**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 |
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 19.30to 11.00 |
11:00to 11:30 |
Lecture 211.30to 13.00 |
13:00to 15:00 |
Tutorials15.00to 16.00 |
16:00to 16.30 |
Tutorials16.30to 17.30 |
17.30 |

Monday | 02-07-2018 | Topic 1 (HKS) |
T B |
Topic 4 (BS) | L U N C H |
Tutorial (Topic 1) |
T B |
Tutorial (Topic 4) |
T a S |

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:**

- BREDON G., Topology and Geometry, Springer 1993. This book introduces many of the tools needed for manifold surgery.
- GAULD David B., Differential Topology: An Introduction, Dover Publications, 2006.
- GREENBERG and HARPER, Algebraic Topology – a first course, 1981.
- 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.
- KOSINSKI, A.A., Differential manifolds, Academic Press, 1993 and Dover publications, 2007.
- MILNOR J., ``Differential Topology'', Volume 3 of his collected papers, Amer. Math. Soc. 2007.
- MILNOR J., Topology from the Differentiable Viewpoint, Univ. Pross of Virginia at Charlotesville, 1965.
- Mukherjee, Amiya, Topics in Differential Topology, Hindistan Book Agency, 2005.
- Shastri, A.R., Elements of Differential Topology, CRC Press, 2011.
- Shastri, A.R., Basic Algebraic Topology, CRC Press, 2013.
- STEENROD Norman J. , Topology of Fibre bundles, Princeton, 1951; Landmarks in Mathematics and Physics; Princeton, 1999.
- WALLACE A.H., Differential Topology: First Steps, Dover Books on Mathematics, 2006.

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