# AIS Differential Topology (2018) - Speakers and Syllabus

Introduction: Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. Differential and integral calculus carry over to differential manifolds in a suitable way. During the 1930s Hassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, the Whitney embedding theorem, that manifolds intrinsically defined by charts could always be embedded in Euclidean space, as in the extrinsic definition, showed that the two concepts of manifold were equivalent. Differential Topology mainly deals with the global properties of differentiable manifolds. In dealing with problems of classification and intersection theory of such manifolds one often has to bring in tools and techniques from algebraic topology.
The purpose of the AIS would be to expose some of these topics in a condensed and yet lucid fashion and help students in digesting the topics through tutorials; this will prepare them to take up research work on the subject.

Resource persons:

 Sr. Name of Speaker Affiliation Legend 1 Amiya Mukherjee ISI, Kolkata AM 2 Anant R. Shastri IIT Bombay, Mumbai ARS 3 Angom Tiken Singh NEHU, Shillong ATS 4 B. Subhash IISER Tirupati BS 5 Himadri Kumar Mukerjee NEHU, Shillong HKM 6 Hemant Kumar Singh Delhi University, Delhi HKS

Syllabus:

 Topic Lectures Topic Topic 1(HKS) 6  (90 Minutes) Topological manifolds, differentiable structures, smooth manifolds, smooth functions and smooth maps, tangent vectors and tangent space at a point, derivatives of a smooth map, inverse, implicit and rank theorems, submanifolds, regular and critical points and values, Sard’s theorem (special cases), immersion and submersion, inverse image of regular values. (First week) Topic 2(ATS) 6 (90 Minutes) Whitney’s embedding theorem, transversality, inverse image of transverse submanifolds, transversality theorem, vector bundles, tangent and normal bundles, Grassmann manifolds, universal bundles, universality of the universal bundle. Topic 3(HKM) 6 (90 Minutes) Thom’s cobordism theory, definition of cobordism groups, Thom spaces, Thom’s isomorphism theorem. Topic 4(BS) 6 (90 Minutes) Review of singular homology of spaces and pairs, Mayer-Vietoris sequence, adjunction spaces, cell complexes, singular cohomology, calculation of (co)homology groups of cell complexes and standard manifolds, Cup and cap products. Topic 5(AM) 6 (90 Minutes) Local (co)homology of manifolds, orientation covers, orientation of manifolds, fundamental class of a compact orientable manifold, the Poincare, Alexander and Lefschetz duality theorems (statements, explanations and applications only). Topic 6(ARS) 6 (90 Minutes) Geometric and algebraic intersection theory, the Lefschetz fixed point theorem, the Hopf index theorem, framed manifolds, framed cobordism, Hopf degree theorem.

Tentative Time Table

 Day Date Lecture 19.30to11.00 11:00to11:30 Lecture 211.30to13.00 13:00to15:00 Tutorials15.00to16.00 16:00to16.30 Tutorials16.30to17.30 17.30 Monday 02-07-2018 Topic 1 (HKS) Tea Break Topic 4 (BS) LUNCH Tutorial (Topic 1) Tea Break Tutorial (Topic 4) Tea and Snacks Tuesday 03-07-2018 Topic 1 (HKS) Topic 4 (BS) Tutorial (Topic 1) Tutorial (Topic 4) Wednesday 04-07-2018 Topic 1 (SPT) Topic 4 (BS) Tutorial (Topic 1) Tutorial (Topic 4) Thursday 05-07-2018 Topic 1 (SPT) Topic 4 (BS) Tutorial (Topic 1) Tutorial (Topic 4) Friday 06-07-2018 Topic 1 (HKS) Topic 4 (BS) Tutorial (Topic 1) Tutorial (Topic 4) Saturday 07-07-2018 Topic 1 (HKS) Topic 4 (BS) Tutorial (Topic 1) Tutorial (Topic 4) Sunday 08-07-2018 Break Break Break Break Monday 09-07-2018 Topic 2 (ATS) Topic 5 (AM) Tutorial (Topic 2) Tutorial (Topic 5) Tuesday 10-07-2018 Topic 2 (ATS) Topic 5 (AM) Tutorial (Topic 2) Tutorial (Topic 5) Wednesday 11-07-2018 Topic 2 (ATS) Topic 5 (AM) Tutorial (Topic 2) Tutorial (Topic 5) Thursday 12-07-2018 Topic 2 (ATS) Topic 5 (AM) Tutorial (Topic 2) Tutorial (Topic 5) Friday 13-07-2018 Topic 2 (ATS) Topic 5 (AM) Tutorial (Topic 2) Tutorial (Topic 5) Saturday 14-07-2018 Topic 2 (ATS) Topic 5 (AM) Tutorial (Topic 2) Tutorial (Topic 5) Sunday 15-07-2018 Break Break Break Break Monday 16-07-2018 Topic 3 (HKM) Topic 6 (ARS) Tutorial (Topic 3) Tutorial (Topic 6) Tuesday 17-07-2018 Topic 3 (HKM) Topic 6 (ARS) Tutorial (Topic 3) Tutorial (Topic 6) Wednesday 18-07-2018 Topic 3 (HKM) Topic 6 (ARS) Tutorial (Topic 3) Tutorial (Topic 6) Thursday 19-07-2018 Topic 3 (HKM) Topic 6 (ARS) Tutorial (Topic 3) Tutorial (Topic 6) Friday 20-07-2018 Topic 3 (HKM) Topic 6 (ARS) Tutorial (Topic 3) Tutorial (Topic 6) Saturday 21-07-2018 Topic 3 (HKM) Topic 6 (ARS) Tutorial (Topic 3) Tutorial (Topic 6)

References:

1. BREDON G., Topology and Geometry, Springer 1993. This book introduces many of the tools needed for manifold surgery.
2. GAULD David B., Differential Topology: An Introduction, Dover Publications, 2006.
3. GREENBERG and HARPER, Algebraic Topology – a first course, 1981.
4. GUILLEMIN Victor and POLLACK Alan, Differential Topology, AMS, 2010.
5. HIRSCH M.W., Differential Topology, Springer-Verlag, 1997.
6. HUSEMULLER Dale, Fibre bundles, GTM-20, Springer, 1994.
7. KOSINSKI, A.A., Differential manifolds, Academic Press, 1993 and Dover publications, 2007.
8. MILNOR J., Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007.
9. MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
10. Mukherjee, Amiya, Topics in Differential Topology, Hindistan Book Agency, 2005.
11. Shastri, A.R., Elements of Differential Topology, CRC Press, 2011.
12. Shastri, A.R., Basic Algebraic Topology, CRC Press, 2013.
13. STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.
14. WALLACE A.H., Differential Topology: First Steps, Dover Books on Mathematics, 2006.

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