1 Motivation and definitions

The Adams–Novikov spectral sequence is a spectral sequence

$$\operatorname{Ext}^{s, t}_{BP_*BP}(BP_*, BP_*) \Rightarrow \pi _{t - s}(\mathbb {S}).$$

We will abbreviate $\operatorname{Ext}^{s, t}_{BP_*BP}(BP_*, M)$ as $\operatorname{Ext}^{s, t}(C)$, and sometimes omit the $t$.

To use this to construct elements in $\pi _{t - s}(\mathbb {S})$, we have to do three things:

1. Find an element in $\operatorname{Ext}^{s, t}(BP_*)$

2. Show that it doesn't get hit by differentials

3. Show that all differentials vanish on it.

All three steps are difficult, except with some caveats.

1. We can do this if $s = 0$.

2. We can do this if $s$ is small enough.

3. We can do this if we know the map of spheres actually exists, and want to show it is non-zero.

Of these three caveats, (1) is perhaps the worst, because the $s = 0$ line is boring. To make better use of our ability to calculate $\operatorname{Ext}^0$, suppose we have a short exact sequence of comodules, such as

$$0 \longrightarrow BP_* \overset {p}{\longrightarrow } BP_* \longrightarrow BP_*/p \longrightarrow 0.$$

We then get a coboundary map

$$\delta : \operatorname{Ext}^0(BP_*/p) \to \operatorname{Ext}^1 (BP_*).$$

So we can use this to produce elements in $\operatorname{Ext}^1(BP_*)$. To understand the geometry of this operation, so that we can do (3), we use the following lemma, whose proof is a fun diagram chase:

The slogan is

If $\delta$ comes from geometry, it sends permanent cycles to permanent cycles.

For the short exact sequence above, we can realize it as the $BP$ homology of

$$\mathbb {S}\overset {p}{\longrightarrow } \mathbb {S}\longrightarrow \mathbb {S}/p \equiv V(0).$$

Recall that $\operatorname{Ext}^0(BP_*/p) = \mathbb {F}_p[v_1]$. If we can find a map $\tilde{v}_1: \mathbb {S}^{2p - 2} \to V(0)$ that gives $v_1 \in \operatorname{Ext}^0(BP_*/p)$, then we know $\delta (v_1) \in \operatorname{Ext}^1(BP_*/p)$ is a permanent cycle. Since this has $s = 1$, no differentials can hit it, and as long as $\delta (v_1) \not= 0 \in \operatorname{Ext}^1(BP_*)$, which is a purely algebraic problem, we get a non-trivial element in the homotopy groups of sphere.

This is actually not a very useful operation to perform, because the way we are going to construct $\tilde{v}_1$ is by analyzing the Adams–Novikov spectral sequence for $V(0)$, which is not very much easier than finding the element in $\operatorname{Ext}^1(BP_*)$ directly.

But if we can promote this to a map $v_1: \Sigma ^{2p - 2} V(0) \to V(0)$ that induces multiplication by $v_1$ on $BP_*$, then we can form the composition

$$\mathbb {S}^{t (2p - 2)} \hookrightarrow \Sigma ^{t(2p - 2)} V(0) \overset {v_1^t}{\longrightarrow } V(0)$$

which represents $v_1^t \in \operatorname{Ext}^0(BP_*/p)$, where the first map is the canonical quotient map $\mathbb {S}\to \mathbb {S}/p = V(0)$. We then know that $\delta (v_1^t) \in \operatorname{Ext}^1(BP_*)$ is permanent, and the above argument goes through. Thus, by constructing a single map $v_1: \Sigma ^{2p - 2} V(0) \to V(0)$, we have found an infinite family of permanent cycles in $\operatorname{Ext}^1$, knowing by magic that all the differentials from it must vanish. It is in fact true that for $p > 2$, the map $v_1$ exists and they are all non-trivial. These elements are known as $\alpha _t$.

The map $v_1$ is known as a $v_1$ self map of $V(0)$. If we are equipped with such a map, we can play the same game with the cofiber sequence

$$\Sigma ^{2p - 2} V(0) \overset {v_1}{\longrightarrow } V(0) \to V(1).$$

We then know that $BP_* V(1) = BP_*/(p, v_1)$, and we have a short exact sequence

$$0 \longrightarrow BP_*/p \overset {v_1}{\longrightarrow } BP_*/p \longrightarrow BP_*/(p, v_1) \longrightarrow 0.$$

Again we know that $\operatorname{Ext}^0(BP_*/(p, v_1)) = \mathbb {F}_p[v_2]$, and we can seek a $v_2$ self map $v_2: \Sigma ^{2p^2 - 2} V(1) \to V(1)$ that induces multiplication by $v_2$ on $BP_*$. If we can do so, then we know that $\delta (v_2^t) \in \operatorname{Ext}^1(BP_*/p)$ is a permanent cycle, and hence $\delta \delta (v_2^t) \in \operatorname{Ext}^2(BP_*)$ is also a permanent cycle. This gives us a second sequence of elements in the stable homotopy group of spheres. Moreover, in this case the non-triviality is again an algebraic problem of showing that $\delta \delta (v_2^t) \not= 0 \in \operatorname{Ext}^2(BP_*)$, since no differentials can hit it. These elements are known as $\beta _t$.

In these notes, I will construct the $v_1$ self maps for $p > 2$ and $v_2$ self maps for $p > 3$. It is true that the corresponding $\alpha _t$ and $\beta _t$ are in fact non-zero, but I will not prove it here. These maps were first constructed by Adams and Smith (for $v_1$ and $v_2$ respectively), but they had to do more work because they didn't have $BP$ and the Adams–Novikov spectral sequence.

We can of course continue this process to seek $v_n$ self maps for larger $n$, and you should be glad to hear that this will become prohibitively difficult way before we run out of Greek letters.