Making sense of holes in spaces using algebra

Abstract

Algebraic topology provides a nice method for detecting holes in topological spaces by the use of algebra. It turns out that algebra and holes are related to each other by groups: the fundamental group and homology group. On the matter of the fundamental group, there is a generalization to higher dimensions, called higher homotopy groups. In practice these are more difficult to compute, leading the discussion to go in the direction of homology groups, which are easier to compute. In doing this, we address the two varieties of homology groups, called simplicial and singular homology groups. Even though homology groups are easier to compute, we have to work hard to construct them. To remedy this, we turn to the Eilenberg-Steenrod approach which takes the properties of homology as axioms.