I'm reading section 2.1 of Lawson's book, Spin Geometry. The book states the following fact. Let $X$ be a manifold and $E$ a vector bundle over it. Equip $E$ with a Riemannian structure. Let $P_O$ be the bundle of orthonormal frames in $E$ which is a principal $O_n$ bundle. The fibration $O_n \rightarrow P_O(E) \rightarrow X$ gives an exact sequence $0 \rightarrow H^{0}(X;\mathbb{Z}_2) \rightarrow H^{0}(P_O(E);\mathbb{Z}_2) \rightarrow H^{0}(O_n;\mathbb{Z}_2) \rightarrow H^{1}(X;\mathbb{Z}_2) $ and the fibration $SO_n \rightarrow P_{SO}(E) \rightarrow X$ gives another exact sequence $0 \rightarrow H^{1}(X;\mathbb{Z}_2) \rightarrow H^{1}(P_{SO}(E);\mathbb{Z}_2) \rightarrow H^{1}(SO_n;\mathbb{Z}_2) \rightarrow H^{2}(X;\mathbb{Z}_2) $. Lawson only says that we can deduce them from Serre spectral sequence but I don't know how. Could someone give an explicit recipe? (By the way, we are around page 79 to page 81.)

Thank you.

Serre spectral sequence. It's deduced from the Serre spectral sequence as in Example 1.A of McCleary's "User's guide to spectral sequences". $\endgroup$ – Mark Grant Oct 13 '13 at 20:152more comments