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 book gives
a deeper account of basic ideas of differential topology than usual in
introductory texts. Also many more examples of manifolds like matrix groups
and Grassmannians are worked out in detail. Topics covered includes:
Continuity, compactness and connectedness, Smooth manifolds and maps, Regular
values and Sard’s theorem, Manifolds with boundary and orientations, Smooth
homotopy and vector bundles, Intersection numbers, vector fields and Euler
characteristic.

The
first half of the book deals with degree theory, the Pontryagin construction,
intersection theory, and Lefschetz numbers. The second half of the book is
devoted to differential forms and deRham cohomology.

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.