# Stiefel Whitney classes on the simplex or the simplicial complex

The Stiefel Whitney classes of the base manifold $$M$$ are characteristic class as $$w_j(M) \in H^j(M,\mathbb{Z}_2),$$

• Puzzle: How do we write $$w_1(M) \in H^1(M,\mathbb{Z}_2)$$ $$w_2(M) \in H^1(M,\mathbb{Z}_2)$$ on the simplex or the simplicial complex of base manifold?

For example, how is it be different from the $$G$$-group cohomology cocycle on the base manifold $$M$$, say $$\omega_1(G) \in H^1(G,\mathbb{Z}_2)$$ $$\omega_2(G) \in H^1(G,\mathbb{Z}_2),$$ if I can pair the $$\omega_j(G)$$ to the base manifold $$M$$ through the fundamental classes?