Given a TFT $Z$, I aim to calculate the partition function $Z(\mathbb{S}^2)$ as discussed by Lurie on page-7, Example 1.2.1 in:
https://arxiv.org/abs/0905.0465
If I am not wrong I do need to show that the value $Z(\mathbb{S}^2)$ is independent of the choice of the cross-section used to decompose $\mathbb{S}^2$ into a 'cup' and a 'cap'. In this case it is easy to see that this is inddeed the case. However, let me approach this problem differently. If we choose a different cross-section $S_2$ ($S_1$ being the first), and if $S_1S_2$ forms the boundary of a cylinder $C = (-S_1) \sqcup S_2$, then the result follows from the fact that fact that $Z$ is a symmetric-monoidal functor to the category of $\mathbb{K}$-vector spaces, using the composite $$ \emptyset \to S_1 \to S_2 \to \emptyset $$ My question is: what will happen if $S_1$ and $S_2$ do not form a cylinder and instead intersects (for instance when both $S_1$ and $S_2$ are great circles)? Definitely, I am viewing $\mathbb{S}^2$ as embedded in $\mathbb{R}^3$, and I think even when $S_1$ and $S_2$ intersects in embedded picture, there exists a representative in the diffeomorphism class of $\mathbb{S}^2$ where $S_1$ and $S_2$ does form a cylinder and the result follows. Am I right? My actual problem is much more serious in nature but while working on that this problem caught my attention. Maybe I am asking a stupid/obvious question.