The idea of algebraic topology is to study slippery geometric objects (topology) using rigid discrete objects (algebra). We look at algebraic shadows of the geometric objects, and see what we can learn. If we are lucky, the algebraic shadows will suggest geometric insights that can then be proved.
Topology −→ Algebra
Algebraic topology is one of the most important creations in mathematics which uses algebraic tools to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism (though usually classify up to homotopy equivalence). The most important of these invariants are homotopy groups, homology groups, and cohomology groups (rings).
Furthermore, Algebraic topology can be roughly defined as the study of techniques for forming algebraic images of topological spaces. Most often these algebraic images are groups, but more elaborate structures such as rings, modules, and algebras also arise. The mechanisms that create these images.
Also, one of the main objectives of topology is to classify spaces up to continuous deformations, i.e. up to homeomorphism. Unfortunately, in general it is a quite difficult problem to show when two spaces are not homeomorphic. A classification which is usually easier to obtain is based on the rougher notion of homotopy.
we define an algebraic group called the fundamental group of a space. This group consists of different equivalence classes of loops in that space.
We use the fundamental group as a tool to tell different spaces apart. For example, what makes a torus different from a double torus, or an n-fold torus? Or how is a coffee cup like a donut?
The main purpose of this course to presentation to the basic materials of algebraic topology