Topological Elliptic Genera I -- The mathematical foundation

TL;DR

This paper develops the mathematical foundation for Topological Elliptic Genera, homotopy-theoretic refinements of elliptic genera for specific manifolds, with applications including divisibility results for Euler numbers.

Contribution

It introduces the construction of Topological Elliptic Genera as homotopy-theoretic refinements and explores their applications in topology.

Findings

Derived divisibility results for Euler numbers of Sp-manifolds.

Established the foundational framework for Topological Elliptic Genera.

Connected the genera to G-equivariant Topological Modular Forms.

Abstract

We construct {\it Topological Elliptic Genera}, homotopy-theoretic refinements of the elliptic genera for $SU$-manifolds and variants including the Witten-Landweber-Ochanine genus. The codomains are genuinely $G$-equivariant Topological Modular Forms developed by Gepner-Meier, twisted by $G$-representations. As the first installment of a series of articles on Topological Elliptic Genera, this issue lays the mathematical foundation and discusses immediate applications. Most notably, we deduce an interesting divisibility result for the Euler numbers of $Sp$-manifolds.