On the double transfer and the f-invariant

TL;DR

This paper explores the algebraic double transfer in the Adams-Novikov spectral sequence, analyzing classes via chromatic factorization and the f-invariant at odd primes, revealing new structural insights.

Contribution

It introduces an algebraic perspective on the double transfer, connecting it with the f-invariant and chromatic methods for the first time.

Findings

Identification of classes in the Adams-Novikov spectral sequence via algebraic double transfer.

Application of chromatic factorization and f'-invariant for p≥5.

Use of Laures' f-invariant to analyze algebraic double transfer classes.

Abstract

The purpose of this paper is to investigate an algebraic version of the double complex transfer, in particular the classes in the two-line of the Adams-Novikov spectral sequence which are the image of comodule primitives of the MU-homology of the product of two copies of infinite complex projective space via the algebraic double transfer. These classes are analysed by two related approaches; the first, p-locally for an odd prime, by using the morphism induced in MU-homology by the chromatic factorization of the double transfer map together with the f'-invariant of Behrens (for p>=5). The second approach uses the algebraic double transfer and the f-invariant of Laures.