Let $X$ and $Y$ be connected smooth manifolds. Let $L$ be a topological real line
bundle over $X\times Y$. Then we know that the isomorphism class of such a line bundle is determined by its first Stiefel-Whitney class $w\_1(L)\in H^1(X\times Y,\mathbf{Z}/2\mathbf{Z})$.


I would like to have an example of a nontrivial line bundle $L$ (so $w\_1(L)\neq 0$), such that


**(1)** $\forall y\in Y$ we have $w\_1(L)|\_{X\times\{y\}}=0\in H^1(X\times\{y\},\mathbf{Z}/2\mathbf{Z})$. 


**(2)** $L$ is not the pullback of a line bundle over $Y$.


I don't see how to construct such a line bundle. May be there is some trick using Kunneth's formula.



Non-trivial topological line bundles over cartesian product of manifolds not coming from a pullback
Denote by $p\_X$ the natural projection $X\times Y\to X$ and define $p\_Y$ similarly. Kunneth formula shows that any $\newcommand{\bZ}{\mathbb{Z}}$ $w\in H^1(X\times Y,\bZ/2)$ has the form


$$ w= p\_X^\* u+p\_Y^\* v,\;\;u\in H^1(X,\bZ/2),\;\;v\in H^1(Y,\bZ/2). $$


This proves that any real line bundle $L\to X\times Y$ has the form


$$ L= p\_X^\*L\_X\otimes p\_Y^\*L\_Y, $$


with


$$w\_1(L)=p\_X^\*w\_1(L\_X)+p^\*\_Y w\_1(L\_Y). $$


Moreover


$$ L|\_{X\times y}= L\_X. $$


The line bundles with the properties you want are all pullbacks of nontrivial line bundles on $Y$.


