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.



Exact sequences of the cohomology induced by fiber bundle
What you are asking about is a consequence of a more general statement about fibrations. Let $p: E \to B$ be a fibration with $B$ path connected and based. Set $F = p^{-1}(\*)$. 
Assume $B$ is $r$-connected and $p$ is $s$-connected. Then there's a exact sequence
$$
0 \to H^0(B) \to H^0(E) \to H^0(F) \to H^1(B) \to \cdots \to H^{r+s}(F) \to H^{r+s+1}(B) 
$$
One way to prove this is to show that the evident map $E \cup CF \to B$, whose domain is the mapping cone of $F\to E$, is $(r+s+2)$-connected. There are a variety of ways to show this, one of which is called the "dual Blakers-Massey Theorem." Then the long exact cohomology sequence of the cofiber sequence $F \to E \to E\cup CF$ combined with the connectivity statement gives what you want.


