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It is a well-known result in homotopy theory that a fibration $F \rightarrow E \rightarrow B$ induces a long exact sequence in the homotopy groups; namely,

$$\pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F)\rightarrow \cdots \pi_1(B) \rightarrow \pi_0(F) \rightarrow \pi_0(E) \rightarrow \pi_0(B).$$

My concern is, what does exactly mean being exact at the level of the $0$-th Homotopy groups? In general these are not groups. In particular, is there a geometric interpretation of the connecting map $\pi_1(B) \rightarrow \pi_0(F)$?

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1 Answer

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$F, E, B$ are all supposed to be pointed spaces here, and so their $\pi_0$ are pointed sets. The definition of exactness for a sequence of pointed sets is the same as for a sequence of groups: it means the kernel of one map (defined as the preimage of the distinguished point) is the image of another.

The map $\pi_1(B) \to \pi_0(F)$ is given by applying the monodromy action of $\pi_1(B)$ on $\pi_0(F)$ to the distinguished point of $\pi_0(F)$.

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