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OverviewThe 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. Full Product DetailsAuthor: Robert R. Bruner , John RognesPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.633kg ISBN: 9781470469580ISBN 10: 1470469588 Pages: 690 Publication Date: 30 April 2022 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: In Print  This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us. Table of ContentsIntroduction 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 IndexReviewsAuthor InformationRobert R. Bruner, Wayne State University, Detroit, MI, and University of Oslo, Norway. John Rognes, University of Oslo, Norway. Tab Content 6Author Website:Countries AvailableAll regions |
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