The Adams spectral sequence for topological modular forms
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Bibliographic Information
The Adams spectral sequence for topological modular forms
（Mathematical surveys and monographs, v. 253）
American Mathematical Society, c2021
- : softcover
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Description based on reprint 2022
Includes bibliographical references (p. 675-682) and index
Description and Table of Contents
Description
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.
Table of Contents
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
by "Nielsen BookData"
