Lectures on differential topology by Alexander Kupers

Lectures on differential topology by Alexander Kupers

Lectures on differential topology by Alexander Kupers

The
contents include: Spheres in Euclidean space, Smooth manifolds, Submanifolds
and tori, Smooth maps and their derivatives, Tangent bundles, Immersions and
submersions, Quotients and coverings, Three further examples of manifolds,
Partitions of unity and the weak Whitney embedding theorem, Transversality
and the improved preimage theorem, Stable and generic classes of smooth
maps, Transverse maps are generic, Knot theory, Orientations and integral
intersection theory, Integration on manifolds, De Rham cohomology, Invariant
forms in de Rham cohomology, First fundamental theorem of Morse theory,
Second fundamental theorem of Morse theory, Outlook.

The
contents include: Spheres in Euclidean space, Smooth manifolds, Submanifolds
and tori, Smooth maps and their derivatives, Tangent bundles, Immersions and
submersions, Quotients and coverings, Three further examples of manifolds,
Partitions of unity and the weak Whitney embedding theorem, Transversality
and the improved preimage theorem, Stable and generic classes of smooth
maps, Transverse maps are generic, Knot theory, Orientations and integral
intersection theory, Integration on manifolds, De Rham cohomology, Invariant
forms in de Rham cohomology, First fundamental theorem of Morse theory,
Second fundamental theorem of Morse theory, Outlook.

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 Sards 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.