I want to prove that all line bundles with the same base $B$ form group with operation $\otimes$ which isomorphic to $H^1(B,\mathbb{Z}_2)$. It's obvious to me, that there is bijection between line bundles and $H^1(B,\mathbb{Z}_2)$ which exists because $B\mathbb{Z}_2 = K(\mathbb{Z}_2,1)$ but I dont understand why that group is isomorphism. I found some proof in icc prop. 3.7.12 but it uses concept of Euler characteristic class, and I want to some prove which uses only concept of Stiefel Whitney class, does that exist? Hope for your help.
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1I would recommend this explanation in terms of sheaf cohomology. – Ted Shifrin Apr 01 '17 at 19:11
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Oh, thanks! Beautiful explanation! – Mykola Pochekai Apr 01 '17 at 19:18