While trying to prove de Rham's theorem on my own, I came up with the following question:
If $M$ is a closed orientable $n-$manifold and $X$ is a closed orientable $p$-submanifold of $M$ which represents a nontrivial element of the $p$-th homology group of $M$, then is there a closed orientable $(n-p)$-submanifold $Y$ of $M$ whose oriented intersection number with $X$ is nonzero?
I wish to know whether the above statement is true, and if it is false, then what some simple counterexamples are.
Thank you in advance!
Well, let me add a few words on how I came up with the question. I was trying to prove de Rham's theorem, which says that the natural map $i:H_{dR}^p(M;\mathbb{R})\rightarrow H_p(M;\mathbb{R})^*\simeq H^p(M;\mathbb{R})$ is an isomorphism. In order to prove surjectivity, I first wanted to show that if $[X]\in H_p(M;\mathbb{R})$ is a nontrivial element of the homology, then there is an element $[\omega]\in H_{dR}^p(M;\mathbb{R})$ that "detects" such nontrivial homology. In attempting to prove the last statement, I divided the statement into two parts:
Find an element $Y\in H_{n-p}(M;\mathbb{R})$ such that $I(X, Y) \neq 0$ where $I(X,Y)$ is the oriented intersection number beween $X$ and $Y$. (Then $I(-,Y)$ is a linear functional on $H_p(M;\mathbb{R})$ under which the image of $X$ is nonzero.)
Find an element $[\omega]\in H_{dR}^p(M;\mathbb{R})$ such that $\omega(X)=I(X,Y)$.
At this moment, I am not even sure if either of the above two statement (finding $Y$ and finding $[\omega]$) is valid or not. But anyway I found this quite interesting, and from this I hope to understand better about the interaction between the oriented intersection number (defined differential topologically) and the de Rham cohomology.