On Relative Homotopy Groups of Modules

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Volume 2007, Article ID 27626,14pages doi:10.1155/2007/27626

Research Article

On Relative Homotopy Groups of Modules

C. Joanna Su

Received 10 May 2007; Accepted 16 August 2007 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 homotopy groups, of modules, from a viewpoint which is closer to that of (absolute) homotopy groups. Then, through the study of various cases, we learn that the classic fibration/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.

Copyright © 2007 C. Joanna Su. This is an open access article distributed under the Cre- ative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. 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, πn(A,β),n≥1, for a mapβ:B1→B2 has a structure which is fairly similar to an (injective) absolute homotopy group, namely,πn(A, coker{ι,β}), whereι:B1CB1is the

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inclusion ofB1into an injective containerCB1that induces a short exact sequence:

B1 {ι,β}

CB1⊕B2 coker{ι,β}. (1.1)

Thereafter, we analyze the phenomena related πn(A,β) and πn(A, coker{ι,β}) through cases. As expected, the two are not always isomorphic; nevertheless, the fact that all rela- tive 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.

2. Relative homotopy groups—from a diﬀerent viewpoint

In the injective relative homotopy theory of modules, for a givenΛ-module homomorphism β:B1→B2 and a givenΛ-moduleA, one computes thenth relative homotopy group,πn(A,β),n≥1, through the diagram

0 Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A ⁿ

σ

ΣⁿA

kerβ B1

β B2 cokerβ

(2.1)

whereι0 is the inclusion map which embedsAinto an injective containerCA, and1

is the quotient map to ΣA, called the suspension of A, as the quotient. We say that the map (ρ,σ) :ιn−1→βisi-nullhomotopic, denoted (ρ,σ)i0, if it can be extended to an injective container ofιn−1, and thatπn(A,β)=Hom(ιn−1,β)/Hom0(ιn−1,β), where Hom(ιn−1,β) is the abelian group of maps ofιn−1toβ, and Hom0(ιn−1,β) the subgroup consisting ofi-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 mapβ:B1→B2is monomorphic so (2.1) is essentially

Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A ⁿ

σ

ΣⁿA

σ

B1

β B2 κ cokerβ.

(2.2)

In (2.2), each pair of maps (ρ,σ) :ιn−1→βinduces a map σ:ΣⁿA→cokerβ. We define RHom_Λ(ΣⁿA, cokerβ) to be the subgroup of HomΛ(ΣⁿA, cokerβ) consisting of such induced maps; it gives the relative homotopy groupπn(A,β) an alternative aspect.

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Theorem 2.1. Suppose given a monomorphismβ:B1B2. For eachA, consider the dia- gram

ρ

CΣⁿ⁻¹A ⁿ

σ

ΣⁿA

σ ιn

CΣⁿA B1

β B2 κ cokerβ

(2.3)

whereι0:ACAis the inclusion ofAinto an injective containerCA,1the quotient map withΣA, called the suspension ofA, as the quotient, andκthe expected quotient map. Then,

πn(A,β)^∼=RHomΛ

ΣⁿA, cokerβκ∗ι^∗_nHomΛ

CΣⁿA,B2

, (2.4)

where

RHom_ΛΣⁿA, cokerβ

=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

σ∈Hom_ΛΣⁿA, cokerβ|σ is the induced map of a commutative square Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A

σ

B1

β B2

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

. (2.5)

To prepare for the proof ofTheorem 2.1, we first state a couple of existing propositions.

Proposition 2.2 ([2]). In Hom(ιn−1,β), whenβis monomorphic, (ρ,σ)i0 if and only if σ=βθ+χιnnfor someθ:CΣⁿ⁻¹A→B1andχ:CΣⁿA→B2;

ρ

CΣⁿ⁻¹A

θ

n

σ

ΣⁿA_σ ^ιⁿ CΣⁿA

χ

B1

β B2 κ cokerβ

(2.6)

Proposition 2.3 [1]. In the commutative diagram of short exact sequences:

A ^μ

α

B

ξ

C

γ

A ^μ B C

(2.7)

αfactors throughμif and only ifγfactors through. Proof ofTheorem 2.1. We define

φ:πn(A,β)→RHom_ΛΣⁿA, cokerβκ∗ι^∗_nHom_ΛCΣⁿA,B2

(2.8)

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byφ([(ρ,σ)])=[σ] and show thatφis an isomorphism; first, suppose given a [(ρ,σ)]∈ πn(A,β) and assume that (ρ,σ)i0. By Proposition 2.2, σ =βθ+χιnn for some θ: CΣⁿ⁻¹A→B1 and χ:CΣⁿA→B2. Thus, σn=κσ=κ(βθ+χιnn)=κβθ+κχιnn= κχιnn, soσ=κχιn, due to the fact thatnis surjective. Hence,σ∈κ∗ι^∗nHom_Λ(CΣⁿA,B2) andφis well defined.

To prove φ monomorphic, suppose given a [(ρ,σ)]∈πn(A,β) and assume that φ([(ρ,σ)])=[σ]=0∈RHom_Λ(ΣⁿA, cokerβ)/κ∗ι^∗nHom_Λ(CΣⁿA,B2). That is,σ=κχιn

for someχ:CΣⁿA→B2, which means thatσfactors through the map κ. Then, by an immediate corollary ofProposition 2.3, namely,γ=0 if and only ifξfactors throughμ, there exists anη:CΣⁿ⁻¹A→B1such thatσ−χιnn=βη;

ρ

CΣⁿ⁻¹A ⁿ

σ−χιnn

η

ΣⁿA

0

B1

β B2 κ

cokerβ

(2.9)

Hence, (ρ,σ)i0 byProposition 2.2, and thusφis monomorphic.

Finally, the definition of RHom_Λ(ΣⁿA, cokerβ) yields that eachσis induced from a commutative square

Σⁿ⁻¹A ^ιⁿ⁻¹ CΣⁿ⁻¹A B1

β B2

(2.10)

Thus,φis epimorphic.

We remark that one can interpret RHom_Λ(ΣⁿA, cokerβ) as the “reversible” subgroup of Hom_Λ(ΣⁿA, cokerβ); suppose given a mapσ∈Hom_Λ(ΣⁿA, cokerβ), we say thatσis reversible if it can pull back and produce a commutative diagram (2.2). Furthermore, it reveals a connection between the relative homotopy groupπn(A,β) and the (absolute) homotopy groupπn(A, cokerβ).

Next, for the general case thatβ:B1→B2is arbitrary, we exploit the mapping cylinder ofβandTheorem 2.5follows immediately afterProposition 2.4.

Proposition 2.4 [2]. Suppose given mapsβ:B1→B2 andι:B1CB1, whereCB1is an injective container ofB1so that{ι,β}:B1CB1⊕B2is a monomorphism, then, for arbi- traryA,πn(A,{ι,β})^∼₌πn(A,β) canonically,n≥1.

Theorem 2.5. Suppose givenβ:B1→B2. For eachA, consider the diagram Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A ⁿ

σ

ΣⁿA ^ιⁿ

σ

CΣⁿA B1

{ι,β}

CB1⊕B2 κ coker{ι,β}

(2.11)

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whereι0:ACAis the inclusion ofAinto an injective containerCA,1is the quotient map withΣA, called the suspension ofA, as the quotient,ι:B1CB1is the inclusion ofB1into an injective containerCB1, andκis the expected quotient map. Then,

πn(A,β)^∼=RHomΛ

ΣⁿA, coker{ι,β}

κ∗ι^∗_nHomΛ

CΣⁿA,CB1⊕B2

, (2.12)

where

RHom_ΛΣⁿA, coker{ι,β}

=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

σ∈Hom_ΛΣⁿA, coker{ι,β}

|σis the induced map of a commutative square Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A

σ

B1 {ι,β}

CB1⊕B2

⎫⎪

⎪⎪

⎪⎬

⎪⎪

⎭

. (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 homomorphismα:A1→A2 and a given Λ- moduleB, one alternatively views the projective relative homotopy groupπ_n(α,B),n≥1, as follows.

Theorem 2.6. Suppose givenα:A1→A2. For eachB, consider the diagram kerα,η ^ι

ρ|

A1⊕PA2 α,η ρ

A2 σ

PΩⁿB ^ηⁿ ΩⁿB ^μⁿ PΩⁿ⁻¹B ^ηⁿ⁻¹ Ωⁿ⁻¹B

(2.14)

whereη0:PBBis the projection of a projective ancestorPBontoB,μ1 is the inclusion map withΩB, called the loop space ofB, as the kernel,η:PA2A2is the projection of a projective ancestorPA2ontoA2, andιis the expected inclusion map. Then,

π_n(α,B)^∼=RHomΛ

kerα,η,ΩⁿBι^∗ηn∗HomΛ

A1⊕PA2,PΩⁿB, (2.15) where

RHom_Λkerα,η,ΩⁿB

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ρ∈Hom_Λkerα,η,ΩⁿB|ρ is the restriction of a commutative square A1⊕PA2

α,η ρ

A2 σ

PΩⁿ⁻¹B ^ηⁿ⁻¹ Ωⁿ⁻¹B

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

. (2.16)

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3. Various cases forβ:B1→B2

Here, we haveTheorem 2.5, which does not only give us an alternative way of computing relative homotopy groups for a mapβ:B1→B2, but also shows a close connection between the (injective) relative homotopy groupsπn(A,β) and the (injective) homotopy groups πn(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(ΣⁿA, coker{ι,β}).

First, we examine the case that the mapβ:B1→B2is the zero map. The homotopy exact sequence of a mapβ:B1→B2(see [1, Theorem 13.15]), thus,

··· ^∂ πn A,B1

β∗

πn A,B2

J

πn(A,β) ^∂ πn−1

A,B1

β∗

···

∂ π1

A,B1

^β^∗ π1

A,B2

J

π1(A,β) ^∂ πA,B1

^β^∗

πA,B2

, (3.1) yields a short exact sequence

πn A,B2

J

A,B1

(3.2)

asβ∗=0. In addition, the special feature of the zero map suggests that (3.2) actually splits, thus, the relative homotopy groupπn(A,β) is the direct sum of the other two.

Theorem 3.1. Assume thatβ:B1→B2is the zero map. Then, for eachA, πn(A,β)^∼₌πn−1

A,B1

⊕πn A,B2

, canonically, n≥1. (3.3) Before proceeding with its proof, we note that the theorem can also be derived using the conventional method, namely, computeπn(A,β) through the commutative square

Σⁿ⁻¹A ^ιⁿ⁻¹ CΣⁿ⁻¹A B1

β=0

B2

(3.4)

Proof. In diagram (2.11), thus, Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A ⁿ

σ

ΣⁿA ^ιⁿ

σ

CΣⁿA B1

{ι,β}

CB1⊕B2 κ

coker{ι,β}

(3.5)

we first note that coker{ι,β} =coker{ι, 0} =ΣB1⊕B2 and that κ= {κ1, 0,0, 1_B2}, whereι:B1CB1is the inclusion ofB1into an injective containerCB1,κ1is the quotient

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map toΣB1, called the suspension ofB1, and 1_B2 is the identity map onB2. So (2.11) is essentially

ρ

CΣⁿ⁻¹A ⁿ

σ={σ1,σ2}

ΣⁿA ^ιⁿ

σ={σ1,σ2}

CΣⁿA B1

{ι,0}

CB1⊕B2 κ ΣB1⊕B2

(3.6)

Moreover, it is the natural combination of the two commutative diagrams Σⁿ⁻¹A ^ιⁿ⁻¹

ρ

CΣⁿ⁻¹A ⁿ

σ1

ΣⁿA ^ιⁿ

σ1

CΣⁿA B1 ι CB1 κ1

ΣB1

(3.7)

ρ

CΣⁿ⁻¹A ⁿ

σ2

ΣⁿA ^ιⁿ

σ2

CΣⁿA B1

β=0

B2 1B2

B2

(3.8)

Thus we define

φ: RHom_ΛΣⁿA, coker{ι,β}

κ∗ι^∗_nHom_ΛCΣⁿA,CB1⊕B2

−→πn−1

A,B1

⊕πn A,B2

(3.9) byφ([{σ1,σ2}])=([ρ], [σ2]) and show that φis an isomorphism. First, suppose given [{σ1,σ2}]∈RHomΛ(ΣⁿA, coker{ι,β})/κ∗ι^∗_nHomΛ(CΣⁿA,CB1⊕B2) and assume that {σ1,σ2}∈κ∗ι^∗_nHom_Λ(CΣⁿA,CB1⊕B2). Then there exists{χ1,χ2}:CΣⁿA→CB1⊕B2such that{σ1,σ2} =κ◦ {χ1,χ2} ◦ιn. Equivalently, one hasσ1=κ1◦χ1◦ιnin (3.7) andσ2=1_B2◦ χ2◦ιn=χ2◦ιnin (3.8). The former says that the mapσ1factors throughκ1; therefore, by Proposition 2.3,ρ=θιn−1for someθ:CΣⁿ⁻¹A→B1. Hence [ρ]=0 inπn−1(A,B1). The latter says that [σ2]=0 inπn(A,B2). Soφis well defined.

To show thatφis monomorphic, suppose given [{σ1,σ2}]∈RHom_ΛΣⁿA, coker{ι,β}

κ∗ι^∗_nHom_ΛCΣⁿA,CB1⊕B2

(3.10)

and assume thatφ([{σ1,σ2}])=([ρ], [σ2])=(0, 0)∈πn−1(A,B1)⊕πn(A,B2). That is,ρ= γιn−1for someγ:CΣⁿ⁻¹A→B1andσ2=ηιnfor someη:CΣⁿA→B2, respectively. The former says that the mapρ factors throughιn−1in (3.7); therefore, byProposition 2.3, σ1=κ1τfor someτ:ΣⁿA→CB1. Moreover,τ=νιnfor someν:CΣⁿA→CB1, due to the facts thatCB1 is injective and thatιnis monomorphic. Therefore,σ1=κ1νιnand hence {σ1,σ2} = {κ1◦ν◦ιn,η◦ιn} = {κ1◦ν◦ιn, 1_B2◦η◦ιn} =κ◦ {ν,η} ◦ιn∈κ∗ι^∗_nHom_Λ (CΣⁿA,CB1⊕B2).

Finally, suppose given ([ρ], [σ2])∈πn−1(A,B1)⊕πn(A,B2). We use the mapρto com- plete a diagram (3.7)—sinceCB1is injective andιn−1is monomorphic, there exists a map

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σ1:CΣⁿ⁻¹A→CB1such thatιρ=σ1ιn−1andσ1is then the induced map:

ρ

CΣⁿ⁻¹A ⁿ

σ1

ΣⁿA

σ1

B1 ι CB1

κ1

ΣB1

(3.11)

Similarly, the mapσ2completes diagram (3.8), precisely,

ρ

CΣⁿ⁻¹A ⁿ

σ2n

ΣⁿA

σ2

B1

0 B2

1_B2

B2

(3.12)

Nowφis epimorphic because of the existence of the commutative diagram

ρ

CΣⁿ⁻¹A ⁿ

{σ1,σ2n}

ΣⁿA

{σ1,σ2}

B1 {ι,0}

CB1⊕B2 κ ΣB1⊕B2

(3.13)

Theorem 3.1also implies a couple of immediate consequences.

Corollary 3.2. IfB1=0, then, for eachA,πn(A,β)^∼=πn(A,B2).

Corollary 3.3. IfB2=0, then, for eachA,πn(A,β)^∼=πn−1(A,B1).

The dual ofTheorem 3.1and its corollaries say that if we assume thatα:A1→A2 is the zero map, then for eachB,π_n(α,B)^∼=π_n₋1(A2,B)⊕π_n(A1,B) forn≥1. Specifically, if A2=0, thenπ_n(α,B)^∼₌π_n(A1,B), and ifA1=0, thenπ_n(α,B)^∼₌π_n₋1(A2,B). Notice that asA2=0, one sees from diagram (2.14) inTheorem 2.6thatπ_n(α,B)^∼=π_n(kerα,η,B);

however, the isomorphism fails whenA1=0. As an example, consider theΛ-mapα: 0→ Z, whereΛis the integral group ring of the finite cyclic groupCkwith generatorτandZ is regarded as a trivialCk-module. Then,

π_n(α,Z)^∼=πn−1(Z,Z)=

⎧⎨

⎩Z/k, fornodd,

0, forneven. (3.14)

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(See [3, Theorem 3.1].) On the other hand, the well-known projective resolution ofZ, thus,

··· ^ρ ZCk σ

ZCk ρ

ZCk Z

ICk Z ICk

(3.15) where the maps,ρ,σare the augmentation ofZCk, multiplication byτ−1, and multiplication byτ^k⁻¹+···+τ+ 1, respectively, gives us that

π_nkerα,η,Z∼=π_n(ΩZ,Z)^∼=π_nICk,Z∼=

⎧⎨

⎩πICk,Z

, forneven πICk,ICk

, fornodd ⁼0, (3.16) because all the maps in HomΛ(ICk,Z) and HomΛ(ICk,ICk) arep-nullhomotopic.

Similarly, asB1=0,πn(A,β)^∼₌πn(A, coker{ι,β}); however, this position alters when B2=0; consider theΛ-mapβ:Q/Z→0, where, again,Λis the integral group ring of the finite cyclic groupCkandQ/Zis regarded as a trivialCk-module. Then,

πn(Q/Z,β)^∼=πn−1(Q/Z,Q/Z)^∼₌

⎧⎨

⎩Z/k, fornodd,

0, forneven. (3.17)

(See [3, Theorem 2.6].) Forπn(A, coker{ι,β}), we adopt the injective resolution ofQ/Z: Q/Z ^Δ (Q/Z)^k ^ρ

∗

(Q/Z)^k ^σ^∗ (Q/Z)^k ^ρ

∗ ···

I(Q/Z)^k Q/Z I(Q/Z)^k

(3.18) whereΔ=^∗is the diagonal map, and obtain that

πn

Q/Z, coker{ι,β}∼=πn

Q/Z,ΣQ/Z∼=πn

Q/Z,I(Q/Z)^k

∼=

⎧⎨

⎩πQ/Z,I(Q/Z)^k, forneven

πI(Q/Z)^k,I(Q/Z)^k, fornodd ⁼0, (3.19) again because all the maps in Hom_Λ(Q/Z,I(Q/Z)^k) and Hom_Λ(I(Q/Z)^k,I(Q/Z)^k) are i-nullhomotopic.

Therefore, as one may expect, the classic fibration/cofibration does not hold for arbitrary maps in module theory. The same phenomena arise again even when we generalize B1andB2, respectively, to injective modules.

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Theorem 3.4. Letβ:B1→B2be arbitrary. Then, for eachA,

(i) ifB1is injective, thenπn(A,β)^∼=πn(A,B2)^∼=πn(A, coker{ι,β});

(ii) ifB2is injective, thenπn(A,β)^∼=πn−1(A,B1).

Proof. The first halves of both parts of the theorem, namely,πn(A,β)^∼₌πn(A,B2) when B1 is injective andπn(A,β)^∼=πn−1(A,B1) whenB2 is injective, come directly from the (injective) homotopy exact sequence of the mapβ:B1→B2, thus,

··· ^∂ πn A,B1

β∗

πn A,B2

J

A,B1

β∗

πn−1

A,B2

J

···

(3.20) To prove thatπn(A,β)^∼=πn(A, coker{ι,β}) whenB1is injective, one considers diagram (2.11), but nowCB1=B1. Thus,

ρ

CΣⁿ⁻¹A ⁿ

σ

ΣⁿA ^ιⁿ

σ

CΣⁿA B1

{ι,β}

B1⊕B2 κ

coker{ι,β}

(3.21)

SinceB1 is injective, the short exact sequence B1 {ι,β}

B1⊕B2 κ

coker{ι,β} splits.

Thus there exists a mapν: coker{ι,β} →B1⊕B2 such that κ◦ν=1coker_{ι,β}. Applying Theorem 2.5, we defineχ:πn(A,β)→πn(A, coker{ι,β}) byχ(σ)=[σ] and show thatχ is an isomorphism.

First, ifσ=0 in πn(A,β), then there is θ:CΣⁿA→B1⊕B2 such thatσ=κ◦θ◦ ιn, which also means thatχ(σ)=[σ]=0 inπn(A, coker{ι,β}). Soχis well defined. To show that χis monomorphic, suppose given σ∈πn(A,β) such thatχ(σ)=[σ]=0, thenσ=ω◦ιnfor someω:CΣⁿA→coker{ι,β}. Thus,σ=ω◦ιn=1coker_{ι,β}◦ω◦ιn= κ◦ν◦ω◦ιn, which forcesσ=0. Thus,χis monomorphic. Finally, the fact that every σ:ΣⁿA→coker{ι,β}yields a commutative diagram

0

CΣⁿ⁻¹A ⁿ

νσn

ΣⁿA

σ

B1 {ι,β}

B1⊕B2 κ coker{ι,β}

(3.22)

allows us to conclude thatχis epimorphic.

Examining the connection betweenπn(A,β) andπn(A, coker{ι,β}), even for the rather simple case thatB2is injective, we find that, for a mapσ:ΣⁿA→coker{ι,β}to be related to an element inπn(A,β),σought to be “reversible” in a diagram such as (2.11), that is, σ must guarantee the existence of a pair (ρ,σ), or equivalently, σ is the induced map of (ρ,σ). Conversely, for a pair (ρ,σ) :ιn−1→ {ι,β} to be related to an element inπn(A, coker{ι,β}), the reversibleσ ought to, simultaneously, factor through not onlyιn but also κ as (ρ,σ) is i-nullhomotopic. These lead precisely to our group