The Adams Spectral Sequence for Topological Modular Forms

ISBN-13: 9781470469580

Robert R. Bruner;John Rognes

Opis

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 $H$-infinity ring structure of the sphere and of $tmf$ are used to determine many differentials and relations.

Spis treści

• Introduction
• The Adams $E_2$-term: Minimal resolutions
• The Davis-Mahowald spectral sequence
• Ext over $A(2)$
• Ext with coefficients
• The Adams differentials: The Adams spectral sequence for $tmf$
• The Adams spectral sequence for $tmf/2$
• The Adams spectral sequence for $tmf/\nu$
• The Adams spectral sequence for $tmf/v$
• The abutment: The homotopy groups of $tmf$
• Duality
• The Adams spectral sequence for the sphere
• Homotopy of some finite cell $tmf$-modules
• Odd primes
• Calculation of $E_r(tmf)$ for $r=3,4,5$
• Calculation of $R_r(tmf/2)$ for $r=3,4,5$
• Calculation of $E_r(tmf/\nu)$ for $r=3,4$
• Calculation of $E_r(tmf/v)$ for $r=3,4,5$
• Bibliography
• Index