Handle slides are not handle cancellations

A handle slide changes the attaching data of one handle and the handle-chain basis by an elementary operation, but it changes the number of handles of no index; a handle cancellation removes two handles of consecutive indices and decreases the handle counts in those two indices by one each. In particular a slide is not a cancellation, the algebraic effect of a slide (an elementary row or column operation on the intersection matrix) is not the removal of a unit entry from that matrix, and the two moves serve different purposes: slides perform basis changes, cancellation reduces the number of handles. Handles can never be removed one at a time, because the Euler characteristic of the pair is independent of the presentation.

For the numerical obstruction, the relative CW model of A handle decomposition gives a relative CW complex has one cell per handle. Its finite relative rational cellular complex computes relative homology by Relative cellular homology computes relative singular homology. Write each chain dimension as the sum of the incoming boundary rank, the homology dimension, and the outgoing boundary rank; in the alternating sum the boundary ranks cancel. Thus $\chi (W,{\mathrm{\partial }}_{0}W)={\sum }_q(-1{)}^q{p}_q$ is independent of the presentation. Removing just one handle changes this integer by $\pm 1$. Balancing that count is a necessary numerical condition, not a geometric cancellation criterion. A slide leaves every $p_q$ unchanged.