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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 
Differential Topology by Andrew Kobin
This note covers Smooth manifolds, Regular values, Transversality and embeddings, Vector bundles.
Author(s): Andrew Kobin
52 Pages
An Introduction to Differential Topology, de Rham Theory and Morse Theory
This note covers the following topics: Basics of Differentiable Manifolds, Local structure of smooth maps, Transversality Theory, IDifferential Forms and de Rham Theory, TIensors and some Riemannian Geometry.
Author(s): Michael Muger
80 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
Differential topology Lecture notes (PDF 20p)
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.
Author(s): Sergei Tabachnikov
20 Pages