Homology, Homotopy and Applications, vol.2, No.8, 2000, pp.115–118 ISSN 1512–0139

ERRATUM TO ”ON SPACES OF THE SAME STRONG n-TYPE” [HHA, V. 1 (1999) NO. 10, PP. 205-217]

YVES FÈLIX and JEAN-CLAUDE THOMAS

(communicated by Lionel Schwartz)

On the Arkowitz-Maruyama conjecture.

The main purpose to this short note is to make a correction to one of the result of the article : On spaces of the same strong n-type which has been published in [7]. We want to thank Ken-Ichi Maruyama who kindly reports to us our mistake. We also add some comments about the Arkowitz-Maruyama conjecture.

1) The AM-conjecture.

Let (X, x_{0}) be a based path connected space and let AutX be the group of based homotopy classes of homotopy self-equivalences of (X, x_{0}). We denote by Aut_{π}^{n}X the subgroup of homotopy classes that induce the identity on the homotopy groups π_{i}(X, x_{0}) for i ⩽ n. Then we obtain the normal series

AutX ⊃ Aut_{π}^{1}X ⊃ ...Aut_{π}^{n−1}X ⊃ Aut_{π}^{n}X ⊃ ...

and we denote by Aut_{π}Z the inverse limit:

lim_{←} Aut_{π}^{n}X ≅ ⋂_{n⩾1} Aut_{π}^{n}X .

M. Arkowitz and K.I. Maruyama, [2] have conjectured that:

A-M. conjecture. Let Z be a simply connected finite complex. There exists an integer N such that the natural monomorphism

ρ_{N} : Aut_{π}Z → Aut_{π}^{N}Z

is an isomorphism, ie. Aut_{π}^{N}Z = Aut_{π}^{n}Z for all n ⩾ N.

At our knowledge, the AM-conjecture is still unsolved for general complexes. It is trivially true for any finite complex Z which admits a finite Postnikov decomposition. In this case, if Z^{(n)} denotes the n^{th}-Postnikov section of Z = Z^{(k)} then for n ⩾ k

AutZ = AutZ^{(n)} = Aut_{π}^{n}Z^{(n)} ≅ Aut_{π}Z .

The conjecture is also known for products of spheres [2] and if Z is an H_{0}-space [6].

2) The localization conjecture.

Now recall that if n ⩾ dim Z then Aut_{π}^{n}Z is a finitely presented nilpotent group [3]. Let P be any set of prime numbers. Given a localization l_{P} : Z → Z_{P}, the natural homomorphism l_{P} : Aut_{π}^{n}Z → Aut_{π}^{n}(Z_{P}) , [f] ↦ [f_{P}] and the localization homomorphism λ_{p} : Aut_{π}^{n}Z →

Received 5 October 2000; published on 24 October 2000.

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(Aut_{π}^{n}Z)_{P} coincides, up to a natural isomorphism [4]:

Aut_{π}^{n}Z
λ_{P}^{n} ↙  ↘ l_{P}^{n}
(Aut_{π}^{n}Z)_{P}  ⟶^{θ_{P}^{n}}_{≅} Aut_{π}^{n}(Z_{P})

Thus, each Aut_{π}^{n}(Z_{P}) , n ⩾ dim Z is P -local and the group Aut_{π}(Z_{P}) = lim_{←} Aut_{π}^{n}(Z_{P}) is also P -local. Universal property of localization defines the natural homomorphisms θ_{P} in the diagram below:

Aut_{π}Z
L_{P} ↙  ↘ lim_{←} l_{P}^{n} = ϕ_{P}
(Aut_{π}Z)_{P}  ⟶^{θ_{P}}  Aut_{π}(Z_{P})

Localization does not necessarily respect inverse limit, nonetheless we conjecture:

P -local conjecture. Let Z be a nilpotent finite complex. Then the natural map ϕ_{P} : Aut_{π}Z → Aut_{π}(Z_{P}) is a P -localization, ie. θ_{P} is an isomorphism.

As usual we denote by Z_{0}, instead of Z_{∅}, the rationalization of the space Z and more generally the subscript _{∅} is replaced by subscript _{0}. In a recent preprint, [5], K-I. Maruyama proves:

If X is a finite nilpotent complex and if Aut_{π}(X_{0}) = {1} then Aut_{π}X_{P} ≅ (Aut_{π}X)_{P} for any set of primes P.

3) Equivalence of the AM-conjecture and of the ∅-local conjecture.

In [7]-(first part of theorem 3), we have proved:

Theorem A. Let Z be a simply connected CW complex of finite type and let Z_{0} its rationalization. If H^{>M}(Z; ℚ) = 0 for some M then there exists an integer N such that the natural map ρ_{0}^{N} : Aut_{π}(Z_{0}) → Aut_{π}^{N}(Z_{0}) is an isomorphism.

Recently K-I. Maruyama [5] has proved theorem A for finite nilpotent complexes.

A consequence of theorem A is

Theorem B. Let Z be a simply connected finite complex. The space Z satisfies the AMconjecture iff Z satisfies the ∅-conjecture.

Proof. Let N as in theorem A and consider the commutative diagram,

(Aut_{π}Z)_{0} ⟶^{θ_{0}} Aut_{π}(Z_{0})
(ρ^{N})_{0} ↓  ≅↓ ρ_{0}^{N}
(Aut_{π}^{N}Z)_{0}  ⟶^{θ_{0}^{n}}_{≅} Aut_{π}^{N}(Z_{0})

If the AM-conjecture holds then (ρ^{N})_{0} is an isomorphism and so is θ_{0}. Thus the ∅-conjecture is satisfied. Conversely, suppose that θ_{0} is an isomorphism then the monomorphism ρ^{N} has finite cokernel C^{N}(Z). If C^{N}(Z) = C^{n}(Z) for all n ⩾ N then Aut_{π}^{N}Z = Aut_{π}^{n}Z and the AMconjecture is proved. If for some N_{1} ⩾ N, C^{N}(Z) ≠ C^{N_{1}}(Z) then C^{N_{1}}(Z) is strictly included in C^{N}(Z). Again with N_{1} playing the role of N the AM-conjecture is satisfied or there exists N_{2} such that ... and so on. At the end we have a sequence N_{1}, N_{2}, ..., N_{k} with C^{N_{k}}(Z) = {1} and the AM-conjecture is proved for Z.

4) Composition of homotopy classes.

Theorem C. The AM-conjecture is true for simply connected finite complexes Z satisfying: for each element [a] ∈ π_{m}(Z) there exists a non torsion element [b] ∈ π_{r}(Z) and a continuous map g : S^{m} → S^{r} such that [bg] = [a].

Proof. Let us denote by Aut_{π/τ}^{n} Z the subgroup of AutZ which consists of elements inducing the identity on each quotient π_{i}(Z)/τ(π_{i}(X)) , i ⩽ n where τ(π_{i}(Z)) denotes the torsion subgroup of π_{i}(Z). By our assumption,

Aut_{π}^{n}Z = Aut_{π/τ}^{n} Z .

This subgroup Aut_{π/τ} Z have been considered in [5]. I.K. Maruyama has observed that these groups are not nilpotent in general and proves (Th. 1.2) that the natural map

ρ_{τ}^{N} : Aut_{π/τ} Z → Aut_{π/τ}^{N} Z

is an isomorphism for some N. Then theorem C is a consequence of theorem A and of the following commutative diagram:

(Aut_{π}Z)_{0} = (Aut_{π/τ} Z)_{0} ⟶^{θ_{0,τ}} Aut_{π/τ} (Z_{0})
(ρ^{N})_{0} ↓  ↓ (ρ_{τ}^{N})_{0}  ≅↓ ρ_{0}^{N}
(Aut_{π}^{N}Z)_{0} = (Aut_{π/τ}^{N} Z)_{0}  ⟶^{θ_{0,τ}^{N}}_{≅} Aut_{π/τ}^{N} (Z_{0}) .

5) Correction to the last assertion of the theorem 3 in [7].

The proof of the last assertion of theorem 3 in [7]:

“Moreover if H^{>M}(Z; ℤ) = 0, then there exists an integer N such that the natural map Aut_{π}Z → Aut_{π}^{N}Z is an isomorphism”

is false, since in fact we have assumed the ∅-local conjecture to be true in our proof.

6) The Ω-conjecture.

Denote by Aut_{Ω}^{n}X the group of homotopy classes of self-homotopy equivalences f of X such that the restriction of Ωf to (ΩX)^{(n−1)} is homotopic to the identity.

Clearly, each Aut_{Ω}^{n}X is a subgroup of Aut_{π}^{n}X.

If Z is a finite simply connected complex then Aut_{π}^{n}Z , n ⩾ dim Z is a finitely generated nilpotent group and thus Aut_{Ω}^{n}Z is a nilpotent group for n ⩾ dim Z.

We denote by Aut_{Ω}X the inverse limit :

lim_{←} Aut_{Ω}^{n}X ≅ ⋂_{n⩾2} Aut_{Ω}^{n}X .

Ω-Conjecture. Let Z be a simply connected finite complex. There exists an integer N such that the natural map

ρ_{Ω}^{N} : Aut_{Ω}Z → Aut_{Ω}^{N}Z

is an isomorphism.

If Z is a finite simply connected complex, the natural injections Aut_{Ω}^{n}Z ↪ Aut_{π}^{n}Z induce isomorphisms

(Aut_{Ω}^{n}Z)_{0} ≅ (Aut_{π}^{n}Z)_{0} ,

for any n ⩾ dim Z. Indeed, if [f] ∈ Aut_{π}Z there are only finitely many obstructions for [f] being in Aut_{Ω}Z.

We do not know if there exists a simply connected finite complex Z such that (Aut_{Ω}Z)_{0} ≅ (Aut_{π}Z)_{0}. Clearly, we obtain:

Theorem D. Let Z be a simply connected finite complex such that

(Aut_{Ω}Z)_{0} ≅ (Aut_{π}Z)_{0} .

Then Z satisfies the AM-conjecture iff Z satisfies Ω-conjecture.

References

[1] M. Arkowitz and C. Curjel, Groups of homotopy classes, Lectures Notes in Mathematics, 4 Springer-Verlag, 1967.

[2] M. Arkowitz and K.I. Maruyama, Self homotopy equivalences which induce the identity on homology, cohomology or homotopy groups, Topology Appl. 87 (1998) 133-154.

[3] E. Dror and A. Zabrodsky. Unipotency and nilpotency in homotopy equivalences. Topology 18 (1979), 187-197.

[4] K.I. Maruyama, Localization of a certain subgroup of self-homotopy equivalences, Pacific J. Math. 136 (1989), 293-301.

[5] K.I. Maruyama, A subgroup of self-homotopy equivalences which satisfies the M-L condition. Bulletin of The Faculty of Education, Chiba University 48 -02/29/2000.

[6] K.I. Maruyama, Stability properties of maps between Hopf spaces. Preprint.

[7] Yves FÈlix and Jean-Claude Thomas, On spaces of the same strong n-type, Homology, Homotopy and Applications 1 No 10 (1999), 205-217.

This article may be accessed via WWW at http://www.rmi.acnet.ge/hha/ or by anonymous ftp at ftp://ftp.rmi.acnet.ge/pub/hha/volumes/2000/n8/n8.(dvi,ps,dvi.gz,ps.gz)

Yves Fèlix felix@agel.ucl.ac.be

Départment de Mathématiques Chemin du Cyclotron 2 1348 Louvain-la-Neuve Belgique

Jean-Claude Thomas jean-claude.thomas@univ-angers.fr

Université d’Angers Faculté des Sciences 2bd Lavoisier, Cedex 01 France