This note explains the
following topics: The concept of a fiber bundle, Morphisms of Bundles, Vector
Bundles, Principal Bundles, Bundles and Cocycles, Cohomology of Lie Algebras,
Smooth G-valued Functions, Connections on Principal Bundles, Curvature and
Perspectives.

This note explains the
following topics: The concept of a fiber bundle, Morphisms of Bundles, Vector
Bundles, Principal Bundles, Bundles and Cocycles, Cohomology of Lie Algebras,
Smooth G-valued Functions, Connections on Principal Bundles, Curvature and
Perspectives.

This note covers the following
topics: Smooth manifolds, The tangent space, Regular values, Vector bundles,
Constructions on vector bundles and Integrability.

This note covers
the following topics: Smooth manifolds and smooth maps, Tangent spaces and differentials ,
Regular and singular values , Manifolds with boundary, Immersions
and embeddings , Degree mod 2 , Orientation of manifolds and
Applications of degree.

This note explains how two standard techniques from the study of
smooth manifolds, Morse theory and Bochner’s method, can be adapted to aid
in the investigation of combinatorial spaces.