Let $(W, \partial W)$ be an $n$-dimensional manifold with boundary. Suppose that $(W', \partial W')$ is obtained from $(W, \partial W)$ by attaching a $k$-handle via an embedding $\phi: S^{k-1}\times D^{n-k} \hookrightarrow \partial W$ and suppose that $(W'', \partial W'')$ is obtained from $(W', \partial W')$ by attaching a $(k+1)$-handle via an embedding $\psi: S^{k}\times D^{n-k-1} \hookrightarrow \partial W'$. If the belt sphere of the $k$-handle intersects the attaching sphere of the $(k+1)$-handle transversely in a single point then the handles cancel.
Does the result still hold if the second handle attachment is of index $(k-1)$ instead of index $(k+1)$?