I am interested in what conditions can I add to path lifting that imply covering. Let $X$ and $Y$ be two topological manifolds, and $p:X\to Y$ be a continuous map for which path lifting holds.
Theorem 4.19 from here tells me that it is sufficient for $X$ to be a manifold, $Y$ to be Hausdorff, and $p$ to be a local homeomorphism.
In my particular case, I am happy with assuming that $X$ and $Y$ are manifolds embedded in Euclidean space (and are thus Hausdorff, I think?). However, I do not want to assume a priori that $p$ is a local homeomorphism.
Given that both are manifolds, as I understand, a local homeomorphism will follow from both manifolds having the same dimensionality. EDIT: As pointed out by JHF in the comments this is not true.
As I understand, path lifting in itself is not sufficient to imply the local homeomorphism as well. What if I assume that the path lifting is unique? Going by this response here, unique path lifting is in general not sufficient either. But does the additional assumption that $X$ and $Y$ are manifolds help?
Could any additional structure on the manifolds or the map be useful (differentiability, smoothness etc.) to make the map a local homeomorphism and thus covering?
EDIT: The construction in answer at the above link (that provides a counter example for unique path lifting implying covering) relies on compositions of cover maps having path lifting, but not necessarily being covering maps themselves. However, going by Theorem 2.11 here, compositions of covering maps will be covers if my spaces are locally path connected and semi-locally simply connected. If my spaces are manifolds, I believe I have enough niceness properties that this type of counter example would not work.
EDIT: Without going into too many details, my problem has its origins in modeling something physical, so I can make more assumptions, but I'm not sure what kinds of assumptions I should be making. I have unique path lifting, so I'm happy assuming that. In my problem, I know $X$ and $Y$ are "nice" manifolds, but I don't know what sense of nice I need to think about.
I don't want to assume a priori that my manifolds have the same dimensionality, would prefer that to follow from some other assumptions.
Best Answer
Here's an attempt under the assumptions that
Moreover, we will assume without loss of generality that $N$ is connected and $M \neq \emptyset$.
For a start, notice that $p$ is surjective since the constant paths at every point in $N$ need to lift through it. Moreover, I claim it is a submersion: Let $m_0 \in M$ be any point and $n_0 = p(m_0)$ its image in $N$. Let $\gamma\colon I \to N$ be a smooth path with $\gamma(1 / 2) = n_0$. We obtain from this two paths $\gamma_1 = \bar{\gamma}|_{[0, 1 / 2]}$ and $\gamma_2 = \gamma|_{[1 / 2, 1]}$ (here $\bar{\gamma}(t) = \gamma(1 - t)$) starting at $n_0$ and therefore two unique smooth lifts $\tilde{\gamma}_1$, $\tilde{\gamma}_2$ to $M$ starting at $m_0$. The concatenation $\tilde{\gamma} = \bar{\tilde{\gamma}}_1 \ast \tilde{\gamma}_2$ is then a lift of $\gamma$ subject to $\tilde{\gamma}(1 / 2) = m_0$ (and smooth by the requirement that lifts of smooth paths be smooth), so since every $v \in T_{n_0} N$ is the velocity vector of such a path $\gamma$ through $n_0$ and $\gamma'(1 / 2) = (p \circ \tilde{\gamma})'(1 / 2) = d p(\tilde{\gamma}'(1 / 2))$, we conclude that $d p$ is surjective and therefore that $p$ is a submersion.
It then remains to show that $p$ is an immersion and therefore a local diffeomorphism which implies it is a covering map. But this is not difficult: Any point in the domain of a submersion is in the image of a smooth local section, so let $m \in M$ be arbitrary and $\sigma\colon U \to M$ be section defined on a connected open neighborhood $U \subseteq N$ of $p(m)$ such that $m \in \sigma(U)$. Assume that $p|_{\sigma(U)}$ is not injective and let $m_0, m_1 \in \sigma(U)$ be two distinct points with $p(m_0) = p(m_1)$. Moreover, pick a smooth path $\gamma\colon I \to \sigma(U)$ from $m_0$ to $m_1$. Then $p \circ \gamma$ is loop at $p(m_0)$, so $\sigma \circ p \circ \gamma$ is a loop in $\sigma(U)$, i.e. $\sigma \circ p \circ \gamma \neq \gamma$ are two distinct lifts of $p \circ \gamma$, contradicting unique path lifting. Hence, $p$ is a locally injective submersion, i.e. a local diffeomorphism.