For k≤3, we saw the space B_k(S^2)=C_k(S^2)/SO(3) of k-configurations
on the sphere (modulo rotations) is topologically trivial: a point, a
half-open interval, and a 3-ball (perhaps shaped like a triagular
prism) for k=1,2 and 3, respectively. We also observed we can use
stereographic projection to identify B_k(S^2) with C_{k-1}(R^2)/SO(2).
Project 3: (i) Show the corresponding space B_k(S^1)=C_k(S^1)/SO(2)
is a disjoint union of (k-1)! copies of a (k-1)-cell (shaped like a
(k-1)-simplex, so a point for k=1, two open intervals for k=2, six
open triangles for k=3...). This means that, aside from permuting
the cyclic order of the points, only one E-critical configuration (by
Project 2, the regular k-gon) is demanded by Morse theory.
(ii) Analyze C_{k-1}(R^2) and C_{k-1}(R^2)/SO(2) for small k. For k=1
they're empty, of course; for k=2 one is R^2 and the other is a half-
open interval [0,∞) corresponding to radius in R^2; this agrees with our
picture for B_2(S^2) as a half-open interval interval (0,π] representing
angular distance between distinct points (antipodal points have angular
distance π). Keep going with this comparison for k=3, 4, 5....
(iii) We worked out in class the homotopy type of B_4(S^2) and found it's
a (fattened) "Theta"-graph: the two vertices (0-cells) come from the two
ways to label a tetrahedron (energy minizers for any comp'ly monotonic
potential) and three edges (1-cells) come from the three ways to to label
a 4-ring (index-1 critical points). We also explored B_5(S^2) this way,
speculating there should be 20=5!/6 vertices (0-cells) which correspond
to triangular bi-pyramid E-minimizers, 30=5!/4 edges (1-cells) correspond
to square pyramids, and 12=5!/10 faces (2-cells) correspond to 5-rings;
that seems combinatorially like the dodecahedron, but we also know that
B_5(S^2) is a twisted product with B_4(S^2) as base and a 4-punctured
sphere as fibre, i.e. has homotopy type "Theta" graph twisted product
with tetrahedron 1-skeleton (i.e. the graph consisting of the edges
and vertices of the tetrahedron). Can you figure out how this product is
twisted [this may be very hard!]?