TITLE: Cohomology of fibrations over the circle
QUESTION [17 upvotes]: Are there any general results on the (integral) cohomology of manifolds that are fibrations over the circle? Any literature references much appreciated.

REPLY [17 votes]: This began as a comment, but it is interesting enough that I decided to make it an answer instead.  Let's consider bundles over the circle whose fibers are closed genus $g$ surfaces $\Sigma_g$.  
The diffeomorphism type of the total space of our bundle only depends on the isotopy class of the monodromy map.  Denote by $M_g$ the mapping class group, ie the group of isotopy classes of orientation-preserving diffeomorphisms of $\Sigma_g$.  For $f \in M_g$, denote by $B_f$ the surface bundle over the circle determined by $f$.
In a comment, Tom Church observed that the homology of $B_f$ will be the same as the homology of the trivial bundle if and only if $f$ acts trivially on $H_1(\Sigma_g)$.  The group of mapping classes that act trivially on $H_1(\Sigma_g)$ is known as the Torelli group.
One could demand more.  Namely, we could require that the cup-product structure on $H^{\ast}(B_f)$ be the same as the cup product structure on the trivial bundle.  As Tom observed, a beautiful theorem of Dennis Johnson gives a precise characterization of the subgroup of $I_g$ consisting of monodromies with this property.  One easy way of describing it is that it is the kernel of the (outer) action of $M_g$ on the second nilpotent truncation of $\pi_1(\Sigma_g)$ (the group $H_1(\Sigma_g)$ is the first nilpotent, ie abelian, trunctation).
The story does not end here.  A topological space has a "higher-order" intersection theory given by the so-called Massey products. They are sort of like generalized cup products. Anyway, Kitano generalized Johnson's work and gave a precise and beautiful description of the monodromies of surface bundles over the circle in which these higher intersection products (up to a certain level) are trivial.  Namely, all the degree at most $k$ Massey products of $B_f$ will be trivial if and only if $f$ acts trivially on the (k+1)st nilpotent truncation of $\pi_1(\Sigma_g)$.  
For the details of this plus references to Johnson's papers, see the following paper:
MR1381688 (97f:57014)
Kitano, Teruaki(J-TOKYTE)
Johnson's homomorphisms of subgroups of the mapping class group, the Magnus expansion and Massey higher products of mapping tori. (English summary)
Topology Appl. 69 (1996), no. 2, 165--172.