Let $\Sigma\_g$ be a Riemannian surface of genus $g$. Let $M^4$ be a surface bundle over surface: $\Sigma\_g \to M^4 \to \Sigma\_h$. $\Sigma\_g$ is the fiber and $\Sigma\_h$ is the base space.


My question: *is there a surface bundle over surface $M^4$ such that it has a 2-cocycle $c \in H^2(M^4;Z)$ satisfying
(1) $\int\_{M^4} c^2 =\pm 1$, and
(2) $\int\_{\Sigma\_g} c =0$*


Also: *How to construct surface bundles with known odd intersection form?* This may help to answer the first question.


See a related question:
[Oddness of intersection form of surface bundle](https://mathoverflow.net/questions/364699/oddness-of-intersection-form-of-surface-bundle/364728?noredirect=1#comment920802_364728)


A further question: *is there a surface bundle over surface $M^4$ such that it has a 2-cocycle $c \in H^2(M^4;Z)$ satisfying
(1) $\int\_{M^4} c^2 =\pm 1$,
(2) $\int\_{\Sigma\_g} c =0$, and (3) $c = w\_2$ mod 2.*
What is the signature and $g$ for such surface bundle?


Note that the condition (1) implies that $M^4$ is not spin and $w\_2$ is non-trivial.



Intersection form of surface bundle over surface
Yes, such a thing exists, but I don't know an explicit example.


To see that it exists, it is clearest to me to consider the universal situation. For any $k \in \mathbb{Z}$ there is a space $\mathcal{S}\_g(k)$ which classifies oriented surface bundles
$$\Sigma\_g \to E \overset{\pi}\to B$$
equipped with a class $c \in H^2(E; \mathbb{Z})$ such that $\int\_{\Sigma\_g} c = k$. Associated to such a family there are characteristic classes
$$\kappa\_{i,j} = \int\_\pi e(T\_\pi E)^{i+1} \cdot c^j \in H^{2(i+j)}(B;\mathbb{Z}),$$
where $T\_\pi E$ denotes the tangent bundle of $E$ along the fibres of $\pi$, and $e(T\_\pi E)$ denotes its Euler class. (The classes $\kappa\_{i,0}$ are the usual Miller--Morita--Mumford classes $\kappa\_i$.)


In



> 
> J. Ebert and O. Randal-Williams, *Stable cohomology of the universal Picard varieties and the extended mapping class group*. Doc. Math. 17 (2012), 417–450.
> 
> 
> 


Johannes Ebert and I studied, among other things, the low-dimensional integral cohomology of $\mathcal{S}\_g(k)$, and showed that as long as $g$ is large enough (I think $g \geq 6$ will do) one has
$$H^1(\mathcal{S}\_g(k);\mathbb{Z})=0 \quad\quad H^2(\mathcal{S}\_g(k);\mathbb{Z})\cong\mathbb{Z}^3$$
where the isomorphism in the second case is given by a basis of cohomology classes $\lambda, \kappa\_{0,1}, \zeta$, where the outer two are related to the $\kappa\_{i,j}$ by the identities
$$12 \lambda = \kappa\_{1,0} \quad\quad 2\zeta = \kappa\_{0,1} - \kappa\_{-1,2}.$$


In particular, applying this with $k=0$ and using that every second homology class is represented by a map from an oriented surface, it follows that there is a surface bundle
$$\Sigma\_g \to E \overset{\pi}\to \Sigma\_h$$
for some $h$ (which is uncontrollable using this method) with a class $c \in H^2(E;\mathbb{Z})$ satisfying $\int\_{\Sigma\_g}c = 0$, and having
$$\int\_{\Sigma\_h}\lambda=\text{whatever you like} \quad\quad \int\_{\Sigma\_h}\kappa\_{0,1}=1 \quad\quad \int\_{\Sigma\_h}\zeta = 0$$
and hence having
$$\int\_E c^2 = \int\_{\Sigma\_h} \int\_\pi c^2 = \int\_{\Sigma\_h} \kappa\_{-1,2} = \int\_{\Sigma\_h} \kappa\_{0,1} = 1.$$


