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Hardcover ISBN:  9781470456740 
eBook: ISBN:  9781470465636 
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Hardcover ISBN:  9781470456740 
Product Code:  SURV/253 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470469580 
Product Code:  SURV/253.S 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
eBook ISBN:  9781470465636 
Product Code:  SURV/253.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Softcover ISBN:  9781470469580 
eBook ISBN:  9781470465636 
Product Code:  SURV/253.S.B 
List Price:  $250.00$187.50 
MAA Member Price:  $225.00$168.75 
AMS Member Price:  $200.00$150.00 
Hardcover ISBN:  9781470456740 
eBook ISBN:  9781470465636 
Product Code:  SURV/253.B 
List Price:  $250.00$187.50 
MAA Member Price:  $225.00$168.75 
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Book DetailsMathematical Surveys and MonographsVolume: 253; 2021; 690 ppMSC: Primary 18; 55;
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 BrownComenetz 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 \(H_{\infty}\) ring structure of the sphere and of \(tmf\) are used to determine many differentials and relations.ReadershipGraduate students and researchers interested in algebraic topology, specifically in stable homotopy theory.

Table of Contents

Chapters

Introduction

The Adams $E_2$term

Minimal resolutions

The DavisMahowald spectral sequence

Ext over $A(2)$

Ext with coefficients

The Adams differentials

The Adams spectral sequence for $tm\!f$

The Adams spectral sequence for $tm\!f/2$

The Adams spectral sequence for $tm\!f/\eta $

The Adams spectral sequence for $tm\!f/\nu $

The abutment

The homotopy groups of $tm\!f$

Duality

The Adams spectral sequence for the sphere

Homotopy of some finite cell $tm\!f$modules

Odd primes

Appendices

Calculation of $E_r(tm\!f)$ for $r=3,4,5$

Calculation of $E_r(tm\!f/2)$ for $r=3,4,5$

Calculation of $E_r(tm\!f/\eta )$ for $r=3,4$

Calculation of $E_r(tm\!f/\nu )$ for $r=3,4,5$


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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 BrownComenetz 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 \(H_{\infty}\) ring structure of the sphere and of \(tmf\) are used to determine many differentials and relations.
Graduate students and researchers interested in algebraic topology, specifically in stable homotopy theory.

Chapters

Introduction

The Adams $E_2$term

Minimal resolutions

The DavisMahowald spectral sequence

Ext over $A(2)$

Ext with coefficients

The Adams differentials

The Adams spectral sequence for $tm\!f$

The Adams spectral sequence for $tm\!f/2$

The Adams spectral sequence for $tm\!f/\eta $

The Adams spectral sequence for $tm\!f/\nu $

The abutment

The homotopy groups of $tm\!f$

Duality

The Adams spectral sequence for the sphere

Homotopy of some finite cell $tm\!f$modules

Odd primes

Appendices

Calculation of $E_r(tm\!f)$ for $r=3,4,5$

Calculation of $E_r(tm\!f/2)$ for $r=3,4,5$

Calculation of $E_r(tm\!f/\eta )$ for $r=3,4$

Calculation of $E_r(tm\!f/\nu )$ for $r=3,4,5$