The Adams Spectral Sequence for Topological Modular Forms (Mathematical Surveys and Monographs, 253) - Hardcover

Robert R. Bruner; John Rognes

Hardcover

ISBN 10: 1470456745 ISBN 13: 9781470456740

Publisher: American Mathematical Society, 2021

Synopsis

The connective topological modular forms spectrum, tmf, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of tmf and several tmf-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature. Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The H∞ ring structure of the sphere and of tmf are used to determine many differentials and relations.

About this title

About the Author

Robert R. Bruner: Wayne State University, Detroit, MI and University of Oslo, Oslo, Norway, John Rognes: University of Oslo, Oslo, Norway

Edition details

• Publisher: American Mathematical Society
• Publication date: 2021
• Language: English
• ISBN 10: 1470456745
• ISBN 13: 9781470456740
• Binding: Hardcover
• Number of pages: 690