October 11, 2010
This one will be a quick post. In effect, we continue with last time, where we defined the relative homotopy groups, and now describe a practical means of determining when something in one of these relative groups is zero or not. This will become useful in the future.
The compression criterion
We have defined the group above, but we still need a good criterion for knowing when something in , represented by , is zero. Or, when , when it represents the base element. The obvious reason is that if there is a homotopy starting with and ending at the constant map. Here is another that will be useful.
Theorem 1 (Compression criterion) A map represents zero in if and only if is homotopic relative to a map .
Proof: This is one of those things which is not really all that hard to prove, but for which pictures help significantly. So I will try to draw pictures. (more…)
October 4, 2010
The Whitehead theorem states that a map of connected CW complexes that induces an isomorphism in homotopy groups is a homotopy equivalence. In particular, isomorphisms in the homotopy category of pointed CW complexes can be detected by homming out of spheres . But the equality of two morphisms cannot. The fact that this “relative Whitehead theorem” fails was the subject of a MO question. Today, I want to discuss another example along these lines. (I will assume a little more familiarity with algebraic topology than I have in previous posts.)
Recall that a common technique to show that a map is not nullhomotopic is to show that it does not induce the trivial morphism on some functor in algebraic topology. For instance, the fact that is used to show that is not contractible; this is probably the most basic example. But the basic invariants of algebraic topology can be insufficient. Here is an example which Eric Larson showed me yesterday.