By Adams J. F.

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**Extra resources for Infinite Loop Spaces - Hermann Weyl Lectures the Institute for Advanced Study**

**Example text**

2) For an point x E X. there exists a local coordinate system a in neighbourhood of x such that U= df Of = ... = — = 0. (3) The mapping ii is oriented. c a distribution K E D'(R) such that f = K(u). Pmof Conditions (2) and (3) imply that for any point xX,E we have = in the vicinity of x (it is a consequence of Schwartz's theorem on solutions of the equation = 0). We only have to show that if u(x) = u(y) = UO, in the vicinity of u0. It follows from condition (I) that there then = exists a path y : [0, II —÷ X such that y(O) = x, y(l) = y and u(y(t)) It is easy to see that = for t and t' close enough.

X and is a fibre bundle whose fibre is R,; as above, the standard orientation of the fibre (the 24 1. Homogeneous functions, Fourier transformation, and contact structures direction of the vector is regarded as positive) defines the orientation of the projection. We have a commutative diagram of oriented mappings: RP' (The upper embedding is oriented, since the orientations of and are chosen and fixed. The corresponding orientations on and define the orientation of the lower embedding). in the following, we often denote simply by 3.

0, o=O, k