# Rational surgery and attaching $2$-handles

It is well-known fact that integral Dehn surgeries on $$3$$-sphere $$S^3$$ are viewed as the result on the boundary of attaching $$2$$-handles $$B^2 \times B^2$$ to the $$4$$-ball $$B^4$$.

Is there an analogue of rational surgeries relating handle attachment of rational framing? If not, which problem occurs?

To attach a 4-dimensional 2-handle to the 4-ball, one requires an attaching region in $$S^3=\partial B^4$$ and a map from the attaching region of the handle (which has a natural parametrization as $$S^1\times D^2\subset \partial(D^2\times D^2)$$) to the attaching region in $$S^3$$. The attaching region in $$S^3$$ is determined by specifying a knot $$K\subset S^3$$ (and then convention dictates that the attaching region $$\nu(K)\cong S^1\times D^2$$ is parametrized by identifying the Seifert longitude $$\lambda$$ for $$K$$ with $$S^1\times\{pt\}$$). Thus the handle may be attached via any orientation reversing homeomorphism from the $$S^1\times D^2$$ in the boundary of the handle to the $$S^1\times D^2$$ neighborhood of $$K$$. There are only an integers worth of such maps up to isotopy (see eg Rolfsen Knots and Links 2D4 and 2E5); in particular $$S^1\times \{pt\}$$ has to be mapped to $$\lambda+n\mu$$ and $$\{pt\}\times \partial D^2$$ has to be mapped to $$\mu$$, where $$\mu$$ denotes a meridian of $$K$$.
The resulting boundary after the handle attachment should be thought of as (the bits of the boundary of the handle that didn't get stuck to anything)$$\cup$$(the bits of the boundary of $$S^3$$ that didn't get something stuck to them) . That's $$(D^2\times S^1) \cup (S^3\smallsetminus\mathring{\nu(K))}$$, so the boundary is some Dehn surgery on $$K$$. And we can see which; we had to send $$\partial D^2\times \{pt\}$$ to $$\lambda+n\mu$$, so the only surgeries we can obtain are integral.