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.

Appendix A Appendix on cobar complexes

In this paper, we occasionally must refer to cocycles in the cobar complex of a Hopf algebroid $(A,\Gamma )$ with coefficients in a left $\Gamma$-comodule M. The standard reference for Hopf algebroids is Appendix 1 of [25], and the (two-sided) cobar complex of $(A,\Gamma )$ is defined in Definition A1.2.11 of [25]. For convenience, we recall the one-sided version of that definition here, as well as a simplification that occurs when the coefficient comodule M is A itself.

Definition A.1

Let $(A,\Gamma )$ be a Hopf algebroid with left unit map $\eta _L:A \rightarrow \Gamma$, right unit map $\eta _R:A \rightarrow \Gamma$, and coproduct $\Delta : \Gamma \rightarrow \Gamma \otimes _A \Gamma$. Let M be a left $\Gamma$-comodule with structure map $\psi : M \rightarrow \Gamma \otimes _A M$. Then the (one-sided) cobar complex of $(A,\Gamma )$ with coefficients in M is the Moore complex²¹

(46)

whose 0th coface map $C^n \rightarrow C^{n+1}$ is $\Delta \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \dots \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A {{\,\mathrm{{id}}\,}}_M$, whose 1st coface map $C^n \rightarrow C^{n+1}$ is ${{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A\Delta \otimes _A{{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \dots \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A {{\,\mathrm{{id}}\,}}_M$, and so on, up through its $(n-1)$st coface map $C^n \rightarrow C^{n+1}$ given by ${{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \dots \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A\Delta \otimes _A {{\,\mathrm{{id}}\,}}_M$, and with its last (that is, nth) coface map $C^n \rightarrow C^{n+1}$ given by ${{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \dots \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \psi$.

When $M = A$, the above simplifies, and (46) is isomorphic to the cosimplicial abelian group

(47)

whose 0th coface map $\tilde{C}^0 \rightarrow \tilde{C}^1$ is $\eta _R$, and whose 1st coface map $\tilde{C}^0 \rightarrow \tilde{C}^1$ is $\eta _L$; and, when $n>0$, whose 0th coface map $\tilde{C}^n \rightarrow \tilde{C}^{n+1}$ sends $x_1\otimes \dots \otimes x_n$ to $1\otimes x_1\otimes \dots \otimes x_n$, whose nth coface map $\tilde{C}^n \rightarrow \tilde{C}^{n+1}$ sends $x_1\otimes \dots \otimes x_n$ to $x_1\otimes \dots \otimes x_n\otimes 1$, and whose intermediate coface maps $\tilde{C}^n \rightarrow \tilde{C}^{n+1}$ are given by letting the 1st coface map be $\Delta \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \dots \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }$, letting the 2nd coface map be ${{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A\Delta \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }\otimes _A \dots \otimes _A {{\,\mathrm{{id}}\,}}_{\Gamma }$, and so on.

The above conventions are standard, and agree with the more well-known standard conventions when working with Hopf algebras. When working with Hopf algebroids, an extra wrinkle is introduced by having two unit maps: the convention when working with Hopf algebroids is that, when we tensor $\Gamma$ over A on the left, we use the A-module structure on $\Gamma$ given by the left unit map $\eta _L: A\rightarrow \Gamma$, and when we tensor $\Gamma$ over A on the right, we use the A-module structure on $\Gamma$ given by the right unit map $\eta _R:A\rightarrow \Gamma$. So, for example, since $\eta _L(v_1) = v_1$ and $\eta _R(v_1) = v_1 + pt_1$ in the Hopf algebroid $(BP_*,BP_*BP)$ of stable co-operations in Brown–Peterson homology, we have that

$$\begin{aligned} v_1 \otimes t_1 = (\eta _R(v_1) - pt_1)\otimes t_1 = 1\otimes v_1t_1\ -\ pt_1 \otimes t_1\in BP_*BP\otimes _{BP_*}BP_*BP\cong \tilde{C}^2. \end{aligned}$$

With these conventions in place, it is an exercise to verify that (32) is a 1-cocycle in $\tilde{C}^{\bullet }$.