Cobordisms

Cobordism

Suppose ${\Sigma }_{-}$ and ${\Sigma }_{+}$ are two closed manifolds of dimension $n-1$, then a cobordism between ${\Sigma }_{-}$ and ${\Sigma }_{+}$ is a compact $n$-manifold $M$ whose boundary is ${\Sigma }_{-}\bigsqcup {\Sigma }_{+}$. If such $M$ exists, we say ${\Sigma }_{-}$ and ${\Sigma }_{+}$ are cobordant to each other.

Oriented Cobordism

Let ${\Sigma }_{-}$ and ${\Sigma }_{+}$ be two oriented closed manifolds of dimension $n-1$. An oriented cobordism from ${\Sigma }_{-}$ to ${\Sigma }_{+}$ is a compact oriented manifold $M$ together with orientation-preserving smooth maps ${\gamma }_{\pm }:{\Sigma }_{\pm }\to M$ such that ${\Sigma }_{\pm }$ maps diffeomorphically onto the in/out-boundary of $M$.

Cobordism Equivalence

Two oriented cobordisms $(M,{\gamma }_{-},{\gamma }_{+})$ and $(M^{\prime },{\gamma }_{-}^{\prime },{\gamma }_{+}^{\prime })$ from ${\Sigma }_{-}$ to ${\Sigma }_{+}$ are equivalent if there is an orientation-preserving diffeomorphism $\varphi :M\to M^{\prime }$ such that the following diagram commutes:

Clearly this is an equivalent relation and we can accordingly define a cobordism class.

Axiomatic TQFTs

Atiyah first gave a set of axioms for a topological quantum field theory in 1988, for which is now characterized as a symmetric monoidal functor. The following is a modernized version of Atiyah’s original axioms:

An $n$-dimensional topological quantum field theory (TQFT) is a rule $\mathcal{A}$ that assigns to each closed $(n-1)$-dimensional manifold $\Sigma$ a vector space $\mathcal{A}(\Sigma )$, and to each $n$-dimensional cobordism $M:{\Sigma }_{-}\to {\Sigma }_{+}$ a linear map $\mathcal{A}(M):\mathcal{A}({\Sigma }_{-})\to \mathcal{A}({\Sigma }_{+})$, such that the following five axioms hold:

1. Two equivalent cobordisms must have the same image;
2. The cylinder $\Sigma \times [0,1]$ must be sent to the identity map on $\mathcal{A}(\Sigma )$;
3. $\mathcal{A}$ preserves the decomposition of cobordisms;
4. Disjoint union of manifolds must be sent to the tensor product of vector spaces, and disjoint union of cobordisms must be sent to the tensor product of linear maps;
5. The empty manifold must be sent to the ground field $\mathbb{K}$.

The first two axioms guarantee that the theory is “topological”, in the sense that it does not depend on any additional structure like metric or curvature. The fourth axiom is a standard principle of quantum mechanics that the state space of two independent systems is the tensor product of the two state spaces.

Proposition

The state space $\mathcal{A}(\Sigma )$ assigned to a closed $(n-1)$-manifold $\Sigma$ is always finite-dimensional.

Proof To show $V:=\mathcal{A}(\Sigma )$ is finite dimensional, it suffices to find a spanning set for $V$. Note that we have the following diagram commutes:

$\gamma$ is a copairing $\text{K}\to W\otimes V$, suppose that $\gamma (1_{\text{K}})$ is the finite sum ${\sum }_i{c}_i{w}_i\otimes v_i$, then for each $u\in V$, there holds $u=(\beta \otimes 1_V)(1_V\otimes \gamma )(u)=(\beta \otimes 1_V)\left({\sum }_i{c}_{i}u\otimes w_i\otimes v_i\right)={\sum }_i{c}_i\beta (u\otimes w_i)v_i.$This means $\{v_i{\}}_i$ forms a spanning set for $V$. $\square$

The Cobordism Category

Cobordism Category

The cobordism category $n\mathsf{Cob}$ is the category whose objects are closed $(n-1)$-manifolds, and morphisms are cobordism classes between them. The composition of morphisms is given by gluing of cobordisms, and the identity morphism on $\Sigma$ is given by the cobordism class represented by the cylinder $\Sigma \times [0,1]$.

Cobordism Classes Induced by Diffeomorphisms

Proposition

Two diffeomorphisms ${\varphi }_0,{\varphi }_1:{\Sigma }_0\to {\Sigma }_1$ induce the same cylinder cobordism class if and only if they are smoothly homotopic.

Proof We have the following diagram commute as ${\varphi }_0$, ${\varphi }_1$ are homotopic:

Let us define $\eta :{\Sigma }_0\times I\to {\Sigma }_1\times I,(x,t)\mapsto (\varphi (x,t),t),$so that the diagram

commute. This shows that the induced cobordisms are equivalent. The other implication follows by composing $\eta :{\Sigma }_0\times I\to {\Sigma }_1\times I$ with the projection ${\Sigma }_1\times I\to {\Sigma }_1$, we get a homotopy from ${\varphi }_0$ to ${\varphi }_1$. $\square$

Disjoint Union and Twists

Twist Map

Functorial TQFTs

Topological Quantum Field

An $n$-dimensional topological quantum field is a symmetric monoidal functor from $(n\mathsf{Cob},\bigsqcup ,\varnothing ,T)$ to $({\mathsf{V}}_{\mathbb{K}},\otimes ,\mathbb{K},\sigma )$.