Abstract

Homology and Cohomology are functors which describe a topological spaces geometry. These invariants tell us loosely speaking about how many holes in each dimension each space possess. We equip these spaces with a group action which takes into advantage certain symmetries of the spaces, thus giving us more invariants associated to each space.

Equivariant Homology and Cohomology

The subject of Topology studies sets with structure that mimics that of distance, the sets are equipped with open sets called a topology which can be used to generalise the notion of convergence without using a metric. This notion can be extended to talk about what it means for a function to be continuous, a very elementary concept one first encounters in high school. Two topological spaces are called homeomorphic if there exists a continuous bijection with continuous inverse between them. This loosely translates (in a geometric sense) that the objects are identical, one might think of the famous example that a coffee mug and a donut are the same. The task of determining whether two spaces are homeomorphic is quite challenging especially in the case that they are not. For example, if they are homeomorphic one can simply just construct a homeomorphism, however, if they are not, how do we prove that there does not exist a homeomorphism? In introductory topology we know that if two spaces have different number of path components, then they are not homeomorphic. This is an example of an invariant used in topology. We can think of this as a “function” which takes each topological space to a natural number denoting its path components, and two spaces mapping to two different numbers cannot be homeomorphic. This “function” is a nice invariant, however, many spaces which aren’t homeomorphic can have the same number of path components. Hence, we may be interested to seek more invariants. This gives motivation to define what is called homology groups, which are studied in algebraic topology. These loosely speaking count the number of n dimensional holes in a space. For each n, we define a function that takes in a topological space and spits out a group. This is the concept of a functor.

Now to the contents of my paper, we look at spaces with group action and then study their homology. This is incredibly amazing as we get more invariants. For a group G acting on a topological space, take subgroups H of G. Then for each H we have a different invariant! Note that equipping a space with a group with many subgroups will allow us to increase the number of invariants. Furthermore, even the ability for a group G to act (with nice constraints) on a space we shouldn’t take for granted. Many spaces and actions are not compatible. Hence, even the action of acting a group on a space tells us more about the space in question.

Kenrick Chung
University of New South Wales