Classification of maps $f:\mathbb{S}^1\times\mathbb{S}^2\to\mathbb{S}^2$ up to homotopy and geometrical meaning.

Question

I have two questions.

First, how can one classify maps $f:\mathbb{S}^1\times\mathbb{S}^2\to\mathbb{S}^2$ up to homotopy?

Second, how to interpret geometrically what distinguishes them? I mean, if two maps are non homotopic (let's say they appear to be non homotopic because some invariants distinguish them), I would be really interested in the geometrical interpretation of these invariants. Thank you in advance.

Answer 1

A map $\Bbb S^1\times \Bbb S^2\to\Bbb S^2$ is a free loop in the space of maps $\Bbb S^2\to\Bbb S^2$.

This means that you have to look at

• the set of connected components of maps $\Bbb S^2\to\Bbb S^2$: they are indexed by $\pi_2(\Bbb S^2)=\Bbb Z$. I write "=" rather than "$\simeq$" because the identity map is your favourite generator.
• for each connected component $C_i,\ i\in \Bbb Z$, consider $\pi_1(C_i)$.

To see in which connected component $C_i$ a map $\Bbb S^1\times \Bbb S^2\to\Bbb S^2$ is, you just need the degree $i$ of the induced map $\{\cdot\}\times \Bbb S^2\to\Bbb S^2$ where $\cdot$ is an arbitrary point of the circle.

Now to compute $\pi_1(C_i)$, I can't see just now how to do that easily for all $i$, but at least it is clear that $\pi_1(C_0)=\{\operatorname{1}\}$, and I believe that one should have $\pi_1(C_1)\simeq\pi_1(SO(3))\simeq \Bbb Z/2\Bbb Z$.