I am very interested in the cohomology ring of the following construction. Let $f: Y\to X$ be a map between (connected) topological spaces. Suppose that the image of the map $f^*:H^*(X) \to H^*(Y)$ is isomorphic as a ring to the quotient $H^*(X)/I$. Recall that the fiberwise join $Y\ast_X Y$ is the homotopy pushout of the diagram:
$$
\require{AMScd}
\begin{CD}
Y\times_X Y @>{p_1}>> Y\\
@V{p_2}VV @VVV \\
Y @>>> Y\ast_X Y
\end{CD} $$
Here $Y\times_X Y$ is the homotopy pullback:
$$
\require{AMScd}
\begin{CD}
Y\times_X Y @>{p_1}>> Y\\
@V{p_2}VV @VV{f}V \\
Y @>{f}>> X
\end{CD} $$
Then there exists the natural map $f*f:Y*_XY \to X$. I strongly believe that the image of the map $(f*f)^*:H^*(X)\to H^*(Y*_XY)$ is isomorphic as a ring to $H^*(X)/I^2$. Unfortunately, I have no idea how to prove this.
I am going to use this in the case of the natural map $f: G/T\to BT$. Here G is a Lie group and T is its maximal torus. In this case we understand a lot (at least in rational cohomology) about the Leray spectral sequence for the map $f$. So maybe, there exists a connection between the Leray spectral sequence for the map $f*f$ and the Leray spectral sequence for the map $f$.
1 Answer
What you wish to prove is not true. Namely, if $I=\operatorname{ker} f^*$ then $I^2\subseteq \operatorname{ker}(f\ast f)^*$ but the inclusion may be strict.
The fibred join construction comes up in the study of sectional category. If $f:Y\to X$ is a fibration, the sectional category of $f$, denoted $\operatorname{secat} f$, is the minimal number of open sets needed to cover $X$, such that $f$ admits a local section on each set in the cover. Taking $p:PX\to X$ to be the Serre based path fibration space over $X$, we see that $\operatorname{secat}(p)=\operatorname{cat}(X)$, so that this invariant includes as a special case the Lusternik-Schnirelmann category of topological spaces.
Now, as was shown originally by A. S. Schwarz (who initiated the study of sectional category under the name genus), a fibration $f:Y\to X$ has $\operatorname{secat} f\le k$ if and only if the $k$-fold iterated fibred join of $f$ admits a section. In particular, $\operatorname{secat} f\le 2$ if and only if $f\ast f:Y\ast_X Y\to X$ admits a section.
This means people who are interested in estimating sectional category from below are interested in finding elements in the kernel $\operatorname{ker}(f\ast f)^*$, because these obstruct the existence of a section of $f\ast f$. In fact, elements in this kernel are precisely the elements of sectional category weight at least 2 with respect to the fibration $f$, see
Michael Farber and Mark Grant, MR 2407101 Robot motion planning, weights of cohomology classes, and cohomology operations, Proc. Amer. Math. Soc. 136 (2008), no. 9, 3339--3349.
This notion generalizes the notion of category weight, due to Fadell and Husseini, and later studied by Rudyak, Strom and others.
A product of elements in $I$ will have weight at least 2. However, there are indecomposable classes of weight 2, and these can be produced using stable cohomology operations or Massey triple products. Examples can be found in the above paper, or in the papers of Fadell-Husseini and Rudyak or the thesis of Jeff Strom on category weight.
To give an explicit example, let $p:PX\to X$ be the based path fibration where $X$ is the complement of the Borromean rings. All cup products of elements in the $\tilde{H}^*(X)=\operatorname{ker} p^*$ are trivial, so $I^2=0$. But there are triple Massey products of category weight 2, and these are in $\operatorname{ker} (p\ast p)^*$, which is non-trivial (hence $\operatorname{cat}(X)\ge 2$).
