Document Type

Book

Abstract

The connective topological modular forms spectrum, π‘‘π‘šπ‘“, 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 π‘‘π‘šπ‘“ and several π‘‘π‘šπ‘“-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 π‘‘π‘šπ‘“ are used to determine many differentials and relations.

Disciplines

Algebraic Geometry | Geometry and Topology

Comments

This is the preliminary version of the book The Adams spectral sequence for topological modular forms, subsequently published by the American Mathematical Society and available at https://bookstore.ams.org/SURV/253. Deposited here under the terms of the author's publishing contract.

Charts and supplemental material for the final publication are available at https://www.ams.org/publications/authors/books/postpub/surv-253. Errata are published at https://www.ams.org/publications/authors/books/postpub/surv-253-errata.pdf.

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