16:640:548 Differential Topology

Fall 2020

Paul Feehan


Differential Topology

Course Description:

Differential Topology of central importance in Mathematics and required background for every research mathematician and theoretical physicist. Differential Topology has core applications in all areas of Complex Analysis and Geometry, Differential Geometry, Geometric Analysis, Geometric Topology, Global Analysis, Mathematical Physics, Partial Differential Equations, and Theoretical Physics. This course will give a broad introduction to Differential Topology, with prerequisites that we shall try to keep to a minimum in order to introduce students to the field while also providing guidance for more advanced students. Topics may vary depending on the audience and their interests but should include:

I. Smooth manifolds and smooth maps

II. Submersions, immersions, and embeddings

III. Transversality and Sard's theorem

IV. Intersection theory

V. Vector fields, Lie derivatives and brackets, distributions, tensor fields, and vector bundles

VI. Riemannian metrics

VII. Differential forms, orientations, integration on manifolds, and de Rham cohomology

VIII. Morse theory

IX. Symplectic manifolds

X. Banach Manifolds


Primary References:
- J. Lee, Introduction to smooth manifolds, Springer, 2013
- V. Guillemin & A. Pollack, Differential topology, AMS Chelsea, 201
Secondary References:
The following references may be useful for more advanced students with specific interests:
- R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer, 1988
- M. Hirsch Differential topology, Springer 1994
- J. W. Milnor, Morse theory, Princeton University Press, 1963
- S. Lang, Introduction to differentiable manifolds, Springer, 2002
- J. Margalef Roig and E. Outerelo Dominguez, Differential topology, North Holland, 1992
- F. Warner, Foundations of differentiable manifolds and Lie groups, Springer, 1986


Analysis, Linear Algebra, and Elementary Topology or permission of the instructor