The Adams Spectral Sequence for Topological Modular Forms

The connective topological modular forms spectrum, $tmf$, is an essential bridge between the homotopy groups of spheres and modular forms. This volume provides a complete account of the homotopy of $tmf$ and several $tmf$-module spectra using the classical Adams spectral sequence, verifying, correcting, and extending existing approaches. It also includes an account of the homotopy groups of spheres through degree 44, with complete proofs except for the Adams conjecture. Techniques from commutative algebra are used to make the calculations precise and finite.

Format: Paperback / softback
Length: 690 pages
Publication date: 30 March 2022
Publisher: American Mathematical Society

The connective topological modular forms spectrum, $TMF$, holds a significant position among elliptic spectra, serving as a vital bridge between the homotopy groups of spheres and modular forms. This volume aims to provide a comprehensive account, accompanied by thorough proofs, of the homotopy of $TMF$ and several $TMF$-module spectra through the classical Adams spectral sequence. By doing so, it seeks to verify, correct, and extend existing approaches while also clarifying and generalizing folklore results.

The book delves into the intricate details of Anderson and Brown-Comenetz duality, along with its corresponding dualities in homotopy groups. It meticulously proves these dualities, contributing to the understanding of these concepts. Furthermore, the volume encompasses an account of the homotopy groups of spheres up to degree 44, offering complete proofs, except for the Adams conjecture, which is employed without proof. Additionally, the book presents modern stable proofs of classical results that can be challenging to extract from the literature.

To accomplish its goals, this book employs a multiplicative spectral sequence, which generalizes a construction by Davis and Mahowald. Furthermore, computer software is utilized to compute the cohomology of modules over the Steenrod algebra and its products. Commutative algebra techniques are employed to ensure precise and finite calculations. The $H$-infinity ring structure of the sphere and $TMF$ is employed to derive various differentials and relations.

In conclusion, this book offers a comprehensive exploration of the homotopy of $TMF$ and its module spectra, employing advanced techniques and providing thorough proofs. It serves as a valuable resource for researchers in the field of mathematics, contributing to our understanding of these complex mathematical structures.

ISBN-13: 9781470469580