1.2 Digression: the Topology of Closed One-Dimensional Submanifolds of the Projective Plane

For brevity, we shall refer to closed one-dimensional submanifolds of the projective plane as topological plane curves, or simply curves when there is no danger of confusion.

A connected curve can be situated in $\mathbb{R}P^2$ in two topologically distinct ways: two-sidedly, i.e., as the boundary of a disc in $\mathbb{R}P^2$, and one-sidedly, i.e., as a projective line. A two-sided connected curve is called an oval. The complement of an oval in $\mathbb{R}P^2$ has two components, one of which is homeomorphic to a disc and the other homeomorphic to a Möbius strip. The first is called the inside and the second is called the outside. The complement of a connected one-sided curve is homeomorphic to a disc.

Any two one-sided connected curves intersect, since each of them realizes the nonzero element of the group $H_1(\mathbb{R}P^2;\mathbb{Z}_2)$, which has nonzero self-intersection. Hence, a topological plane curve has at most one one-sided component. The existence of such a component can be expressed in terms of homology: it exists if and only if the curve represents a nonzero element of $H_1(\mathbb{R}P^2;\mathbb{Z}_2)$. If it exists, then we say that the whole curve is one-sided; otherwise, we say that the curve is two-sided.

Two disjoint ovals can be situated in two topologically distinct ways: each may lie outside the other one--i.e., each is in the outside component of the complement of the other--or else they may form an injective pair, i.e., one of them is in the inside component of the complement of the other--in that case, we say that the first is the inner oval of the pair and the second is the outer oval. In the latter case we also say that the outer oval of the pair envelopes the inner oval.

A set of $h$ ovals of a curve any two of which form an injective pair is called a nest of depth $h$.

The pair $(\mathbb{R}P^2,X)$, where $X$ is a topological plane curve, is determined up to homeomorphism by whether or not $X$ has a one-sided component and by the relative location of each pair of ovals. We shall adopt the following notation to describe this. A curve consisting of a single oval will be denoted by the symbol $\langle 1\rangle$. The empty curve will be denoted by $\langle 0\rangle$. A one-sided connected curve will be denoted by $\langle J\rangle$. If $\langle A\rangle$ is the symbol for a certain two-sided curve, then the curve obtained by adding a new oval which envelopes all of the other ovals will be denoted by $\langle 1\langle A\rangle\rangle$. A curve which is a union of two disjoint curves $\langle A\rangle$ and $\langle B\rangle$ having the property that none of the ovals in one curve is contained in an oval of the other is denoted by $\langle A\amalg B\rangle$. In addition, we use the following abbreviations: if $\langle A\rangle$ denotes a certain curve, and if a part of another curve has the form $A\amalg A\amalg \cdots\amalg A$, where $A$ occurs $n$ times, then we let $n\times A$ denote $A\amalg \cdots \amalg A$. We further write $n\times 1$ simply as $n$.

When depicting a topological plane curve one usually represents the projective plane either as a disc with opposite points of the boundary identified, or else as the compactification of $\mathbb{R}^2$, i.e., one visualizes the curve as its preimage under either the projection $D^2\to \mathbb{R}P^2$ or the inclusion $\mathbb{R}^2\to \mathbb{R}P^2$. In this book we shall use the second method. For example, 1.2 shows a curve corresponding to the symbol $\langle J\amalg 1\amalg 2\langle 1\rangle\amalg 1\langle 2\rangle\amalg 1\langle 3 \amalg 1\langle 2\rangle\rangle\rangle$.

Figure 1: