(office: Math 4-119, tel. 632-8358, e-mail
A basic introduction to geometry/topology, such as
and MAT 531.
Thus prior exposure to basic point set topology, homotopy, fundamental group, covering spaces is assumed, as well as
some acquaintance with differentiable manifolds and maps, differential
forms, the Poincaré Lemma, integration and volume on manifolds,
Stokes' Theorem. We will briefly review some of this material in the
first week of classes.
Differential forms in algebraic topology, by Raoul Bott and Loring W. Tu, GTM 82, Springer Verlag 1982.
The guiding principle of the book is to use differential forms and
in fact the de Rham theory of differential forms as a prototype of all cohomology
thus enabling an easier access to the machineries of algebraic topology in the realm of smooth manifolds.
The material is structured around four core sections: de Rham theory, the Cech-de
Rham complex, spectral sequences, and characteristic classes, and
includes also some applications to homotopy theory.
Other recommended texts:
- Algebraic Topology: A first Course, W. Fulton, GTM 153, Springer Verlag 1995
- Topology from the Differentiable Viewpoint, J. Milnor, U. of Virginia Press 1965
- Algebraic Topology, A. Hatcher (on-line), Cambridge University Press, to appear
- Characteristic classes, J. Milnor and J. Stasheff, Princeton University Press 1974
The book contains more material than can be resonably covered in a
one-semester course. We will hopefully cover the following sections:
- de Rham theory: the de Rham complex, orientation and
integration, Poincaré lemmas, the Mayer-Vietoris argument,
Poincaré duality on an orientable manifold, Thom class and the
Thom isomorphism (orientable vector bundle case)
- the Cech-de Rham complex: the generalized Mayer-Vietoris argument,
sheaves and Cech cohomology, the de Rham theorem, sphere bundles,
Euler class, the Hopf index theorem, the Thom isomorphism in general,
- characteristic classes: Chern classes of complex vector
bundles, the splitting principle, Whitney's product formula explicit
computations of Chern classes, Pontrjagin classes of real vector
bundles, the universal bundle, infinite grassmannians
- spectral sequences: spectral sequence of a double complex, products, applications
and explicit computations (all these only if time permits)
I will assign problems in each lecture, ranging in difficulty from
routine to more challenging. There will be also a take-home midterm and
a final exam. Course grades will be based on these problems (and any other participation); solving at least half of them
will be considered a perfect score.
Here are some pointers to software that may be used to visualize
- "The Topological Zoo" at the Geometry Center: a visual dictionary of surfaces and other mathematical objects.
- Images of "classical" topological "objects" from the Geometry Center.
- "A Knot Zoo". Here is another one. Or Hyperbolic knots. All these sites are part of an exciting collection of knots and links available at
"The KnotPlot Site". Very instructive are also the
VRML knot models.
- Raytraced images: Sphere, Torus, projective plane: a Crosscap, a Steiner surface, a Boy surface, and a genus 3 orientable surface.
- VRML models: a Möbius band, a Klein bottle and a Trefoil Knot. Download here a vrml viewer for Linux.
- David Eppstein's "Geometry Junkyard": a collection of pointers, clippings, research blurbs, and other stuffs related to discrete, computational geometry, and topology.
- Paul Bourke's collection of raytraced surfaces. Here is for instance the animation of a transition from a Steiner surface into a Boy surface.
- A picture of the Hopf fibration
created by Ken Shoemake. Click here for a
better quality TIFF version of the picture. The picture visualizes well the remarkable
geometric fact that any two fibres (=circles) of the Hopf fibration are linked.
Here is another page
and an mpeg
animation of the Hopf fibration (created with Knotplot).