Kernels of Signature Operators Twisted by Almost Flat Bundles Are Not Homotopy Invariant

At an Oberwolfach problem session in 2006, T. Schick asked the following. Fix closed Riemannian manifolds and a homotopy equivalence f. Do the signature operators twisted by a bundle E and by f*E have isomorphic kernels once the curvature of E is small enough? By a theorem of Hilsum and Skandalis, their indices agree. Sauer and Schick later asked the same for the twisted Betti numbers. We show that the answer is no, by an elementary example. On the flat square torus let f(x,y) = (x + (c/2π) sin 2πy, y), a diffeomorphism isotopic to the identity. Let E_ε be the trivial Hermitian line bundle with the connection d + iε cos(2πy) dx, whose curvature has norm 2π|ε|. The twisted signature operator D = d_∇ + d_∇* has a four-dimensional kernel for E_ε, for every ε, and a zero kernel for f*E_ε whenever εc ∉ 4πℤ; for its chiral half D⁺ the dimensions are 2 and 0. The pull-back moves the harmonic part of the connection form by the curvature flux εc/2. Products give examples in every dimension ≥ 2, for instance kernels of dimensions 16 and 0 on the 4-torus (8 and 0 for D⁺). The answer stays negative for the underlying Euclidean rank-two bundle, for the spin Dirac operator, and for the natural readings of degree-wise twisted Betti numbers; for non-flat bundles the kernel is in general graded only by parity, not by degree. By contrast, the number of eigenvalues near zero is stable: if r = |ε|(1 + c²)^{1/2} < π, both operators have exactly four eigenvalues in [−r, r]. This is an unrefereed note.