The group of self-homotopy equivalences of A
               
                  n
               
               2-polyhedra

Cristina Costoya; David Méndez; Antonio Viruel

doi:10.1515/jgth-2018-0203

Article Publicly Available

The group of self-homotopy equivalences of A _n ²-polyhedra

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Published/Copyright: March 20, 2020

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From the journal Journal of Group Theory Volume 23 Issue 4

Abstract

Let X be a finite type A n 2 -polyhedron, n ≥ 2 . In this paper, we study the quotient group ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) , where ℰ ⁢ ( X ) is the group of self-homotopy equivalences of X and ℰ * ⁢ ( X ) the subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. We show that not every group can be realised as ℰ ⁢ ( X ) or ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) for X an A n 2 -polyhedron, n ≥ 3 , and specific results are obtained for n = 2 .

1 Introduction

Let ℰ ⁢ ( X ) denote the group of homotopy classes of self-homotopy equivalences of a space X, and let ℰ * ⁢ ( X ) denote the normal subgroup of self-homotopy equivalences inducing the identity on the homology groups of X. Problems related to ℰ ⁢ ( X ) have been extensively studied, with Kahn’s realisability problem deserving a special mention, having been placed first to solve in [2] (see also [1, 11, 12, 14]). It asks whether an arbitrary group can be realised as ℰ ⁢ ( X ) for some simply connected X, and though the general case remains an open question, it has recently been solved for finite groups [7]. As a way to approach Kahn’s problem, in [9, Problem 19], the question of whether an arbitrary group can appear as the distinguished quotient ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) is raised.

In this paper, we work with ( n - 1 ) -connected ( n + 2 ) -dimensional CW-complexes for n ≥ 2 , the so-called A n 2 -polyhedra. Homotopy types of these spaces have been classified by Baues in [4, Ch. I, § 8] using the long exact sequence of groups associated to simply connected spaces introduced by J. H. C. Whitehead in [15]. The author of [6] uses that classification to study the group of self-homotopy equivalences of an A 2 2 -polyhedron X. He associates to X a group ℬ 4 ⁢ ( X ) that is isomorphic to ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) and asks if any group can be realised as such a quotient in this context, that is, if A 2 2 -polyhedra provide an adequate framework to solve the realisability problem.

Here, in the general setting of an A n 2 -polyhedra X, n ≥ 2 , we also construct a group ℬ n + 2 ⁢ ( X ) (see Definition 2.4) that is isomorphic to ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) (see Proposition 2.5). We show that there exist many groups (for example ℤ / p , p odd, Corollary 1.2) for which the question above does not admit a positive answer. This fact should illustrate that A n 2 -polyhedra might not be the right setting to answer [9, Problem 19].

We show, for instance, that under some restrictions on the homology groups of X, ℬ n + 2 ⁢ ( X ) is infinite, which in particular implies that ℰ ⁢ ( X ) is infinite (see Proposition 3.6 and Proposition 3.9). Or for example, in many situations the existence of odd order elements in the homology groups of X implies the existence of involutions in ℬ n + 2 ⁢ ( X ) (see Lemma 3.4 and Lemma 3.5).

In this paper, we prove the following result.

Theorem 1.1.

Let X be a finite type A n 2 -polyhedron, n ≥ 3 . Then B n + 2 ⁢ ( X ) is either the trivial group or it has elements of even order.

As an immediate corollary, we obtain the following.

Corollary 1.2.

Let G be a non-trivial group with no elements of even order. Then G is not realisable as B n + 2 ⁢ ( X ) for X a finite type A n 2 -polyhedron, n ≥ 3 .

The case n = 2 is more complicated. Detailed group-theoretical analysis shows that a finite type A 2 2 -polyhedra might realise finite groups of odd order only under very restrictive conditions. Recall that, for a group G, rank ⁡ G is the smallest cardinal of a set of generators for G [13, p. 91]. We have the following result.

Theorem 1.3.

Suppose that X is a finite type A 2 2 -polyhedron with a non-trivial finite B 4 ⁢ ( X ) of odd order. Then the following holds:

rank ⁡ H 4 ⁢ ( X ) ≤ 1 ,
π 3 ⁢ ( X ) and H 3 ⁢ ( X ) are 2 -groups, H 2 ⁢ ( X ) is an elementary abelian 2 -group,
rank ⁡ H 3 ⁢ ( X ) ≤ 1 2 ⁢ rank ⁡ H 2 ⁢ ( X ) ⁢ ( rank ⁡ H 2 ⁢ ( X ) + 1 ) - rank ⁡ H 4 ⁢ ( X ) ≤ rank ⁡ π 3 ⁢ ( X ) ,
the natural action of ℬ 4 ⁢ ( X ) on H 2 ⁢ ( X ) induces a faithful representation

ℬ 4 ⁢ ( X ) ≤ Aut ⁡ ( H 2 ⁢ ( X ) ) .

All our attempts to find a space satisfying the hypothesis of Theorem 1.3 were unsuccessful. We therefore make the following conjecture.

Conjecture 1.4.

Let X be an A 2 2 -polyhedron. If ℬ 4 ⁢ ( X ) is a non-trivial finite group, then it necessarily has an element of even order.

This paper is organised as follows. In Section 2, we give a brief introduction to Whitehead’s and Baues’s results for the classification of homotopy types of A n 2 -polyhedra, or equivalently, isomorphism classes of certain long exact sequences of abelian groups (see Theorem 2.3). In Section 3, we study how restrictions on X affect the group ℬ n + 2 ⁢ ( X ) . Finally, Section 4 is devoted to the proof of our main results, Theorem 1.1 and Theorem 1.3.

2 The Γ-sequence of an A n 2 -polyhedron

Let Ab denote the category of abelian groups. In [15], J. H. C. Whitehead constructed a functor Γ : Ab → Ab , known as Whitehead’s universal quadratic functor, and an exact sequence, which are useful to our purposes and introduced in this section. The Γ-functor is defined as follows. Let A and B be abelian groups and η : A → B a map (of sets) between them. The map η is said to be quadratic if

η ⁢ ( a ) = η ⁢ ( - a ) for all a ∈ A ,
the map A × A → B taking ( a , a ′ ) to η ⁢ ( a + a ′ ) - η ⁢ ( a ) - η ⁢ ( a ′ ) is bilinear.

For an abelian group A, Γ ⁢ ( A ) is the only abelian group for which there exists a quadratic map γ : A → Γ ⁢ ( A ) such that every other quadratic map η : A → B factors uniquely through γ. This means that there is a unique group homomorphism η □ : Γ ⁢ ( A ) → B such that η = η □ ⁢ γ . The quadratic map γ : A → Γ ⁢ ( A ) is called the universal quadratic map of A.

The Γ-functor acts on morphisms as follows. Let f : A → B be a group homomorphism, and γ : A → Γ ⁢ ( A ) and γ : B → Γ ⁢ ( B ) the universal quadratic maps. Then γ ⁢ f : A → Γ ⁢ ( B ) is a quadratic map, so there exists a unique group homomorphism ( γ ⁢ f ) □ : Γ ⁢ ( A ) → Γ ⁢ ( B ) such that ( γ ⁢ f ) □ ⁢ γ = γ ⁢ f . Define Γ ⁢ ( f ) = ( γ ⁢ f ) □ .

We now list some of its properties that will be used later in this paper.

Proposition 2.1 ([5, pp. 16–17]).

The Γ functor has the following properties:

Γ ⁢ ( ℤ ) = ℤ .
Γ ⁢ ( ℤ n ) is ℤ 2 ⁢ n if n is even, or ℤ n if n is odd.
Let I be an ordered set and A i an abelian group for each i ∈ I . Then

Γ ⁢ ( ⊕ I A i ) = ( ⊕ I Γ ⁢ ( A i ) ) ⊕ ( ⊕ i < j A i ⊗ A j ) .

Moreover, the groups Γ ⁢ ( A i ) and A i ⊗ A j are respectively generated by elements γ ⁢ ( a i ) and a i ⊗ a j , with a i ∈ A i , a j ∈ A j , i < j , and

γ ⁢ ( a i + a j ) = γ ⁢ ( a i ) + γ ⁢ ( a j ) + a i ⊗ a j 𝑓𝑜𝑟 ⁢ a i ∈ A i , a j ∈ A j , i < j

(see [ 15 , § 5, § 7] ).

We now introduce Whitehead’s exact sequence. Let X be a simply connected CW-complex. For n ≥ 1 , the n-th Whitehead Γ-group of X is defined as

Γ n ⁢ ( X ) = Im ⁡ ( i * : π n ⁢ ( X n - 1 ) → π n ⁢ ( X n ) ) .

Here, i : X n - 1 → X n is the inclusion of the ( n - 1 ) -skeleton of X into its n-skeleton. Then Γ n ⁢ ( X ) is an abelian group for n ≥ 1 . This group can be embedded into a long exact sequence of abelian groups

(2.1) ⋯ → H n + 1 ⁢ ( X ) → b n + 1 Γ n ⁢ ( X ) → i n - 1 π n ⁢ ( X ) → h n H n ⁢ ( X ) → ⋯ ,

where h n is the Hurewicz homomorphism and b n + 1 is a boundary representing the attaching maps.

For each n ≥ 2 , a functor Γ n 1 : Ab → Ab is defined as follows. Let Γ 2 1 = Γ be the universal quadratic functor, and for n ≥ 3 , Γ n 1 = - ⊗ ℤ 2 . It turns out that if X is ( n - 1 ) -connected, then Γ n 1 ⁢ ( H n ⁢ ( X ) ) ≅ Γ n + 1 ⁢ ( X ) (see [5, Theorem 2.1.22]). Thus the final part of the long exact sequence (2.1) can be written as

(2.2) H n + 2 ⁢ ( X ) → b n + 2 Γ n 1 ⁢ ( H n ⁢ ( X ) ) → i n π n + 1 ⁢ ( X ) → h n + 1 H n + 1 ⁢ ( X ) → 0 .

Now, for each n ≥ 2 , we define the category of A n 2 -polyhedra as the category whose objects are ( n + 2 ) -dimensional ( n - 1 ) -connected CW-complexes and whose morphisms are continuous maps between objects. Homotopy types of these spaces are classified through isomorphism classes in a category whose objects are sequences like (2.2) [4, Ch. I, § 8].

Definition 2.2 ([3, Ch. IX, § 4]).

Let n ≥ 2 be an integer. We define the category Γ-sequences n + 2 as follows. Objects are exact sequences of abelian groups

H n + 2 → Γ n 1 ⁢ ( H n ) → π n + 1 → H n + 1 → 0 ,

where H n + 2 is free abelian. Morphisms are triples of group homomorphisms f = ( f n + 2 , f n + 1 , f n ) , f i : H i → H i ′ , such that there exists a group homomorphism Ω : π n + 1 → π n + 1 ′ making the diagram

commutative. Objects in Γ ⁢ -sequences n + 2 are called Γ-sequences, and morphisms in the category are called Γ-morphisms.

On the one hand, we can assign to an A n 2 -polyhedron X an object in Γ-sequences n + 2 by considering the associated exact sequence (2.2). We call such an object the Γ-sequence of X . On the other hand, to a continuous map α : X → X ′ of A n 2 -polyhedra, we can assign a morphism between the corresponding Γ-sequences by considering the induced homomorphisms

Therefore, we have a functor A n 2 ⁢ -polyhedra → Γ ⁢ -sequences n + 2 which clearly restricts to the homotopy category of A n 2 -polyhedra, ℋ ⁢ o ⁢ A n 2 ⁢ -polyhedra . It is obvious that this functor sends homotopy equivalences to isomorphisms between the corresponding Γ-sequences. Thus we can classify homotopy types of A n 2 -polyhedra through isomorphism classes of Γ-sequences.

Theorem 2.3 ([4, Ch. I, § 8]).

The functor H ⁢ o ⁢ A n 2 ⁢ -polyhedra → Γ ⁢ -sequences n + 2 previously defined is full. Moreover, for any object in Γ ⁢ -sequences n + 2 , there exists an A n 2 -polyhedron whose Γ-sequence is the given object in Γ ⁢ -sequences n + 2 . In fact, there exists a 1–1 correspondence between homotopy types of A n 2 -polyhedra and isomorphism classes of Γ-sequences.

Following the ideas of [6], we introduce the following.

Definition 2.4.

Let X be an A n 2 -polyhedron. We denote by ℬ n + 2 ⁢ ( X ) the group of Γ-isomorphisms of the Γ-sequence of X.

Let Ψ : ℰ ⁢ ( X ) → ℬ n + 2 ⁢ ( X ) be the map that associates to α ∈ ℰ ⁢ ( X ) the Γ-isomorphism Ψ ⁢ ( α ) = ( H n + 2 ⁢ ( α ) , H n + 1 ⁢ ( α ) , H n ⁢ ( α ) ) . Then Ψ is a group homomorphism: its kernel is the subgroup of self-homotopy equivalences inducing the identity map on the homology groups of X, that is, ℰ * ⁢ ( X ) . Also, Ψ is onto as a consequence of Theorem 2.3. Hence, we immediately obtain the following result.

Proposition 2.5.

Let X be an A n 2 -polyhedron, n ≥ 2 . Then

ℬ n + 2 ⁢ ( X ) ≅ ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) .

3 Self-homotopy equivalences of finite type A n 2 -polyhedra

Henceforth, an A n 2 -polyhedron will mean an ( n - 1 ) -connected, ( n + 2 ) -dimensional CW-complex of finite type. Recall that, for simply connected and finite type spaces, the homology and homotopy groups H n ⁢ ( X ) and π n ⁢ ( X ) are finitely generated and abelian for n ≥ 1 .

The Γ-sequence tool introduced in Section 2 will help us to illustrate, from an algebraic point of view, how different restrictions on an A n 2 -polyhedron X affect the quotient group ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) . We devote this section to that matter. We also obtain several results that are needed in the proof of Theorem 1.1 and Theorem 1.3. The following result is a generalisation of [6, Theorem 4.5].

Proposition 3.1.

Let X be an A n 2 -polyhedron and suppose that the Hurewicz homomorphism h n + 2 : π n + 2 ⁢ ( X ) → H n + 2 ⁢ ( X ) is onto. Then every automorphism of H n + 2 ⁢ ( X ) is realised by a self-homotopy equivalence of X.

Proof.

As part of the exact sequence (2.1) for X, we have

⋯ → π n + 2 ⁢ ( X ) → h n + 2 H n + 2 ⁢ ( X ) → b n + 2 Γ n 1 ⁢ ( H n ⁢ ( X ) ) → π n + 1 ⁢ ( X ) → ⋯ .

Then, since h n + 2 is onto by hypothesis, b n + 2 is the trivial homomorphism. Thus, for every f n + 2 ∈ Aut ⁡ ( H n + 2 ⁢ ( X ) ) , we have b n + 2 ⁢ f n + 2 = b n + 2 = 0 , so if Ω = id , ( f n + 2 , id , id ) ∈ ℬ n + 2 ⁢ ( X ) . Then there exists f ∈ ℰ ⁢ ( X ) with H n + 2 ⁢ ( f ) = f n + 2 , H n + 1 ⁢ ( f ) = id , H n ⁢ ( f ) = id . ∎

We can easily prove that automorphism groups can be realised; a result that can also be obtained as a consequence of [14, Theorem 2.1].

Example 3.2.

Let G be a group isomorphic to Aut ⁡ ( H ) for some finitely generated abelian group H. Then, for any integer n ≥ 2 , there exists an A n 2 -polyhedron X such that G ≅ ℬ n + 2 ⁢ ( X ) : take the Moore space X = M ⁢ ( H , n + 1 ) , which in particular is an A n 2 -polyhedron. The Γ-sequence of X is

H n + 2 ⁢ ( X ) = 0 → Γ n 1 ⁢ ( H n ⁢ ( X ) ) = 0 → H ⁢ → = ⁢ H → 0 .

Then, for every f ∈ Aut ⁡ ( H ) , taking Ω = f , we see that ( id , f , id ) ∈ ℬ n + 2 ⁢ ( X ) , and those are the only possible Γ-isomorphisms. Thus ℬ n + 2 ⁢ ( X ) ≅ Aut ⁡ ( H ) ≅ G .

The use of Moore spaces is not required in the n = 2 case.

Example 3.3.

Let G be a group isomorphic to Aut ⁡ ( H ) for some finitely generated abelian group H. Consider the following object in Γ-sequences 4 :

(3.1) ℤ ⁢ ↠ b 4 ⁢ Γ ⁢ ( ℤ 2 ) = ℤ 4 → H → = H → 0 .

By Theorem 2.3, there exists an A 2 2 -polyhedron X realising this object. In particular, H 4 ⁢ ( X ) = ℤ , H 3 ⁢ ( X ) = π 3 ⁢ ( X ) = H and H 2 ⁢ ( X ) = ℤ 2 . It is clear from (3.1) that ( id , f , id ) is a Γ-isomorphism for every f ∈ Aut ⁡ ( H ) . Now, Aut ⁡ ( ℤ 2 ) is the trivial group while Aut ⁡ ( ℤ ) = { - id , id } . It is immediate to check that ( - id , f , id ) is not a Γ-isomorphism since id ⁢ b 4 ≠ b 4 ⁢ ( - id ) . Then we obtain ℬ 4 ⁢ ( X ) ≅ Aut ⁡ ( H ) .

Observe that not every group G is isomorphic to the automorphism group of an abelian group (for example ℤ p if p is odd). Hence, examples from above only provide a partial positive answer to the realisability problem for ℬ n + 2 ⁢ ( X ) . Indeed, the automorphism group of an abelian group (other than ℤ 2 ) has elements of even order. The following results go in that direction.

Lemma 3.4.

Let X be an A n 2 -polyhedron, n ≥ 2 . If H n ⁢ ( X ) is not an elementary abelian 2-group, then B n + 2 ⁢ ( X ) has an element of order 2.

Proof.

Since H n ⁢ ( X ) is not an elementary abelian 2-group, it admits a non-trivial involution - id : H n ⁢ ( X ) → H n ⁢ ( X ) . But Γ n 1 ⁢ ( - id ) = id for every n ≥ 2 , so we have ( id , id , - id ) ∈ ℬ n + 2 ⁢ ( X ) , and the result follows. ∎

Notice a key difference between the n = 2 and the n ≥ 3 cases: Γ 2 1 ⁢ ( A ) = Γ ⁢ ( A ) is never an elementary abelian 2-group when A is finitely generated and abelian, as can be deduced from Proposition 2.1. However, for n ≥ 3 , Γ n 1 ⁢ ( A ) = A ⊗ ℤ 2 is always an elementary abelian 2-group. Taking advantage of this fact we can prove the following result.

Lemma 3.5.

Let X be an A n 2 -polyhedron, n ≥ 3 . If any of the homology groups of X is not an elementary abelian 2-group (in particular, if H n + 2 ⁢ ( X ) ≠ 0 ), then B n + 2 ⁢ ( X ) contains a non-trivial element of order 2.

Proof.

Under our assumptions, Γ n 1 ⁢ ( H n ⁢ ( X ) ) is an elementary abelian 2-group. For Ω = - id , the triple ( - id , - id , - id ) is a Γ-isomorphism of order 2 unless H n + 2 ⁢ ( X ) , H n + 1 ⁢ ( X ) and H n ⁢ ( X ) are all elementary abelian 2-groups. ∎

We remark that this result does not hold for A 2 2 -polyhedra. Indeed, if we consider the construction in Example 3.3 for H = ℤ 2 , then ℬ 4 ⁢ ( X ) ≅ Aut ⁡ ( ℤ 2 ) = { * } does not contain a non-trivial element of order 2 although H 4 ⁢ ( X ) = ℤ is not an elementary abelian 2-group.

We now prove some results regarding the finiteness of ℬ n + 2 ⁢ ( X ) .

Proposition 3.6.

Let X be an A n 2 -polyhedron, n ≥ 2 , with rank ⁡ H n + 2 ⁢ ( X ) ≥ 2 and every element of Γ n 1 ⁢ ( H n ⁢ ( X ) ) of finite order. Then B n + 2 ⁢ ( X ) is an infinite group.

Proof.

Since rank ⁡ H n + 2 ⁢ ( X ) ≥ 2 , we may write H n + 2 ⁢ ( X ) = ℤ 2 ⊕ G , G a (possibly trivial) free abelian group. Consider the Γ-sequence of X,

ℤ 2 ⊕ G → b n + 2 Γ n 1 ⁢ ( H n ⁢ ( X ) ) → i n π n + 1 ⁢ ( X ) → h n + 1 H n + 1 ⁢ ( X ) → 0 .

Since b n + 2 ⁢ ( ℤ 2 ) ≤ Γ n 1 ⁢ ( H n ⁢ ( X ) ) is a finitely generated ℤ -module with finite order generators, it is a finite group. Define k = exp ⁡ ( b n + 2 ⁢ ( ℤ 2 ) ) , and consider the automorphism of ℤ 2 given by the matrix

( 1 k 0 1 ) ∈ GL 2 ⁡ ( ℤ ) ,

which is of infinite order. If we take f ⊕ id G ∈ Aut ⁡ ( ℤ 2 ⊕ G ) , then we have b n + 2 ⁢ ( f ⊕ id ) = b n + 2 , thus ( f ⊕ id G , id , id ) ∈ ℬ n + 2 ⁢ ( X ) , which is an element of infinite order. ∎

As previously mentioned, Γ n 1 ⁢ ( H n ⁢ ( X ) ) is an elementary abelian 2-group for n ≥ 3 . Hence, from Proposition 3.6 we get:

Corollary 3.7.

Let X be an A n 2 -polyhedron, n ≥ 3 , with rank ⁡ H n + 2 ⁢ ( X ) ≥ 2 . Then B n + 2 ⁢ ( X ) is an infinite group.

This result does not hold, in general, for n = 2 . However, if A is a finite group, Proposition 2.1 implies that Γ ⁢ ( A ) is finite as well, so from Proposition 3.6, we get the following.

Corollary 3.8.

Let X be an A 2 2 -polyhedron with rank ⁡ H 4 ⁢ ( X ) ≥ 2 and H 2 ⁢ ( X ) finite. Then B 4 ⁢ ( X ) is an infinite group.

We end this section with one more result on the infiniteness of ℬ n + 2 ⁢ ( X ) .

Proposition 3.9.

Let X be an A n 2 -polyhedron, n ≥ 3 . If H n ⁢ ( X ) = Z 2 ⊕ G for a certain abelian group G, then B n + 2 ⁢ ( X ) is an infinite group.

Proof.

If H n ⁢ ( X ) = ℤ 2 ⊕ G , then

Γ n 1 ⁢ ( H n ⁢ ( X ) ) = H n ⁢ ( X ) ⊗ ℤ 2 = ℤ 2 2 ⊕ ( G ⊗ ℤ 2 ) .

Hence GL 2 ⁡ ( ℤ ) ≤ Aut ⁡ ( H n ⁢ ( X ) ) and GL 2 ⁡ ( ℤ 2 ) ≤ Aut ⁡ ( H n ⁢ ( X ) ⊗ ℤ 2 ) . Moreover, for every f ∈ GL 2 ⁡ ( ℤ ) , we have f ⊕ id G ∈ Aut ⁡ ( H n ⁢ ( X ) ) , which yields, through Γ n 1 , an automorphism ( f ⊕ id G ) ⊗ ℤ 2 = ( f ⊗ ℤ 2 ) ⊕ id G ⊗ ℤ 2 ∈ Aut ⁡ ( H n ⁢ ( X ) ⊗ ℤ 2 ) . This means that the functor Γ n 1 restricts to GL 2 ⁡ ( ℤ ) → GL 2 ⁡ ( ℤ 2 ) . Moreover, we have that - ⊗ ℤ 2 : GL 2 ( ℤ ) → GL 2 ( ℤ 2 ) has an infinite kernel. Hence, there are infinitely many morphisms f ∈ Aut ⁡ ( H n ⁢ ( X ) ) such that f ⊗ ℤ 2 = id . For any such a morphism f, ( id , id , f ) is an element of ℬ n + 2 ⁢ ( X ) . Therefore, ℬ n + 2 ⁢ ( X ) is infinite. ∎

4 Obstructions to the realisability of groups

We have seen in Section 3 that the group ℬ n + 2 ⁢ ( X ) contains elements of even order unless strong restrictions are imposed on the homology groups of the A n 2 -polyhedron X. Since we are interested in realising an arbitrary group G as ℬ n + 2 ⁢ ( X ) for X a finite type A n 2 -polyhedron, in this section, we focus our attention on the remaining situations and prove Theorems 1.1 and 1.3. We first give some previous results.

Lemma 4.1.

For G an elementary abelian 2-group, Γ ⁢ ( - ) : Aut ⁡ ( G ) → Aut ⁡ ( Γ ⁢ ( G ) ) is injective.

Proof.

Let us show that the kernel of Γ ⁢ ( - ) is trivial. Assume that G is generated by { e j ∣ j ∈ J } , J an ordered set. If f ∈ Aut ⁡ ( G ) is in the kernel of Γ ⁢ ( - ) , then, for each j ∈ J , there exists a finite subset I j ⊂ J such that f ⁢ ( e j ) = ∑ i ∈ I j e i , and

γ ⁢ ( e j ) = Γ ⁢ ( f ) ⁢ γ ⁢ ( e j ) = γ ⁢ f ⁢ ( e j ) = γ ⁢ ( ∑ i ∈ I j e i ) = ∑ i ∈ I j γ ⁢ ( e i ) + ∑ i < k e i ⊗ e k ,

as a consequence of Proposition 2.1 (3), so I j = { j } and f ⁢ ( e j ) = e j for every j ∈ J . ∎

Lemma 4.2.

Let H 2 = ⊕ i = 1 n Z 2 , and let χ ∈ Γ ⁢ ( H 2 ) be an element of order 4. If there exists a non-trivial automorphism of odd order f ∈ Aut ⁡ ( H 2 ) such that Γ ⁢ ( f ) ⁢ ( χ ) = χ , then there exists g ∈ Aut ⁡ ( H 2 ) of order 2 such that Γ ⁢ ( g ) ⁢ ( χ ) = χ .

Proof.

Notice that according to [15, p. 66], we can write h ⊗ h = 2 ⁢ γ ⁢ ( h ) for any element h ∈ H 2 . Therefore, given a basis { h 1 , h 2 , … , h n } of H 2 , and replacing 3 ⁢ γ ⁢ ( h i ) by γ ⁢ ( h i ) + h i ⊗ h i if needed, we can write

χ = ∑ i = 1 n a ⁢ ( i ) ⁢ γ ⁢ ( h i ) + ∑ i , j = 1 n a ⁢ ( i , j ) ⁢ h i ⊗ h j ,

where every coefficient a ⁢ ( i ) , a ⁢ ( i , j ) is either 0 or 1. We now inductively construct a basis { e 1 , e 2 , … , e n } of H 2 as follows. Without loss of generality, assume a ⁢ ( 1 ) = 1 , and define e 1 = ∑ i = 1 n a ⁢ ( i ) ⁢ h i . Then { e 1 , h 2 , … , h n } is again a basis of H 2 and

χ = γ ⁢ ( e 1 ) + α 1 ⁢ e 1 ⊗ e 1 + β 1 ⁢ e 1 ⊗ ( ∑ s = 2 n b ⁢ ( 1 , s ) ⁢ h s ) + ∑ i , j > 1 n a 1 ⁢ ( i , j ) ⁢ h i ⊗ h j ,

where every coefficient in the equation is either 0 or 1. Assume a basis

{ e 1 , … , e r , h r + 1 , … , h n }

has been constructed such that

χ = γ ⁢ ( e 1 ) + ∑ j = 1 r α j ⁢ e j ⊗ e j + ∑ j = 1 r - 1 β j ⁢ e j ⊗ e j + 1 + β r ⁢ e r ⊗ ( ∑ s = r + 1 n b ⁢ ( r , s ) ⁢ h s ) + ∑ i , j > r n a r ⁢ ( i , j ) ⁢ h i ⊗ h j ,

where every coefficient is either 0 or 1. We may assume b ⁢ ( r , r + 1 ) = 1 and define e r + 1 = ∑ s = r + 1 n b ⁢ ( r , s ) ⁢ h s . Thus { e 1 , … , e r + 1 , h r + 2 , … , h n } is again a basis of H 2 and

χ = γ ⁢ ( e 1 ) + ∑ j = 1 r + 1 α j ⁢ e j ⊗ e j + ∑ j = 1 r β j ⁢ e j ⊗ e j + 1 + β r + 1 ⁢ e r + 1 ⊗ ( ∑ s = r + 2 n b ⁢ ( r + 1 , s ) ⁢ h s ) + ∑ i , j > r + 1 n a r + 1 ⁢ ( i , j ) ⁢ h i ⊗ h j .

Finally, we obtain a basis { e 1 , e 2 , … , e n } of H 2 such that

(4.1) χ = γ ⁢ ( e 1 ) + ∑ j = 1 n α j ⁢ e j ⊗ e j + ∑ j = 1 n - 1 β j ⁢ e j ⊗ e j + 1

for some coefficients

α j ∈ { 0 , 1 } , j = 1 , 2 , … , n , and β j ∈ { 0 , 1 } , j = 1 , 2 , … , n - 1 .

Now, for n = 1 , H 2 = ℤ 2 has a trivial group of automorphisms, so the result holds. For n = 2 , assume that there exists f ∈ Aut ⁡ ( H 2 ) such that Γ ⁢ ( f ) ⁢ ( χ ) = χ . From equation (4.1), χ = Γ ⁢ ( f ) ⁢ ( γ ⁢ ( e 1 ) ) + Γ ⁢ ( f ) ⁢ ( P ) , where

P ∈ Ω 1 ( Γ ( H 2 ) ) = { h ∈ Γ ( H 2 ) : ord ( h ) ∣ 2 } .

Then Γ ⁢ ( f ) ⁢ ( γ ⁢ ( e 1 ) ) has a multiple of γ ⁢ ( e 1 ) as its only summand of order 4, which implies f ⁢ ( e 1 ) = e 1 . Then either f ⁢ ( e 2 ) = e 2 , so f is trivial, or f ⁢ ( e 2 ) = e 1 + e 2 , so f has order 2.

For n ≥ 3 , we define g ∈ Aut ⁡ ( H 2 ) by g ⁢ ( e j ) = e j for j = 1 , 2 , … , n - 2 , and g ⁢ ( e n - 1 ) and g ⁢ ( e n ) , depending on α n - j and β n - 1 - j , for j = 0 , 1 , in equation (4.1), according to the following table.

α n	β n - 1	α n - 1	β n - 2	g ⁢ ( e n - 1 )	g ⁢ ( e n )
0	0	0 or 1	0 or 1	e n - 1	e n - 1 + e n
0	1	0	0	e n	e n - 1
0	1	0	1	e n - 2 + e n	e n - 2 + e n - 1
0	1	1	0	e n - 1 + e n	e n
0	1	1	1	e n - 2 + e n - 1 + e n	e n
1	0	0	0	e n - 2 + e n - 1	e n
1	0	0	1	e n - 2 + e n - 1	e n - 2 + e n
1	0	1	0	e n	e n - 1
1	0	1	1	e n - 2 + e n - 1	e n
1	1	0 or 1	0 or 1	e n - 1	e n - 1 + e n

A simple computation shows that, in all cases, g has order 2 and Γ ⁢ ( g ) ⁢ ( χ ) = χ , so the result follows. ∎

Definition 4.3.

Let f : H → K be a morphism of abelian groups. We say that a non-trivial subgroup A ≤ K is f-split if there exist groups B ≤ H and C ≤ K such that H ≅ A ⊕ B , K = A ⊕ C and f can be written as

id A ⊕ g : A ⊕ B → A ⊕ C for some ⁢ g : B → C .

Henceforward, we will make extensive use of this notation applied to

h n + 1 : π n + 1 ⁢ ( X ) → H n + 1 ⁢ ( X ) ,

the Hurewicz morphism. We prove the following.

Lemma 4.4.

Let X be an A n 2 -polyhedron, n ≥ 2 . Let A ≤ H n + 1 ⁢ ( X ) be an h n + 1 -split subgroup; thus H n + 1 ⁢ ( X ) = A ⊕ C for some abelian group C. Then, for every f A ∈ Aut ⁡ ( A ) , there exists f ∈ E ⁢ ( X ) inducing ( id , f A ⊕ id C , id ) ∈ B n + 2 ⁢ ( X ) .

Proof.

By hypothesis, H n + 1 ⁢ ( X ) = A ⊕ C , π n + 1 ⁢ ( X ) ≅ A ⊕ B for some abelian group B, and h n + 1 can be written as id A ⊕ g for some morphism g : B → C . Thus, for every f A ∈ Aut ⁡ ( A ) , we have a commutative diagram

Hence ( id , f A ⊕ id C , id ) ∈ ℬ n + 2 ⁢ ( X ) , and by Theorem 2.3, there exists f ∈ ℰ ⁢ ( X ) such that H n + 1 ⁢ ( f ) = f A ⊕ id C , H n + 2 ⁢ ( f ) = id and H n ⁢ ( f ) = id . ∎

The following lemma is crucial in the proof of Theorems 1.1 and 1.3.

Lemma 4.5.

Let X be an A n 2 -polyhedron, n ≥ 2 . Suppose that there exist h n + 1 -split subgroups of H n + 1 ⁢ ( X ) .

If n ≥ 3 , then ℬ n + 2 ⁢ ( X ) is either trivial or it has elements of even order.
If ℬ 4 ⁢ ( X ) is finite and non-trivial, then it has elements of even order.

Proof.

First of all, observe that we just need to consider when H n ⁢ ( X ) is an elementary abelian 2-group. Otherwise, the result is a consequence of Lemma 3.4.

Let A be an arbitrary h n + 1 -split subgroup of H n + 1 ⁢ ( X ) . If A ≠ ℤ 2 , there is an involution ι ∈ Aut ⁡ ( A ) that induces an element ( id , ι ⊕ id , id ) ∈ ℬ n + 2 ⁢ ( X ) of order 2 by Lemma 4.4, and the result follows. Hence we can assume that every h n + 1 -split subgroup of H n + 1 ⁢ ( X ) is ℤ 2 .

Both assumptions, namely H n ⁢ ( X ) being an elementary abelian 2-group and every h n + 1 -split subgroup of H n + 1 ⁢ ( X ) being ℤ 2 , imply that H n + 1 ⁢ ( X ) is a finite 2-group. Indeed, since H n ⁢ ( X ) is finitely generated, Γ n 1 ⁢ ( H n ⁢ ( X ) ) is a finite 2-group and so is coker ⁡ b n + 2 . Then, since H n + 1 ⁢ ( X ) is also finitely generated, any direct summand of H n + 1 ⁢ ( X ) which is not a 2-group would be h n + 1 -split, contradicting our assumption that every h n + 1 -split subgroup of H n + 1 ⁢ ( X ) is ℤ 2 .

To prove our lemma, we start with the case A = H n + 1 ⁢ ( X ) is h n + 1 -split. When H n + 2 ⁢ ( X ) = 0 , the Γ-sequence of X becomes then the short exact sequence

0 → Γ n 1 ⁢ ( H n ⁢ ( X ) ) → Γ n 1 ⁢ ( H n ⁢ ( X ) ) ⊕ ℤ 2 → ℤ 2 → 0 .

Notice that any automorphism of order 2 in H n ⁢ ( X ) yields an automorphism of order 2 in Γ n 1 ⁢ ( H n ⁢ ( X ) ) since Γ n 1 is injective on morphisms: it is immediate for n ≥ 3 , and for n = 2 , apply Lemma 4.1. As our sequence is split, any f ∈ Aut ⁡ ( H n ⁢ ( X ) ) induces the Γ-isomorphism ( id , id , f ) of the same order. Hence, for H n ⁢ ( X ) ≠ ℤ 2 , it suffices to consider an involution. For H n ⁢ ( X ) = ℤ 2 , since by hypothesis

H n + 1 ⁢ ( X ) = ℤ 2 and H n + 2 ⁢ ( X ) = 0 ,

the only Γ-isomorphism is ( id , id , id ) , and therefore ℬ n + 2 ⁢ ( X ) is trivial as claimed.

When H n + 2 ⁢ ( X ) ≠ 0 , for n ≥ 3 , the result follows directly from Lemma 3.5. For n = 2 , we also assume that ℬ 4 ⁢ ( X ) is finite and non-trivial. Hence, since H 2 ⁢ ( X ) is an elementary abelian 2-group, Proposition 3.6 implies that H 4 ⁢ ( X ) = ℤ . Then, if a Γ-isomorphism of the form ( - id , f , id ) exists, it is of even order. In particular, if Im ⁡ b 4 is a subgroup of Γ ⁢ ( H 2 ⁢ ( X ) ) of order 2, ( - id , id , id ) is a Γ-isomorphism of even order.

Assume otherwise that Im ⁡ b 4 is a group of order 4. If a Γ-isomorphism ( id , f , id ) of odd order exists, then Γ ⁢ ( f ) ∘ b 4 = b 4 . In this situation, by Lemma 4.2 for χ = b 4 ⁢ ( 1 ) , there exists g ∈ Aut ⁡ ( H 2 ⁢ ( X ) ) , an automorphism of order 2 such that Γ ⁢ ( g ) ⁢ b 4 ⁢ ( 1 ) = b 4 ⁢ ( 1 ) . Moreover, as we are in the case A = H 3 ⁢ ( X ) being h 3 -split, ( id , g , id ) ∈ ℬ 4 ⁢ ( X ) is a Γ-isomorphism of order 2.

We deal now with the case A ≨ H n + 1 ⁢ ( X ) . Since A = ℤ 2 is a proper h n + 1 -split subgroup of H n + 1 ⁢ ( X ) , there exist non-trivial groups B and C such that

π n + 1 ⁢ ( X ) = ℤ 2 ⊕ B → h n + 1 ℤ 2 ⊕ C = H n + 1 ⁢ ( X ) ,

( t , b ) ↦ ( t , g ⁢ ( b ) )

for some group morphism B → 𝑔 C . Moreover, H n + 1 ⁢ ( X ) is a finite 2-group; thus C is a (non-trivial) finite 2-group, and there exists an epimorphism C → 𝜏 ℤ 2 .

Define

f ∈ Aut ⁡ ( ℤ 2 ⊕ C ) = Aut ⁡ ( H n + 1 ⁢ ( X ) ) ,

Ω ∈ Aut ⁡ ( ℤ 2 ⊕ B ) = Aut ⁡ ( π n + 1 ⁢ ( X ) )

to be the non-trivial involutions given by

f ⁢ ( t , c ) = ( t + τ ⁢ ( c ) , c ) and Ω ⁢ ( t , b ) = ( t + τ ⁢ ( g ⁢ ( b ) ) , b ) .

By construction, h n + 1 ⁢ Ω = f ⁢ h n + 1 , and if ( t , b ) ∈ coker ⁡ b n + 2 = ker ⁡ h n + 1 (thus g ⁢ ( b ) = 0 ), then Ω ⁢ ( t , b ) = ( t , b ) . In other words, ( id , f , id ) ∈ ℬ n + 2 ⁢ ( X ) , and it has order 2. ∎

We now prove our main results.

Proof of Theorem 1.1.

Assume that H n ⁢ ( X ) and H n + 1 ⁢ ( X ) are elementary abelian 2-groups, and H n + 2 ⁢ ( X ) = 0 . Otherwise, there would already be elements of order 2 in ℬ n + 2 ⁢ ( X ) as a consequence of Lemma 3.5.

Write H n ⁢ ( X ) = ⊕ I ℤ 2 , I an ordered set. Since n ≥ 3 , we have Γ n 1 = - ⊗ ℤ 2 , so Γ n 1 ⁢ ( H n ⁢ ( X ) ) = H n ⁢ ( X ) . We can also assume that there are no subgroups in H n + 1 ⁢ ( X ) that are h n + 1 -split. Otherwise, we would deduce from Lemma 4.5 that there are elements of order 2 in ℬ n + 2 ⁢ ( X ) . Thus H n + 1 ⁢ ( X ) = ⊕ J ℤ 2 with J ⊂ I , and the Γ-sequence corresponding to X is

0 → ⊕ I ℤ 2 → 𝑏 ( ⊕ I - J ℤ 2 ) ⊕ ( ⊕ J ℤ 4 ) → ℎ ⊕ J ℤ 2 → 0 .

We may rewrite the sequence as

0 → ( ⊕ I - J ℤ 2 ) ⊕ ( ⊕ J ℤ 2 ) → 𝑏 ( ⊕ I - J ℤ 2 ) ⊕ ( ⊕ J ℤ 4 ) → ℎ ⊕ J ℤ 2 → 0

and assume that b ⁢ ( x , y ) = ( x , 2 ⁢ y ) and h ⁢ ( x , y ) = y ⁢ mod ⁢ 2 . It is clear that any f ∈ Aut ⁡ ( ⊕ I - J ℤ 2 ) induces a Γ-isomorphism ( 0 , id , f ⊕ id ) of the same order.

On the one hand, for | I - J | ≥ 2 , ⊕ I - J ℤ 2 has an involution, and therefore ℬ n + 2 ⁢ ( X ) has elements of even order. On the other hand, for | I - J | < 2 , we consider the remaining possibilities.

Suppose that | I - J | = 1 . Then π n + 1 ⁢ ( X ) = ℤ 2 ⊕ ( ⊕ J ℤ 4 ) . If J is trivial, then ℬ n + 2 ⁢ ( X ) is clearly trivial as well. Otherwise, suppose that I - J = { i } and choose j ∈ J . Define

f ∈ Aut ⁡ ( ℤ 2 ⊕ ℤ 2 ⊕ ( ⊕ I - { i , j } ℤ 2 ) ) by f ( x , y , z ) = ( x , x + y , z ) ,

g ∈ Aut ⁡ ( ℤ 2 ⊕ ℤ 4 ⊕ ( ⊕ I - { i , j } ℤ 4 ) ) by g ( x , y , z ) = ( x , 2 x + y , z ) .

Then ( id , id , f ) is a Γ-isomorphism of order 2 since we have a commutative diagram

Suppose that I = J . If H n ⁢ ( X ) = H n + 1 ⁢ ( X ) = ℤ 2 , ℬ n + 2 ⁢ ( X ) is trivial. If not, choose i , j ∈ I , and define maps

f ∈ Aut ⁡ ( ℤ 2 ⊕ ℤ 2 ⊕ ( ⊕ I - { i , j } ℤ 2 ) ) by f ( x , y , z ) = ( y , x , z ) ,

g ∈ Aut ⁡ ( ℤ 4 ⊕ ℤ 4 ⊕ ( ⊕ I - { i , j } ℤ 4 ) ) by g ( x , y , z ) = ( y , x , z ) .

We have the following commutative diagram:

Then ( 0 , f , f ) is a Γ-isomorphism of order 2. ∎

As a consequence, we obtain a negative answer to the problem of realising groups as self-homotopy equivalences of A n 2 -polyhedra.

Corollary 4.6.

Let G be a non-nilpotent finite group of odd order. Then, for any n ≥ 3 and for any A n 2 -polyhedron X, we have G ≇ E ⁢ ( X ) .

Proof.

Assume that there exists an A n 2 -polyhedron X such that ℰ ⁢ ( X ) ≅ G . Then, if ℰ ⁢ ( X ) ≠ ℰ * ⁢ ( X ) , the quotient ℰ ⁢ ( X ) / ℰ * ⁢ ( X ) is a finite group of odd order, which contradicts Theorem 1.1. Thus G ≅ ℰ ⁢ ( X ) = ℰ * ⁢ ( X ) . However, since X is a 1-connected and finite-dimensional CW-complex, ℰ * ⁢ ( X ) is a nilpotent group, [8, Theorem D], which contradicts the fact that G is non-nilpotent. ∎

We end this paper by proving our second main result.

Proof of Theorem 1.3.

By hypothesis, ℬ 4 ⁢ ( X ) is a finite group of odd order. From Lemma 3.4, we deduce that H 2 ⁢ ( X ) is an elementary abelian 2-group, and from Proposition 2.1, we deduce that Γ ⁢ ( H 2 ⁢ ( X ) ) is a 2-group. In particular, every element of Γ ⁢ ( H 2 ⁢ ( X ) ) is of finite order, and therefore rank ⁡ H 4 ⁢ ( X ) ≤ 1 by Proposition 3.6, so we have Theorem 1.3 (1). Now, any element in ℬ 4 ⁢ ( X ) is of the form ( 0 , f 2 , f 3 ) if H 4 ⁢ ( X ) = 0 , or ( id , f 2 , f 3 ) if H 4 ⁢ ( X ) = ℤ . Notice that a Γ-morphism of the form ( - id , f 2 , f 3 ) has even order thus it cannot be a Γ-isomorphism under our hypothesis. Therefore, if H 4 ⁢ ( X ) = ℤ , then b 4 ⁢ ( 1 ) generates a ℤ 4 factor in Γ ⁢ ( H 2 ⁢ ( X ) ) , and under our hypothesis, the equation

rank ⁡ Γ ⁢ ( H 2 ⁢ ( X ) ) = rank ⁡ H 4 ⁢ ( X ) + rank ⁡ ( coker ⁡ b 4 )

holds for rank ⁡ H 4 ⁢ ( X ) ≤ 1 .

Observe that any Γ-isomorphism of X induces a chain morphism of the short exact sequence

0 → coker ⁡ b 4 → π 3 ⁢ ( X ) → missing H 3 ⁢ ( X ) → 0 .

We will draw our conclusions from this induced morphism, which can be seen as an automorphism of π 3 ⁢ ( X ) that maps the subgroup i 2 ⁢ ( coker ⁡ b 4 ) to itself, thus inducing an isomorphism on the quotient, H 3 ⁢ ( X ) .

As we mentioned above, Γ ⁢ ( H 2 ⁢ ( X ) ) is a 2-group. Then coker ⁡ b 4 is a quotient of a 2-group so a 2-group itself. We claim that H 3 ⁢ ( X ) is also a 2-group; otherwise, H 3 ⁢ ( X ) has a summand whose order is either infinite or odd, and therefore this summand would be h 3 -split, which from Lemma 4.5 implies that ℬ 4 ⁢ ( X ) has elements of even order, leading to a contradiction. Since coker ⁡ b 4 and H 3 ⁢ ( X ) are 2-groups, so is π 3 ⁢ ( X ) , proving thus Theorem 1.3 (2).

Moreover, as a consequence of Lemma 4.5, no subgroup of H 3 ⁢ ( X ) can be h 3 -split, and thus rank ⁡ H 3 ⁢ ( X ) ≤ rank ⁡ ( coker ⁡ b 4 ) = rank ⁡ Γ ⁢ ( H 2 ⁢ ( X ) ) - rank ⁡ H 4 ⁢ ( X ) . We can compute rank ⁡ Γ ⁢ ( H 2 ⁢ ( X ) ) using Proposition 2.1 and immediately obtain Theorem 1.3 (3).

Now, for a 2-group G, define the subgroup Ω 1 ( G ) = { g ∈ G : ord ( g ) ∣ 2 } . One can easily check that Ω 1 ⁢ ( π 3 ⁢ ( X ) ) ≤ i 2 ⁢ ( coker ⁡ b 4 ) , and from [10, Ch. 5, Theorem 2.4], we obtain that any automorphism of odd order of π 3 ⁢ ( X ) acting as the identity on i 2 ⁢ ( coker ⁡ b 4 ) must be the identity.

Then, if ( id , f 3 , f 2 ) ∈ ℬ 4 ⁢ ( X ) is a Γ-morphism with f 3 non-trivial, f 3 has odd order, so we may assume that Ω : π 3 ⁢ ( X ) → π 3 ⁢ ( X ) (see Definition 2.2) has odd order too. By the argument above, it must induce a non-trivial homomorphism on i 2 ⁢ ( coker ⁡ b 4 ) , and therefore f 2 is non-trivial as well. So the natural action of ℬ 4 ⁢ ( X ) on H 2 ⁢ ( X ) must be faithful since any Γ-automorphism ( id , f 3 , f 2 ) ∈ ℬ 4 ⁢ ( X ) induces a non-trivial f 2 ∈ Aut ⁡ ( H 2 ⁢ ( X ) ) . Then Theorem 1.3 (4) follows. ∎

Communicated by Radha Kessar

Funding source: Ministerio de Economía y Competitividad

Award Identifier / Grant number: MTM2016-79661-P

Award Identifier / Grant number: MTM2016-78647-P

Funding source: Ministerio de Educación, Cultura y Deporte

Award Identifier / Grant number: FPU14/05137

Funding statement: The first author was partially supported by Ministerio de Economía y Competitividad (Spain), grant MTM2016-79661-P. The second author was partially supported by Ministerio de Educación, Cultura y Deporte grant FPU14/05137, and by Ministerio de Economía y Competitividad (Spain) grants MTM2016-79661-P and MTM2016-78647-P. The third author was partially supported by Ministerio de Economía y Competitividad (Spain) grant MTM2016-78647-P.

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Received: 2018-10-26

Revised: 2020-02-27

Published Online: 2020-03-20

Published in Print: 2020-07-01

Articles in the same Issue

https://doi.org/10.1515/jgth-2018-0203

The group of self-homotopy equivalences of A n 2-polyhedra

Article

Abstract

1 Introduction

Theorem 1.1.

Corollary 1.2.

Theorem 1.3.

Conjecture 1.4.

2 The Γ-sequence of an A n 2 -polyhedron

Proposition 2.1 ([5, pp. 16–17]).

Definition 2.2 ([3, Ch. IX, § 4]).

Theorem 2.3 ([4, Ch. I, § 8]).

Definition 2.4.

Proposition 2.5.

3 Self-homotopy equivalences of finite type A n 2 -polyhedra

Proposition 3.1.

Proof.

Example 3.2.

Example 3.3.

Lemma 3.4.

Proof.

Lemma 3.5.

Proof.

Proposition 3.6.

Proof.

Corollary 3.7.

Corollary 3.8.

Proposition 3.9.

Proof.

4 Obstructions to the realisability of groups

Lemma 4.1.

Proof.

Lemma 4.2.

Proof.

Definition 4.3.

Lemma 4.4.

Proof.

Lemma 4.5.

Proof.

Proof of Theorem 1.1.

Corollary 4.6.

Proof.

Proof of Theorem 1.3.

References

Articles in the same Issue

Articles in the same Issue

Articles in the same Issue

The group of self-homotopy equivalences of A _n ²-polyhedra