Classification of surface bundles and an isomorphism between $[S^n, BDiff(X)]$ and $\pi_{n-1}(Diff(X))/\pi_0(Diff(X))$

Question

I'm trying to learn about the classification of surface bundles (in the smooth case, over a circle), and I might be missing some prerequisites. I am somewhat familiar with classifying spaces and mapping class groups, but when I tried to read chapter 4 in the book "Geometry of Characteristic Classes" by Morita, I couldn't understand where the following "canonical identifications" in page 136 come from:

$[S^n, BDiff(X)] \cong \pi_n(BDiff(X))/\pi_1(BDiff(X)) \cong \pi_{n-1}(Diff(X))/\pi_0(Diff(X))$

I'm quite certain that the first isomorphism is the one from the following post, but I have no idea about the second one.

Firstly, an explanation/reference for why these identifications are true will be very much appreciated.

Secondly, since I suspect that I might need to learn more material before jumping into the details of the classification, it would be useful if you could recommend what I should know well before continuing and, ideally, provide some useful references (either for the preliminaries, or for the classification itself).

Answer 1

You're correct that the first isomorphism is the one that you linked. For the second, one, this follows from the general fact that $\pi_n(BG) \cong \pi_{n-1}(G)$ for any topological group $G$.

$BG$ is the quotient of a contractible space $EG$ by a free action of $G$, so there is a fiber sequence $G \to EG \to BG$, from which the long exact sequence in homotopy gives the isomorphism.

Alternatively, you can use the loop-suspension adjunction together with a homotopy equivalence $G \simeq \Omega BG$ (although this equivalence basically follows from the long exact sequences of homotopy anyway so it's really not much of an "alternate" way to see the result, it's worth knowing that this homotopy equivalence exists):

$[S^n, BG] = [\Sigma S^{n-1}, BG] = [S^{n-1}, \Omega BG] = [S^{n-1}, G]$.

Answer 2

It is important to note that the quotients are orbit spaces. Namely, there is a right-action of $\pi_1(B\text{Diff}(X))$ on $\pi_n(B\text{Diff}(X))$ given by relation that $[f][\gamma]=[g]$ if there exists a homotopy $h:I\times S^n \to B\text{Diff}(X)$ from $f$ to $g$ such that $h(t,s_0)\simeq \gamma$ for all $t\in I$ and where $s_0$ is the basepoint of $S^n$. Then the first isomorphism is a proposition. For example, see Hatcher's book and Proposition 4A.2.

The second isomorphism comes from the universal bundle $\text{Diff}(X) \to E\text{Diff}(X)\to B\text{Diff}(X)$ and the long exact sequence in homotopy. Namely, $\pi_n(B\text{Diff}(X))\cong \pi_{n-1}(\text{Diff}(X))$.