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This fact ties in well with Theorem 14: the lower hemisphere of S3 given by u -- 0 is clearly homeomorphic to D3 by the projection (x,y,z,u) -+ (x,y,z). Every point of S3/x -- -x has a representative in the lower hemisphere, and this representative is unique unless u = 0. A point of S3/x -- -x with u = 0 has two representatives in the lower hemisphere and they are antipodal points of the boundary of D3, that is, points x, -x with If x11 = 1. This discussion gives an alternative proof of Theorem 14.

The group S, has an important (normal) subgroup of index 2 called the alternating group A. It consists of those permutations that can be written as a product of an even number of transpositions. It is not quite obvious that this is a well defined condition on a permutation (and hence that the set of all such permutations forms a subgroup of Sr). , xn) by a(xi) = x,pi then A. = { or e S1Iaf = f where f = n (x-x1)). As an alternative one has: Exercise Define sign: by sign (a) =n sign (transposition) = -1 ii) sign is a homomorphism.

What is its image in SO(3)? 40. ) The spaces S3 and SO(3) are both connected, compact, closed, smootn manifolds of dimension three. The map p: S3 -. SO(3) is smooth, everywhere regular and Jp 'xI - 2 for each x E SO(3). Deduce that p is onto. Part II Projective Geometry Projective geometry was invented in the 17th century, the first important contributions to the subject being made by a French architect Gerard Desargues. These studies arose from attempts to understand the geometrical properties of perspective drawing.

### Algorithmic Geometry [auth. unkn.]

by Kenneth

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