In geometric topology, what does it mean for a curve to be homotopic to a puncture?
Let $F$ be a hyperbolic surface of finite type, i.e. $F$ is a surface of genus $g$ with $b$ boundary components and $n$ punctures such that, $2-2g-b-n< 0.$ Let $\pi_1(F)$ be the fundamental group of $F$. We identify $\pi_1(F)$ with a discrete subgroup of $PSL_2(R)$, the group of orientation preserving isometries of the upper half plane $H$. The action of $\pi_1(F)$ on $H$ is properly discontinuous and does not fix any point. Therefore the quotient space is isometric to $F.$ Henceforth by an isometry of $H$, we mean an orientation preserving isometry and by a closed curve we mean an oriented close curve.
A homotopically non-trivial closed curve in $F$ is called essential if it is not homotopic to a puncture. By a lift of a closed curve $g$ to $H$, we mean the image of a lift $R\rightarrow H$ of the map $g\circ\pi$ where $\pi:R\rightarrow S^1$ is the usual covering map.
For a curve $g : S^1 \to F$ to be homotopic to a puncture means that there exists a closed subset $A \subset F$ homeomorphic to $\mathbb{D}^2 - \{(0,0)\}$ such that $g$ is homotopic in $F$ to a curve in $A$. Here $\mathbb{D}^2 \subset \mathbb{R}^2$ denotes the closed unit disc.
If $F$ is explicitly given as $F = \overline F-P$ where $\overline F$ is a compact surface-with-boundary and $P$ is a finite subset of the interior of $F$, then $g$ is homotopic to a puncture if and only if there exists $p \in P$ such that for every neighborhood $U \subset \overline F$ of $p$ the curve $g$ is homotopic in $F$ to a curve in $U-p$.