studies `geometric' shapes, spaces and maps
between them by algebraic means
An example of a space is a circle, or a doughnut-shaped figure, or a Möbius band. A little
more precisely, the objects we want to study belong to a certain
geometric `category' of topological spaces (the appropriate
definition will be given in due course). This category is hard to
study directly in all but the simplest cases. The objects involved
could be multidimensional, or even have infinitely many dimensions
and our everyday life intuition is of little help. To make any
progress we consider a certain `algebraic' category and a
`functor' or a `transformation' from the geometric category to the
algebraic one. We say `algebraic category' because its objects have
algebraic nature, like natural numbers, vector spaces, groups etc.
This algebraic category is more under our control. The idea is to
obtain information about geometric objects by studying their image
under this functor. Now the basic problem of algebraic topology is to find a system of
algebraic invariants of topological spaces which would be powerful
enough to distinguish different shapes. On the other hand these
invariants should be computable. Over the decades people have come
up with lots of invariants of this sort. In this course we will
consider the most basic, but in some sense, also the most
important ones, the so-called homotopy
Algebra: Groups, rings, fields, homomorphisms, examples
Standard point-set topology: topological spaces, continuous maps, subspaces, product spaces, quotient spaces, examples
Homotopy: fundamental group and covering spaces, sketch of higher homotopy groups.
Singular homology: construction, homotopy invariance, relationship with fundamental
Basic properties of cohomology (not excision or Mayer-Vietoris yet), motivated by
Connecting homomorphisms and exact sequences.
The Mayer-Vietoris sequence.
Betti numbers and the Euler characteristic.
Options for additional content:
Thom spaces and the Thom isomorphism theorem, Cohomology
of projective spaces and projective bundles, Chern classes.