Algebraic topology 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 and homology groups.
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 group.
- Basic properties of cohomology (not excision or Mayer-Vietoris yet), motivated by singular cohomology.
- Relative (co)homology.
- 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.