nLab puncture

Idea

In (differential) topology and differential geometry, a punctured manifold is the complement $M \setminus S$ of a finite set $S \subset M$ of points of a given manifold $M$.

This is most prominent in the case where $M \equiv \Sigma^b_g$ is a surface of genus $g \in \mathbb{N}$ with $b \in \mathbb{N}$ boundary components: For $\mathbf{n} = \{s_1, \cdots, s_n\} \subset \Sigma^b_g$ any finite subset of interior points, there is the punctured surface $\Sigma^b_{g,n} \,\simeq\, \Sigma^b_{g,n} \setminus \mathbf{n}$.

Accordingly one speaks of punctured disks, punctured spheres, etc.

Literature

In the context of mapping class groups and braid groups:

• Benson Farb, Dan Margalit, p 18 of: A primer on mapping class groups, Princeton Mathematical Series, Princeton University Press (2012) [ISBN:9780691147949, jstor:j.ctt7rkjw, pdf]

See also:

• Wikipedia, Puncture (topology)