2024/10/19 by Liu, Hongzhi, Wang, Hang, Wang, Zijing +1 · 1 citation
#Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2410.15103
In this paper, we employ the framework of localization algebras to compute the equivariant K-homology class of the Euler characteristic operator, a central object in studying equivariant index theory on manifolds. This approach provides a powerful algebraic language for analyzing differential operators on equivariant structures and allows for the application of Witten deformation techniques in a K-homological context. Utilizing these results, we establish an equivariant version of the Poincaré-Hopf theorem, extending classical topological insights to the equivariant case, inspired by the results of Lück-Rosenberg. This work thus offers a new perspective on the localization techniques in the equivariant K-homology, highlighting their utility in deriving explicit formulas for index-theoretic invariants.