Even index pairings are integer-valued homotopy invariants combining an even Fredholm module with a -class specified by a projection. Numerous classical examples are known from differential and non-commutative geometry and physics. Here it is shown how to construct a finite-dimensional self-adjoint and invertible matrix, called the spectral localizer, such that half its signature is equal to the even index pairing. This makes the invariant numerically accessible. The index-theoretic proof heavily uses fuzzy spheres.
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Terry A. Loring, Hermann Schulz-Baldes, The spectral localizer for even index pairings. J. Noncommut. Geom. 14 (2020), no. 1, pp. 1–23DOI 10.4171/JNCG/357