Hearing Exotic Smooth Structures

This paper explores the existence and properties of \emph{basic} eigenvalues and eigenfunctions associated with the Riemannian Laplacian on closed, connected Riemannian manifolds featuring an effective isometric action by a compact Lie group. Our primary focus is on investigating the potential existence of homeomorphic yet not diffeomorphic smooth manifolds that can accommodate invariant metrics sharing common basic spectra. We establish the occurrence of such scenarios for specific homotopy spheres and connected sums. Moreover, the developed theory demonstrates that the ring of invariant admissible scalar curvature functions fails to recover the smooth structure in many examples. We show the existence of homotopy spheres with identical rings of invariant scalar curvature functions, irrespective of the underlying smooth structure.