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J-1 D'J = D" is a distribution defined in the unit sphere of n-dimensional Euclidean space, and where h,Ax) is the coordinate expression of the tensor h e V. 1 subset of the unit sphere. Let ,n be a smooth non-negative function in R", of total integral 1, vanishing outside the unit circle. )) are a family of C°° functions, defined in the sphere of radius 1 - e, and converging as e - 0 to D'J in the sense of the theory of distributions. From the statement i D'J au au 5 0. J-t (ax, ax1) it is easily verified that - f DQJ(x) 1*1 '.

Say positive definite at each point of M. Clearly K' is a convex cone, open for the- S' topology. For every C'+1 embedding zof M in some Euclidean space, f(z) belongs to K'. Let E' c K' be the set of all such f(z). Lemma: E°° is a convex cone dense in K°° for the S0°- topology. Proof: (i) E' is a convex cone. For every A z 0 we have Rf(z) = f(Jir) (f is a quadratic form). then the embedding t = z ® u : M -+ Rm ® R' (defined in the obvious way) satisfies f(t) = f(z) + f(u). Both properties together define a convex cone.

D. Back to the Definition . . . . . E. The Continuous Case . . . . . . . F. The Multiplicative Property and Consequences . . . . . . . . . . . G. Borsuk's Theorem . H. Preliminaries: Degree Theory in an Arbitrary Finite Dimensional Space . . . . . . . 1. Preliminaries: Restriction to a Subspace . . J. Degree of Finite Dimensional Perturbations of the Identity . . . . K. Properties . . . . . . . . . . . . L. Limits . . . . .

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