at.algebraic topology - Cohomology of fibrations over the circle: how to compute the ring structure?

This is a continuation of Ryan's answer above, but it has become too large for a comment. I wanted to work out the details of Ryan's example explicitly, so that we can see explicitly where your conditions fail to determine the cohomology; perhaps this can help you to pin down precisely what conditions you want. It doesn't seem that we actually need Kitano here, just Johnson's classic results.

Let $S_gto M^3to S^1$ be a mapping torus of an element of the Torelli group, i.e. a diffeomorphism $S_gto S_g$ acting trivially on homology. Such a bundle admits cohomology classes satisfying the Leray-Hirsch condition [this is a fun exercise], implying that as $H^{ast}(S^1)$-modules, $H^ast(M^3) = H^ast(S_g) otimes H^ast(S^1)$. Thus the following do not depend on the monodromy:

• $Q = H^ast(S_g)$,


• $I$, which is $Q$ with grading shifted by 1 (if $H^ast(S^1) = mathbb{Z}[t]/t^2$, this is $tQ$)


• the action of $Q$ on $I$ (just the action of $Q$ on $tQ$),


• and the Massey products on $Q = H^ast(S_g)$ [although perhaps I misunderstand what you mean here].

However, Johnson's work implies that your 3-manifold has the same cohomology ring as the product $S_g times S^1$ iff the monodromy lies in the kernel of a certain homomorphism called the Johnson homomorphism; in particular, the ring $H^ast(E)$ depends on the monodromy. It seems this shows that the answers to 1) and 2) are both "No".

Now we can compare this with your conditions to see exactly what information we're missing; it turns out to be exactly the "Johnson homomorphism". The exact sequence above $0to Ito H^ast(E)to Qto 0$ has a splitting as abelian groups $H^ast(E) = Qoplus tQ$ coming from the Leray-Hirsch theorem as above. The only information we don't know automatically is the cup product on $Q$ in this splitting with itself. We know when we project back to the $Q$ factor we recover the cup product there, which means that the missing information is the projection onto the $tQ$ factor. Letting e.g. $Q(1)$ denote the degree 1 part, the cup product is a map $Q(1) wedge Q(1) to H^2(E)$. Projecting onto the $tQ$ factor, we have $Q(1) wedge Q(1) to tQ(2)$. But both $Q(1)$ and $tQ(2)$ are isomorphic to $H^1(S_g)$, so this projection of cup product is a map $bigwedge^2 H^1(S_g) to H^1(S_g)$. This exactly encodes the data that is not determined by your conditions; Johnson's beautiful result is that this map is exactly the Johnson homomorphism, originally defined from the algebraic properties of the monodromy. In particular he showed that this missing data could be zero or nonzero, and in fact can be anything in the subspace $bigwedge^3 H^1<textrm{Hom}(bigwedge^2 H^1,H^1)$.

This was first laid out in Johnson's survey "A survey of the Torelli group" (MR0718141), and the details are worked out carefully in Hain, "Torelli groups and geometry of moduli spaces of curves" (MR1397061). What Kitano is doing is different, or rather a generalization of this: showing that just as the cup product on $H^ast(E)$ detects the Johnson homomorphism, the higher Massey products on $H^ast(E)$ detect "higher Johnson homomorphisms" measuring deeper algebraic invariants. (If any of this is useful, please consider it a partial repayment for your beautiful summary of Hodge theory in this answer.)