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Differential Algebraic Topology
This book presents some basic concepts and results from algebraic topology. Topics covered includes: Smooth manifolds revisited, Stratifolds, Stratifolds with boundary: c-stratifolds, The Mayer-Vietoris sequence and homology groups of spheres, Brouwer’s fixed point theorem, separation and invariance of dimension, Integral homology and the mapping degree, A comparison theorem for homology theories and CW-complexes, Kunneth’s theorem, Singular cohomology and Poincare duality, Induced maps and the cohomology axioms, The Chern classes, Pontrjagin classes and applications to bordism, Constructions of stratifolds.
Author(s): Matthias Kreck
168 Pages 
Differential Topology by Andrew Kobin
This note covers Smooth manifolds, Regular values, Transversality and embeddings, Vector bundles.
Author(s): Andrew Kobin
52 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
Introduction To Differential Topology
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.
Author(s): Joel W. Robbin and Dietmar A. Salamon
306 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