# (Topological quantum field theory) identifying objects of cobordism category

I am beginning the study of Topological quantum field theory(TQFT) and I am confused with the basic notions. Before writing down the question, to check if I understood the definition correctly, I state what I learned as follows. Also I will only deal with the 2-dimensional TQFT here.

1. TQFT assigns a $\mathbb{C}$-vector space Z(M) to each 1-dim closed oriented manifold M.

2. TQFT assigns a vector Z(L)$\in$Z($\partial$L) to each 2-dim closed oriented manifold with boundary L.

3. Z(M1$\cup$M2)=Z(M1)$\otimes$Z(M2), Z(M$^*$)=Z(M)$^*$ (M$^*$ is the orientation reversal of M, Z(M)$^*$ is the dual of Z(M))

4. For orientation preserving diffeomorphism $\phi : M_1\rightarrow M_2 (M_1,M_2 : \text{1 dim closed manifold})$ there is an induced vector space isomorphism $\phi_{*} : Z(M_1)\rightarrow Z(M_2)$.(plus some naturality)

I omitted other axioms because these seems to be enough for my question. Using 3, we can interpret a vector as a linear map between vector spaces. With this, The theory says that the TQFT assigns Id$_{Z(S^1)}$to the cylinder S$^1\times$I, whose boundary is disjoint union of two S$^1$. Of course they are diffeomorphic, but for later convenience, let's call them S$^1_a$and $S^1_b$. Next I thought as below.

1. "every closed oriented 1 dim manifold is diffeomorphic to each other(~$S^1$), even up to orientation preserving diffeomorphism. so there exists isomorphism Z($S^1_a$)~Z($S^1_b$)~Z($S^1$)~Z($S^{1*}$). But we are not identifying the objects itself via orientation preserving diffeomorphism because both notion Z($S^1$) and Z($S^{1*}$) are appearing in the texts."

2. "A choice of the orientation of the cylinder induces orientation($\mu$) on its boundary. since $S^1_a$ and $S^1_b$ are already oriented manifold, comparing this with induced orientation $\mu$ will determine wheter to write each $S^1_a, S^1_b$ as oriented the same way($S^1_i$), or reversely oriented($S^{1*}_i$)."

Because $S^1_a$ and $S^1_b$ are not equal to $S^1$ set theoretically,(yet we do not mod out by diffeomorphism because of "1"?!) so to say this cylinder(S$^1\times$I) gives a vector in Z($S^1$)$\bigotimes$Z($S^{1*}$), I thought that I need to choose a diffeomorphism of cylinder and use some naturality axiom to identify Z($S^1_a$)$\bigotimes$Z($S^1_b$) with Z($S^1$)$\bigotimes$Z($S^{1*}$). But I'm not sure this method makes sense since there are lots of (orientation preserving)diffeomorphisms between closed oriented 1 dim manifold. Considering all this, I am also suspicious that why it corresponds to an element of Z($S^{1*}$)$\bigotimes$Z($S^1$) but not an element of Z($S^1$)$\bigotimes$Z($S^1$).

I hope my word was understandable. In short, would anyone help me with the identification problem of Z($S^1_a$)$\bigotimes$Z($S^1_b$) and determining its orientation compared to specified $S^1$? Great Thanks.

I followed the defintion of TQFT in

Atiyah M. , Topological quantum field theory , p177-181

Lawrence R.J. , Introduction to topological quantum field theory , p5-7.