Anyone answer with good explanation is appreciated.
In differential geometry, we discuss about topological quantities like characteristic classes.
For example, the first Chern character of some curvature 2-form $F=dA$ in $2D$, $$ \int F = \frac{1}{2}\int F_{\mu\nu}dx^{\mu}\wedge dx^{\nu}=\frac{1}{2}\int(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})dx^{\mu}\wedge dx^{\nu} $$
Another example is the "winding number" in $3D$, $$ \frac{1}{24\pi^2}\int(U^{-1}dU)^3$$
My questions is, does the relative signature of the metric affect these quantities?
If yes, How?
If no, why and how to understand it intuitively?
For instance, one can choose a metric $[g_{\mu\nu}]=\mbox{diag}\{-1,1\}$ or $\mbox{diag}\{1,1\}$ in $2D$. (Also different relative signs in $3D$.) It seems that it does not change the explicit expression of the Chern character. (Please tell me if I am wrong here.)
A topological property does not depend on higher level structures like a metric or a connection. In fact the Chern number is even independent of the connection you use to calculate it (there are connections which are not induced by a metric). One way of seeing this would be to look at a purely topological definition of the quantities you are looking at. Another is to realise that in the expressions you are using the metric does not appear anywhere.