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.

There is a map

defined by quotienting by the 2-skeleton. One way to see this is that is a cube with opposite faces identified. is a cube with all the faces collapsed to a point. So we can define the quotient map by crushing the boundary of the cube defining . Then, we compose this with the Hopf fibration

The composite is a map , which induces the trivial morphism on the homotopy groups. Indeed,

is zero except in dimension one, since admits a covering map from the contractible space . is zero for (by a corollary of the simplicial approximation theorem). So the two spaces don’t have nonvanishing homotopy groups in the same dimension.

It is also true that induces the trivial morphism in reduced homology. The reason is that is zero except in dimension 2. But maps the 2-skeleton of into zero. Since the map induces a surjection on , we see that is zero.

Proposition 1is not nullhomotopic.

The reason is that the Hopf fibration is, well, a fibration. If were nullhomotopic, then the map would be nullhomotopic. This is because of the homotopy lifting property is precisely what defines a fibration. If there were a nullhomotopy of to the constant map at , then we could lift this nullhomotopy to get a homotopy of to a map of into one of the fibers ; the fibers are contractible, though.

But does not induce the trivial map in homology. In terms of cellular homology in dimension 3, the map is actually just the identity. So is not nullhomotopic. Another way to see this is to look at simplicial homology. The third homology class of the torus is represented by the cube itself. The same is true for .

October 6, 2010 at 2:51 am

Who is Eric Larson? Is he the same guy as here – http://arxiv.org/find/math/1/au:+Larson_E/0/1/0/all/0/1 ?

October 9, 2010 at 10:04 pm

Eric Larson is a Harvard Undergraduate.He was awarded the grand prize in the 2009 Intel Science Talent Search. He also won the silver medal at the International Math Olympiad in 2007.He study at South Eugene High School in Eugene and MIT before.He Oct. 12 at Harvard’s math table to report,Report entitled is “Tannakian Duality for Finite Groups”.

October 14, 2010 at 12:00 am

I didn’t want the guy’s life story! Just who he was. Anyway thanks. How do you know he is giving a report at Harvard’s math table? Are you at Harvard or something?