ATMW Classification of Manifolds (2013)
|Dates:||17th to 26th Jun|
|Convener(s)||Speakers, Syllabus and Time table||Applicants/Participants|
|Name||Himadri Kumar Mukerjee|
Department of Mathematics,
Classification of manifolds is a very old and natural problem in topology. Classification of one and two dimensional manifolds up to diffeomorphism has been done more than a century ago. The basic technique underlying these works have been “surgery” (cutting and gluing). The same technique have been extended to attack classification problems of manifolds of dimension 3, 4 and > 4. In dimension 3 and 4 one could achieve only a very little owing to the limitations of algebraic topological tools developed in the late twentieth century. Bigger success in these dimensions had to rely heavily on geometry and analysis. I dimension > 4 complete classification is ruled out owing to unsolvability of word problem, consequently classification of manifolds within a homotopy type have been tried. Surgery theory had a great success in this domain as exemplified by works of Milnor, Kervaire, Browder, Novikov, Wall, Kirby, Siebenmann, and numerous other very renowned mathematicians.
The purpose of the ATM workshop/conference on surgery theory and classification of manifolds would be to expose the participants to the basic techniques of surgery and related works leading to the surgery exact sequence for the classification of manifolds of dimension > 4 within a homotopy type, together with many applications, in a condensed and yet lucid fashion, help participants in digesting the topics through tutorials, and to familiarize them with the current problems through the papers to be presented in the applications and conference section.
|1||James F. Davis||Indiana University Bloomington, USA|
|2||L.C. Siebenmann||Université de Paris-sud, Centre d’Orsay, France|
|3||Parameswaran Sankaran||IMSC, Chennai, India|
|4||S. K. Roushon||TIFR, Mumbai, India|
|5||C.S. Aravinda||TIFR, Bangalore, India|
|6||Goutam Mukherjee||Indian Statistical Institute, Kolkata, India|
|7||Samik Basu||Rankrishna Mission Vivekananda University, Howrah, India|
|8||Himadri Kumar Mukerjee||NEHU, Shillong, India|
|9||S.S. Khare||Dept. of Maths, NEHU, Shillong|
Syllabus (Some addition/subtraction may be made in the topics)
|Topic 1 (TNB)||1 (75 minutes)||Tangent bundle, immersion and embedding, normal bundle.|
|Topic 2 (FR)||1 (75 m)||Homotopy classification of vector bundles, framing of a normal bundle,
parallelizable and s-parallelizable manifolds.
|Topic 3 (HM)||1 (75 m)||Handle decompositions and Morse theory|
|Topic 4 (HS)||1 (75 m)||h-cobordism and s-cobordism theorems|
|Topic 5 (SBMD)||2 (75 m each)||Surgery below the middle dimension|
|Topic 6 (SMD0)||2 (75 m each)||Surgery below the middle dimension|
|Topic 7 (SMD2)||2 (75 m each)||Surgery in the middle dimension of a 4k-manifold|
|Topic 8 (ALTS)||2 (75 m each)||Surgery in the middle dimension of a (4k+2)-manifold|
|Topic 9 (SES)||2 (75 m each)||L-groups and the algebraic theory of surgery|
|Topic 10 (ES)||1 (75 m)||Surgery exact sequence|
|Topic 11 (CSSAM)||1 (75 m)||Exotic spheres|
|Topic 12 (CLSH)||1 (75 m)||Classifying homotopy complex projective spaces and homotopy lens spaces|
|Topic 13 (BNC)||1 (75 m)||The Borel and Novikov Conjectures|
|Topic 14 (STM)||1 (75 m)||Splitting theorems in manifold theory|
|Topic 15 (FJC-1)||1 (75 m)||The Farrell-Jones Conjectures|
|Topic 16 (FJC-2))||1 (75 m)||Applications of the Farrell-Jones Conjecture|
|Topic 17 (TUPS)||12 (75 m)||Tutorials and problem sessions|
|Topic 18 (CFTK)||12 (435 m total)||Conference talks|
Time-Table (First week)
| Day, Date &
|9.00 - 10.15||TNB||HS||SMD0-1||SMD2-2||SES-1||CSSAM|
|10.15 - 10.30||Tea/Coffee|
|10.30 - 11.45||FR||SBMD-1||SMD0-1||ALTS-1||SES -2||TUPS|
|11.45 - 12.00||Tea/Coffee|
|12.00 - 13.15||HM||SBMD-2||SBMD-1||ALTS-2||ES||TUPS|
|13.15 - 14.45||LUNCH|
|14.45 - 16.00||TUPS||TUPS||TUPS||TUPS||TUPS||Break|
|16.00 - 16.15||Tea/Coffee||Break|
|16.15 - 17.30||TUPS||TUPS||TUPS||TUPS||TUPS||Break|
|Day, Date &
|9.00 - 10.00||CLSH||BNC||FJC-2|
|10.10 - 11.10||STM||FJC-1||CFTK-11|
|11.10 - 11.25||Tea/Coffee|
|11.25 - 12.10||CFTK-1||CFTK-6||CFTK-12|
|12.20 - 13.05||CFTK-2||CFTK-7||Closing|
|13.05 - 14.35||LUNCH|
|14.35 - 15.00||CFTK-3||CFTK-8||--|
|15.00 - 15.25||CFTK-4||CFTK-9||--|
|15.35 - 16.00||CFTK-5||CFTK-10||--|
|16.00 - 16.15||Tea/Coffee||--|
|16.15 - 17.30||TUPS||TUPS||--|
References: (These form a very small portion of the literature on this topic but contains the original papers and books on the subject, more is available in the internet, however some of the books, which contain quite a bit of the materials to be covered, are highlighted with bold letters)
- BREDON G., Topology and Geometry, Springer 1993. This book introduces many of the tools needed for manifold surgery.
- BROWDER W., Surgery on simply connected manifolds, Springer-Verlag, 1972.
- DAVIS, J.F. and KIRK, P., Lectures Notes in Algebraic Topology, Department of Mathematics, Indiana University, Bloomington, IN 47405, E-mail address: email@example.com, firstname.lastname@example.org
- GAULD David B., Differential Topology: An Introduction, Dover Publications, 2006.
- GUILLEMIN Victor and POLLACK Alan, Differential Topology, AMS, 2010.
- HIRSCH M.W., Differential Topology, Springer-Verlag, 1997.
- HUSEMULLER Dale, Fibre bundles, GTM-20, Springer, 1994.
- KERVAIRE M. and MILNOR J., *Groups of homotopy spheres-I*, Annals of Mathematics, Vol. 77 (1963), 504-537.
- KIRBY R.C. and SIEBENMANN L.C., Foundational Essays on the Topological manifolds, smoothing and triangulations, Annals of Mathematics Series No. 88, Princeton University Press (1977).
- KISTER, J.M., Microbundles and Fibre bundles, Annals of Mathematics, Second series Vol. 80, No. 1, 1964, pp. 190-199.
- KOSINSKI, A.A., Differential manifolds, Academic Press, 1993 and Dover publications, 2007.
- LÜCK Wolfgang , A basic introduction to Surgery theory, 2003.
- MILNOR J., ``Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007. Surgery theory was initiated by Milnor in the period 1956-1960 for the purpose of classifying smooth manifolds that are homotopy equivalent to a sphere of dimension \geq 5. In addition to published research articles, this volume contains four sets of notes for lectures he gave in this period; these lectures had remained in mimeographed form until their formal publication in this volume. They still provide excellent instruction in differential topology.
- MILNOR J., A procedure for killing homotopy groups of differentiable manifolds, Differential Geometry, Proc. Sympos. Pure Math. Vol. 3, pp. 39-55, Amer. Math. Soc. Providence, R. I., 1961.
- MILNOR J., Morse theory, (based on lecture notes by M. Spivak and R. Wells) Annals of Mathematical Studies No. 51, Princeton University Press, 1963.
- MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
- MILNOR J., Lectures on the h-cobordism theorem, (notes by L. Siebenmann and J. Sondow), Princeto Mathematical Notes, Princeton University Press, 1965.
- MILNOR J., Whitehead Torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426.
- MILNOR J. and STASHEFF J., Characteristic classes, Annals of Mathematics Studies No. 76, Princeton University Press, 1974. This monograph is based on more succinct mimeographed notes by Milnor and Stasheff for lectures of Milnor given in 1959 and still available at <http://www.maths.ed.ac.uk/~aar/papers/milnorcc.pdf>.
- RANICKI A. A., Algebraic L-theory and topological manifolds, Cambridge Tracts in Math., vol. 102,Cambridge Univ. Press, Cambridge, 1992.
- RANICKI A. A., Exact sequences in the algebraic theory of surgery, Princeton Univ. Press, Princeton, NJ, 1980. MR 82h:57027
- RANICKI A. A., Algebraic and Geometric Surgery, Oxford University Press, 2002.
- ROURKE R.C. and SANDERSON B.J., Introduction to Piecewise Linear Topology, Springer-Verlag, 1972.
- STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.
- WALL C.T.C., Surgery on Compact manifolds, 2nd Edition, Edited by Andrew Ranicki, 1998.
- WALLACE A.H., Differential Topology: First Steps, Dover Books on Mathematics, 2006.
|Sr||SID||Fullname||Gender||Affiliation||State||Position in College/Univ.||Univ./ institute M.Sc./ M.A.||Year of Passing M.Sc./
|Ph.D. Degree Date|
|1||3640||Dr. Somnath Basu||Male||Binghamton University||U.S.A||Visiting Assistant Professor||Indian statistical institute||2005||15 Aug 2011|
|2||2874||Mr. Surojit Ghosh||Male||Ramakrisna Mission Vivekananda University||Phd student||IIT Kanpur||2012|
|3||2912||Mr. Ramen Ghosh||Male||CMI, Chennai||Tamil nadu||Pass out Msc student||CMI, Chennai||2011|
|4||2967||Dr. Abhijit Pal||Male||CMI, Chennai||Tamil nadu||Visiting fellow||M.Sc||2004||24 Mar 2011|
|5||3271||Mr. Sanjay Kumar Singh||Male||TIFR, Mumbai||Maharashtra||Research scholar||University of Allahabad||2005|
|6||3514||Mr Arghya Mondal||Male||The Institute Of Mathematical Sciences||Tamil nadu||Phd student||The Institute Of Mathematical Sciences||2012|
|7||3545||Mr. Punya Plaban Satpathy||Male||IISER, Pune||Maharashtra||4th year student||Awaiting Result|
|8||3565||Mr. Apurv Nakade||Male||CMI, Chennai||Tamil nadu||Ph.d||CMI, Chennai||2012|
|9||3587||Mr. Devendra Tiwari||Male||TIFR, Mumbai||Maharashtra||Ph.d||BHU, Varanasi||2010|
|10||3616||Mr. Karthik Chandrasekhar||Male||CMI, Chennai||Tamil nadu||Ph.d||Chennai mathematical institute||2011|
|11||3628||Mr Vidyasagar Mysoremath||Male||Rani Chennamma University||Karnataka||Asst. professor||Karnatak University/M.Sc||2008|
|12||3525||Mr. Sudipta Kolay||Male||CMI, Chennai||Tamil Nadu||Integrated phd||CMI, Chennai||Awaiting Result|
|13||3507||Mr. Anand Bhushan Deshpande||Male||Fergusson College, Pune||Maharashtra||Research student||University of Pune||2011|
|14||3249||Mr Dipendu Maity||Male||IIT, Patna||Bihar||Research scholar (phd)||Indian institute of technology kanpur||2007|
|15||3522||Mr. Pradip Kumar||Male||HRI, Allahabad||Uttar Pradesh||Phd student||IIT, Kanpur||2009|
|16||3117||Dr. Priyavrat Deshpande||Male||CMI, Chennai||Tamil Nadu||Postdoctoral fellow||The University Of Western Ontario||2007||31 May 2011|
|17||3341||Mr. Anurag Singh||Male||IIT Kanpur||Uttar Pradesh||Student||IIT-Guwahati||2012|
|18||3353||Mr. Kushal Lalwani||Male||Department. of Mathematics Delhi||New Delhi||M.Phil||D.U. M.A.||2011|
|19||3250||Mr. Anand Kumar Tiwari||Male||IIT Patna||Bihar||Ph.D.||B.H.U||2007|
|20||3104||Dr. Kamlesh Kumar Rai||Male||D.N.G.College, Itanagar, A.P.||Arunachal Pradesh||Sr Lecturer||Purvanchal / U.P.College Varanasi, UP||1995||29 Dec 2005|