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Can anyone recommend one place or a few places that describe what is known about the classification of (real) surface bundles over (real) surfaces?

Now, if the fibre F and the base B are both Hausdorff and paracompact surfaces, then:

What is the classification of F bundles over the base B ? ...

... where F and/or B may or may not be compact ...

and:

a) the group of the bundle is Diff∞(F),

or

b) F and B are fixed Riemann surfaces and the group of the bundle is the group of conformal automorphisms of F

?

c) Ideally, it would be good to know the mixed cases where F is C∞ and B is a Riemann surface, as well as vice versa.

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  • $\begingroup$ I'm pretty sure that except for some small cases, the identity component of the diffeomorphism group of a surface (compact or noncompact) is contractible. The compact case was proved by Earle-Eells, but I wasn't able to find the noncompact case in the literature. If that is true, then bundles are classified by maps $\pi_1(B)\to Mod(F)$, where $Mod(F)$ is the mapping class group of $F$. $\endgroup$
    – Ian Agol
    Commented Apr 1, 2023 at 6:10
  • $\begingroup$ @IanAgol: I remember that Earle had a later paper (maybe with his student) on noncompact case as well. $\endgroup$
    – Moishe Kohan
    Commented Apr 1, 2023 at 20:15
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    $\begingroup$ @MoisheKohan: I think you’re thinking of his work with Schatz, which if I remember correctly covered compact surfaces with nonempty boundary. $\endgroup$
    – Andy Putman
    Commented Apr 1, 2023 at 21:54

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For a), when $\chi(F)<0$ $F$-bundles over $B$ are classified by maps $\pi_1(B)\to Mod(F)$, where $Mod(F)$ is the mapping class group of $F$. This boils down to the fact that $Diff_0(F)$ is contractible, which was proved by Earle-Eells in the compact case and Yagasaki in the non-compact case.

The point is that $F$-bundles are classified by homotopy classes of maps $B\to BDiff(F)$. We have a fibration $$Diff(F) \to EDiff(F) \to BDiff(F).$$ Since $Diff(F)\simeq Mod(F)$, $BDiff(F)\simeq BMod(F) \simeq K(Mod(F),1) $. And homotopy classes of maps $B\to K(Mod(F),1)$ are in bijection with homomorphisms $\pi_1(B)\to Mod(F)$.

For the other cases when $\chi(F)\geq 0$, the homotopy type of $BDiff(F)$ is known, eg Smale showed that $Diff(S^2)\simeq O(3)$.

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I wrote some papers recently with Campagnolo, Ranicki and Rovi on this subject. Please see MR3892239 and MR4104487. The references in those point you to many more interesting (sic) papers on the subject.

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About 6 years ago there was an Oberwolfach meeting on surface bundles, and most of the talks were recorded and can be seen here. If I remember correctly, Benson Farb’s overview talk was particularly good.

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    $\begingroup$ I really enjoyed that conference. It was good to meet Benson! He suggested we write a paper together. My reply was that that was fine, as long as we just used our first names. It was good to meet you, too. It was also a different era, with Andrew Ranicki still alive. I miss him terribly. $\endgroup$
    – Dave Benson
    Commented Apr 1, 2023 at 20:17
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    $\begingroup$ I miss Andrew a lot too. The last time I saw him was when we organized a conference together in Edinburgh. He was pretty sick at that point, but he came to all the talks and was just so full of life that it was hard imagine a world without him. $\endgroup$
    – Andy Putman
    Commented Apr 1, 2023 at 21:51
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    $\begingroup$ I do wonder what it must have been like to be under the shadow of his father, Marcel Reich-Ranicki, whom he absolutely idolised. But he managed to transcend that in a most individual and charming way. - I apologise to the original poster for these reminiscences, which are really not germane to the discussion. $\endgroup$
    – Dave Benson
    Commented Apr 1, 2023 at 22:10

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