2 Construction of v1 self maps
We wish to construct a map Σ2p−2V(0)→V(0) inducing multiplication by v1 on BP∗ homology. The strategy is to construct a map S2p−2→V(0) that induces multiplication by v1 on BP∗ homology, and then extend it to a map Σ2p−2V(0)→V(0) by obstruction theory.
First consider the BP Adams–Novikov spectral sequence for V(0). In degrees up to 2p−2, the spectral sequence looks like
where v1∈(2p−2,0) and t1∈(2p−3,1). If p=2, then we have an extra t12 which will be right above v1.
In either case, we see that there is no room for extra differentials. So we see that
Lemma
There is a map v~1:S2p−2→V(0) that induces multiplication by v1 on BP∗. If p>2, then this map has order p and is unique.
Since
V(0)=S/p, the map
v~1 having order
p is the same as it extending to a map
Σ2p−2V(0). Thus, we deduce that
Theorem
If p>2, then there is a map v1:Σ2p−2V(0)→V(0) that induces multiplication by v1 on BP∗.
In the case p=2, we know π2V(0)=Z/4Z or Z/2⊕Z/2. If it is Z/4Z, then this map has order 4 and does not lift to a map Σ2V(0)→V(0). This is indeed the case, as one can check using the HF2 Adams spectral sequence, so we do not have a v1 self map at p=2.