[GH] Problem 3.7. The quotient space is a stack of shrinking
circles - a cone or disc. See Proposition 1.6 [H] if the
second hint is perplexing (it was at first to me). In [GH], "s"
is the path parameter and "t" seems to be the homotopy parameter. Keep
these straight.
[GH] 3.8. Make a cartoon sketch to see why this is a
"cone". Recall X/A is the quotient space with 2 points distinct unless
both in A.
[GH] 3.9. p and f are homotopy (equivalence) inverses.
[GH] 3.10. For each x in S^n, there is a UNIQUE great circle
subarc (try n=2) from f(x) to g(x) whose length varies
continuously.