Topological Modular Forms ( Mathematical Surveys and Monographs )

Publication series ：Mathematical Surveys and Monographs

Author： Christopher L. Douglas;John Francis;André G. Henriques

Publisher： American Mathematical Society‎

Publication year： 2014

E-ISBN: 9781470420024

P-ISBN(Paperback): 9781470418847

Subject： O189.1 general topology

Keyword： Geometry and Topology

Language： ENG

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Topological Modular Forms

Description

The theory of topological modular forms is an intricate blend of classical algebraic modular forms and stable homotopy groups of spheres. The construction of this theory combines an algebro-geometric perspective on elliptic curves over finite fields with techniques from algebraic topology, particularly stable homotopy theory. It has applications to and connections with manifold topology, number theory, and string theory. This book provides a careful, accessible introduction to topological modular forms. After a brief history and an extended overview of the subject, the book proper commences with an exposition of classical aspects of elliptic cohomology, including background material on elliptic curves and modular forms, a description of the moduli stack of elliptic curves, an explanation of the exact functor theorem for constructing cohomology theories, and an exploration of sheaves in stable homotopy theory. There follows a treatment of more specialized topics, including localization of spectra, the deformation theory of formal groups, and Goerss–Hopkins obstruction theory for multiplicative structures on spectra. The book then proceeds to more advanced material, including discussions of the string orientation, the sheaf of spectra on the moduli stack of elliptic curves, the homotopy of topological modular forms, and an extensive account of the construction of the spectrum of topological modular forms. The book concludes with the three original, pioneering and enormously i

Chapter

Cover

Title page

Contents

Preface and Acknowledgements

Introduction

Chapter 1. Elliptic genera and elliptic cohomology

Chapter 2. Ellliptic curves and modular forms

Chapter 3. The moduli stack of elliptic curves

Chapter 4. The Landweber exact functor theorem

Chapter 5. Sheaves in homotopy theory

Chapter 6. Bousfield localization and the Hasse square

Chapter 7. The local structure of the moduli stack of formal groups

Chapter 8. Goerss–Hopkins obstruction theory

Chapter 9. From spectra to stacks

Chapter 10. The string orientation

Chapter 11. The sheaf of 𝐸_{∞}-ring spectra

Chapter 12. The construction of tmf

Chapter 13. The homotopy groups of tmf and of its localizations

Ellitpic curves and stable homotopy I

From elliptic curves to homotopy theory

𝐾(1)-local 𝐸_{∞}-ring spectra

Glossary

Back Cover

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