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
a reasonable 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.
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.
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,
Spectral sequences: basics, spectral sequence of a
double complex, products, applications and some explicit computations
Homotopy theory: homotopy groups, long homotopy
sequence of a fibration, loop spaces, Eilenberg-MacLane spaces,
the Hurewicz isomorphism, a few low dimensional homotopy
groups of spheres (Hopf invariant, etc)
(all of these, perhaps more only if time permits)
Homework & Exams
I will assign problems in each lecture, ranging in difficulty from
routine to more challenging. Course grades will be based on these problems,
class participation, and final exam.
Here are some pointers to software that may be used to visualize
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).