Differential Topology - Differential Topology (Lecture III); Differential Topology (Lecture III)

Credits (2011)

The following was videotaped August 30 to September 2, 1965 at Cornell University during the 46th Summer Meeting of the Mathematical Association of America, celebrating the association's 50th anniversary. The lectures were part of the Individual Lectures Project of the Committee on Educational Media of the Mathematical Association of America.

The Mathematical Association of America presents

Differential Topology (Lecture III)

with Professor John W. Milnor, Princeton University, Earle Raymond Hedrick Lectures 1965

Credits (1965)

An Individual Lectures Project, Committee on Educational Media, Mathematical Association of America Production

Differential Topology with Professor John W. Milnor, Princeton University

The Earle Raymond Hedrick Lectures, 1965, Lecture III

Beginning of Milnor lecture

Grassman manifold

To each plane of dimension n through the origin

Important use of the Grassman manifold: generalized Gauss map associated with a manifold

Illustrates the construction with a problem

Look at homology theory

Fundamental class of the manifold

Fundamental theorem of oriented cobordism theory

Notion of tangent bundle and notion of a vector bundle

Tangent bundle of a smooth manifold

Grassman bundle

2 examples of vector bundles (Grassman and tangent) related by the Gauss mapping

G exists and is unique up to homotopy

Examines 3 categories simultaneously: differentiable, piecewise linear, topological

For each x in our manifold, there exists a neighborhood use of x

Impose restriction that these neighborhoods' use of x will vary nicely with x

Impose condition of local triviality

Notion of tangent bundle in any one of the 3 categories

End of lecture

End credits (1965)

An Individual Lectures Project, Committee on Educational Media, Mathematical Association of America Production

Produced for CEM by Professor A.N. Feldzamen, University of Wisconsin

Directed by Louis P. Levine

Financial support from the National Science Foundation; ©1965 Mathematical Association of America

Grateful appreciation is expressed to Professor Alex Rosenberg of the Mathematics Department, Cornell University and the Staff of the Statler Auditorium

Technical facilities by Tele-Tape Productions, New York City

Recorded 1965 by videotape in the Statler Auditorium, Cornell University during the 50th Anniversary Meeting of the Mathematical Association of America

End credits (2011)

John Milnor was awarded the 2011 Abel Prize for pioneering discoveries in topology, geometry and algebra.

Digital re-mastering of this film in 2011 was a joint project of the Mathematical Sciences Research Institute and the Mathematical Association of America with generous support from the Simons Foundation.

Technical support and production: Monaco Labs, Zala Films

©1965, 2011 Mathematical Association of America, All Rights Reserved