# 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:

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

Show that it doesn't get hit by differentials

Show that all differentials vanish on it.

All three steps are difficult, except with some caveats.

We can do this if $s = 0$.

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

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:

Suppose $A \to B \to C \to \Sigma A$ is a cofiber sequence, and suppose that the map $BP_* A \to BP_*B$ is injective, so that we have a short exact sequence

$0 \to BP_* A \to BP_* B \to BP_* C \to 0.$Suppose $f: \mathbb {S}^? \to C$ is a map, whose corresponding element in the Adams spectral sequence is $\hat{f} \in \operatorname{Ext}^s (BP_*C)$. Then the composition $\mathbb {S}^? \to C \to \Sigma A$ corresponds to $\delta \hat{f} \in \operatorname{Ext}^{s + 1} (BP_* C)$. In particular, $\delta \hat{f}$ is a permanent cycle.

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

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.