Index stability and homotopy invariance

For bounded operators between fixed Hilbert spaces, the Fredholm set is open in operator norm and the integer-valued index is locally constant there. Therefore a norm-continuous path T_t of Fredholm operators, with t in a connected interval, has constant index. A familiar finite-dimensional matrix path is a special case, but infinite-dimensional spaces allow components labeled by different indices. Pointwise or strong convergence is weaker than norm convergence and does not by itself give this conclusion.

Differential operators of a fixed order can be compared as bounded maps between fixed Sobolev spaces. Smoothly varying coefficients and a continuously elliptic principal symbol on a compact closed manifold give an appropriate continuous Fredholm family. For unbounded L² realizations one instead needs a controlled common domain or a suitable graph topology; merely writing a parameter in a differential expression does not specify such a family. Boundary conditions that vary must be included in the continuity analysis.

If K is compact and T is Fredholm, then every T+tK is Fredholm and has the same index as T. One proof passes to operators modulo compact operators: T has an inverse in that quotient, and adding K does not change its quotient class. Finite-rank corrections are especially concrete examples. On a closed manifold, changing only lower-order terms of an order-m elliptic operator gives a compact perturbation from H^s to H^(s−m), because the extra regularity feeds through a compact Sobolev inclusion.

An arbitrary bounded perturbation is different. Small perturbations preserve Fredholmness by openness, while a large one can leave the Fredholm set. A nonzero index obstructs invertibility throughout its component because an invertible operator has both defects zero. The converse fails: index zero does not produce an inverse automatically. It removes this particular numerical obstruction, leaving a finite-dimensional obstruction that a suitable finite-rank correction can remove for an individual Fredholm map.

A kernel dimension is upper semicontinuous in a suitable Fredholm family: additional zero modes can appear at exceptional parameters. The cokernel adjusts so that their difference remains fixed. In finite dimensions rank-nullity makes this cancellation transparent. In infinite dimensions the same phenomenon occurs inside the finite-dimensional blocks left after one splits off uniformly invertible complements. It is precisely these finite blocks that underlie both local constancy and the later construction of a families index.

The unilateral shift gives an infinite-dimensional example with a genuinely nonzero index. Sending the first basis vector to the second and continuing upward is an isometry with no kernel, but its image misses the first coordinate. Its cokernel has dimension one. Repeated shifts miss more coordinates, and compact modifications can move the missing-coordinate conditions without changing their number minus any new kernel. This makes an abstract quotient argument visible as ordinary linear constraints on sequences.