Recalling from Part I the notions of topological space and homeomorphism, we will now use these concepts to rigorously define an -dimensional manifold. This is a notion central to numerous fascinating branches of mathematics and physics but we shall restrict out attention to the details which will allow us to define a topological quantum field theory.
We do, however, require one more detail before we can proceed. We say that is a compact subset of if every cover of by open subsets of admits a finite subcover. This means that if is a collection of open sets with , then there is a finite collection with . Note that the existence of a finite cover for a set does not imply compactness.
We say that a topological space is a -dimensional manifold if it is a second countable Hausdorff space which is locally homeomorphic to -dimensional Euclidian space . We will not go over the notion of second countability here, since it would merely distract from the point, but interested readers are urged to consult any available text on point set topology for the definition. A Hausdorff space is a topological space such that if are distinct points, then there exist open sets with and and . A continuous function between topological spaces and is called a local homeomorphism if, for any point , there exists an open set such that is open and the restriction is a homeomorphism.
Informally, what this means is that if one chooses a point in a manifold , then it can always be separated from any other point and, if one looks closely enough, there is a neighborhood of which “looks like” ordinary -dimensional space. A smooth -manifold is a manifold which “looks enough like ” that you can “do calculus” on it. Additionally, a (smooth) manifold with boundary is defined in the same way as above with the half-space replacing .
We are now equipped with enough information to define a -cobordism. Formally a –cobordism is a quintuple consisting of a -dimensional compact smooth manifold , two compact -manifolds and with boundary, and two embeddings (“ways of inserting”) and of and into the boundary of such that , where denotes the union of disjoint sets.
For the sake of intuition, one may think of a cobordism as a “soap bubble” between two manifolds represented visually as loops of wire.
Cobordisms, or more precisely the category of cobordisms, make up one half of the machinery we need.
In Part III, I will go over Hilbert spaces and basic category theory.