2 ENRIQUE ANTONIANO PROPOSITION 2: There exists an inmersion PK-14-Rn-1• if and only if there is a map s making the following diagram commutative: ·s,'"xn,k ,' -}f pK-1 c:...-, poo We would like to apply K-theory and Adams operations looking for obstructions to the existence of s in the next diagram: ,K(Xn,k) s*,, tf* , ... K(Pm) ~ K(P00 ) Here, the values of K(pffi), K(P00 ), and i* are well known and it is our objec- tiveto describe K(Xn,k) and f*. With this purpose, we will make use of the following theorem due to Hodgkin, [6], [9]. Let R{G) and R(H) be the representation rings of G and H. Then, Z and R{H) are R(G) modules via the augmentation and the homomorphism induced by the inclusion of H in G. THEOREM 3: Let G be a compact Lie group and H c G a closed subgroup. Then there is a spectral sequence {E~'q} such that: I Tor:(G) (R{H) Z) if q is even 1) Ep,q = 2 o if q is odd 2) {Er} converges to K(G/H) To apply such a theorem we will restrict ourwelves to the case x4n,2k_1• since then: x4n,2k-l = Spin(4n)/Spin(4n-2k+l) X z2 so the involved Lie groups have easier representation rings. in fact. [5], [8]: R(Spin(4n)) = Z['lfp· .. ,'lf2n_2.X40 .o n] R(Spin(4n-2k+l) x z2) = Z['IT1, •.. ,'lf2n-k-l"o,y]/(y2+2y) To determine the algebra. TorR(G)(R(H) Z), we have the following lemma that desc.ri bes : j*: R{Spin(4n))-+ R(Spin(4n-2k+l)x z2) LEMMA 4: The homomorphism j* is given by: j*{'lf.) = (-l)i-1 I 22j-1 (2n-~+j) Y 1T .. +(l+y)i1T 1 1 j=l J 1-J j*(X4n) = -2k-loy-22n-1 Y j t(o+ ) = 2k-lo 4n
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