Essential closed curves and orientation preserving homeomorphism

Question

Q1. In the book "A primer on mapping class groups", the author gives a definition of essential closed curve as "A closed curve is called essential if it is not homotopic to a point, puncture, or a boundary component."

Being homotopic is a property of two (continuous) maps, so I'm not able to understand this definition. Also can you provide me an example of an essential closed curve?

Q2. What is the definition of "orientation - preserving" homeomorphism?

Thanks in advance.

Answer 1

1. A curve is a continuous map $[0,1]\to M$, where $M$ is the manifold on which you're considering it. In the same way one can define a point (a constant curve) and a boundary component if the manifold is of dimension 2 (and therefore the boundary is of dimension 1, meaning that it is a disjoint union of curves). An example of essential closed curve is given in the following picture taken from Basic results on braid groups by Juan González-Meneses.

2. That depends first of all on how you define orientations. Since I don't want to assume any smooth structure, I'll say it in terms of local homology. An orientation on a manifold $M$ of dimension $n$ is a choice of generator $[M]_x$ of $H_n(M,M-x;\mathbb{Z})$ for each $x\in M$ in such a way that this choice is continuous (See Definition 3.1, and the others if you want another notion). A homeomorphism is orientation preserving if the induced maps on local homology takes orientation to orientations.

Answer 2

To answer your first question in detail, let's represent the closed curve by a continuous function $f : S^1 \to F$, where $F$ is the given surface.

In a setting where a particular set $A$ is given, I might speak about whether "$f$ is homotopic to a function with values in $A$". Each of the three cases you ask about -- point, puncture, and boundary component -- can be reworded to fit this setting.

To say "the closed curve is not homotopic to a point" means that $f$ is not homotopic to a function which maps $S^1$ to a point, in other words $f$ is not homotopic to a constant function.

To say "the closed curve is not homotopic to a puncture" means that $f$ is not homotopic to a function which maps $S^1$ to an open annulus neighborhood of a puncture.

To say "the closed curve is not homotopic to a boundary component" means that $f$ is not homotopic to a function which maps $S^1$ to a component of $\partial F$.

If the surface $F$ has no puncture nor boundary point, then to be essential is equivalent to saying that the curve is not homotopic to a point. You can use the fundamental group or the first homology group to provide many examples. For instance, in the torus $S^1 \times S^1$, the closed curve $\{\text{point}\} \times S^1$ represents a nonzero element of $H_1(S^1 \times S^1)$ and is therefore not homotopic to a point, hence is essential.