ATMW Classification of Manifolds (2013)

Venue: NEHU, Shillong
Dates: 17th to 26th Jun


Convener(s) Speakers, Syllabus and Time table Applicants/Participants


School Convener(s)

Name Himadri Kumar Mukerjee
Mailing Address

Department of Mathematics,
North-Eastern Hill University,
NEHU Campus, Shillong-793022, Meghalaya, India.



Speakers and Syllabus 

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.



Sr. Name Affiliation
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)



Detailed Syllabus
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
 17.30 SNACKS Break

Second week

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  --
17.30 Snacks Snacks  --

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:,
  • 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 <>.
  • 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.


Selected Applicants


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