A virtual bundle with compact support is a triple $E$={$E_1$,$E_2$,$a$}, where $E_1$ and $E_2$ are bundles over $M$ of the same dimension, $M$ is a closed manifold, $a$ is a bundle map from $E_1$ to $E_2$, which is an isomorphism of bundles over $M \smallsetminus X$, where $X$ is a compact set in M.

Let $\nabla_1$ be a connection on $E_1$, $\nabla_2$ a connection on $E_2$. If $X$ is finite number of points in $M$, and Ch($E$) is defined by $\text{tr}(\exp(-\nabla_1)^{2})-\text{tr}(\exp(-\nabla_2)^{2})$, we will find Ch($E$) integrated on $M$ is an integer multiple of a power of $2\pi i$.

I want to know, if $X$ is a compact submanifold, then how to get the value of the integration? Maybe we can't get the value but how can we analyze the information about $X$ revealed by the integration?