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Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology

$$\mathcal{H}(E) \simeq H(E).$$

A classical example with differential forms ($E = (\Omega,d)$) identifies the space of harmonic forms with de Rham cohomology. This shows that the operator $(\Delta -0)$ is able to recover topological information of the space.

Question How about other eigenspaces? Namely, can $(\Delta -\lambda)$ tell us anything interesting (preferably in the topological aspect)?

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    $\begingroup$ This wouldn't be invariant of the metric which would make it, at least naively, hard to recover topological information from it. $\endgroup$
    – Will Sawin
    Commented Apr 10, 2022 at 20:30
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    $\begingroup$ I meant the dimension wouldn't be invariant. Consider the case of a torus, with the flat metric. Eigenspaces can be computed by Fourier decomposition. $\endgroup$
    – Will Sawin
    Commented Apr 10, 2022 at 22:03
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    $\begingroup$ As @WillSawin comments, the dimensions of eigenspaces of Laplace-Beltrami operators are not invariant under perturbations of metric. For example, the "square" torus has Laplacian eigenvalues essentially equal to sums of integer squares. The multiplicities are understandable via Gaussian integers. Slightly changing one of the lengths makes all the multiplicities $1$, instead (for incommensurate side lengths this follows from the obvious lazy proof.) :) $\endgroup$
    – paul garrett
    Commented Apr 11, 2022 at 0:03
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    $\begingroup$ You may be interested in Bott, "Morse theory indomitable", which explains an idea of Witten's for how the eigenspaces of a deformed Laplacian, $\Delta_s$, can be related to Morse theory. $\endgroup$
    – Jesse Silliman
    Commented Apr 11, 2022 at 1:19
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    $\begingroup$ While the dimensions of individual eigenspaces depend on the choice of metric, one can cook up invariants like Ray-Singer torsion by considering all these dimensions at once. To get something truely topological, one needs to twist with an acyclic flat vector bundle. $\endgroup$
    – Sebastian Goette
    Commented Apr 11, 2022 at 6:50

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