On the double transfer and the \(f\)-invariant (Q2902678)

The classical \(S^1\)-transfer \(\tau\) is a stable map from \(\mathbb{C}\mathbb{P}^{\infty}_+\) to \(\mathbb{S}_{-1}\) and has extensively been studied in stable homotopy theory. The double \(S^1\)-transfer which is investigated in the paper under review simply is the smash product \(\tau \wedge \tau\). While the classical \(S^1\)-transfer has been used to detect classes in the \(1\)-line of the Adams-Novikov spectral sequence for the sphere spectrum (e.g. see [\textit{H. Miller}, in: Current trends in algebraic topology, Semin. London/Ont. 1981, CMS Conf. Proc. 2, 2, 437--449 (1982; Zbl 0552.55007)], Theorem 4.6.) it is not clear yet what the precise image of the double \(S^1\)-transfer on the \(2\)-line of the Adams-Novikov spectral sequence for the sphere spectrum corresponds to. On the level of \(MU_*MU\)-comodules the double \(S^1\)-transfer also yields a class in \(\mathrm{Ext}^2_{MU*MU}(MU_*(\mathbb{C}\mathbb{P}^{\infty}_+ \wedge \mathbb{C}\mathbb{P}^{\infty}_+), MU_{*-4})\), so that taking the Yoneda product with this particular class yields a homomorphism \(\varphi\) from \(\Hom_{MU_*MU}(MU_*, MU_*(\mathbb{C}\mathbb{P}^{\infty}_+ \wedge \mathbb{C}\mathbb{P}^{\infty}_+))\) to the \(2\)-line of the Adams-Novikov spectral sequence for the sphere spectrum. Note that the domain of the homomorphism is just the group of \(MU_*MU\)-primitives in \(MU_*(\mathbb{C}\mathbb{P}^{\infty}_+ \wedge \mathbb{C}\mathbb{P}^{\infty}_+)\), and that the stably spherical classes in \(MU_*(\mathbb{C}\mathbb{P}^{\infty}_+ \wedge \mathbb{C}\mathbb{P}^{\infty}_+)\) map to these primitive elements. Hence the study of the homomorphism \(\varphi\) yields some partial information about what classes in the \(2\)-line of the Adams-Novikov spectral sequence of the sphere can be detected by the image of the double \(S^1\)-transfer.NEWLINENEWLINE In the paper under review, the author now provides for any odd prime \(p\) and any \(MU_*MU\)-primitive \(x\in MU_*(\mathbb{C}\mathbb{P}^{\infty}_+ \wedge \mathbb{C}\mathbb{P}^{\infty}_+)\) an explicit formula for the \(p\)-primary part of its image \(\varphi(x) \in \mathrm{Ext}^2_{MU*MU}(MU_*, MU_{*-4})\). The formulas achieved are given in terms of so-called generalized Bernoulli numbers. Generalized Bernoulli numbers can be defined for any formal group law over a torsion-free commutative ring and they were first introduced in [\textit{H. Miller}, ibid.] (Definition 1.1) in order to detect the elements on the \(1\)-line of the Adams-Novikov of the sphere spectrum which correpond to classes in the image the classical \(S^1\)-transfer. The ring used here of course is \(MU_*\) and the corresponding formal group law is the universal one. Using change of rings from \(MU_*\) to \(KU_*\) and \(TMF[1/6]\) respectively the new formulas also yield explicit formulas for the \(f\)-invariant (in the sense of [\textit{G. Laures}, Topology 38, No.2, 387--425 (1999; Zbl 0924.55004)]) and the \(f'\)-invariant (in the sense of [\textit{M.~Behrens}, Geom. Topol. 13, No. 1, 319-357 (2009; Zbl 1205.55012)]) in terms of the generalized Bernoulli numbers of \(KU\)- and \(TMF[1/6]\)-theory.