# Inflation of $w_j(V_{SO(N)})$ and $w_j(M)$ from $SO(N)$ to $Spin(N)$ or Spin geometry

We know well this short exact sequence $$1 \to \mathbb{Z}_2 \to Spin(N) \to SO(N) \to 1.$$ The $$j$$-th Stiefel-Whitney class of the associated vector bundle of $$SO(N)$$, as $$w_j(V_{SO(N)})$$, can be nontrivial in general,

$$w_j(V_{SO(N)}) \in H^j(BG,\mathbb{Z}_2)\overset{?}{=}H^j(BSO(N),\mathbb{Z}_2).$$ Here I suppose the $$G=SO(N)$$. We can either use the topological cohomology $$H^j(BG,\mathbb{Z}_2)$$ of classifying space $$BG$$, or consider the group cohomology H$$^j(G,\mathbb{Z}_2)$$ of the group $$G$$.

If the corresponding vector bundle of is the tangent bundle of the base manifold $$M$$, we have $$w_j(M) \in H^j(M,\mathbb{Z}_2).$$

Questions: In either case,

when we lift from

• $$V_{SO(N)}$$ to $$V_{Spin(N)}$$.

or

• The base manifold $$M$$ with the SO-structure ($$SO(N)$$) to the $$M'$$ with the Spin-structure ($$Spin(N)$$),

how do we see explicitly that

• the nontrivial $$w_j(V_{SO(N)}) \in H^j(BG,\mathbb{Z}_2)\overset{?}{=}H^j(BSO(N),\mathbb{Z}_2)$$ and
• the nontrivial $$w_j(M) \in H^j(M,\mathbb{Z}_2)$$,

when the $$w_j$$ is inflated from $$SO$$ to $$Spin$$, and $$w_j$$ becomes its trivialization (a coboundary or just vanish)? Namely, how to show:

• $$w_j(V_{Spin(N)})=0$$.

• $$w_j(M')=0$$.

I know the statements are almost obviously true by definition, but can we precisely write down a (de Rham?) manifold-cocycle or group-cocycle on the simplicial complex (well-triangulated), such that we can show both - $$w_j$$ becomes a coboundary (instead of a cocycle) when lifting to $$Spin(N)$$? Namely,

• $$w_j(V_{Spin(N)})=\delta \beta_{j-1}(V_{Spin(N)})$$.

• $$w_j(M')=\delta \beta_{j-1}(M')$$.

The best answer to me is also to introduce the proper way to write down the $$j$$-cocycle $$w_j(V_{SO(N)})$$ and $$w_j(M)$$ on the SO or Spin manifold; and see how they can be split to the $$(j-1)$$-cochains when pulling backward from $$Spin(N) \to SO(N)$$.