Homotopy groups of generalized \(E(2)\)-local Moore spectra at the prime three (Q2488030)

A spectrum \(X\) in the \(E(n)\)-local category \({\mathcal L}_n\) is called invertible if there is a local spectrum \(Y\) with \(X\wedge Y=L_nS\). These spectra form the Picard group \(\text{Pic} ({\mathcal L}_n)\). Let \(\text{Pic} ({\mathcal L}_n)^0\) be the subgroup of invertible spectra with \(H \mathbb Q_0 (X) \cong \mathbb Q\). Then it holds that \[ \text{Pic} {\mathcal (L}_n) = \text{Pic} ({\mathcal L}_n)^0 \oplus Z. \] In [J. Lond. Math. Soc. 60, 284-302 (1999; Zbl 0947.55013)], \textit{M. Hovey} and \textit{H. Sadofsky} showed that \(X\) lies in \(\text{Pic} ({\mathcal L}_n)^ 0\) if and only if \[ E(n)_* (X) \cong E(n)_* (S) \] as an \(E(n)_* E(n)\) comodule. A generalized \(k\)-th Smith-Toda spectrum is a \(E(n)\)-local spectrum \(X\) with \[ E(n)_* X=E(n)_*/ {(p, v_1 ,\dots , v_k)} \] as an \(E(n)_* E(n)\)-comodule. It is known that the localization of the classical Smith-Toda spectrum \(V(k)\) is the only generalized one for \(n^2 +n<2p.\) In particular, for \(p=3\) the first open case for a generalized mod 3 Moore spectrum appears for \(n=2.\) This is the case considered in the paper. The authors consider the spectra \(L_2 V_0, L_2 V_1 , L_2 V_5 , WP\) (and eventually \(WQ, WQ^2\)), show that all of these are different generalized Moore spectra and compute their homotopy groups. Moreover, they conjecture that this is a complete list of generators over the Picard group \(\text{Pic} ({\mathcal L}_2)^0.\) The main technical tool to show their result is the \(E(n)\)-based Adams-Novikov spectral sequence. For a strictly invertible spectrum \(X\) the \(E_2\)-term of the spectral sequence coincides with the one for the sphere. By [\textit{Y. Kamiya} and \textit{K. Shimomura}, Contemp. Math. 346, 321--333 (2004; Zbl 1097.55007)], an invertible spectrum is characterized by the value of the differential \(d_5\) on the generator \(g\) in \(E^{0,0} (X).\) The spectrum \(P\) of the list comes with a map \[ P\longrightarrow \Sigma^{-21} L_2 V (1\frac{1}{2}) \] which realizes the projection \(E(2)_* \longrightarrow E(2)_*/3\) and is characterized by \(d_5 (g) = v^{-1}_2 h_{11} b^2_{10}.\) The generalized Moore spectrum \(WP^k\) is part of a cofiber sequence \[ S \longrightarrow P^k \longrightarrow WP^k. \] Finally, the spectrum \(L_2 V_i\) is up to suspension given by the cofiber of a map \[ B^{(i)} :\Sigma^{16i} V(0) \longrightarrow V(1) \] which induces \[ v^i_2 :BP_*/3 \longrightarrow BP_*/{(3, v_1)}. \] In particular, \(V_0=V(0)\) and \(V_1 =\Sigma^{-21} V(1\frac{1}{2}).\)