Licensed to Univ of Rochester. Prepared on Tue Jan 12 07:38:01 EST 2021for download from IP 128.151.13.58. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms MEMOIRS of the American Mathematical Society This journal is designed particularly for long research papers (and groups of cognate papers) in pure and applied mathematics. It includes, in general, longer papers than those in the TRANSACTIONS. Mathematical papers intended for publication in the Memoirs should be addressed to one of the editors. Subjects, and the editors associated with them, follow: Real analysis (excluding harmonic analysis) and applied mathematics to ROBERT T. SEELEY, Department of Mathematics, University of Massachusetts-Boston, Harbor Campus, Boston, MA 02125. Harmonic and complex analysis to R. O. WELLS, Jr., School of Mathematics, Institute for Advanced Study, Princeton, N. J. 08540 Abstract analysis to W.A.J. LUXEMBURG, Department of Mathematics 253 -37, California Institute of Technology, Pasadena, CA 91125. Algebra and number theory (excluding universal algebras) to MICHAEL ARTIN, Department of Mathematics, Room 2—239, Massachusetts Institute of Technology, Cambridge, MA 02139. Logic, foundations, universal algebras and combinatorics to SOLOMON FEFERMAN, Department of Mathematics, Stanford University, Stanford, CA 94305. Topology to JAMES D. STASHEFF, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514. Global analysis and differential geometry to ALAN D. WEINSTEIN, Department of Mathematics, California Institute of Technology, Pasadena, CA 91125. Probability and statistics to STEVEN OREY, School of Mathematics, University of Minnesota, Minneapolis, MN 55455. All other communications to the editors should be addressed to the Managing Editor, JAMES D. STASHEFF, Department of Mathematics, University of North Carolina, Chapel Hill, NC 27514. MEMOIRS are printed by photo-offset from camera-ready copy fully prepared by the authors. Prospective authors are encouraged to request booklet giving detailed instructions regarding reproduction copy. Write to Editorial Office, American Mathematical Society, P.O. Box 6248, Providence, Rhode Island 02940. For general instructions, see inside back cover. SUBSCRIPTION INFORMATION. MEMOIRS of the American Mathematical Society is published bi­ monthly. The 1979 subscription begins with Number 210 and consists of six mailings, each containing one or more numbers. Subscription prices for 1979 are $65.00 list; $32.50 member. Each number may be ordered separately; please specify number when ordering an individual paper. For prices and titles of recently released numbers, refer to the New Publications sections of the NOTICES of the American Mathematical Society. BACK NUMBER INFORMATION. Prior to 1975 MEMOIRS was a book series; for back issues see the AMS Catalog of Publications. TRANSACTIONS of the American Mathematical Society This journal consists of shorter tracts which are of the same general character as the papers published in the MEMOIRS. The editorial committee is identical with that for the MEMOIRS so that papers intended for publication in this series should be addressed to one of the editors listed above. Subscriptions and orders for publications of the American Mathematical Society should be addressed to American Mathematical Society, P.O. Box 1571, Annex Station, Providence, R.I. 02901. All orders must be accompanied by payment. Other correspondence should be addressed to P.O. Box 6248, Providence, R.I. 02940 Copyright © 1979 American Mathematical Society Printed in the United States of America Licensed to Univ of Rochester. Prepared on Tue Jan 12 07:38:01 EST 2021for download from IP 128.151.13.58. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Memoirs of the American Mathematical Society Number 221 Victor P. Snaith Algebraic cobordism and K-theory Published by the AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island, USA September 1979 • Volume 21 • Number 221 (second of 3 numbers) Licensed to Univ of Rochester. Prepared on Tue Jan 12 07:38:01 EST 2021for download from IP 128.151.13.58. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms ABSTRACT A decomposition is given of the S-type of the classifying spaces of the classical groups. This decomposition is in terms of Thorn spaces and by means of it cobordism groups are embedded into the stable homotopy of classifying spaces. This is used to show that each of the classical cobordism theories, and also complex K-theory, is obtainable as a localisation of the stable homotopy ring of a classifying space. Similar decompositions are given for classical groups over 3F . The new construction of cobordism generalises immediately to define the algebraic cobordism of any ring. The familiar prop­ erties of cobordism are described in terms of the new formulation. Also the (p-adic) algebraic cobordism is computed for several IF -algebras and schemes. AMS (MPS) Subject Classification (1970): Primary: 55B20, 55E10, 57D75, 18F25, 55B15 Secondary: 57A70, 55G25, 14F15. Keywords and Phrases: transfer, S-type, classifying space, cobordism, K-theory, algebraic K-theory, spectrum, Pontrjagin-Thom construction, Conner-Floyd map, Landweber-Novikov operations, Adams operations, p-adic algebraic cobordism. Library of Congress Cataloging in Publication Data Snaith, Victor Percy, 19^4- Algebraic cobordism and K-theory. (Memoirs of the American Mathematical Society ; no. 221) "Volume 21 ... second of 3 numbers.,r Bibliography: p. 1. Cobordism theory. 2. K-theory. 3. Homotopy. I. Title. II. Series: American Mathematical Society. Memoirs ; no. 221. QA3-A57 no. 221 [QA612.3] 510'.8s [51^'.23] ISBN 0-8218-2221-7 79-17981 Copyright © 1979,. American Mathematical Society Licensed to Univ of Rochester. Prepared on Tue Jan 12 07:38:01 EST 2021for download from IP 128.151.13.58. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms TABLE OF CONTENTS Page Prologue - global summary of results and motivation Part I: Cobordism and the stable homotopy of classifying space 1 §1: Introduction, statement of results of Part I 1 §2: Computations with the transfer 3 §3: Factorisations of QBU(l), QB0(2) and QBSp(l) 12 §4: Stable decompositions of BU, BSp, BO and BSO 19 S 22 §5: The first connections of TT^(BG) with cobordism g §6: A computation - application to 7r^(BU(n)) 25 §7: The deviation from additivity of the real transfer and of the 30 Becker-Gottlieb solution to the real Adams conjecture §8: Stable decompositions of BGL3F and B03F„ 35 q 3 Part H: A new construction of unitary and symplectic cobordism 42 §0: Introduction, statement of results of Part II 42 §1: The homology of the stable decomposition of & \ BU(1) 44 o ° §2: AU (X), ASp(X) and their relation with cobordism 47 §3: The spectra AU and ASp 55 §4: Adams operations in MU- and AU-theory 59 §5: Idempotents in MU- and AU-theory - construction from a genus 64 and Hansenfs formula §6: The complexification homomorphism in AU- and ASp-theory 69 §7: Landweber - Novikov operations and the Thorn isomorphism 72 §8: Two descriptions of MU-bordism in terms of AU-theory 75 §9: A new identity for BU from (CP , a nilpotence result for 78 S °° TT^(CP ) and a description of the Conner-Floyd map as the determinant (iii) Licensed to Univ of Rochester. Prepared on Tue Jan 12 07:38:01 EST 2021for download from IP 128.151.13.58. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms Part HI: Unoriented cobordism, algebraic cobordism and the 85 X(b)-spectrum §0: Introduction, statement of results of Part IE 85 §1: The spectrum X(b) - examples including cobordism, K-Theory 86 and algebraic cobordism S §2: The stable homotopy of BO, some new elements in TT^(BO) 94 §3: Unoriented cobordism as an X(b)-spectrum 100 §4: New elements in Tr^(ImJ) 107 §5: The algebraic cobordism of Z and its epimorphism onto TT^(MO) 113 Part IV: Algebraic cobordism and geometry 120 §0: Epilogue, statement of results and the advocation of a 120 cobordismic viewpoint §1: Algebraic vector bundles over number fields, a remark on a 121 problem of Atiyah §2: An analogue of the Pontrjagin-Thom construction on the etale 124 site, with examples §3: Some computations of the p-adic algebraic cobordism of schemes 128 over Spec IF q §4: Units, p-adic cobordism of IF -algebras and their Quillen 133 K-theory q §5: Twelve problems arising from Parts I - IV 141 §6: Bibliography 144 §7: Footnotes 150 (iv) Licensed to Univ of Rochester. Prepared on Tue Jan 12 07:38:01 EST 2021for download from IP 128.151.13.58. License or copyright restrictions may apply to redistribution; see https://www.ams.org/publications/ebooks/terms PROLOGUE Each of the four parts of this paper has its own introduction. In this prologue I wish, therefore, to describe the global mathematical and philoso­ phical theme which will be pursued in this paper. We seek simultaneously to satisfy the following motivating demands. (i) To find invariants for use in algebraic geometry which are at least as powerful as (and related to) Quillen K-theory. (ii) To achieve (i) within the framework of stable homotopy theory. (iii) To achieve (i) by a method which recogniseably generalises classical cobordism theories. A few remarks on (i)-(iii) are in order. Recall first that the Chow ring of a smooth variety A*(X) may be obtained from algebraic K-theory as the sheaf cohomology # H (X;K ) [Q4]. This recommends K-theory and its more powerful relatives as a source of suitable invariants to study. Secondly the computational machinery available in stable homotopy theory is superior to that of ordinary homotopy theory, whence (ii). Also in the topologists* natural area of geometry - manifold theory - cobordism theories have been very important invariants, whence (iii).