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P. But 1'*(J2 = (1'*(J)2 = (F*(J)2 = ((Jl + (J2)2 = (Ji + (J~ + (Jl(J2 + (J2(Jl = 0+0 +p + (_1)"2 p =2p so that H(cp)p = 2p and H(cp) = 2, as was to be proved. Proposition 2 The transformation H: 1l"2"-I(S")-Z is a homomorphism. We first recall the definition of the additive group structure in 1l"2"-1 (S"). We denote by Q the" pinching map" Q: (e 2",S2"-I)_(e 2" v e 2", S2"-1 V S2"-I) obtained by collapsing an equatorial (2n - I)-cell to a point. Let F denote the folding map S" v S"- So, as before.

In this case we may also refer to SqI as admissible. The length of any sequence I is the number of terms, r in the above notation. The degree d(I) is the sum of the terms, Lj ij . ) For an admissible sequence I, the excess e(I) is 2i[ - d(I). ) The last expression justifies the name, but the first two are more convenient in practice. Recall that S denotes the symmetric polynomial subring of the polynomial ring H*(Kn;Z2)=Z2[X 1, ••• ,xn]. We define an ordering on the monomials of S as follows: given any such monomial, write it as III = aj:aj~' ..

3. - and D. B. A. Epstein [1]. The Hop! invariant one problem 1. J. F. Adams [1,2]. 2. J. Adem [1]. 3. H. Cartan [6]. Whitehead products 1. P. J. Hilton [1]. 2. G. W. Whitehead [2,4]. CHAPTER 5 APPLICATION: VECTOR FIELDS ON SPHERES We consider the (n - I)-sphere sn-1 imbedded in Rn in the usual way. A tangent vector field, or simply a vector field on sn- 1, is a function assigning to each point of 5 n - 1 a tangent vector at that point such that the vector thus defined varies continuously with the point.

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Artin L-functions (2005)(en)(1s) by Garrett P.

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