Topology of Manifolds

Fundamental group and covering spaces. Chain complexes. Mayer-Vietoris sequence. De Rham complex; compact support. Homotopic invariance of de Rham cohomology. Maximal dimension cohomology. Poincaré’s duality theorem. Jordan-Brouwer theorem. Open invariance. Alexander’s duality theorem in the sphere. Singular homology. Singular cohomology. De Rham’s theorem; Applications. Introduction to Morse theory. Morse-type singularities, passing through critical levels, Morse inequalities.

References:
BREDON, G. – Topology and Geometry , Springer-Verlag, 1993.
LEE,J. – Introduction to Smooth Manifolds, Springer-Verlag, 2002.
LIMA, E. – Homologia Básica, Projeto Euclides, 2009.
WARNER, F. – Foundations of Differentiable Manifolds and Lie Groups,Springer-Verlag, 1983.