The cohomology of lambda-rings and K-theory

THE COHOM OLOGY OF λ -RINGS AND K -THEOR Y MICHAEL ROBINSON Abstract. W e int ro duce t he Andr ´ e-Quillen cohomology of λ -r i ngs and Ψ- rings, this is di fferen t to the λ -ri ng cohomology defined by Y au in 2005. W e sho w that there is a natural transformation connecting the cohomology of the K -theory of spheres to the homotop y groups of spheres. 1. Introduction λ -rings were first introduced in an algebraic- geometry s e tting b y Grothendieck in 1958, then later used in g roup theory by A tiyah and T all. In 1962 Adams int r o duced the o p e r ations Ψ i to study vector fields of spher es. These op eratio ns give us another type of ring, the Ψ-rings, which a r e clo s ely related to the λ -rings. The main example of λ -rings and Ψ-rings are in algebra ic top olo gy; the K -theory of a top olog ical space is a λ -ring and Ψ-ring. F or more detaile d information on the cohomolog y of λ -r ings and Ψ-r ings, see my thesis [5 ]. In 2005, Dona ld Y au defined a co homology for λ -rings . W e are using the Andr´ e-Quillen coho mology o f λ -r ings and Ψ - rings which is different to Y au’s cohomolog y . In this pap er , w e le t N deno te the non-zero natura l nu m be rs a nd N 0 denote N ∪ { 0 } . 2. λ -rings and Ψ -rings In this section, we intro duce the definitions of a λ -r ing and a Ψ-ring. F or more information on λ -r ings and Ψ- rings, se e Atiy ah and T all [1] o r Knutson [4]. 2.1. λ -rings. A λ -ring is a unital co mm uta tive ring, R , together with a seq uenc e of op er ations λ i : R → R , for i ∈ N 0 , satisfying (1) λ 0 ( r ) = 1 , (2) λ 1 ( r ) = r, (3) λ i ( r + s ) = Σ i k =0 λ k ( r ) λ i − k ( s ) , (4) λ i ( r ) = 0 for i > 1, (5) λ i ( rs ) = P i ( λ 1 ( r ) , λ 2 ( r ) , . . . , λ i ( r ) , λ 1 ( s ) , . . . , λ i ( s )), (6) λ i ( λ j ( r )) = P i,j ( λ 1 ( r ) , . . . , λ ij ( r )) , where P i and P i,j are universal p olynomia ls with in teger co efficients, see the cited material for precis e definitions. Note that what w e refer to as a λ -ring is ca lled a sp ecial λ -ring in the mater ials. 2.2. λ -mo dul es. M is a λ -mo dule o ver the λ -ring R if M is an R -mo dule together with a seq uence of g roup homomo rphisms Λ i : M → M , for i ∈ N , s atisfying (1) Λ 1 ( m ) = m, (2) Λ i ( rm ) = Ψ i ( r )Λ i ( m ) , 1 2 MICHAEL R OBINSON (3) Λ ij ( m ) = ( − 1) ( i +1)( j +1) Λ i Λ j ( m ) , for a ll m ∈ M , r ∈ R and i, j ∈ N . 2.3. λ -deriv ations. A λ -derivation of R with v alues in M is a n additive homo- morphism d : R → M such that (1) d ( rs ) = rd ( s ) + d ( r ) s, (2) d ( λ i ( r )) = Λ i ( d ( r )) + P i − 1 j =1 Λ j ( d ( r )) λ i − j ( r ) , for a ll r , s ∈ R , and i ∈ N . W e let D e r λ ( R, M ) denote the set of all λ -deriv a tio ns of R with v alues in M . 2.4. Ψ -rings . The λ - op erations are neither additive nor multiplicativ e which makes them difficult to use. F rom these we ca n obtain the Adams op eratio ns whic h are ring homomorphisms . W e define the Ada ms op erations using the Newto n formula. Ψ i ( r ) − λ 1 ( r )Ψ i − 1 ( r ) + ... + ( − 1) i − 1 λ i − 1 ( r )Ψ 1 ( r ) + ( − 1 ) i iλ i ( r ) = 0 . A Ψ -ring is a unit al commutativ e ring, R , together with a sequence of ring homomorphisms Ψ i : R → R , for i ∈ N , satisfying (1) Ψ 1 ( r ) = r , (2) Ψ i (Ψ j ( r )) = Ψ ij ( r ) , for all r ∈ R and i, j ∈ N . W e say that a Ψ-ring R is sp e cial if it a lso satisfies the prop erty Ψ p ( r ) ≡ r p mo d pR for all pr imes p and r ∈ R . All of the Ψ-r ings which come fro m λ -rings are sp ec ia l. Wilkerson [8 ] gives us a condition for when the conv er se is true. Theorem 2.1. (Wilkerson) If R is a Z torsion fr e e sp e cial Ψ -ring, t hen ther e exists a unique λ -ring structur e on R whose ad ams op er ations ar e pr e cisely the Ψ -op er ations. There is a unique λ -r ing s tructure on the ring of the integers Z given by λ i ( x ) =  x i  , for i ∈ N . The corre s p o nding sp ecial Ψ-o p erations on Z a re given by Ψ i ( x ) = x for i ∈ N . The definition of the free λ -ring is well-kno wn, see [4 ]. W e a re now going to construct the free Ψ-ring on one genera to r a . Let A be the free c ommut ative ring generated by { a i | i ∈ N } . Let the opera tio ns Ψ i : A → A b e g iven by Ψ i ( a j ) = a ij , for i , j ∈ N . Then A is the fr e e Ψ -ring on one gener ator . Lemma 2.2. If R and S ar e Ψ -rings, then R ⊗ S with Ψ i : R ⊗ S → R ⊗ S given by Ψ i ( r , s ) = (Ψ i ( r ) , Ψ i ( s )) is the c opr o duct in the c ate gory of Ψ - rings. Pr o of. The copro duct of tw o commutativ e rings is giv en by the tens or pro duct, s o we only need to chec k the Ψ-o p e rations. There is a unique Ψ-ring s tructure on R ⊗ S s uch that R → R ⊗ S, r 7→ r ⊗ 1 , S → R ⊗ S, s 7→ 1 ⊗ s, THE COHOMOLOGY OF λ -RINGS AND K -THEOR Y 3 are homo morphisms of Ψ - rings g iven by Ψ i ( r ⊗ s ) = Ψ i (( r ⊗ 1)(1 ⊗ s )) = Ψ i ( r ⊗ 1 )Ψ i (1 ⊗ s ) = (Ψ i ( r ) ⊗ 1)(1 ⊗ Ψ i ( s )) = Ψ i ( r ) ⊗ Ψ i ( s ) .  Corollary 2. 3. L et A b e the fr e e c ommut ative ring gener ate d by { a i , b i , . . . , x i | i ∈ N } . L et the op er ations Ψ i : A → A b e given by Ψ i ( a j ) = a ij , Ψ i ( b j ) = b ij , . . . , Ψ i ( x j ) = x ij for i, j ∈ N . Th en A is the fr e e Ψ -ring gener ate d by { a, b , . . . , x } . 2.5. Ψ -mo dules. M is a Ψ -mo dule ov er the Ψ-r ing R if M is a n R -mo dule toge ther with a seq uence of g roup homomo rphisms ψ i : M → M , for i ∈ N , s atisfying (1) ψ 1 ( m ) = m, (2) ψ i ( rm ) = Ψ i ( r ) ψ i ( m ) , (3) ψ i ( ψ j ( m )) = ψ ij ( m ) , for all m ∈ M , r ∈ R , a nd i, j ∈ N . W e let R − mo d Ψ denote the categor y of all Ψ-mo dules over R . W e say that M is sp e cial if R is sp ecial and ψ p ( m ) ≡ 0 mo d pM for a ll primes p and m ∈ M . 2.6. Ψ -deriv ations. A Ψ -derivation of R with v alues in M is an additiv e homo- morphism d : R → M such that (1) d ( rs ) = rd ( s ) + d ( r ) s, (2) ψ i ( d ( r )) = d (Ψ i ( r )) , for all r, s ∈ R, and i ∈ N . W e let D er Ψ ( R, M ) denote the set of all Ψ- deriv ations of R with v alues in M . 3. Coh omology of λ -rings It is known that there is an adjoint pair of functors Sets F / / λ − rings U o o where U is the for getful functor a nd F ta kes a set S to the free λ -ring genera ted b y S . The adjoint pair gives rise to a comonad G on λ − rings whic h is monadic. Let R be a λ -ring a nd M b e a λ -mo dule ov er R . W e define the cohomo lo gy of the λ -ring R with co efficients in M , deno ted by H ∗ λ ( R, M ), to b e the co monad coho mology [3] of R with coefficients in D e r λ ( − , M ). Note that D er λ ( − , M ) is a functor fr o m the category of λ -ring s to the catego ry of ab elian groups. Corollary 3. 1. F or any λ -ring R and M ∈ R − mo d λ , we have H 0 λ ( R, M ) ∼ = D er λ ( R, M ) . F urthermor e, if R is fr e e as a λ -ring then H i λ ( R, M ) = 0 for i > 0 . 4 MICHAEL R OBINSON Let R b e a λ -ring and M ∈ R − m od λ . A λ -ring ex t ension of R b y M is an exact sequence 0 / / M α / / X β / / R / / 0 where X is a λ -ring , β is a map of λ -rings, α is an additive homomorphism suc h that αλ i = Λ i α for all i ∈ N and α ( m ) x = α ( mβ ( x )) for all m ∈ M and x ∈ X . The map α identifies M wit h an idea l o f squa re-zero in X . Two λ -ring extensions ( X ) , ( X ′ ) with R, M fixed a r e sa id to be e quivalent if there exists a map o f λ -rings φ : X → X ′ such that the following dia gram co mmu tes. 0 / / M / / X / / φ   R / / 0 0 / / M / / X ′ / / R / / 0 W e denote the set of equiv alence class es of λ -ring extensions of R by M by E xtalg λ ( R, M ). Lemma 3.2. F or any λ -ring R and M ∈ R − mod λ , we have H 1 λ ( R, M ) ∼ = E xtalg λ ( R, M ) . The pro of can b e found in my thesis [5]. 4. Cohomol ogy of Ψ -rings Let I deno te the catego ry with o ne ob ject ass o ciated to the multiplicativ e monoid of the nonzero na tural num b ers. W e can consider Ψ-rings as diagrams of comm u- tative r ings; Ψ-rings ar e functor s from I to the categ ory of commutativ e rings . R : I → Com . rings . It is well known that there is an adjoint pair of functor s Sets F / / Com . rings U o o where U is the forgetful functor and F takes a s et S to the free commutativ e ring generated by S . The adjoint pair gives rise to a co monad G o n Com . rings w hich is monadic a nd the cohomo logy with resp ect to this como nad is the Andr´ e-Quillen cohomolog y of commutativ e ring s. The adjoint pair g ives rise to a nother adjoint pair Sets F I / / Com . rings I U I o o where U I is the forgetful functor and F I takes a set S to the free Ψ -ring g enerated by S . This a djoint pair y ields a como na d G I on Com . rings I = Ψ − rings whic h is monadic. Let R b e a Ψ-ring and M be a Ψ -mo dule over R . W e define the cohomolo gy of a Ψ-r ing R with co efficients in M , denoted by H ∗ Ψ ( R, M ), to b e the co monad cohomolog y of R with coefficients in Der Ψ ( − , M ). Note tha t Der Ψ ( − , M ) is a functor fro m the categ ory o f Ψ -rings to the categ ory of ab elian gro ups . Corollary 4. 1. F or any Ψ -ring R and M ∈ R − mod Ψ , we have H 0 Ψ ( R, M ) ∼ = D er Ψ ( R, M ) . F urthermor e, if R is fr e e as a Ψ -ring then H i Ψ ( R, M ) = 0 for i > 0 . THE COHOMOLOGY OF λ -RINGS AND K -THEOR Y 5 Let R b e a Ψ-ring and M ∈ R − mod Ψ . A Ψ -ring extens ion of R by M is an exact sequence 0 / / M α / / X β / / R / / 0 where X is a Ψ-r ing , β is a map of Ψ-r ings, α is an additiv e homomorphism such that α Ψ i = ψ i α for all i ∈ N a nd α ( m ) x = α ( mβ ( x )) for all m ∈ M a nd x ∈ X . The map α identifies M wit h an idea l o f squa re-zero in X . Two Ψ-ring extensio ns ( X ) , ( X ) with R, M fixed are said to b e e quivalent if there exists a map o f Ψ-r ings φ : X → X such tha t the following diagram c ommutes. 0 / / M / / X / / φ   R / / 0 0 / / M / / X / / R / / 0 W e denote the set of equiv alence class es of Ψ-ring extensions o f R by M by E xtalg Ψ ( R, M ). If R a nd M are s pe cial, then we s ay that an extens ion 0 / / M α / / X β / / R / / 0 is sp e cial if X is a lso sp ecial. Lemma 4.2. F or any Ψ -ring R and M ∈ R − mod Ψ , we have H 1 Ψ ( R, M ) ∼ = E xtalg Ψ ( R, M ) . F or e a ch n ∈ N 0 , there is a natural system [2] on I as follows D f := H n AQ ( R, M f ) where M f is an R -mo dule with M as an abelia n group with the following actio n of R ( r , m ) 7→ Ψ f ( r ) m, for r ∈ R, m ∈ M . F or a ny mo rphism u ∈ I , we have u ∗ : D f → D uf which is induced b y Ψ u : M f → M uf . F or any morphism v ∈ I , w e hav e v ∗ : D f → D f v which is induced by Ψ v : R → R . Theorem 4.3. Ther e exists a sp e ctr al se quenc e E p,q 2 = H p B W ( I , H q ( R, M )) ⇒ H p + q Ψ ( R, M ) wher e H q ( R, M ) is t he natur al system on I wh ose value on a morphism α in I is given by H q AQ ( R, M α ) and H p B W ( I , H q ( R, M )) is the Baues-Wirsching c ohomolo gy [2] of the smal l c at e gory I with c o efficients in the natur al system H q ( R, M ) . The pro of of this theor em can b e found in my thesis [5] or the pa p e r [6 ]. 5. Na tural transf o rma tion W e let K ( − ) denote the complex K -theory and e K ( − ) denote the re duced complex K-theory . Let X , Y be top ologic a l spaces such that e K ( Y ) = 0 and e K (Σ X ) = 0. Let f : Y → X b e a contin uo us map, then we can consider the P uppe s equence Y f / / X / / C f / / Σ Y / / Σ X / / Σ C f / / . . . 6 MICHAEL R OBINSON where C f is the mapping cone of f , and Σ X is the susp ens io n of X . After applying the functor e K ( − ) we get the long exact s equence. . . . / / e K (Σ X ) / / e K (Σ Y ) / / e K ( C f ) / / e K ( X ) / / e K ( Y ) How ever, since e K (Σ X ) = 0 and e K ( Y ) = 0 we obtain the s ho rt ex act s e quence. 0 / / e K (Σ Y ) / / K ( C f ) / / K ( X ) / / 0 This gives us the following pro p o sition. Prop ositi on 5 .1. If X and Y ar e top olo gic al sp ac es as ab ove t hen ther e exist natur al tra n sformations τ λ : [ Y , X ] → E xtal g λ ( K ( X ) , e K (Σ Y )) , τ Ψ : [ Y , X ] → E xtal g Ψ ( K ( X ) , e K (Σ Y )) . Corollary 5.2. If X is a t op olo gic al sp ac e such that e K (Σ X ) = 0 then ther e ex- ist natur al tr ansformations τ λ : π 2 n − 1 ( X ) → E xtal g λ ( K ( X ) , e K ( S 2 n )) and τ Ψ : π 2 n − 1 ( X ) → E xtalg Ψ ( K ( X ) , e K ( S 2 n )) . 6. The Hopf inv ariant of an extension Consider the co mmut ative r ing R gene r ated by x and y as an ab elian gro up, R ∼ = Z [ x ] ⊕ Z [ y ], where x is the unit of the ring and y 2 = 0 . The r ing R is known as the r ing of dua l n umbers . Let M ∼ = Z [ z ] b e the R -mo dule such that y · z = 0. W e can cons ider the ex tens ions of R b y M in the category of commutativ e rings. All the ex tensions hav e the following form (6.0.1) 0 / / M / / X ⊕ Z [ γ ] / / R / / 0 where X ∼ = Z [ α ] ⊕ Z [ β ] as an ab elia n group with α being the image of the gener ator z , the image of the unit γ is the unit x and the image of β being the gener ator y . Since M 2 = 0 w e get that α 2 = 0. Since y 2 = 0, w e get that αβ = 0 a nd β 2 = hα for s ome integer h . W e de fine h to b e the Hopf invaria nt of the extension (6.0.1). W e are going to co nsider the extensio ns of K ( S 2 n ) b y e K ( S 2 n ′ ) in the category of Ψ- rings. W e are going to pr ov e the following theorem Theorem 6.1. E xtalg Ψ ( K ( S 2 n ) , e K ( S 2 n ′ )) ∼ =  Z ⊕ Z G n,n ′ if n 6 = n ′ ; Z ⊕ Q p prime Z if n = n ′ . wher e G n,n ′ denotes the gr e atest c ommon diviso r of al l the inte gers in the set { l n − l n ′ | l ∈ Z , l ≥ 2 } Corollary 6. 2. If n 6 = n ′ then E xtalg λ ( K ( S 2 n ) , e K ( S 2 n ′ )) ∼ = { ( h, ν ) ∈ Z ⊕ Z G n,n ′ | h ≡ ν (2 n − 2 n ′ ) G n,n ′ mo d 2 . } If n = n ′ then E xtalg λ ( K ( S 2 n ) , e K ( S 2 n ′ )) ∼ = { ( h, ν 2 , ν 3 , . . . ) ∈ Z ⊕ Y p prime Z | h ≡ ν 2 mo d 2 , ν p ≡ 0 mo d p, p > 2 . } THE COHOMOLOGY OF λ -RINGS AND K -THEOR Y 7 All the Ψ-r ing extensio ns of K ( S 2 n ) by e K ( S 2 n ′ ) have the form (6.0.1). The Ψ-op era tions o n Ψ k : X → X are given by ψ k ( m, r ) = ( k n ′ m + ν k r , k n r ) for s ome ν k ∈ Z . Ψ k (Ψ l ( m, r )) = ( k n ′ l n ′ m + k n ′ ν l r + ν k l n r , k n l n r ) Ψ l (Ψ k ( m, r )) = ( l n ′ k n ′ m + l n ′ ν k r + ν l k n r , l n k n r ) Since the Ψ-o p erations commut e, we g et that ν l r ( k n ′ − k n ) = ν k r ( l n ′ − l n ) If n = n ′ then ther e is no r estriction on the choice of ν p for p prime. Other wise we can r earra nge the ab ove to get that ν l = ν k ( l n ′ − l n ) ( k n ′ − k n ) . By se tting k = 2 we get that for all l ≥ 2 ν l = ν 2 ( l n ′ − l n ) (2 n ′ − 2 n ) . W e c a n wr ite all the ν l ’s as multiples of ν 2 since ν l = ν 2 ( l n ′ − l n ) (2 n ′ − 2 n ) = ν 2 ( k n ′ − k n ) (2 n ′ − 2 n ) ( l n ′ − l n ) ( k n ′ − k n ) = ν k ( l n ′ − l n ) ( k n ′ − k n ) . Since ν 2 is an integer, we g et that ν 2 = z (2 n ′ − 2 n ) G n,n ′ for some integer z. If we repla ce the generator β by β + N α , note that ( β + N α ) 2 = hα , then we hav e to replace ν k by ν k + N ( k n ′ − k n ). W e get that ν k + N ( k n ′ − k n ) = ν 2 k n ′ − k n 2 n ′ − 2 n + N ( k n ′ − k n ) = ( ν 2 + N (2 n ′ − 2 n ))( k n ′ − k n ) (2 n ′ − 2 n ) So we only ha ve to b e concerned with replacing ν 2 by ν 2 + N (2 n ′ − 2 n ), then our usual for mula for ν k holds. Hence we are replacing z (2 n ′ − 2 n ) G n,n ′ by z (2 n ′ − 2 n ) G n,n ′ + N (2 n ′ − 2 n ) = ( z + N G n,n ′ )(2 n ′ − 2 n ) G n,n ′ This proves theorem 6 .1. The isomor phism dep ends on n and n ′ . B y restricting to the sp ecial Ψ -ring extensio ns, we get that ν 2 r ≡ hr 2 mo d 2 and ν p r ≡ 0 mo d p for p ≥ 3. Since all the Ψ-r ings in o ur extensions ar e Z torsio n free, the theorem of Wilkerson 2 .1 g ives us co rollary 6 .2. Prop ositi on 6.3. I f t her e exists an ext ension in E xtal g λ ( K ( S 2 n ) , e K ( S 2 n ′ )) whose Hopf invariant is o dd, then either n = n ′ or mi n ( n, n ′ ) ≤ g 2 | n − n ′ | , wher e g p j denotes the multiplicity of t he prime p in the prime factorisatio n of the gr e atest c ommon divisor of the set of inte gers { ( k j − 1) | k ∈ N − { 1 , q p |∀ q ∈ N }} . 8 MICHAEL R OBINSON Pr o of. The case when n = n ′ is clear . Assume that n 6 = n ′ , then the sp ecia l Ψ-ring extensio ns ar e given by a pa ir ( h, ν ) wher e h is the Hopf inv a riant. By 6.2, h can only b e o dd if 2 n divides G n,n ′ . Assume that n < n ′ , since the other case is ana logous. The m ultiplicity of 2 in the prime factorisation of G n,n ′ is n if n ≤ g 2 | n − n ′ | or g 2 | n − n ′ | if g 2 | n − n ′ | < n . It follows tha t if n ≤ g 2 | n − n ′ | then 2 n divides G n,n ′ .  Note that g 2 2 n − 1 = 1 for all n ∈ N . Since ( k 2 n − 1) = ( k n + 1)( k n − 1) it follows that g 2 2 n =  3 , n o dd g 2 n + 1 , n even. Theorem 6.4. If ther e exists an extension in E xtal g λ ( K ( S 2 n ) , e K ( S 2 n ′ )) whose Hopf invariant is o dd, then one of the fo l lowing is satisfie d (1) n = n ′ . (2) n = 1 or n ′ = 1 . (3) n ′ − n is even and either n = 2 or n ′ = 2 . (4) n ′ > n ≥ 3 and n ′ = n + 2 n − 2 b for some b ∈ N 0 . (5) n > n ′ ≥ 3 and n = n ′ + 2 n ′ − 2 b fo r s ome b ∈ N 0 . Pr o of. 1. is cle a r. 2. follows from g 2 n ≥ 1 for all N . 3. follows from g 2 2 n ≥ 3 for all n ∈ N . 4. and 5. follows fro m g 2 | n − n ′ | being 2 plus the mult iplic ity o f 2 in the prime factorisatio n of | n − n ′ | .  Lemma 6.5. If ther e exists an extension in E xtal g λ ( K ( S 2 n ) , e K ( S 2 an )) for a ∈ N whose Hopf invaria nt is o dd, then one of the following is satisfie d (1) n = 1 , 2 or 4 . (2) n = 3 and a is even. (3) n ≥ 5 and ( a − 1) n = n + 2 n − 2 b fo r s ome b ∈ N 0 . Corollary 6 .6. If ther e ex ists an extension in E xtalg λ ( K ( S 2 n ) , e K ( S 4 n )) whose Hopf invariant is o dd, then n = 1 , 2 or 4 . Corollary 6.7 (Adams) . If f : S 4 n − 1 → S 2 n is a c ont inuous map whose Hopf invariant is o dd, then n = 1 , 2 or 4 . 7. S t able Ext al g gr oups of spheres Prop ositi on 7.1. If n > k + 1 then G n,n + k = G n +1 ,n + k +1 . Pr o of. Let n > k + 1 . W e know tha t G n,n + k = G n +1 ,n + k +1 if and only if the m ultiplicit y of an y prime p in the pr ime factoriz ation of G n,n + k is g p k . F or all primes p > 2 we ge t that p n > 2 k − 1, so the mu ltiplicit y of p in the prime factorisation of G n,n + k is g p k . W e ca n eas ily see tha t g 2 k ≤ k + 1 for all k . It follows that the m ultiplicit y of 2 in the prime facto risation o f G n,n + k is g 2 k .  Corollary 7. 2. If n > k + 1 then Extalg λ ( K ( S 2 n ) , e K ( S 2( n + k ) )) ∼ = Extalg λ ( K ( S 2( n +1) ) , e K ( S 2( n + k +1) )) . The gr oups Extal g λ ( K ( S 2 n ) , e K ( S 2( n + k ) )) are indep endent of n for n > k + 1, we call these the stable Extalg gr oups of spher es which we denote by Extalg s 2 k . THE COHOMOLOGY OF λ -RINGS AND K -THEOR Y 9 Prop ositi on 7.3. Th er e ar e natur al tr ansformations Υ k : π s 2 k − 1 → Extalg s 2 k wher e π s 2 k − 1 denotes the stable homotopy gr oups of spher es. F or s mall k thes e groups lo ok as follows. k π s 2 k − 1 Extalg s 2 k 1 Z 2 2 Z ⊕ Z 2 2 Z 24 ⊕ Z 3 2 Z ⊕ Z 24 3 0 2 Z ⊕ Z 2 4 Z 240 2 Z ⊕ Z 240 5 Z 2 ⊕ Z 2 ⊕ Z 2 2 Z ⊕ Z 2 6 Z 504 2 Z ⊕ Z 504 7 Z 3 2 Z ⊕ Z 2 8 Z 480 ⊕ Z 2 2 Z ⊕ Z 480 References [1] M.F. A tiy ah an d D.O. T all . Group representa tions, λ - rings and the J - homomorphism, T op ology 8, 1969. p253-297. [2] H.J. Baues and G. Wirsching . Cohomology of small categories, Journal of pure and applied algebra 38, 1984. [3] J.M . Beck . T riples, algebras and cohomology , Ph.D. thesis, Columbia Unive rsity , 1967. [4] D. Kn utson . λ -Rings and the Represen tation Theory of the Symmetric Group, Springer, 1973, v ol . 308. [5] M. R obinson . The cohomology of λ -rings and Ψ- rings, Ph.D. thesis, Universit y of Leicester, 2010. [6] M. Robinson . Cohomology of diagrams of algebras. arXiv:0802.3651v1 [math.KT]. [7] G. Whitehead . Recen t Adv ances in Homotop y Theory , The M IT Press, 1971. [8] C. Wilkerson . Lambda rings, binomial domains and vec tor bundles ov er C P (), Comm. Algebra 10 (1982) , 311-328. [9] D. Y au . Cohomology of λ -ri ngs. J. Al gebra 284 (2005) , 37-51. Dep ar tment of Ma them a tics, University of Leic ester, University R oad, Leicester, LE1 7RH, UK E-mail addr ess : mr85@alumn i.le.ac.uk