Let $X$ be a manifold and let $Y,Z$ be two compact submanifolds of $K=X\times S^1$. dim $Y+$ dim $Z=$ dim $K$. If $I_2(Y,Z)\neq 0$ prove either $\pi\circ i_Y$ or $\pi\circ i_Z$ is onto where $\pi$ is the projection map from $K$ to $S^1$ and $i_Y, i_Z$ are inclusion maps from $Y$ to $K$ and $Z$ to $K$.
So if I assume both are not surjective then there are $s_Y, s_Z\in S^1$ such that $s_Y\neq s$ for all $(x,s)\in Y$ and $s_Z\neq s$ for all $(x,s)\in Z$. To get a contradiction I wanted to get an even no. of elements in $Y\cap Z$ but I don't see just the above argument do so. Any hint is appreciated.