Differential Topology by Bjorn Ian Dundas

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

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

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 Pages