Course MT3818 Topics in Geometry

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## Topology of projective spaces

We may use the representation of projective space via homogeneous coordinates to get a topological picture of these spaces.

1. When we write RP1 = R2 - {(0, 0)}/~ with ~ as before, the equivalence classes are lines through the origin in R2.
Each line meets the "upper semicircle" in a unique point except for the horizontal line which meets it twice.
Thus RP1 is the space we get from the semi-circle by identifying its end points. This is a circle S1.

One can see the same thing by mapping the line to the circle by stereoscopic projection from the top point of the circle.

2. The space RP2 is the set of lines through the origin in R3 - 0. Each line meets the "upper hemisphere" in a unique point except for any horizontal line which meets it twice in opposite points on the equator. Thus can be made out of a hemisphere (topologically equivalent to a 2-dimensional disc) by identifying opposite points on the boundary.
This can be represented by one of the pictures:

Alternatively, each line in through the origin in R3 - 0 meets the unit sphere S2 in a pair of antipodal points.
Thus RP2 is the space we get from the sphere by identifying antipodal points.

As a topological space, RP2 is non-orientable.

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JOC March 2003