briefly introduce the fundamental group. We will only consider closed paths since we will not prove anything.

Definition 3.2.1. Let X be a topological space and let a, b ∈ X. A path p from a to b is a continuous map p : [0, 1] −→ X with p(0) = a and p(1) = b. A path p is closed if p(0) = p(1). Then x0= p(0) is called the base point.

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3.2. TORSION IN HOMOLOGY 33 Definition 3.2.2. Let x0∈ X. Let p, q : [0, 1] −→ X be two closed paths with

base point x0, i.e. p(0) = p(1) = q(0) = q(1) = x0. We say that the paths p

and q are homotopic, if there exists a continuous map H : [0, 1] × [0, 1] −→ X such that H(x, 0) = f (x) and H(x, 1) = g(x) for all x ∈ [0, 1] and H(0, t) = H(1, t) = x0 for all t ∈ [0, 1].

Let x0∈ X. We define a relation on the set of clothed paths with base point

x0 as follows. Two paths p and q are in relation if p and q are homotopic to

each other. This relation is an equivalence relation, see Proposition 1.2 in [16, Section 1.1]. We define π1(x0, X) to the set of all classes of this relation.

Let p and q be two closed paths with the same base point. We define the product p ◦ q of p and q as follows.

p ◦ q : [0, 1] −→ X t ↦−→ {︄ p(2t) for 0 ≤ t ≤ 1 2 q(2t − 1) for 1₂ ≤ t ≤ 1

For two classes [p], [q] ∈ π1(x0, X) we set [p] ◦ [q] := [p ◦ q]. This is well

defined. Let

c : [0, 1] −→ X t ↦−→ x0

be the constant path. Then we have [c] ◦ [p] = [p] = [p] ◦ [c] for all [p] ∈ π(x0, X).

π1(x0, X) together with the binary operation ◦ is group, where [c] is the

neutral element.

Proposition 3.2.3. Let x0, x1∈ X, then π1(x0, X) and π1(x1, X) are

isomor-phic if X is pathwise connected.

Since for a pathwise connected topological space the fundamental group is unique up to isomorphism, we simply denote its fundamental group by π1(X).

Proposition 3.2.4. Let X be a pathwise connected topological space, G a group acting on X such that there exists some x0 ∈ X with gx0 = x0 for all g ∈ G.

Assume X is simply connected, then X/G is also simply connected. Proof. This is a special case of Corollary 6.3 in [5].

Proposition 3.2.5. Let X and Y be topological spaces such that X is homotopy equivalent to Y . Then π1(X) and π1(Y ) are isomorphic.

Proof. See Proposition 1.18 in [16, Section 1.1].

3.2.2 Covering Spaces

Definition 3.2.6. A covering space of a topological space X is a topological space Y together with a continuous map p : Y −→ X with the following property: Each x ∈ X has a neighborhood U such that its preimage p−1(U ) can be written as

p−1(U ) = ⋃︂

j∈J

Vj

where the Vj are disjoint open subsets of Y and for each j ∈ J the restriction

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p

−1

(U )

p

U

Figure 3.1: A covering space locally

Definition 3.2.7. Let G be a group acting on a topological space Y . We call this action even if each y ∈ Y has a neighborhood U such that g1(U )∩g2(U ) ̸= ∅

implies g1= g2 for all g1, g2∈ G.

In [16] an even action is called a covering space action.

Proposition 3.2.8. Any free action of a finite group on a Hausdorff space is even.

Proof. Exercise 23 in [16, Section 1.3].

Proposition 3.2.9. Assume we have an even action of a group G on a topo-logical space Y , then:

1. The quotient map p : Y −→ Y /G is a covering space. 2. G is isomorphic to π1(Y /G) if Y is simply connected.

Proof. Proposition 1.40 in [16].

3.2.3 Free actions on trisps

Let ∆ be a trisp and G a group that acts freely on ∆, then the quotient map ∆ −→ ∆/G is a covering space by Proposition 3.2.9 and Proposition 3.2.8. Assume ∆ is simply connected, then π1(∆/G) is isomorphic to G by Proposition

3.2.9. Furthermore, if G is additionally an abelian group, then H1(∆/G;Z) is

isomorphic to G and we may apply Proposition 3.2.5. We obtain the following results:

Proposition 3.2.10. Let ∆ be a simply connected trisp and G ̸= 0 a finite group that acts freely on ∆, then ∆/G is not homotopy equivalent to a wedge of spheres.

Lemma 3.2.11. Let ∆ be a simply connected trisp and G an abelian group that acts freely on ∆, then H1(∆/G;Z) is isomorphic to G.

Lemma 3.2.12. Let ∆ be a trisp that is homotopy equivalent to a wedge of k spheres of dimension d > 0. Let G be a finite group that acts freely on ∆. Then

Hi(∆/G;Q) ∼= ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Q for i = 0

Qk+1|G|−1 _{for i = d and d even}

Qk−1|G|+1 _{for i = d and d odd}

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3.2. TORSION IN HOMOLOGY 35 Proof. We have Hi(∆;Q) ∼= 0 for i ̸= 0 and i ̸= d. By applying Theorem 2.6.12

we obtain Hi(∆/G;Q) ∼= 0 for i ̸= 0 and i ̸= d. By Theorem 2.6.11 we have

χ(∆) =∑︂

n≥0

(−1)nβQ_n(∆) = 1 + (−1)dk

which implies χ(∆/G) = 1+(−1)_|G|dk, since G acts freely. We apply Theorem 2.6.11 again and obtain χ(∆/G) = 1 + (−1)d_βQ

d (∆/G). Hence β Q d (∆/G) = (−1)d₍1+(−1)d_k |G| − 1) = (−1)d_+k |G| − (−1) d_.

A prime period action on the reduced subset lattice

Now let p ≥ 5 be a prime number. We consider the subgroup of Sp that is

generated by a cycle of length p, which we denote by Cp. Set

Lp:= P([p]) \ {∅, [p]}

Here, P([p]) denotes the set of subsets of [p]. Sp acts on Lp in a natural way.

Lemma 3.2.13. Let p > 0 be a prime number, then Cp acts freely on Lp. In

particular Cp acts freely on ∆(Lp).

Proof. We have to show gv = v implies g = id for all v ∈ Lp and g ∈ Cp. Let

g ∈ Cp with g ̸= id, then g generates Cp. Let v ∈ Lp. Since Cpacts transitively

on the set [p] and v ̸= ∅, we obtain v = [p] ̸∈ Lp which is a contradiction.

Proposition 3.2.14. Let p ≥ 5 be a prime number, then

H1(∆(Lp)/Cp;Z) ∼= Zp (3.1)

Hp−2(∆(Lp)/Cp;Z) ∼= Z (3.2)

Proof. Cpacts freely on ∆(Lp) by Lemma 3.2.13. By applying Lemma 3.2.11 we

obtain H1(∆(Lp)/Cp;Z) ∼=Zp, since ∆(Lp) is simply connected. The abstract

simplicial complex Lp, where the vertices are the singleton sets, is homotopy

equivalent to a sphere of dimension p − 2. This can be verified via Discrete Morse Theory for example. Since ∆(Lp) is the barycentric subdivision of Lp,

∆(Lp) is also homotopy equivalent to a sphere of dimension p − 2. Via Lemma

3.2.12 we obtain Hp−2(∆(Lp)/Cp;Q) ∼=Q.

A prime period action on the reduced partition lattice

Let n ≥ 3 be a fixed natural number. The symmetric group Sn operates on

∆(Πn) in a natural way. Let p ≥ 5 be a prime number, then ∆(Πp) is homotopy

equivalent to a wedge of (p − 1)! spheres of dimension p − 3. We consider the quotient ∆(Πp)/Cp.

Lemma 3.2.15. Let p ≥ 3 be a prime number, then Cp acts freely on ∆(Πp).

Proof. It suffices to show that Cpacts freely on the set of vertices of ∆(Πp). We

have to show gv = v implies g = id for all v ∈ Πpand g ∈ Cp. Assume we have

gv = v with g ̸= id. Since g generates Cp, we have gv = v for all g ∈ Zp. In

particular, Cp acts on the blocks of v and this action is free by Lemma 3.2.13.

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Proposition 3.2.16. Let p ≥ 5 be a prime number, then

H1(∆(Πp)/Cp;Z) ∼= Zp (3.3)

Hp−3(∆(Πp)/Cp;Z) ∼= Z

(p−1)!−(p−1)

p _(3.4)

In particular ∆(Πp)/Cp is not homotopy equivalent to a wedge of spheres.

Proof. Cpacts freely on ∆(Πp) by Lemma 3.2.15. By applying Lemma 3.2.11 we

obtain H1(∆(Πp)/Cp;Z) ∼=Zp, since ∆(Πp) is simply connected. Via Lemma

3.2.12 we obtain Hp−3(∆(Πp)/Cp;Q) ∼=Q

(p−1)!+1 p −1∼₌Q

(p−1)!−(p−1)

p _.

3.3 Forbidden block sizes

In this section we consider subtrisps of ∆(Πn)/S1× Sn−1, that we obtain by

forbidding certain block sizes, and determine their homotopy type, as well as bases of their cohomology. Our arguments make use of trisp closure maps, which were introduced by Kozlov. A similar result, where ∆(Πn)/Sn is considered,

has already been solved by Kozlov.

Let n > k ≥ 2. Let Πn,k denote the poset obtained from Πn by removing

all partitions where some block has more than 1 and less than k elements. We define ∆(Πn,k) to the nerve of the acyclic category Πn,k, which is a regular

trisp. Since Πn,k is a geometric lattice, ∆(Πn,k) is shellable, hence homotopy

equivalent to a wedge of spheres, see [4]. We notice Πn= Πn,2.

The symmetric group Snoperates on ∆(Πn,k) in a natural way. We consider

quotients ∆(Πn,k)/G, where G ⊂ Snis a subgroup. The following theorem is the

first result concerning the topology of such a quotient, where G is a non-trivial subgroup of Sn.

Theorem 3.3.1 (Kozlov [18]). Let n > k ≥ 2, then ∆(Πn,k)/Sn is collapsible.

In this section we study the case, where G is the Young subgroup S1×

Sn−1:= {σ ∈ Sn| σ(1) = 1}. The main result is stated in Proposition 3.3.5.

3.3.1 The main result

Let n ≥ 3 and 2 ≤ k < n be a fixed natural numbers throughout this section. The simplices of ∆(Πn,k)/S1× Sn−1 are represented by sequences of

parti-tions, where the partitions refine each other. Such a sequence can be considered as a leveled forest, where each level corresponds to a partition. Each node on a particular level corresponds to a block of the corresponding partition in the sequence. Now we modify such a forest as follows. We replace each node, which is a subset of [n], by its cardinality and put a mark onto the number if this subset contains 1. It is easy to see that the simplices of ∆(Πn,k)/S1× Sn−1

can be indexed with such forests. The vertices can be indexed with number partitions, which we may write as v0⊕ v1+ · · · + vr, of n that distinguish the

first number, i.e. ⊕ is non-commutative. The number on the left side of ⊕, that is v0, corresponds to the block that contains 1. We call this number the

num-ber on the left side. All other numnum-bers are on the right side. The boundary operators are obtained by deleting entire levels from forests and reconnecting vertices transitively through the deleted levels.

Combinatorial Topology of Quotients of Posets