Families index and parameter dependence

Let B be a compact parameter space and T_b a norm-continuous family of Fredholm maps between Hilbert bundles, locally trivialized continuously. Around each parameter a finite-dimensional target correction makes T_b surjective. Compactness permits finitely many such corrections to be combined into a continuous map J from a fixed trivial bundle B×C^N into the targets, so the augmented family (T,J) is surjective everywhere. Its kernels then form a genuine finite-rank vector bundle K.

The index bundle is [K]−[B×C^N] in complex K⁰(B). Changing the correction or enlarging it changes both terms by compatible stabilizations and leaves the virtual class unchanged. When the original kernels and cokernels have locally constant dimensions, they do form bundles and give the familiar difference. Without that hypothesis, the notation [ker T]−[coker T] can only be shorthand for the stabilized class, not a literal subtraction of two ordinary bundles.

For a smooth proper submersion π:Z→B with closed even-dimensional fibers, choose a smoothly varying vertical Riemannian spin structure and compatible bundle connections. The vertical twisted chiral Dirac operators form an elliptic family. Their analytic K-class equals the topological family pushforward. After applying the rational Chern character, its class is integration along the fibers of vertical  times ch(E). The vertical tangent bundle and fiber orientation are essential: they are not the full tangent bundle of the total space.

The cohomological identity determines all rational even-degree components on B, not only degree zero. Positive-degree components can be nonzero even when every numerical index vanishes. Conversely, rational differential forms may miss torsion in the integral K-class. Superconnections provide differential-form representatives and control variation, but the K-theoretic index is defined without requiring constant-dimensional eigenspaces. Smoothness and the closed-fiber assumption prevent boundary domains or noncompact ends from being silently introduced into this theorem.

For a map between finite-rank bundles E and F, each fiber map is Fredholm and its index bundle is [E]−[F], whether or not its rank is constant. Thus even finite-dimensional examples can have jumping kernels and a nontrivial family index. Adding an identity on a trivial infinite-dimensional Hilbert bundle embeds such an example into the usual Hilbert-space Fredholm setting without changing the finite defects or virtual class.

The ranks of the virtual class are the ordinary indices at the parameters. They are locally constant, but equal rank does not mean equal K-class. A line bundle minus a trivial line has rank zero and may have a nonzero first Chern class. This is the family counterpart of the earlier relative-symbol example: topology lives in how vector spaces fit together, rather than just their dimensions. It is also why inspecting zero eigenvalues one parameter at a time cannot recover the complete index bundle.