Construction of v1v_1 and v2v_2 self-maps — Construction of v1v_1 self maps

2 Construction of v1v_1 self maps

We wish to construct a map Σ2p2V(0)V(0)\Sigma ^{2p - 2} V(0) \to V(0) inducing multiplication by v1v_1 on BPBP_* homology. The strategy is to construct a map S2p2V(0)\mathbb {S}^{2p - 2} \to V(0) that induces multiplication by v1v_1 on BPBP_* homology, and then extend it to a map Σ2p2V(0)V(0)\Sigma ^{2p - 2}V(0) \to V(0) by obstruction theory.

First consider the BPBP Adams–Novikov spectral sequence for V(0)V(0). In degrees up to 2p22p - 2, the spectral sequence looks like

\begin{tikzpicture} 
    \draw [->] (0, 0) -- (5, 0) node [right] {$t - s$};
    \draw [->] (0, 0) -- (0, 3) node [above] {$s$};

    \node [circ] at (0, 0) {};
    \node [below] at (0, 0) {$1$};
    
    \node [opacity = 0, fill, circle, inner sep = 0, minimum size = 9] (v1) at (3.5, 0) {}; \node [circ] at (3.5, 0) {};
    \node [below] at (3.5, 0) {$v_1$};

    \node [circ] at (2.8, 0.7) {};
    \node [left] at (2.8, 0.7) {$t_1$};

    \draw [red, ->] (v1) -- (2.8, 1.4) node [pos=0.5, right] {$d_2$};
  \end{tikzpicture}

where v1(2p2,0)v_1 \in (2p - 2, 0) and t1(2p3,1)t_1 \in (2p - 3, 1). If p=2p = 2, then we have an extra t12t_1^2 which will be right above v1v_1.

In either case, we see that there is no room for extra differentials. So we see that

Lemma

There is a map v~1:S2p2V(0)\tilde{v}_1: \mathbb {S}^{2p - 2} \to V(0) that induces multiplication by v1v_1 on BPBP_*. If p>2p > 2, then this map has order pp and is unique.

Since V(0)=S/pV(0) = \mathbb {S}/p, the map v~1\tilde{v}_1 having order pp is the same as it extending to a map Σ2p2V(0)\Sigma ^{2p - 2} V(0). Thus, we deduce that
Theorem

If p>2p > 2, then there is 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_*.

In the case p=2p = 2, we know π2V(0)=Z/4Z\pi _2 V(0) = \mathbb {Z}/4\mathbb {Z} or Z/2Z/2\mathbb {Z}/2 \oplus \mathbb {Z}/2. If it is Z/4Z\mathbb {Z}/4\mathbb {Z}, then this map has order 44 and does not lift to a map Σ2V(0)V(0)\Sigma ^2 V(0) \to V(0). This is indeed the case, as one can check using the HF2H\mathbb {F}_2 Adams spectral sequence, so we do not have a v1v_1 self map at p=2p = 2.