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Differential Topology of Fiber Bundles
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.
Author(s): Karl-Hermann Neeb
146 Pages 
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.
Author(s): Alexander Kupers
279 Pages
Topology from the Differentiable Viewpoint
This PDF covers the following topics related to Differential Topology : Smooth manifolds and smooth maps, Tangent spaces and derivatives, Regular values, The fundamental theorem of algebra, The theorem of Sard and Brown, Manifolds with boundary, The Brouwer fixed point theorem, Proof of Sard's theorem, The degree modulo 2 of a mapping, Smooth homotopy and smooth isotopy, Oriented manifolds, The Brouwer degree, Vector fields and the Euler number, Framed cobordism, the Pontryagin construction, The Hopf theorem, Exercise.
Author(s): John W. Milnor, Princeton University
77 Pages
Introduction to Differential Topology by Uwe Kaiser
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.
Author(s): Uwe Kaiser
110 Pages
Combinatorial Differential Topology and Geometry (PDF 30p)
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.
Author(s): Robin Forman
30 PagesAdvertisement
