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Intermediate Algebraic Topology

MAT 540 - Fall 2006

MAT 541 - Spring 2007

Dennis Sullivan

Assuming the fundamental group has been developed we will see how
this piece of algebraic topology controls two dimensional
manifolds and three dimensional manifolds.
More algebraic topology is needed to begin to treat four and
higher dimensional manifolds...higher homotopy groups and homology
groups.

One way to understand the homology groups of a space is as a functor
on the homotopy category of cell complexes satisfying exactness and
excision. Without assuming a normalising dimension axiom this allows
other functors like K theory and bordism theory as well.
All of these may be described covariantly by geometric cycles
mapping into the target spaces. They may also be described
contravariantly by geometric objects over base spaces like vector
bundles or block bundles.

We will discuss how differential forms give a purely algebraic model
of the homotopy category.
And how differential forms provide a representation of the real
characteristic classes of vector bundles.

Some of the planned discussion may only appear in the spring
semester ... like the refinement of the Chern Weil characteristic
differential forms of vector bundles with connection and the integral
characteristic classes of vector bundles provided by the Cheeger
Simons differential characters of vector bundles with connection.