Topological Hecke eigenforms

Abstract

We study the eigenforms of the action of A. Baker’s Hecke operators on the holomorphic elliptic homology of various topological spaces. We prove a multiplicity one theorem (i.e., one-dimensionality of the space of these “topological Hecke eigenforms” for any given eigencharacter) for some classes of topological spaces, and we give examples of finite CW-complexes for which multiplicity one fails. We also develop some abstract “derived eigentheory” whose motivating examples arise from the failure of classical Hecke operators to commute with multiplication by various Eisenstein series, or non-cuspidal holomorphic modular forms in general. Part of this “derived eigentheory” is an identification of certain derived Hecke eigenforms as the obstructions to extending topological Hecke eigenforms from the top cell of a CW-complex to the rest of the CW-complex. Using these obstruction classes together with our multiplicity one theorem, we calculate the topological Hecke eigenforms explicitly, in terms of pairs of classical modular forms, on all 2-cell CW complexes obtained by coning off an element in \(\pi _n(S^m)\) which stably has Adams–Novikov filtration 1.