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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?

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