Long exact sequence of a fibration, center

Question

Let $p:E \rightarrow B$ be a fibration with fiber $F$ . Associated to this we have a long exact sequence $$\cdots \rightarrow \pi_n(F) \rightarrow \pi_n(E) \rightarrow \pi_n(B) \rightarrow \pi_{n-1}(F) \rightarrow \cdots.$$

I am trying to show that the image of $\pi_2(B) $ in $\pi_1(F)$ is in the center of $\pi_1(F)$, but with no luck. Any help, solution or reference is welcome!

Answer 1

If $\alpha \colon S^2 \to B$, then the image $[\psi]$of $[\alpha]$ under the connecting homomorphism $\pi_2(B) \to \pi_1(F)$ is defined by the diagram below:

$$\require{AMScd} \begin{CD} S^1 @>{i}>> D^2 @>{q}>> S^2\\ @VV{\psi}V @VV{\hat{\alpha}}V @VV{\alpha}V\\ F @>{i}>> E @>{p}>> B \end{CD} $$

where the map $i \colon S^1 \to D^2$ is the inclusion of the boundary, the map $q \colon D^2 \to S^2$ is the quotient map taking $\partial D^2$ to a point, $p\hat{\alpha} = \alpha q$, and $\hat{\alpha} i = i \psi$. The map $\psi$ is well-defined up to homotopy.

Let $[\beta] \in \pi_1(F)$ have representative $\beta \colon S^1 \to F$ and consider $\beta.\psi.\bar{\beta}$ where $.$ denotes path concatenation and $\bar{\beta}$ is the reverse of the path $\beta$. Then $i(\beta.\psi.\bar{\beta}) = (i \beta).(i\psi).(i\bar{\beta}) = (i \beta).(\hat{\alpha}i).(i\bar{\beta})$.

From here, I think you can use a trivializing neighborhood for the basepoint in $B$ to argue that the terms in the latter expression commute (up to homotopy) and have that $i(\beta.\psi.\bar{\beta}) = \hat{\alpha}i$ (up to homotopy), but these details are eluding me at the moment. Hence $[\beta.\psi.\bar{\beta}] = [\psi] \in \pi_1(F)$.

Answer 2

There is a general result due to Quillen that for a (pointed) fibration $F \to E \to B$ the map $\pi_1(F) \to \pi_1(E)$ may be given the structure of crossed module. This is a variation of the fact due to J.H.C. Whitehead that for a pointed pair of spaces $(X,A)$ the boundary map $\partial: \pi_2(X,A) \to \pi_1(A)$ may be given the structure of crossed module. Recall that a crossed module is a morphism of groups $\mu: M \to P$ together with an action of the group $P$ on say the right of the group $M$ written $(m,p) \mapsto m^p$ satisfying the two rules

1. $\mu(m^p)= p^{-1} \mu(m) p$;

2. $n^{-1}mn= m^{\mu n}$

for all $m,n \in M, p \in P$. A standard property of such a crossed module is that the kernel of $\mu$ lies in the center of $M$.

For more see the paper

Loday, J.-L. "Spaces with finitely many nontrivial homotopy groups". J. Pure Appl. Algebra 24 (1982) 179--202. and also Section 2.6 of the book Nonabelian Algebraic Topology, EMS Tract Vol 15, (2011). (Loday uses left actions. Beware that Mac Lane's book CFTWM, second edition, in its last section, omits the second axiom for a crossed module.)

Another way to think of this is to use the fibration property to show that there is an action of $\Omega E$ on $F$ satisfying the crossed module axioms up to homotopy, but I do not have a reference for that.