My apology for the uninformative title; I don't think my question can be compressed into one line.
I'm trying to understand the relation between handle attaching and spin$^c$ structures. A particular case that I want to consider in this post is the following:
Let $X$ be a 3-manifold endowed with a spin$^c$ structure $\mathfrak{s}_1$, and consider a 1-handle attaching. It would give a cobordism $W$ between $X$ and $X'=(X$ with one 1-handle attached). And it is obvious that $X'\cong X \# (S^1\times S^2)$, hence they will be identified for now.
- In current situation, an obstruction theoretic argument shows that there exists a unique spin$^c$ structure, say $\mathfrak{t}$, on $W$ which extends $\mathfrak{s}$, i.e., $\mathfrak{t}|_{Y}=\mathfrak{s}$ holds. This is because the only (possibly) nontrivial obstruction to extend $\mathfrak{s}$ to $W$ lies in $H^3(W,Y;\mathbb{Z})=0$ (as the fiber of the fibration $Bspin^c(3)\to BSO(3)$ is $BU(1)\cong K(\mathbb{Z},2)$), and the difference cocyle lies in $H^2(W,Y;\mathbb{Z})=0$. Thus, we can say for sure that the 1-handle attaching gives a unique spin$^c$ structure, say $\mathfrak{s}'$ on $X'$.
- On the other hand, we have that $X'\cong X\#(S^1\times S^2)$, which is also obtained by a 1-handle attaching $X\sqcup (S^1\times S^2)\to X\# (S^1\times S^2)$. Here the 1-handle is connecting two different components $X$ and $S^1\times S^2$ along the connected sum regions. Again by the same obstruction theoretic argument, we know that for any pair of spin$^c$ structures $(\mathfrak{s},\tilde{\mathfrak{s}})$ where $\mathfrak{s}$ is on $X$ and $\tilde{\mathfrak{s}}$ is on $S^1\times S^2$, we obtain a unique spin$^c$ structure, often called $\mathfrak{s}\#\tilde{\mathfrak{s}}$, on $X'$ which is the restriction of the spin$^c$ structure on the cobordism $X\sqcup (S^1\times S^2)\to X'$ which extends $\mathfrak{s}\sqcup \mathfrak{s}'$.
Now the question: From the Kunneth formula on $X\# (S^1\times S^2)$ and the $H^2$-action on spin$^c$ structures it is expected that there exists a unique $\tilde{\mathfrak{s}}$ such that $\mathfrak{s'}=\mathfrak{s}\# \tilde{\mathfrak{s}}$ holds. Is it correct? If right, then for what $\tilde{s}$?
(And a small aside, correct me if wrong: if you consider the handles with higher index, then most of the question becomes nonsense; for example, for 2-handle attachings, we can always extend a spin$^c$ structure on $X$ to $W$, but the extension is not unique, so we cannot choose a unique spin$^c$ structure on $X'$ merely from $\mathfrak{s}$. For 3-handle attachings, it is not always possible to extend a spin$^c$ structure on $X$ to $W$, e.g. by reversing the cobordism $X\to X'$ as above.)