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Let $f:M\rightarrow N$ be a surjective map from a fourmanifold $M$ to a surface $N$, with connected fibers (each fiber is connected). Assume that $f$ admits a multiple section $s:N\rightarrow M$.

Suppose now that we are given a finite group $G$ acting freely on $M$ in such a way that the fiber class and the multiple section class in $H^2(M,\mathbb{Z})$ are invariant or multiplied by $-1$. Can we conclude that the map $f$ is $G$-equivariant and descends to $f':M/G\rightarrow N/G$ as topological spaces (as $G$ action may not be free on $N$)?

A point I am confused by is that the fiber class and the section class are preserved (invariant or multiplied by -1) as classes, not as cycles.

Edit My question turns out to be non-sense in its original form above. So let me change my question. I am reading this paper by M. Gross and P.H.M. Wilson, where they study SYZ conjecture for certain CY3s. I now try to understand the proof of Theorem 2.1. where certain holomorphic map $f$ is claimed to be anti-holomorphic $C_2$-action equivariant. Here the assumption in my question is satisfied. I now understand the (anti-)holomorphicity of the map $f$ ($C_2$-action) is crucial. But it is not yet clear to me why $f$ is $C_2$-action equivariant.

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    $\begingroup$ Koopa -- what is the action of $G$ on $N$? $\endgroup$
    – algori
    Commented Sep 30, 2012 at 22:41
  • $\begingroup$ @algori We don't know whether or not the $G$-action descends to $N$. Rather that's exactly what I am asking; if $G$ preserves the fiber class and the multiple section class, can one conclude that $f$ preserves the fiber and thus the $G$-action on $M$ induces that of $N$? $\endgroup$
    – Koopa
    Commented Oct 1, 2012 at 0:24
  • $\begingroup$ Koopa -- what do you mean by "$f$ preserves the fiber"? $\endgroup$
    – algori
    Commented Oct 1, 2012 at 0:55
  • $\begingroup$ .. if you meant to say that $G$ takes fibers to fibres, than this is certainly false, at least in this generality (at least when $G\neq\{e\}$): e.g., take a map $f:M\to N$ for which everything is fine and perturb it slightly to make sure the image of some fiber under some element of $G$ is not a fiber. $\endgroup$
    – algori
    Commented Oct 1, 2012 at 1:03
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    $\begingroup$ What is a "multiple section" or "multisection" of a surjective map? (My guess is it assigns to each point in the base a subset of the fibre, but then I have several guesses for what continuity means in this context.) $\endgroup$
    – Mark Grant
    Commented Oct 1, 2012 at 15:31

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