The equivariant degree and an enriched count of rational cubics
Candace Bethea, Kirsten Wickelgren
TL;DR
The paper develops an equivariant degree theory for proper $G$-maps between smooth $G$-manifolds relative to genuine $G$-spectra $E$, with a local degree at regular points and a local-to-global principle that expresses the global degree as a transfer-weighted sum of local degrees. This framework is then used to compute equivariant Euler characteristics and Euler numbers, providing a principled way to obtain enriched, group-valued counts in equivariant enumerative geometry. As a major application, the authors give an equivariantly enriched count of rational plane cubics through a $G$-invariant set of eight general points, yielding results in both the Burnside ring $A(G)$ and the representation ring $R(G)$; in the special case $G=\mathbb{Z}/2$ acting by complex conjugation, the construction recovers Welschinger-style real counts. The final sections present an explicit Euler-number computation for a Hom-bundle, delivering a concrete $R(G)$-valued formula with rank 12, illustrating the framework’s concrete computability and potential for explicit equivariant enumerative predictions.
Abstract
We define the equivariant degree and local degree of a proper $G$-equivariant map between smooth $G$-manifolds when $G$ is a compact Lie group and prove a local to global result. We show the local degree can be used to compute the equivariant Euler characteristic of a smooth, compact $G$-manifold and the Euler number of a relatively oriented $G$-equivariant vector bundle when $G$ is finite. As an application, we give an equivariantly enriched count of rational plane cubics through a $G$-invariant set of 8 general points in $\mathbb{C}\mathbb{P}^2$, valued in the representation ring and Burnside ring of a finite group. When $\mathbb{Z}/2$ acts by pointwise complex conjugation this recovers a signed count of real rational cubics.