The Order of the Image of the J-homomorphism
نویسندگان
چکیده
The set üS can be identified with the set of all base point preserving maps of $ into itself. SO(n)> acting on S as R with a point a t infinity, is also a set of base point preserving maps of S onto itself. This defines SO(n) C.tiS. The induced map in homotopy is called the /-homomorphism. If we allow n to go to infinity we have the stable /-homomorphism. By Bott 's results [3] 7Ty(50)=Z, i = — 1 mod 4, and = Z 2 j = 0, 1 mod 8, i > 0 , and zero otherwise. Adams [ l ] showed that the Z2 summand maps monomorphically and Milnor and Kervaire [ô] showed that the Z group in dimension 4/ —1 maps non trivially and its image generates a subgroup of at least a certain order XJ. Adams [ l ] showed that the order was either Xj or 2Xj and if j = s l (2) it was Xj. Thus only the two primary part is in question and there only for j==0 (2). Let Xy be the two primary part of X;. If 4 / s 2 ^ m o d 2'<» (which defines p(j)) then Xy = 2 ^ + 1 . We prove:
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