The \(\mathsf{P}^1_2\) margolis homology of connective topological modular forms (Q2240621)

The Steenrod algebra \(\mathcal{A}\) at \(p=2\) contains the Adams-Margolis elements \(P^s_t\) for \(s<t\), which are dual to \(\xi_t^{2^s}\). They square to zero. Thus they act on any \(\mathcal{A}\)-module \(M\) as a differential and one can consider the Margolis homology \(H^*(M; P^s_t)\) with respect to this differential. Margolis homology has many classic applications. For example, their vanishing implies a vanishing line of a corresponding Adams spectral sequence, which was used by Hopkins and Smith to prove their periodicity theorem. Moreover, an \(\mathcal{A}(n)\)-module \(M\) is free if and only if the \(P^s_t\)-Margolis homologies vanish for \(s+t \leq n+1\). This was used by Lellmann and Mahowald for their study of the \(bo\)-based Adams spectral sequence; here it was necessary to identify free \(\mathcal{A}(1)\)-summands in the cohomology of \(bo^{\wedge r}\). Of all Adams-Margolis elements, the \(P^0_t = Q_{t-1}\) are easiest to study since they are primitive. These suffice for the detection of free \(\mathcal{A}(1)\)-modules, but not of free \(\mathcal{A}(2)\)-modules. Recently, there has been an effort to extend the successes of the \(bo\)-based Adams spectral sequence to one height higher, in particular with an eye on the telescope conjecture. Here, one replaces \(bo\) by the spectrum \(tmf\) of connective topological modular forms. Since \(H^*(tmf; \mathbb{F}_2) \cong \mathcal{A} // \mathcal{A}(2)\), it becomes important for that purpose to identify free \(\mathcal{A}(2)\)-summands in \(M = H^*(tmf^{\wedge r}; \mathbb{F}_2)\) and in particular to compute the Margolis homology of \(M\) with respect to all \(P^s_t\) for \(s+t \leq 3\). In the case of \(s=0\), the computation is rather straightforward. The main result of the article under review is the computation of \(H^*(M; P^1_2)\). Other potential applications (than to the \(tmf\)-based Adams spectral sequence) of this computation are discussed as well. The main technique is to define a length filtration on the homology of \(tmf^{\wedge r}\) and consider the associated spectral sequence. Maybe surprisingly, the authors obtain as a result that \(H^*(tmf; \mathbb{F}_2)\) and \(H^*(tmf^{\wedge r}; \mathbb{F}_2)\) have isomorphic \(P^1_2\)-Margolis homologies. The techniques are applicable to the computation of \(P^1_2\)-Margolis homology of other \(\mathcal{A}\)-modules as well, as the authors demonstrate in the case of \(H^*((\mathbb{RP}^{\infty})^{\times r}; \mathbb{F}_2)\).