Suppose that $A$ is an affine algebraic variety and $P$, and $Q$ are subvarieties.  It is easy
to see that the coordinate rings $C[P]$ and $C[Q]$ of $P$ and $Q$ are modules over $C[A]$ and
$C[P]\otimes_{C[A]} C[Q]$ is frequently the coordinate ring of $P\cap Q$.  For instance,
if the sum of the ideals of $P$ and $Q$ is its own radical.



If $X$ is a topological space, and $P$ and $Q$ are subspaces then their singular cohomology
groups  $ H^{\*}(P) $ and $ H^{\*}(Q) $ are modules over $H^{|*}(X)$.  

To make it simple, assume that
$X$ is a smooth manifold and $P$ and $Q$ are smooth submanifolds.  Are there simple conditions
that imply that 
$H^{\*}(P\cap Q)=H^{\*}(P)\otimes_{H^{\*}(X)} H^{\*}(Q)?$

Here is an example.  Let $X=(S^2)^4$.  Let $P\subset X$ of all pairs of the form $(x,x,y,y)$.
Let $Q\subset X$ of all pairs of the form $(x,y,y,x)$.  Notice that $P\cap Q$ is all pairs
of the form $(x,x,x,x)$ so it is homeomorphic to $S^2$.

With a little work you can see that
 
$H^{\*}(X)=\mathbb{Z}[a,b,c,d]/(a^2,b^2,c^2,d^2)$, 

that is, integer polynomials in
$4$ variables where the square of any variable is zero.  
We get $H^{\*}(P)$ is the quotient of $H^{\*}(X)$ by the ideal generated by $a-b,c-d$ and
$H^{\*}(Q)$ is the quotient of $H^*(X)$ by the ideal generated by $a-d,b-c$.  

Its easy to check that 

$$H^{\*}(P)\otimes_{H^{\*}(X)} H^{\*}(Q)=\mathbb{Z}[a,b,c,d]/(a^2,b^2,c^2,d^2,a-b,b-c,c-d)$$ 

which is just $\mathbb{Z}[x]/(x^2)$ which is the cohomology group of the sphere.