For context I am working on Weibel's K-book, chapter IV, my question comes from the proof of proposition 1.7.
In this he claims that for the long exact sequence of a fibration with acyclic fiber $F\to BG\to BG^+$, the image of the connecting map $\partial: \pi_2(BG^+)\to\pi_1(F)$ is in the center of $\pi_1(F)$. According to him this is a consequence of a propostion in "Elements of homotopy theory, George W. Whitehead" (specifically corrollary 3.5 in chapter IV).
What that result says is that for the homotopy long exact sequence of a pair $(X,A)$ we have $Im(\pi_2(X)\to\pi_2(X,A))\leq Z(\pi_2(X,A))$, which is not quite what we need. I have tried for a while now to prove the result I want from the result in Whitehead's book, namely trying to loop the fibration to shift the homotopy long exact sequence, but it shifts the wrong way.
Any help is appreciated, is there a trick I am not seeing? Is Weibel's proof method wrong? Is the result just completely wrong, and so another method to prove proposition 1.7 is needed?
In any case, all help is appreciated.