A brief Description of the school

One can trace the history of Algebraic topology back to the Konesberg’s seven bridges problem and Euler’s solution to that. Poincare used the notion of Betti number for studying manifolds. Once the notion of Algebraic Topology was put in sound modern algebraic setting by Noether the subject advanced by leaps and bounds. Fundamental groups was instrumental in distinguishing R^2 and R^3 upto homeomorphism where as homology groups were instrumental in distinguishing R^3 and R^4 and so on, For distinguishing more and more topological objects more and more sophisticated tools like cohomology, Steenrod operations and more and more different notions of generalized cohomology theories, primary , secondary and higher cohomology operations etc. have been invented. An obstruction theory for extending and lifting of maps have been developed, in a way the operations mentioned above are offshoots of this. Characteristic classes and numbers were developed to classify manifolds upto cobordism. These developments also helped in developing powerful surgery techniques to classify manifolds upto diffeomorphism, pl-homeomorphism and homeomorphism. Another major development is the use of spectral sequence to compute (generalized) (co)homology of a space in terms of the knowledge of (generalized) (co)homology of some associated spaces or associated (co)homology theories of the given space. It also helped in computing stable homotopy groups classifying spaces and sphere. Algebraic topology together with the techniques from geometry and analysis and physics has give rise to new (co)homology theories which have helped in settling old conjectures like the triangulation conjecture. String topology is another promising field attracting attention of a large number of researchers.

The purpose of the AIS would be to expose these topics in a condensed and yet lucid fashion, help students in digesting the topics through tutorials, to direct the development towards some interesting outstanding problems in the subject and also towards applications which are under intensive current development process.

Prerequisite:

It is expected that the participants of the AIS-Algebraic topology 2016 have undergone the material covered in the AFS-III, in algebraic topology as given below. These topics will be assumed and further topics built on these will be taken up.

AFS-III syllabus: Algebraic Topology:

1. Statements of basic problems in algebraic topology extension problems and lifting problems; homotopy, relative homotopy, deformation,contraction, retracts etc. Typical constructions: Adjunction spaces, Mapping cones, Mapping cylinder, Smash-product, reduced cones reduced suspension etc. Categories and Functors. Definition and examples. Equivalence of functors, adjoint functors, examples. Computation of fundamental group of the circle and applications.
2. CW-Complexes and Simplicial complexes basic topological properties of CW complexes. Products of CW complexes (especially the CW-structure on X ×[0, 1]). Homotopy theoretic properties of CW complexes.Abstract simplicial complexes and geometric realization, barycentric sub division and simplicial approximation theorem. Applications: cellular Approximation theorem, Brouwer’s invariance of domains etc.
3. Covering spaces and Fundamental groups lifting properties,relation with fundamental group. Classification of covering spaces(proof of existence may be skipped), Computation of Fundamental groups: simpler cases of Van-Kampen theorem. Effect of attaching n-cells.
4. Singular and Simplicial Homology Chain complexes, exact sequences of complexes, snake lemma, four lemma and five lemma,homology long exact sequence. Axioms for homology, construction of singular chain complex, verification of axioms (except homotopy axiom and excision axiom). Simplicial and singular simplicial homologies. Statement of equivalence of all these homologies. Computations and applications: Separation theorems, Invariance of Domain. Euler characteristic.

References:

1. A. Hatcher: Algebraic Topology, Cambridge University Press.
2. C. R. F Maunder: Algebraic Topology, Van Nostrand Reinhold Company, London.
3. E. H. Spanier: Algebraic Topology, Tata McGraw-Hill.
4. Anant R. Shastri: Basic Algebraic Topology, CRC Press, Taylor and Francis group, 2013.]

Syllabus

 Speakers

• AM – Amiya Mukherjee, ISI, Kolkata.
• ATS – Angom Tiken Sing, NEHU, Shillong.
• GM – Goutam Mukherjee, ISI, Kolkata.
• HKM – Himadri Kumar Mukerjee, NEHU, Shillong.
• MD – Mahuya Dutta - ISI, Kolkata.
• SB – Samik Basu, Ramkrishna Mission Vivekananda Univ, Howrah.
• SNB – Somnath Basu, Ramkrishna Mission Vivekananda Univ, Howrah.

Tentative Time-Table

References:

1. Algebraic Topology – a first course, by Marvin J. Greenberg and John R. Harper, Benjamin, 1981
2. Algebraic Topology, by Edwin H. Spanier, Tat-Mc Graw Hill
3. A basic Course in Algebraic Topology, by William S. Massey, Springer, 1991
4. Algebraic Topology, by Allen Hatcher, 2001
5. Foundations of Algebraic Topology, by Samuel Eilenberg and Norman E. Steenrod, Princeton Univ. Press, 1952.
6. Topology, J.G. Hocking and G.S. Young, Addison-Wesley, 1961
7. The topology of CW-complexes, by A.T Lundell and S. Weingram, van Nostrand Reinhold, 1969
8. Combinatorial Homotopy I, by J.H.C.Whitehead, Bull.Amer. Math. Soc., 55 (1949), 213-245.
9. Lectures on Algebraic Topology, by Albrecht Dold, Springer-Verlag, 1972.
10. Homology, by Sounders Mac Lane, Springer-Verlag, 1963
11. Elements of Homotopy Theory, by G.W. Whitehead, Springer-Verlag, 1978.
12. Algebraic Topology- Homotopy and Homology, by Robert M. Switzer, Sringer-Verlag, 1976.
13. Anant R. Shastri: Basic Algebraic Topology, CRC Press, Taylor and Francis group, 2013.
14. J. R. F Maunder: Algebraic Topology, Van Nostrand Reinhold Company, London.
15. SYMPLECTIC HOMOLOGY AS HOCHSCHILD HOMOLOGY
16. STRING TOPOLOGY AND THE BASED LOOP SPACE- phd thesis, by Eric James Malm, Stanford Univ. (2010)
17. Notes from the Awesome Joint Berkeley-Stanford String Tpology Seminar, by A.P.M. Kupers.
18. NONNEGATIVELY AND POSITIVELY CURVED MANIFOLDS, by BURKHARD WILKING.
19. Curve shortening and the topology of closed geodesics on surfaces, by Sigurd B. Angenent.
20. Short geodesic loops on complete Riemannian manifolds with finite volume, by Regina Rotman .
21. SHORTENING CURVES ON SURFACES , by Jel Hass and Peter Scott.
22. OPEN PROBLEMS AND QUESTIONS ABOUT GEODESICS, by Keith Burns and Vladimir S. Matveev.
23. https://en.m.wikipedia.org/wiki/Differential_geometry_of_surfaces#Shape_operator
24. Computing Shortest Words via Shortest Loops on Hyperbolic Surfaces, by Xiaotian Yin , Yinghua Li , Wei Han , Feng Luo , Xianfeng David Gu , Shing-Tung Yau.
25. Closed Geodesics and the Free Loop Space, by Hans-Bert Rademacher (beamer presentation).
26. Morse theory, closed geodesics, and the homology of free loop spaces, by Alexandru Oancea.
27. ON THE FORMAL GROUP LAWS OF UNORIENTED AND COMPLEX COBORDISM THEORY, by Daniel Quillen.
28. Quillen's work on formal group laws and complex cobordism theory, by D. Ravenel (beamer).
29. FORMAL GROUPS, COMPLEX COBORDISM AND QUILLEN’S THEOREM, by DAVID ROE (minors thesis).
30. Adams, J.F. Stable Homotopy and Generalised Homology. Chicago: University of Chicago Press, 1974 .
31. Strickland, N.P. Formal Groups. Course Notes, http://neil-strickland.staff.shef.ac.uk/courses/formalgroups/
32. FORMAL GROUPS, by N. P. STRICKLAND.
33. Complex cobordism and stable homotopy groups of spheres, D. Ravenel. Academic Press.
34. Specific references:
    • For Topic 5,
      J.F. Adams - Stable homotopy and Generalized homology
      D. Ravenel - Complex Cobordism and the stable homotopy groups of spheres.
    • For Topic 6,
      Milnor - Morse Theory.Gromoll,
      Meyer - Periodic geodesics on compact Riemannian manifolds.
      Bott - On the iteration of closed geodesics and Strum intersection theory.

    
    Further reading:
    Gromoll, Meyer - On differentiable functions with isolated critical points.
    Abraham - Transversality in manifold of mappings.

    There are many more new books and resources in the internet.