TITLE: Exact sequences of the cohomology induced by fiber bundle
QUESTION [6 upvotes]: 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.

REPLY [8 votes]: 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.