Topology = Idealization of space
   projective geometry

Klein-Erlanger Program -> Transformation key! not object!
  1. Determine topological equivalence of spaces (Homeomorphics)
  2. Determine if space x has fixed point
   Classify Topology spaces - equivalence classes/cannonical instances
   1. Spheres w/ g (genus) handles (All closed surface topologically equivalent to)
   2. Cross caps (All non-orientable closed surface topologically equivalent to)
      Mobius band - open surface w/ 1 edge ; non-orientable
      Cross cap = sew mobius band into sphere w/ hole in it


Fixed Points
   Brouwer fix point theorem
         X= topological space compact + convex
         Compact = finite dim topo space bounded w/ all points of boundary
         Convex = pt-to-pt enclosure (triangle,circle but not horshoe,torus)
         f = continuous map of X to itself then f has fixed pt in X,
         i.e. there exists x* in X s.t. f(x*)=x*
   Sperner's Lemma
         Chop up space
             |    ._.         A/____|
             |   /A  \   _____--    |
             |  /   __\_-    /      |  lim del X find fixed point
             |------   \____/ B     |
             |                      |
               A  A  A   B  B   A  A  

1. Cyclones - brouwer's fix pt -> can't comb wind to eliminate pt of zero vertical movement
2. Hohmann path - find map f whose fixed pts  = solution optimal to newton eqn (hohmann path)
3. Equilibria class divisions - occupational mobility
4. Perron-Frobenius Theorem Ar=r Team score vector, pt of constancy
5. Adams (humanitraianistic economics)- L. Walrus (supply/Demand) - V. Neumann(equilib HI out Lo$, HI rate)
   - Arrow (Gen equilib thy)
6.       /\
        /_p\  po->p1 in economic brouwer this is equilibrium pt 
       /p   \  p1 s.t. apply T to p0 get to p1.

S Home
S Next ... Critical Points