Course MT3818 Topics in Geometry

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(Topology of projective spaces)
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The projective space of a vector space

Given any vector space V over a field F, we can form its associated projective space P(V) by using the construction above.
P(V) = V - {0}/~ where ~ is the equivalence relation u ~ v if u = lambdav for u, v belongs V - {0} and lambda belongs F.

Examples

  1. With this definition RP1= P(R2) and RP2= P(R3) so one needs to be careful about dimensions.

  2. Let V be vector space C2 over the complex numbers C. Then P(C2) is the complex projective line CP1 which (arguing as in the real case) consists of C together with a single point at infinity.
    Topologically, this is the Riemann sphere S2.

  3. Let V be a vector space over a finite field F. Such a finite field has pk elements where p is a prime number. Then if V has dimension n over F, we have |V - {0}| = pkn - 1 and since each line through 0 in this has pk - 1 elements on it we get |P(V)| = (pkn - 1)/(pk - 1).

    1. For any finite field, with n = 2 we get a projective line with |F| + 1 points consisting of F together with one extra point.

    2. If |F| = 2 and n = 3 we get a projective plane with 7 points.
      The points lie in 3's on 7 lines and each point is the intersection of 3 lines.
      Every pair of points determines a line and every pair of lines meet in a unique point.
      (One of the lines is represented by a circle on the picture.)


    3. If |F| = 3 and n = 3 we get a projective plane with 13 points.
      The points lie in 4's on 13 lines and each point is the intersection of 4 lines.
      Once again every pair of points determines a line and every pair of lines meet in a unique point.


    Examples like these last two are very interesting combinatorial objects and have many nice properties.


Previous page
(Topology of projective spaces)
Contents Next page
(The projective groups)

JOC March 2003