Characteristic class integral: when does the equality hold $\int c_1 \wedge w_2 = \int c_1 \wedge c_1$, on what manifolds?
Here $c_1$ is the first Chern class. Here $w_2$ is the 2nd Stiefel-Whitney class.
I know that on a complex manifold (say ${\mathcal{M}^4}$ here), we have $w_2=c_1 \text{ mod } 2$, and that the following is true: $$\exp[i \pi (\int_{\mathcal{M}^4} c_1 \wedge w_2)] = \exp[i \pi (\int_{\mathcal{M}^4} c_1 \wedge c_1)],$$
effectively $[\int_{\mathcal{M}^4} c_1 \wedge w_2] \text{ mod } 2=[\int_{\mathcal{M}^4} c_1 \wedge c_1] \text{ mod } 2$
My question is that on what manifold does $$(1) \;\;\; [\int_{\mathcal{M}^4} c_1 \wedge w_2] \text{ mod } 2= [\int_{\mathcal{M}^4} c_1 \wedge c_1] \text{ mod } 2?$$ or $$(2) \;\;\;\int_{\mathcal{M}^4} c_1 \wedge w_2 = \int_{\mathcal{M}^4} c_1 \wedge c_1?$$ hold in general? Just for a complex manifold, or for any manifold? or $$(3) \;\;\; \exp[i \pi (\int_{\mathcal{M}^4} c_1 \wedge w_2)] = \exp[i \pi (\int_{\mathcal{M}^4} c_1 \wedge c_1)]?$$
Is there any Reference for a proof on when the equality (1) or (2) or (3) holds?
[Note add | Nov 18] : Let us write $c_1=F/2\pi$ where $F=dA$ is the field strength of a U(1) gauge field, so that we have an integration done by treating $c_1$ in terms of a differential form.