Handle decompositions are not canonical

Handle decompositions are not canonical: different Morse functions, gradient-like fields, orderings of equal-index handles and choices of attaching data on the same triad can give different presentations of the same diffeomorphism type. Rearranging critical values, grouping equal-index handles, dualizing, and eliminating endpoint handles all change the presentation while preserving the underlying manifold. Consequently no invariant may be read from a presentation without an invariance argument, and the elementary moves that compare presentations (slides, cancellation, introduction of complementary pairs) are not constructed on this page.

The non-canonicity is exhibited piece by piece by the items of this page. The correspondence between Morse functions and presentations, Morse functions and handle decompositions correspond, produces a presentation from any adapted excellent Morse function, and different functions, fields and admissible attaching embeddings can give different presentations of the same $W$. Negating the function replaces every presentation by its dual, Handle duality from negating a Morse function, reversing the order of the handles and exchanging attaching and belt spheres. Rearranging the critical levels by index, Rearrangement of critical levels by index, and passing to a self-indexed function, Self-indexing Morse functions exist, changes the order in which the handles are attached and groups the handles of one index at a single level, without changing $W$; reordering equal-index handles and replacing an attaching embedding by an isotopic one is covered by Handles of equal index can be attached on one level. The two endpoint eliminations, Connected cobordisms admit presentations without superfluous zero handles and Dual elimination of top-index handles, delete or retain handles of extreme index according to the incoming and outgoing boundary, so even the number of handles in a presentation depends on the boundary data. Finally the empty presentation of a product, the product presentation, and the corner-rounding conventions of Attaching a smooth handle with corner rounding show that the attaching data themselves are a choice: the framing and the rounding of a handle are part of its presentation. Compatible changes of rounding preserve the diffeomorphism type; changing a framing is different attaching data and can change that type.

The consequence recorded here is a warning, not a theorem: whenever an invariant is computed from a presentation — a matrix, a chain complex, a torsion class — its definition must be accompanied by invariance under the moves that compare presentations. The construction of those elementary moves (handle slides, cancellation of complementary pairs, introduction of a cancelling pair, and the addition of handles) is not carried out on this page; it belongs to the later development of handle calculus, and this remark only records that such a calculus is necessary before any presentation-dependent quantity can be called an invariant of the manifold. No new proof is given here.