John Lee, Introduction to smooth manifolds; Victor Guillemin & Alan Pollack, Differential topology
Mathematical analysis, linear algebra, general topology
This course is an introduction on both differentiable manifolds and differential topology. In the first part of the course, we will introduce basic objects on differentiable manifolds: differentiable manifolds and smooth maps, tangent and cotangent vectors, differential forms, integration and stokes theorem, de Rham cohomology. In the second part, we will study differential topology (topology of differentiable manifolds): Whitney immersion and embedding theorems, approximation theorem, Sard theorm, transversality, intersection numbers, Morse functions. If there are more time, I will talk more on the Morse theory and the h-cobordism theorem.