On Relative Homotopy Groups of Modules

Recommended by M ´ onica Clapp In his book " Homotopy Theory and Duality, " Peter Hilton described the concepts of relative homotopy theory in module theory. We study in this paper the possibility of parallel concepts of fibration and cofibration in module theory, analogous to the existing theorems in algebraic topology. First, we discover that one can study relative homo-topy groups, of modules, from a viewpoint which is closer to that of (absolute) homo-topy groups. Then, through the study of various cases, we learn that the classic fibra-tion/cofibration relation does not come automatically. Nonetheless, the ability to see the relative homotopy groups as absolute homotopy groups, in a stronger sense, promises to justify our ultimate search. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Introduction
In [1], Peter Hilton developed homotopy theory in module theory, parallel to the existing homotopy theory in topology.However, unlike homotopy theory in topology, there are two types of homotopy theory in module theory, the injective theory and the projective theory.They are dual but not isomorphic.In this paper, we emphasize the injective relative homotopy groups (of modules) and approach the proofs in a way that does not refer to elements of sets, so one can proceed with the dual, in projective relative homotopy theory, without further arguments.
During the search for the analogy between the relative homotopy groups in module theory and those in topology, we realize that the (injective) relative homotopy group, π n (A,β), n ≥ 1, for a map β : B 1 → B 2 has a structure which is fairly similar to an (injective) absolute homotopy group, namely, π n (A,coker{ι,β}), where ι : B 1 CB 1 is the inclusion of B 1 into an injective container CB 1 that induces a short exact sequence: Thereafter, we analyze the phenomena related π n (A,β) and π n (A,coker{ι,β}) through cases.As expected, the two are not always isomorphic; nevertheless, the fact that all relative homotopy groups are isomorphic to certain "strong (absolute) homotopy groups" gives rise to the possibility of developing parallel concepts of fibration and cofibration in projective and injective homotopy theories, respectively, in module theory, corresponding to the existing fibration/cofibration relation in algebraic topology.

Relative homotopy groups-from a different viewpoint
In the injective relative homotopy theory of modules, for a given Λ-module homomorphism β : B 1 → B 2 and a given Λ-module A, one computes the nth relative homotopy group, where ι 0 is the inclusion map which embeds A into an injective container CA, and 1 is the quotient map to ΣA, called the suspension of A, as the quotient.We say that the map (ρ,σ) : ι n−1 → β is i-nullhomotopic, denoted (ρ,σ) i 0, if it can be extended to an injective container of ι n−1 , and that π n (A,β) = Hom(ι n−1 ,β)/ Hom 0 (ι n−1 ,β), where Hom(ι n−1 ,β) is the abelian group of maps of ι n−1 to β, and Hom 0 (ι n−1 ,β) the subgroup consisting of i-nullhomotopic maps.
The computation of such diagrams, as (2.1), is rather challenging at times, especially during the search for suitable definitions of fibration and cofibration in module theory, analogous to those in topology.Therefore we examine the diagram, of relative homotopy groups, from another viewpoint: First assuming that the map β : B 1 → B 2 is monomorphic so (2.1) is essentially In (2.2), each pair of maps (ρ,σ) : ι n−1 → β induces a map σ : Σ n A → coker β.We define RHom Λ (Σ n A,cokerβ) to be the subgroup of Hom Λ (Σ n A,cokerβ) consisting of such induced maps; it gives the relative homotopy group π n (A,β) an alternative aspect.
Theorem 2.1.Suppose given a monomorphism β : B 1 B 2 .For each A, consider the diagram where ι 0 : A CA is the inclusion of A into an injective container CA, 1 the quotient map with ΣA, called the suspension of A, as the quotient, and κ the expected quotient map.Then, where To prepare for the proof of Theorem 2.1, we first state a couple of existing propositions.
Finally, the definition of RHom Λ (Σ n A,cokerβ) yields that each σ is induced from a commutative square (2.10) Thus, φ is epimorphic.
We remark that one can interpret RHom Λ (Σ n A,cokerβ) as the "reversible" subgroup of Hom Λ (Σ n A,cokerβ); suppose given a map σ ∈ Hom Λ (Σ n A,cokerβ), we say that σ is reversible if it can pull back and produce a commutative diagram (2.2).Furthermore, it reveals a connection between the relative homotopy group π n (A,β) and the (absolute) homotopy group π n (A,cokerβ).
Next, for the general case that β : B 1 → B 2 is arbitrary, we exploit the mapping cylinder of β and Theorem 2.5 follows immediately after Proposition 2.4.Proposition 2.4 [2].Suppose given maps β : where ι 0 : A CA is the inclusion of A into an injective container CA, 1 is the quotient map with ΣA, called the suspension of A, as the quotient, ι : B 1 CB 1 is the inclusion of B 1 into an injective container CB 1 , and κ is the expected quotient map.Then, where (2.13) As we mentioned earlier, our argument does not involve references to elements of sets, so one can proceed with the dual, in projective relative homotopy theory, automatically.As an illustration, for a given Λ-module homomorphism α : A 1 → A 2 and a given Λmodule B, one alternatively views the projective relative homotopy group π n (α,B), n ≥ 1, as follows.
Theorem 2.6.Suppose given α : where η 0 : PB B is the projection of a projective ancestor PB onto B, μ 1 is the inclusion map with ΩB, called the loop space of B, as the kernel, η : PA 2 A 2 is the projection of a projective ancestor PA 2 onto A 2 , and ι is the expected inclusion map.Then, where (2.16)

Various cases for
Here, we have Theorem 2.5, which does not only give us an alternative way of computing relative homotopy groups for a map β : B 1 → B 2 , but also shows a close connection between the (injective) relative homotopy groups π n (A,β) and the (injective) homotopy groups π n (A,coker{ι,β}).The latter indicates the possibility of developing analogous concepts of fibration and cofibration in module theory to those existing ones in algebraic topology.Before further commenting on this matter, we demonstrate a few calculations through analyzing these phenomena on RHom Λ (Σ n A,coker{ι,β}).First, we examine the case that the map β : B 1 → B 2 is the zero map.The homotopy exact sequence of a map β : B 1 → B 2 (see [1,Theorem 13.15]), thus, yields a short exact sequence as β * = 0.In addition, the special feature of the zero map suggests that (3.2) actually splits, thus, the relative homotopy group π n (A,β) is the direct sum of the other two.
To show that φ is monomorphic, suppose given . That is, ρ = γι n−1 for some γ : CΣ n−1 A → B 1 and σ 2 = ηι n for some η : CΣ n A → B 2 , respectively.The former says that the map ρ factors through ι n−1 in (3.7); therefore, by Proposition 2.3, σ 1 = κ 1 τ for some τ : Σ n A → CB 1 .Moreover, τ = νι n for some ν : CΣ n A → CB 1 , due to the facts that CB 1 is injective and that ι n is monomorphic.Therefore, σ 1 = κ 1 νι n and hence Finally, suppose given . We use the map ρ to complete a diagram (3.7)-since CB 1 is injective and ι n−1 is monomorphic, there exists a map σ 1 : CΣ n−1 A → CB 1 such that ιρ = σ 1 ι n−1 and σ 1 is then the induced map: Similarly, the map σ 2 completes diagram (3.8), precisely, Now φ is epimorphic because of the existence of the commutative diagram The dual of Theorem 3.1 and its corollaries say that if we assume that α : ).Notice that as A 2 = 0, one sees from diagram (2.14) in Theorem 2.6 that π n (α,B) ∼ = π n (ker α,η ,B); however, the isomorphism fails when A 1 = 0.As an example, consider the Λ-map α : 0 → Z, where Λ is the integral group ring of the finite cyclic group C k with generator τ and Z is regarded as a trivial C k -module.Then, for n even.
C. Joanna Su 9 (See [3, Theorem 3.1].)On the other hand, the well-known projective resolution of Z, thus, where the maps , ρ, σ are the augmentation of ZC k , multiplication by τ − 1, and multiplication by because all the maps in Hom Λ (IC k ,Z) and Hom Λ (IC k ,IC k ) are p-nullhomotopic.