2022/07/27 by Block, Jonathan, Higson, Nigel, Sanchez, Jesus
#19K56 #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT) #Primary 19D55 #Secondary 58J40
paper · doi:10.48550/arxiv.2207.13411
We shall present a simplified version of a construction due to Denis Perrot that recovers the Todd class of the complexified tangent bundle from a JLO-type cyclic cocycle. The construction takes place within an algebraic framework, rather than the customary functional-analytic framework for the JLO theory. The series expansion for the exponential function is used in place of the heat kernel from the functional-analytic theory; the Dirac operator chosen is far from elliptic; and a remarkable new trace discovered by Perrot replaces the operator trace. In its full form Perrot's theory constitutes a wholly new approach to index theory. The account presented here covers most but not all of this approach.