I'm interested in doing $2$-surgeries to $\sharp^k S^1 \times S^3$. That is to the manifold obtained from applying $1$-surgeries to $S^4$.

Since $\pi_1(O(3)) = \mathbb{Z}_2$, there are two possible framings up to equivalence. Is it possible to distinguish between them in a natural way (similar to how orientability of the resulting manifold works in some other cases)? Maybe something coming from complex geometry?

One more or less natural way I'm thinking about is using Akbulut's convention in the following way. Let $M$ be the manifold obtained by $1$ and $2$ surgeries applied to $S^4$. The handle decomposition of $M$ can be described as follows. There are $k$ $1$-handles. There is a $2$-handle for every $2$-surgery. Then we double this $2$-handlebody to obtain $M$, thus getting twice as many $2$-handles: for every original $2$-handle there will be another handle whose attaching circle (nullhomotopic in $1$-handlebody) is linked with the attaching circle of the original handle. Using Akbulut's dotted circle convention we can assign an integer to every framing of $2$-handles. The handles corresponding to doubling will have framing $0$. If there are no $1$-handles, then even integers will correspond to the same manifold, and odd to the other one (as explained in Gompf and Stipsicz). Does the same hold in the presence of $1$-handles?

If we obtain the answer, can we see which framing corresponds to the manifold obtained by taking the boundary of a neighbourhood of a $2$-complex embedded in $\mathbb{R}^5$? Like in the following construction: finite generated group realized as fundamental group of manifolds , Constructing 4-manifolds with fundamental group with a given presentation.