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Noncommutative Localization in Algebra and Topology

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Noncommutative Localization in Algebra and Topology Noncommutative localization in algebra and topology ICMS Edinburgh 2002 Edited by Andrew Ranicki Electronic version of London Mathematical Society Lecture Note Series 330 Cambridge University Press (2006) Contents Dedication . vii Preface . ix Historical Perspective . x Conference Participants . xi Conference Photo . .xii Conference Timetable . xiii On atness and the Ore condition J. A. Beachy ......................................................1 Localization in general rings, a historical survey P. M. Cohn .......................................................5 Noncommutative localization in homotopy theory W. G. Dwyer . 24 Noncommutative localization in group rings P. A. Linnell . 40 A non-commutative generalisation of Thomason's localisation theorem A. Neeman . 60 Noncommutative localization in topology A. A. Ranicki . 81 v L2-Betti numbers, Isomorphism Conjectures and Noncommutative Lo- calization H. Reich . 103 Invariants of boundary link cobordism II. The Blanch¯eld-Duval form D. Sheiham . 143 Noncommutative localization in noncommutative geometry Z. Skoda· ........................................................220 vi Dedicated to the memory of Desmond Sheiham (13th November 1974 ¡ 25th March 2005) ² Cambridge University (Trinity College), 1993{1997 B.A. Hons. Mathematics 1st Class, 1996 Part III Mathematics, Passed with Distinction, 1997 ² University of Edinburgh, 1997{2001 Ph.D. Invariants of Boundary Link Cobordism, 2001 ² Visiting Assistant Professor, Mathematics Department, University of California at Riverside, 2001{2003 ² Research Instructor, International University Bremen (IUB), 2003{2005 vii Publications: 1. Non-commutative Characteristic Polynomials and Cohn Localization Journal of the London Mathematical Society (2) Vol. 64, 13{28 (2001) http://arXiv.org/abs/math.RA/0104158 2. Invariants of Boundary Link Cobordism Memoirs of the American Mathematical Society, Vol. 165 (2003) http://arXiv.org/abs/math.AT/0110249 3. Whitehead Groups of Localizations and the Endomorphism Class Group Journal of Algebra, Vol. 270, 261{280 (2003) http://arXiv.org/abs/math.KT/0209311 4. Invariants of Boundary Link Cobordism II. The Blanch¯eld-Duval Form in this volume, pages 142{218, http://arXiv.org/abs/math.AT/0404229 5. Universal localization of triangular matrix rings to appear in the Proceedings of the American Mathematical Society http://arXiv.org/abs/math.RA/0407497 6. (with A.R.) Blanch¯eld and Seifert algebra in high-dimensional bound- ary links I. Algebraic K-theory, http://arXiv.org/abs/math.AT/0508405 I was Des's Ph.D. supervisor at the University of Edinburgh. I prefer the German name \Doktorvater", and Des was in fact very much a mathemati- cal son to me. We quickly found that we shared the same passion for math- ematics. I suggested that he work on the computation of high-dimensional boundary link cobordism. Des's Ph.D. thesis was a complete solution of this 30-year old problem. The solution was beautiful, clever and original { as indeed was Des himself. Des and I kept in close touch in the years after he left Edinburgh, and the relationship naturally evolved into one of mathematical brothers and collabo- rators. Des was not only a wonderful student but also a wonderful teacher. I certainly learned a lot from him myself! The students of his topology course at IUB were especially inuenced by his teaching. Des made such an im- pression at IUB that he was invited to apply for a more permanent position there. In my letter of recommendation I wrote: \Dr. Sheiham is committed to both teaching and research in mathematics. He is a highly talented young mathematician who has a bright future ahead of him." It is a tragedy that he denied himself this future. A.R. viii Preface Noncommutative localization is a powerful algebraic technique for construct- ing new rings by inverting elements, matrices and more generally morphisms of modules. The applications to topology are via the noncommutative lo- calizations of the fundamental group rings. The volume is the proceedings of a workshop on `Noncommutative lo- calization in algebra and topology' held at the International Centre for the Mathematical Sciences in Edinburgh on April 29 and 30, 2002, with 25 participants. The collection could be used as an introduction to noncom- mutative localization and its applications, but it is not an encyclopedia. Neither is it just a record of the talks at the meeting: the papers submitted by the participants are much more extensive than their talks, and in addi- tion the volume contains papers commissioned from non-participants. The `Historical Perspective' overleaf sets the papers in a historical context. The meeting and the proceedings have the aim of bringing together the algebraists and topologists interested in noncommutative localization. I was particularly pleased that Professor Paul Cohn attended the meeting and contributed to the proceedings. He invented the universal `Cohn noncom- mutative localization' technique which has turned out to be so useful both in its original algebraic setting and in its applications to topology. The meeting was supported by a Scheme 1 Grant of the London Mathe- matical Society, and was an activity of the European Union TMR Network ERB FMRX CT-97-0107 `Algebraic K-Theory, Linear Algebraic Groups and Related Structures'. The papers in the proceedings were refereed individ- ually. I am grateful to all the participants, to the speakers, to the authors of the papers, to the referees, to the LMS and the EU, and to the sta® at ICMS, for their contributions to the success of both the meeting and the proceedings. Additional material (such as errata) will be posted on http://www.maths.ed.ac.uk/~aar/books Andrew Ranicki Edinburgh, July 2005 ix Historical Perspective The localization of commutative rings is a classic technique of commuta- tive algebra, notably in the construction of the quotient ¯eld of an integral domain. The localization of noncommutative rings was pioneered by Ore in 1931, who introduced the `Ore condition' which allows noncommutative fractions. However, the ¯rst general reference to noncommutative localiza- tion in Mathematical Reviews only dates back to 1967; since then, the topic has grown in importance, and by now there are 525 references in total. The history of noncommutative localization in algebra is outlined in the article of Cohn in this volume, including both the Ore localization and the more general universal localization technique he himself introduced in 1971. The article of Beachy characterizes Ore localizations as at universal localizations. In the last 10 years the universal localization of group rings has been used to investigate the rationality of the L2-Betti numbers and the Isomorphism Conjecture in algebraic K-theory, as described in the articles of Linnell and Reich. Commutative localization entered algebraic topology with Serre's funda- mental contributions to homotopy theory in the early 1950's. Some 20 years later the work of Sullivan, Quillen, Kan and Bous¯eld developed the local- ization of spaces, which is now a standard method. The article of Dwyer gives a homotopy-theoretic interpretation of the Cohn universal localization, making a direct link between localization in algebra and topology. In the last 20 years noncommutative localization has been applied to the topology of manifolds via the Cappell-Shaneson homology version (1974) of the Browder-Novikov-Sullivan-Wall surgery theory (1962-1970), as well as to the circle-valued Morse-Novikov theory, and to codimension 1 splitting obstruction theory. The article of Ranicki surveys these applications. The article of Sheiham uses noncommutative localization to give a new interpre- tation of the high-dimensional boundary link cobordism invariants obtained in his 2001 Edinburgh Ph.D. thesis. Noncommutative localization is closely related to the quotient construc- tion of categories, particularly the Verdier quotient of triangulated categories developed in the 1960's. The article of Neeman makes this relation quite explicit, and also explains the connections with algebraic geometry via the work of Thomason. The Ore noncommutative localization is a useful tool in the recent devel- opment of noncommutative geometry, as described in the article of Skoda.· x Conference Participants 1. Pere Ara (Universitat Autonoma de Barcelona) 2. Andrew Baker (Glasgow University) 3. John Beachy (Northern Illinois University) 4. Tom Bridgeland (University of Edinburgh) 5. Jeremy Brookman (University of Edinburgh) 6. Ken Brown (Glasgow University) 7. Paul Cohn (University College London) 8. Diarmuid Crowley (Max Planck Institute, Bonn) 9. Matyas Domokos (University of Edinburgh) 10. Eivind Eriksen (University of Warwick) 11. Ben Franklin (Bristol University) 12. Michael Hudson (Bristol University) 13. Thomas Huettemann (Aberdeen University) 14. Colin Ingalls (University of Warwick) 15. Olav Arn¯nn Laudal (University of Oslo) 16. Tom Lenagan (University of Edinburgh) 17. Amnon Neeman (Australian National University) 18. Francesc Perera (Queen's University Belfast) 19. Andrew Ranicki (University of Edinburgh) 20. Holger Reich (UniversitÄatMÄunster) 21. Jeremy Rickard (Bristol University) 22. Aidan Scho¯eld (Bristol University) 23. Des Sheiham (University of California, Riverside) 24. James Shepherd (Bristol University) 25. Marco Varisco (UniversitÄatMÄunster) xi Conference Photo ICMS, 14 India Street, Edinburgh Bridgeland, Huettemann, Varisco, Crowley Ara, Domokos, Laudal, Eriksen, Brown, Beachy, Shepherd Perera, Neeman, Ranicki, Cohn, Lenagan,
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