Construction of v1v_1 and v2v_2 self-mapsMotivation and definitions

1 Motivation and definitions

The Adams–Novikov spectral sequence is a spectral sequence

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

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

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

  1. Find an element in Exts,t(BP)\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=0s = 0.

  2. We can do this if ss 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=0s = 0 line is boring. To make better use of our ability to calculate Ext0\operatorname{Ext}^0, suppose we have a short exact sequence of comodules, such as

0BPpBPBP/p0. 0 \longrightarrow BP_* \overset {p}{\longrightarrow } BP_* \longrightarrow BP_*/p \longrightarrow 0.

We then get a coboundary map

δ:Ext0(BP/p)Ext1(BP). \delta : \operatorname{Ext}^0(BP_*/p) \to \operatorname{Ext}^1 (BP_*).

So we can use this to produce elements in Ext1(BP)\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 ABCΣAA \to B \to C \to \Sigma A is a cofiber sequence, and suppose that the map BPABPBBP_* A \to BP_*B is injective, so that we have a short exact sequence

0BPABPBBPC0. 0 \to BP_* A \to BP_* B \to BP_* C \to 0.

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

The slogan is

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

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

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

Recall that Ext0(BP/p)=Fp[v1]\operatorname{Ext}^0(BP_*/p) = \mathbb {F}_p[v_1]. If we can find a map v~1:S2p2V(0)\tilde{v}_1: \mathbb {S}^{2p - 2} \to V(0) that gives v1Ext0(BP/p)v_1 \in \operatorname{Ext}^0(BP_*/p), then we know δ(v1)Ext1(BP/p)\delta (v_1) \in \operatorname{Ext}^1(BP_*/p) is a permanent cycle. Since this has s=1s = 1, no differentials can hit it, and as long as δ(v1)0Ext1(BP)\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 v~1\tilde{v}_1 is by analyzing the Adams–Novikov spectral sequence for V(0)V(0), which is not very much easier than finding the element in Ext1(BP)\operatorname{Ext}^1(BP_*) directly.

But if we can promote this to a map v1:Σ2p2V(0)V(0)v_1: \Sigma ^{2p - 2} V(0) \to V(0) that induces multiplication by v1v_1 on BPBP_*, then we can form the composition

St(2p2)Σt(2p2)V(0)v1tV(0) \mathbb {S}^{t (2p - 2)} \hookrightarrow \Sigma ^{t(2p - 2)} V(0) \overset {v_1^t}{\longrightarrow } V(0)

which represents v1tExt0(BP/p)v_1^t \in \operatorname{Ext}^0(BP_*/p), where the first map is the canonical quotient map SS/p=V(0)\mathbb {S}\to \mathbb {S}/p = V(0). We then know that δ(v1t)Ext1(BP)\delta (v_1^t) \in \operatorname{Ext}^1(BP_*) is permanent, and the above argument goes through. Thus, by constructing a single map v1:Σ2p2V(0)V(0)v_1: \Sigma ^{2p - 2} V(0) \to V(0), we have found an infinite family of permanent cycles in Ext1\operatorname{Ext}^1, knowing by magic that all the differentials from it must vanish. It is in fact true that for p>2p > 2, the map v1v_1 exists and they are all non-trivial. These elements are known as αt\alpha _t.

The map v1v_1 is known as a v1v_1 self map of V(0)V(0). If we are equipped with such a map, we can play the same game with the cofiber sequence

Σ2p2V(0)v1V(0)V(1). \Sigma ^{2p - 2} V(0) \overset {v_1}{\longrightarrow } V(0) \to V(1).

We then know that BPV(1)=BP/(p,v1)BP_* V(1) = BP_*/(p, v_1), and we have a short exact sequence

0BP/pv1BP/pBP/(p,v1)0. 0 \longrightarrow BP_*/p \overset {v_1}{\longrightarrow } BP_*/p \longrightarrow BP_*/(p, v_1) \longrightarrow 0.

Again we know that Ext0(BP/(p,v1))=Fp[v2]\operatorname{Ext}^0(BP_*/(p, v_1)) = \mathbb {F}_p[v_2], and we can seek a v2v_2 self map v2:Σ2p22V(1)V(1)v_2: \Sigma ^{2p^2 - 2} V(1) \to V(1) that induces multiplication by v2v_2 on BPBP_*. If we can do so, then we know that δ(v2t)Ext1(BP/p)\delta (v_2^t) \in \operatorname{Ext}^1(BP_*/p) is a permanent cycle, and hence δδ(v2t)Ext2(BP)\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 δδ(v2t)0Ext2(BP)\delta \delta (v_2^t) \not= 0 \in \operatorname{Ext}^2(BP_*), since no differentials can hit it. These elements are known as βt\beta _t.

In these notes, I will construct the v1v_1 self maps for p>2p > 2 and v2v_2 self maps for p>3p > 3. It is true that the corresponding αt\alpha _t and βt\beta _t are in fact non-zero, but I will not prove it here. These maps were first constructed by Adams and Smith (for v1v_1 and v2v_2 respectively), but they had to do more work because they didn't have BPBP and the Adams–Novikov spectral sequence.

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