I am very new to topology and geometry, so please excuse me if my statements are crude.
According to https://en.wikipedia.org/wiki/Foliation#Submersions, if $f: \mathbb{R}^n \to \mathbb{R}^p$ is a submersion, the fibers are $(n-p)$-dim submanifolds. Here, if I understand the concept of foliation correctly, it's basically decomposing a $n$-manifold into a $p$-dim space of $(n-p)$-dim submanifolds, right?
My question is, since we could decompose $n$-manifold in the other way, i.e., $(n-p)$-dim space of $p$-dim submanifolds, I assume some sort of a complement (or a quotient maybe?) of those fibers should be also submanifolds of dimension $n$. Is this true? If it is, is there a term to call such complement?
Thank you in advance for helping me out.