A Slapdash Introduction to TQFTs from the Ground Up, Part II.

Recalling from Part I the notions of topological space and homeomorphism, we will now use these concepts to rigorously define an n-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 U\subset X is a compact subset of X if every cover of U by open subsets of X admits a finite subcover. This means that if \{V_\alpha\}_{\alpha \in A}\subset \mathcal{T} is a collection of open sets with U\subset\bigcup_{\alpha\in A}V_\alpha, then there is a finite collection V_{\alpha_1},\dots,V_{\alpha_n}\in\{V_\alpha\}_{\alpha\in A} with U\subset\bigcup_{i=1}^nV_{\alpha_i}. Note that the existence of a finite cover for a set does not imply compactness.

We say that a topological space M is a n-dimensional manifold if it is a second countable Hausdorff space which is locally homeomorphic to n-dimensional Euclidian space \mathbb{R}^n. 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 X such that if x,y\in X are distinct points, then there exist open sets U,V\subset X with x\in U and y\in V and U\cap V=\varnothing. A continuous function f:X\rightarrow Y between topological spaces X and Y is called a local homeomorphism if, for any point x\in X, there exists an open set U\subset X such that f(U)\subset Y is open and the restriction f|_U:U\rightarrow f(U) is a homeomorphism.

Informally, what this means is that if one chooses a point x in a manifold M, then it can always be separated from any other point y and, if one looks closely enough, there is a neighborhood of x which “looks like” ordinary n-dimensional space. A smooth n-manifold is a manifold which “looks enough like \mathbb{R}^n” 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 \{\boldsymbol{x}\in\mathbb{R}^n:x_i\geq 0\} replacing \mathbb{R}^n.

We are now equipped with enough information to define a n-cobordism. Formally a ncobordism is a quintuple (B;M,N,e,e^\prime) consisting of a (n+1)-dimensional compact smooth manifold B, two compact n-manifolds M and N with boundary, and two embeddings (“ways of inserting”) e and e^\prime of M and N into the boundary \partial B of B such that \partial B=e(M)\sqcup e^\prime(N), where \sqcup 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.


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