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The topological Petersson product

The nondegeneracy of the Petersson inner product on cusp forms, and the fact that Hecke operators are self-adjoint with respect to the Petersson product, together imply that the cusp forms have a basis consisting of Hecke eigenforms. In the literature on topological modular forms, no topological analogue of the Petersson product is to be found, and it is not known which topological spaces have the property that their topological cusp forms admit a basis consisting of eigenforms for the action of Baker's topological Hecke operators. In this note we define and study a natural topological Petersson product on complexified topological cusp forms, whose value on a one-point space recovers the classical Petersson product. We find that the topological Petersson product is usually degenerate: in particular, if $X$ is a space with nontrivial rational homology in any positive degree, then the Petersson product on complexified $tcf^*(X)$ is degenerate. Despite $tcf$-cohomology being a stable invariant, the topological Petersson product is an unstable invariant, vanishing on all suspensions. Nevertheless, we demonstrate nontriviality of the topological Petersson product by giving an explicit calculation of the topological Petersson product on complexified $tcf$-cohomology of the complex projective plane. We show that, for a compact Kahler manifold $X$, the Petersson product is nondegenerate on complexified $tcf$-cohomology of $X$ in a range of Atiyah-Hirzebruch filtrations (essentially one-third of the possibly-nonzero Atiyah-Hirzebruch filtrations in the $tcf$-cohomology of $X$).