  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 $X$).

Algebraic Topology;Number Theory