Newsgroups: sci.math
From: jbaez@riesz.mit.edu (John C. Baez)
Subject: Tough topology puzzles
Date: Thu, 22 Oct 92 04:59:34 GMT

Okay, here are some nice tough topology puzzles.  I certainly couldn't do them
myself without peeking at the answers, but some of you macho topologists
out there may enjoy taking a whack at them, and the rest of you folks
may still enjoy reading them, because they're pretty intense.  

Let's see how good you are at visualizing 100-dimensional space....

1)  As we all know, two S^1's (circles) can be linked in S^3 (or R^3),
but all link circles can be unlinked, and all knotted circles unknotted,
in S^4.  So show that two S^50's (fifty-dimensional spheres):

are unlinked in S^102
can link nontrivially in S^101
can link nontrivially in S^100
can link nontrivially in S^99
can link nontrivially in S^98
are unlinked in S^97
are unlinked in S^96
can link nontrivially in S^95
can link nontrivially in S^94

2)  Let M be a compact connected manifold of dimension 100.  Suppose
that any n-sphere mapped into M can be squished down to a point for
n = 1, 2, ..., 33.  (I.e., suppose pi_n(M) is zero for n = 1 to
33.)  Then show that M can be covered by only 3 embedded balls!  This is
rather surprising since the topology of M can be very complicated.

(Note: the above are true in the PL (piecewise linear) category,
and I don't know whether they are true in other categories.)