# Relative Chern character number and super-connection

## TL;DR

This paper derives formulas to compute the relative Chern character number for complex vector bundles with specific singularities and applies it to express the index of a twisted Dirac operator on spin manifolds with sphere bundle structures.

## Contribution

It introduces a new formula for the relative Chern characteristic number in the presence of singularities on odd-dimensional spheres and relates it to the index of twisted Dirac operators on spin manifolds.

## Key findings

- Formula for relative Chern characteristic number with singularities
- Expression for the index of twisted Dirac operators on sphere bundles
- Application to spin manifolds with specific bundle structures

## Abstract

For two complex vector bundles admitting a homomorphism, whose singularity locates in the disjoint union of some odd--dimensional spheres, we give a formula to compute the relative Chern characteristic number of these two complex vector bundles. In particular, for a spin manifold admitting some sphere bundle structure, we give a formula to express the index of a special twisted Dirac operator.