ON KAN EXTENSION OF HOMOLOGY AND ADAMS COCOMPLETION

Transcription

1 Bulletin o the Institute o Mathematics Academia Sinica (New Series) Vol 4 (2009), No 1, pp ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION B AKRUR BEHERA AND RADHESHAM OTA Abstract Under a set o conditions, it is shown that the Kan extension o an additive homology theory over smaller admissible subcategory is again a homology theory Furthermore, using a Serre class o abelian groups, it is shown that a homology theory over the category o simply connected based topological spaces and continuous maps arising through Kan extension process rom an additive homology theory over a smaller subcategory always admits global Adams cocompletion 1 Introduction Let T be the category o based topological spaces and base point preserving maps A subcategory J o T is said to be admissible i it is nonempty, ull, closed under the ormation o mapping cones and contains (based) homotopy types [12] It is evident that an admissible subcategory o T contains singletons and is closed under suspensions We denote by J the homotopy category o J A homology theory h on an admissible category J is a sequence o unctors h n : J a where a is the category o abelian groups, together with natural transormations σ n : h n h n+1 Σ (Σ denoting the suspension) satisying the homotopy, suspension and exactness axioms: (i) I 0 1 then h n ( 0 ) = h n ( 1 ) Received September 12, 2007 and in revised orm March 18, 2008 AMS Subject Classiication: 18A40, 55P60 Key words and phrases: Kan extension, Adams cocompletion 47

2 48 AKRUR BEHERA AND RADHESHAM OTA [March (ii) σ n : h n = hn+1 Σ (iii) I : is in J and C is the mapping cone o and P : C is the canonical embedding then is exact h n () h n (P ) h n () h n ( ) h n (C ) Let J 0 and J 1 be two admissible subcategories o T with J 0 J 1 and let h be a homology theory deined on J 0 The let Kan extension o h over J 1, say h, is a unctor on J 1 having values in the category o abelian groups In this note we give a set o conditions under which the unctor h is a homology unctor It may be recalled that Deleanu and Hilton have considered this question or cohomology theories [1, 2, 9] and Piccinini [12] has considered the same question or homology theories on stable categories It will be evident rom the conditions imposed and the method o proo o the main result that the description given here is largely dualization o the results obtained in [2] 2 The Extension Procedure We assume that the categories J 0 and J 1 have the ollowing three properties: (i) J 0 has weak local push-outs relative to J 1 ; that is, given a commutative diagram in J 1 u 0 0 u with 0, 1 and in J 0, there exists a diagram

3 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 49 u 1 u v 0 0 Z in J 1 with Z in J 0 (ii) The suspension map Σ : v 1 1 J 1 g J 1 is locally let which means that condition (a) and (b) below are satisied (a) Given : Σ in in J 0 and g : Z in = (Σg)u u J 0 -adjunctable, J 1 with in J 0, there exists an object Z J 1 and u : ΣZ in ΣZ Σ g Σg Z J 0 such that (b) Given diagrams u 1 ΣZ 1 Σg 1 Σ Z 1 g 1 u 2 Σg 2 g 2 ΣZ 2 Z 2 with Z 1, Z 2 and in J 0 and (Σg 1 )u 1 = (Σg 2 )u 2 in v 1 : Z Z 1, v 2 : Z Z 2 and u : ΣZ in J 1, there exist J 0 with g 1 v 1 = g 2 v 2, (Σv 1 )u = u 1 and (Σv 2 )u = u 2 Thus we have the commutative diagrams

4 50 AKRUR BEHERA AND RADHESHAM OTA [March v 1 Z 1 Z g 1 u 1 u ΣZ 1 ΣZ Σv 1 Σg 1 Σ v 2 g 2 u 2 Σv 2 Σg 2 Z 2 ΣZ 2 (iii) J 0 is closed under inite sums ie, i 1 and 2 are in J 0, then so is 1 2 We now describe the extension procedure Let be an object o J 1 Form the category J 01 () o all J 0 -objects over : An object in this category is a morphism : in J 1 with in J 0 ; a morphism u : 1 2 in this category is a morphism u : 1 2 in J 0 such that the diagram is commutative in u J 1 It is easy to check that J 01 () is a category The let Kan extension h n o the homology unctor h n is deined as ollows: h n() = lim (h n ( ),h n (u)) It is obvious that h n is an extension o h n and deines a covariant unctor on J 1 with values in the category o abelian groups An alternative description o the groups h n () can now be given Let P n () = {(α,) : in J 1 with in J 0, α h n ( )} In P n () deine a relation by the rule: (α 1, 1 ) (α 2, 2 ) i and only i there is a commutative diagram

5 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 51 1 u 1 u J 1 with in J 0, u 1, u 2 in in h n (u 2 )α 2 J 0, u 1 = 1, u 2 = 2 and h n (u 1 )α 1 = Proposition 21 Under the assumptions (i), (ii) and (iii), is an equivalence relation on P n () Proo Only transitivity is in question Suppose that (α 1, 1 ) (α 2, 2 ) and (α 2, 2 ) (α 3, 3 ) We then have two commutative diagrams in J 1 as ollows: 1 2 u 1 u 2 1 v 2 v with and in J 0 Moreover, h n (u 1 )α 1 = h n (u 2 )α 2 and h n (v 2 )α 2 = h n (v 3 )α 3 By condition (i), the commutative diagram v 2 2 u 2 can be embedded in a diagram

6 52 AKRUR BEHERA AND RADHESHAM OTA [March u 2 v 2 2 v v Z w in J 1 with Z in J 0 Now consider the diagram 1 vu 1 v v 3 Z 3 w 1 3 which is clearly commutative Moreover h n (vu 1 )α 1 = h n (v)h n (u 1 )α 1 = h n (v)h n (u 2 )α 2 = h n (vu 2 )α 2 = h n (v v 2 )α 2 = h n (v )h n (v 2 )α 2 = h n (v )h n (v 3 )α 3 = h n (v v 3 )α 3 Hence (α 1, 1 ) (α 3, 3 ) This completes the proo o Proposition 21 We denote the equivalence class o (α,) by [α,] and denote the set o such equivalence classes by E n () Now deine addition on E n () as ollows: Given [α 1, 1 ] and [α 2, 2 ] in E n () with 1 : 1 and 2 : 2, let i 1 : and i 2 : denote the usual injections Deine [α 1, 1 ] + [α 2, 2 ] = [h n (i 1 )α 1 + h n (i 2 )α 2, 1 2 ] where 1 2 : 1 2 is deined as usual This addition is easily checked to be well-deined, associative and commutative The additive identity o the group is easily seen to be the element [0,i] where i : is the map that takes the singleton to the base point o We note that [α,] = [0,i] i and only i we have a commutative diagram

7 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 53 u 0 0 in J 1 with 0 J 0 and h n (u)α = 0 The additive inverse o an element [α,] is easily seen to be [ α,] E n () is thus a commutative group For a given : in J 1 with in J 0, deine i : h n ( ) E n () by the rule i (α) = [α,] E n () together with the maps {i } has the required universal property; thus, E n () = h n() Note also that i g : is in J 1, then h n(g) = g : h n() h n( ) is deined by the rule g [α,] = [α,g] 3 The Suspension Axiom We now show that h satisies the suspension axiom First identiy the suspension map σ : h n ( ) h n+1 (Σ ) in J 0 as ollows: σ[α,] = [σα,σ] We then deine the suspension isomorphism σ : h n() h n+1 (Σ) or any in J 1 by the rule σ [α,] = [σα,σ] The map σ is onto: I we take [β,g] h n+1 (Σ), then we have a map g : Σ or some in J 0 and β h n+1 ( ) By assumption (ii) we can actorize this map g = (Σu)k : k ΣZ Σu Σ with Z in J 0 Then the commutative diagram k 1 ΣZ ΣZ ΣZ g Σu Σu Σ shows that [β,g]=[h n+1 (k)β,σu]=[σσ 1 (h n+1 (k)β),σu]=σ [σ 1 (h n+1 (k)β), u] Thus σ is onto

8 54 AKRUR BEHERA AND RADHESHAM OTA [March To show that σ is one-one, assume that σ [α,] = 0, : and α h n ( ) We then have a commutative diagram Σ u 1 1 Σ g 1 with 1 in J 0 and h n+1 (u 1 )(σα) = 0 We can now actorize g 1 = (Σg)u 2 : u 2 Σg 1 ΣZ Σ We thus have a diagram Σ u 1 u 2 Σ 1 ΣZ Σ g 1 Σg Σ Let u = u 2 u 1 ; then we have a commutative diagram u Σ ΣZ Σ Σg Moreover, h n+1 (u)(σα) = h n+1 (u 2 )h n+1 (u 1 )(σα) = 0 Consider the commutative diagram Σ 1 Σ u Σ Σ ΣZ Σ Σg Σ By assumptions (ii)(b), we have commutative diagrams

9 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 55 Σ 1 Σ u v Σ ΣT ΣZ Σk Σm Σ Σg Σ k m T Z g with T in J 0 Consider the element h n+1 (v)(σα) h n+1 (ΣT) Since σ : h n (T) h n+1 (ΣT) is an isomorphism, it ollows that h n+1 (v)(σα) = σβ or some β h n (T) Moreover, 1 Σ = (Σk)v, so that σα = h n+1 (1 Σ )(σα) = h n+1 (Σk)h n+1 (v)(σα) = h n+1 (Σk)h n+1 (σβ) = σh n (k)(β), showing that α = h n (k)(β) Consideration o the diagram k T Σk shows that [β,k] = [α,] On the other hand we have h n+1 (Σm)(σβ) = h n+1 (Σm)h n+1 (v)(σα) = h n+1 (u)(σα) = σ(h n (u)(α)) = 0 Thus, h n (m)(β) = 0 Considering the diagram T m k g we arrive at the conclusion that [β,k] = 0, so that [α,] = 0 Thus, σ is also one-one and this completes the proo o the suspension axiom 4 The Exactness Axiom We shall show now, using the suspension axiom (proved above), that

10 56 AKRUR BEHERA AND RADHESHAM OTA [March the unctor h n satisies the exactness axiom, ie, h n carries every cokernel sequence A g C Pg g in J 1 into an exact sequence Let [α,] ker h n (P g) with : This implies that [α,p g ] = 0 so that we have a map u : 0 in J 0 and a map 0 : 0 C g in J 1 such that h n (u) = 0 Consider the diagram g A P g C g ΣA Σg Σ 0 1 Σ u 0 P u C u Q u Σ Σu Σ 0 where 1 and Σ are induced maps between the corresponding terms o the Puppe sequences Since σ : h n ( ) h n+1 (Σ ) is an isomorphism, it ollows that h n+1 (Σu)(σα) = 0 But the homology unctor on the bottom Puppe sequence is exact; so there is an element β h n+1 (C u ) such that h n+1 (Q u )β = σα From the third sequence, we thereore deduce that [β,(σg) 1 ] = [σα,σ] I γ = [β, 1 ] h n+1 (ΣA), then h n+1 (Σg)(γ) = h n+1 (Σg)[β, 1] = [β,(σg) 1 ] = [σα,σ] = σ [α,] Since we have an isomorphism σ : h n (A) h n+1 (A) we take δ 1 = (σ ) 1 (γ), so that h n (g)(δ) = h n(g)(σ ) 1 (γ) = (σ ) 1 h n+1 (Σg)(γ) = (σ ) 1 [σα,σ] = (σ ) 1 σ [α,] = [α,] This shows that ker h n (P g) Image h n (g) To show that Image h n (g) ker h n(p g ), it is enough to show that or any element [α,] h n(a), h n (P g) h n (g)[α,] = 0, ie, [α,p gg] = 0 In the ollowing commutative diagram A g P g C g g k 1 P 1 C 1 it is clear that [α,p g g] = [h n (P(1 ))h n (1 )α,k] But the bottom Puppe sequence is carried by h n into an exact sequence so that h n (P(1 ))h n (1 )α = 0 Thus we have the desired result

11 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 57 5 Examples We present some examples where the assumptions made on the categories J 0 and J 1 are valid Example 51 In the stable categories, the suspension unctor is an isomorphism, so that there is no diiculty in proving that the three axioms hold in such categories (see [9, 14]) Example 52 Let J 0 to be the set o all based topological spaces having the homotopy type o inite CW-complexes and J 1 to be the set o all spaces having the homotopy type o CW-complexes It is easy to prove that the three axioms are true in this case Example 53 Let J 1 be the category o 1-connected based topological spaces having the homotopy type o a CW-complex and J 0 be the subcategory o spaces whose homotopy groups are all initely generated and P-local, where P denotes a ixed set o primes Then any cohomolgy theory h on J 0 extends to a cohomolgy theory h 1 on J 1 through the Kan extension process [10] Example 54 Let C be a Serre class o abelian groups and T 0 (C) be the subcategory T S (the category o 1-connected spaces) consisting o spaces whose homotopy groups belong to C in the sense o Serre [15] Then any cohomology theory h on T 0 (C) extends to a cohomology theory h 1 on T S through Kan extension process ([9], Theorem 41) Note 55 Example 53 is a sort o variant o Example 54 Example 56 Let CW CF and CW C be the categories o connected inite based CW-complexes and connected based CW-complexes respectively Let h be a homology theory on CW CF Then the Kan extension h 1 o h to CW C is also a homology theory; indeed, it is naturally equivalent to a homology theory deined by a spectrum [13] Example 57 Let T be a triangulated category and let h, k be two homology theories on T Let in T admit the Adams h-completion h, and orm the triangle e i j h C e Σ in T Let T 0 be the ull subcategory o T whose objects are those such that h( ) = 0 It is plain that T 0 is a

12 58 AKRUR BEHERA AND RADHESHAM OTA [March triangulated subcategory o T Let k 0 be the restriction o k to T 0 Then the Kan extension k 1 o k 0 is given by k 1 n () = k n+1(c e ) ([3], Theorem 41) Example 58 Let S be the stable CW-category and let S 0 be the ull subcategory consisting o spaces in the Serre class C P o P-torsion groups Let k be a homology theory on S The Kan extension k 1 to S o the restriction o k to S 0, or in S, is given by k 1 n () = k n+1(; Z P ) ([3], Theorem 44) 6 Adams Cocompletion or the Homology Theory h Next we show that a homology theory over the category o simply connected based topological spaces and continuous maps arising through Kan extension process rom an additive homology theory over a smaller subcategory always admits global Adams cocompletion We do it in the context o Serre class o abelian groups We recall the ollowing Let C be a category and S a set o morphisms o C Let C[S 1 ] denote the category o ractions o C with respect to S and F : C C[S 1 ] the canonical unctor Let S denote the category o sets and unctions Then or a given object o C, C[S 1 ](, ) : C S deines a covariant unctor I this unctor is representable by an object S o C, that is, C[S 1 ](, ) = C( S, ), then S is called the (generalized) Adams cocompletion o with respect to the set o morphisms S or simply the S-cocompletion o We shall oten reer to S simply as the cocompletion o [5] Let T, J and J be as in Section 1 Let h be a generalized homology (cohomology) theory deined on J Let S be the set o morphisms o J which are carried into isomorphisms by h I every object o J admits a cocompletion with respect to S, then we say that the homology theory h admits global Adams cocompletion Deleanu [6] has shown that any additive theory h on the homotopy category o based CW-complexes and based continuous maps admits global Adams cocompletion In this note, we show that every homology theory on an admissible category arising rom an additive homology theory on a smaller admissible category through Kan extension process (see Deleanu and Hilton [1, 2], Hilton [11]) always admits global Adams cocompletion More precisely, let J 0 and J 1 be admissible complete categories with J 0 J 1 and J 1 be

13 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 59 small U -category, where U is a ixed Grothendieck universe Let h be an additive homology theory on J 0 such that its Kan extension h over J 1 is also a homology theory; then we show that h admits global Adams cocompletion The proo o this result depends mainly on the particularly nice description o the homology group h n() as described in Section 2 and additivity o the unctor h We shall use the ollowing theorem or showing the global existence o the Adams cocompletion o the homology theory h ; the result is essentially Theorem 1 in [7] (also see, Theorem [12]) Theorem 61 Let C be a complete small U -category (U is a ixed Grothendieck universe) and S a set o morphisms o C that admits a calculus o right ractions Suppose that the ollowing compatibility condition with product is satisied: (P) I each s i : i i, i I, is an element o S where the index set I is an element o U, then is an element o S s i : i i Then every object o C has an Adams cocompletion S with respect to the set o morphisms S We now apply this result to the category J 1 Let S be the set o morphisms o J 1 which are carried into C-isomorphisms in all dimensions by the homology unctor h where C is a Serre class o abelian groups which is moreover an acyclic ideal o abelian groups It is well known that J 1 is complete We prove the ollowing propositions Proposition 62 S is saturated Proo This is evident rom Proposition 11 ([5], p 63) Proposition 63 S admits a calculus o right ractions Proo Clearly S is closed under composition We shall veriy conditions (a) and (b) o Theorem 13 [5] For this, it is enough to prove that every diagram o the orm

14 60 AKRUR BEHERA AND RADHESHAM OTA [March in C J 1 with γ S, can be imbedded in a weak pull-back diagram γ A B α β D C δ γ A B α with δ S The essential idea, as has been explained in [8], is to actorize these maps in terms o ibrations and then taking the pull-back o these ibrations Suppose α = [] and γ = [s] We replace the maps and s by ibrations to get the ollowing diagram A D p r P r C t t q P s s B where and s are ibrations; r and t are mod-c homotopy equivalences; r and t are mod-c homotopy inverses o r and t; P and P s are mapping tracks o and s; D is the usual pull-back o and s and p, q are the respective projections So we have = r and s = s t Let δ = [rp], β = [tq] Thus αδ = [][rp] = [rp] = [ rrp] = [ p] = [s q] = [s ttq] = [stq] = [s][tq] = γβ Moreover, i αµ = rλ, let u : U A, v : U C be in the classes µ, λ respectively so that u sv or ru sv Let F : U I B be a homotopy with F 0 = ru and F 1 = sv Consider the ollowing diagram

15 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 61 G t U F t Since is a ibration there exists a homotopy G : U I P such that (a) G t = F t, (b) G 0 = ru Thus G 1 = F 1 = sv = s tv Consider the ollowing diagram ru P B U k tv G 1 D p P q P s s By the pull-back property o D, there exists a map k : U D such that pk = G 1 ru and qk = tv Thus i ρ = [k] : U D, then δρ = [rp][k] = [rpk] = [rru] = [u] = µ and βρ = [tq][k] = [tqk] = [ttv] = [v] = λ It now remains to be shown that δ S We assume that the map α : A B is a ibration with iber F; then F is also the iber o the map β : B D and rom the commutative diagram F = F B we have the ollowing commutative diagram π m+1 (C) γ π m+1 (B) β D C π m (F) π m (F) δ γ A B α π m (D) π m (A) π m (C) δ γ π m (B) π m 1 (F) π m 1 (F) The mod-c Five lemma [15] implies that δ is a C-isomorphism or all m 0 Thus δ S

16 62 AKRUR BEHERA AND RADHESHAM OTA [March Proposition 64 Let {s i : i i, i I} be a subset o S Then s i : i i is an element o S, where the index set I is an element o U Proo First we prove that the homology theory h satisies the wedge axiom For each i I, let p i denote the homology class o the projection map j i Thus we have group homomorphisms h (p i ) : h ( j ) h ( i ) We need to prove that the group homomorphism {h (p i )} : h ( j ) h ( i ) is a C-isomorphism We irst show that {h (p i )} is a C-monomorphism, ie, Ker {h (p i )} C (see [15]; p 505) Let [α,] be class in h ( j ) such that {h (p i )}([α,]) = 0, ie, [α,p i ] = 0 or each i I, where : j is in J 1 with J 0 and α h( ) Hence or each i I, there exists a space i in J 0 and maps u i : i in J 0 and i : i i such that the ollowing diagram u i i p i i commutes, ie, i u i = p i and h 0 (u i )(α) = 0 Let q i : j i denote the homotopy class o the projection map By the universal property o the product, there exists a map u : j making the diagram i u i q i u i commutative, ie, q i u = u i Hence or each i I, h(q i )h(u)(α) = h(u i )(α) = 0 Since h is an additive homology theory, we note that {h(q i )} : h( i ) i

17 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 63 h( i ) is an isomorphism and hence h(u)(α) = 0 In the ollowing diagram j q i v i i j by the deinition o the product, there exists a map v : j j in J 1 such that p i v = i q i Thus p i vu = i q i u = i u i = p i ie, the ollowing diagram u j v j u i is commutative and by the universal property o the product we have vu = Thus we have the ollowing commutative diagram u j v p i i p i j with u J 0 and h(u)(α) = 0 Hence [α,] = 0 Thus Ker{h (p i )} is a trivial group and hence by Theorem 961 [15], Ker{h (p i )} C Next we show that {h (p i )} is a C-epimorphism, ie, Coker{h (p i )} C Consider an arbitrary element {[α i, i ]} in h ( i ) where or each i I, the class [α i, i ] is represented by i : i i with i J 0 and i i

18 64 AKRUR BEHERA AND RADHESHAM OTA [March α i h( i ) Since {h(q i )} : h( i ) h( i ) is an isomorphism, the element {α i } h( i ) corresponds to some element α h( i ) such that {h(q i )}(α) = {α i } Thus or each i I, h(q i )(α) = α i In the ollowing diagram j q i q i i i i i i 1 i i i the two triangles commute and or each i I, h(1 i )(α i ) = α i = h(q i )(α) Hence [α i, i ] = [α, i q i ], or each i I By the universal property o q i in the ollowing diagram j g j i q i p i there exists a unique map g : j j such that p i g = i q i Consider the class [α,g] h ( j ) For each i I, we have h (p i )([α,g]) = [α,p i g] = [α, i q i ] = [α i, i ] Hence {h (p i )}([α,g]) = {[α i, i ]} So Im{h (p i )} = h ( i ); thus Coker{h (p i )} is a trivial group and it is in C by Theorem 961 [15] Now we prove that h satisies the compatibility axiom with products Consider the commutative diagram j

19 2009] ON KAN ETENSION OF HOMOLOG AND ADAMS COCOMPLETION 65 ( ) h s i ( ) h i ( ) h i {h (p i )} {h (p i )} h ( i ) h ( i ) h (s i ) Since h (s i ) is a C-isomorphism, so is h ( s i ) By the wedge axiom, as proved above, it ollows that {h (p i )} and {h (p i )} are C-isomorphisms Thus it ollows that h ( s i ) is a C-isomorphism This completes the proo o Proposition 64 Hence rom Propositions 62, 63 and 64, it ollows that all the conditions o Theorem 61 are satisied and so we obtain the ollowing theorem Theorem 65 The homology theory h admits global Adams cocompletion Acknowledgments It is a pleasure to thank the anonymous reeree or his comments which resulted in an improved presentation o the paper Reerences 1 A Deleanu and P Hilton, Some remarks on general cohomolgy theories, Math Scand, 22(1968), A Deleanu and P Hilton, On Kan extension o cohomology theories and Serre class o groups, Fund Math, 73(1971), A Deleanu and P Hilton, Localization, Homology and a construction o Adams, Trans Amer Math Soc, 179(1973), B Eckmann and P J Hilton, Unions and intersections in homotopy theory, Comm Math Helv, 38(1964),

20 66 AKRUR BEHERA AND RADHESHAM OTA [March 5 A Deleanu, A Frei and P Hilton, Generalized Adams completion, Cahiers Topologie Geom Dierentielle, V-1(1973), A Deleanu, Existence o Adams completion or CW-complexes, J Pure Appl Algebra, 4(1974), A Deleanu, Existence o the Adams completion or objects o cocomplete categories, J Pure Appl Algebra, 6(1975), B Eckmann and P J Hilton, Unions and intersections in homotopy theory, Comm Math Helv, 38(1964), P Hilton, On the construction o cohomolgy theories, Rend di Math, 1(1968), No 1-2, S Nanda, On Kan extension o cohomology on the category o p-local spaces, Rend Di Math, (1)-11, Series VI (1978), S Nanda, A note on the universe o a category o ractions, Canad Math Bull, 23(1980), No 4, R A Piccinini, Kan extension o cohomology theories on stable categories, Rend Di Math, 6 Series VI (1973), R A Piccinini, CW-complexes, Homology Theory, Queen s Papers in Pure and Appl Math, 34, Queen s University, Kingston (1973) 14 P Preyd, Stable homotopy, Proceedings, Conerence on Categorical Algebra, La Jolla, Springer-Verlag (1965), E H Spanier, Algebraic Topology, Springer-Verlag, New ork, 1982, McGraw-Hill, New ork, 1966 Department o Mathematics, National Institute o Technology, Rourkela , India akrurbehera@yahoocouk, abehera@nitrklacin Department o Mathematics, SG Women s Junior College, Rourkela , India