Dimension of a preimage

Question

Suppose we have a differentiable function $f:\mathbb{R}^{k}\to\mathbb{R}^{\ell}$ where $k>\ell$. How can we formalize the fact that the "inverse" of a point $\mathbf{y}\in\mathbb{R}^{\ell}$, $f^{-1}(\{\mathbf{y}\})$ will be a $(k-\ell)$-dimensional set ?

Answer 1

This is not always true. For example if $f$ is a constant map then the preimage of any point is either empty or $\Bbb R^k$.

If you add the assumption that $y\in f(\Bbb R^k)$ and that for any $x\in f^{-1}(\{y\})$, $f$ is a submersion at $x$, then you have that $f^{-1}(\{y\})$ is a $(k-l)$-dimensional submanifold of $\Bbb R^k$ (in this case we say that $y$ is a regular value of $f$). In particular it has a structure of $(k-l)$-submanifold, so it's a "$(k-l)$-dimensional object".