3.2 The Most Elementary Prohibitions on Real Topology of a Flexible Curve

The simplest prohibitions are not related to the position of $\mathbb{R}S$ in $\mathbb{R}P^2$, but deal with the following purely topological situation: a surface $S$, which is homeomorphic to a sphere with $g$ ( $=(m-1)(m-2)/2$) handles, and an involution $c$ ($=conj$) of $S$ reversing orientation with fixed point set $F$ ( $=\mathbb{R}S$).

The most important of these prohibitions is Harnack's inequality. Recall that it is

$$L\le\frac{(m-1)(m-2)}2+1,$$

where $L$ is the number of connected components of the real part a curve and $m$ is its degree. Certainly, this formulation given in Section 1.3 can be better adapted to the context of flexible curves. The number $\frac{(m-1)(m-2)}2$ is nothing but the genus. Therefore the Harnack inequality follows from the following theorem.

3.2.A   For a reversing orientation involution $c:S\to S$ of a sphere $S$ with $g$ handles, the number $L$ of connected components of the fixed point set $F$ is at most $g+1$.

In turn, 3.2.A can be deduced from the following purely topological theorem on involutions:

3.2.B Smith-Floyd Theorem.   For any involution $i$ of a topological space $X$,

$$\dim_{\mathbb{Z}_2}H_*(fix(i); \mathbb{Z}_2)\le \dim_{\mathbb{Z}_2}H_*(X; \mathbb{Z}_2).$$

This theorem is one of the most famous results of the Smith theory. It is deduced from the basic facts on equivariant homology of involution, see, e. g., [Bre-72, Chapter 3].

Theorem 3.2.A follows from 3.2.B, since

$$\dim_{\mathbb{Z}_2}H_*(S; \mathbb{Z}_2)=2+2g,$$

and

$$\dim_{\mathbb{Z}_2}H_*(F; \mathbb{Z}_2)=2L.$$

Smith - Floyd Theorem can be applied to high-dimensional situation, too. See Sections 5.3 and . In the one-dimensional case, which we deal with here, Theorem 3.2.B is easy to prove without any homology tool, like the Smith theory. Namely, consider the orbit space $S/c$ of the involution. It is a connected surface with boundary. The boundary is the image of the fixed point set. The Euler characteristic of the orbit space is equal to the half of the Euler characteristic of $S$, i.e. it is $\frac{2-2g}2=1-g$. Cap each boundary circle with a disk. The result is a closed connected surface with Euler characteristic $1-g+L$. On the other hand, as it is well known, the Euler characteristic of a connected closed surface is at most 2. (Remind that such a surface is homeomorphic either to the sphere, which has Euler characteristic 2, or the sphere with $h$ handles, whose Euler characteristic is $2-2h$, or sphere with $h$ Möbius strips having Euler characteristic $2-h$.) Therefore $1-g+L\le2$, and $L\le g+1$.$\qedsymbol$

These arguments contain more than just a proof of 2.3.A. In particular, they imply that

3.2.C   In the case of an M-curve (i.e., if $L=g+1$) and only in this case, the orbit space is a sphere with holes.

Similarly, in the case of an $(M-1)$-curve, the orbit space is homeomorphic to the projective plane with holes.

If $F$ separates $S$ (i.e., $S\smallsetminus F$ is not connected), the involution $c$ is said to be of type I, otherwise it is said to be of type II. The types correspond to the types of real algebraic curves (see Section 2.1).

Note that $F$ separates $S$ at most into two pieces. To prove this, we can use the same arguments as in a footnote in Section 2.1: the closure of tne union of a connected component of $S\smallsetminus F$ with its image under $c$ is open and close in $S$, but $S$ is connected.

3.2.D   The orbit space $S/c$ is orientable if and only if $F$ separates $S$.

Proof. Assume that $F$ separates $S$. Then the halves are homeomorphic, since the involution maps each of them homeomorphically onto the other one. Therefore, each of the halves is homeomorphic to the orbit space. The halves are orientable since the whole surface is.

On the other hand, if $F$ does not separate $S$, then one can connect a point of $S\smallsetminus F$ to its image under the involution by a path in the complement $S\smallsetminus F$. Such a path covers a loop in the orbit space. This is an orientation reversing loop, since the involution reverses orientation. $\qedsymbol$

3.2.E (Cf. 2.6.C)   If the curve is of type I, then $L\equiv \left[\frac{m+1}2\right]\mod 2$.

Proof. This theorem follows from 3.2.C and the calculation of the Euler characteristic of $S/c$ made in the proof of the Harnack inequality above. Namely, $\chi(S/c)=1-g$, but for any orientable connected surface with Euler characteristic $\chi$ and $L$ boundary components $\chi+L\equiv 0\mod2$. Therefore $1-g+L\equiv 0\mod2$. Since $g=(m-1)(m-2)/2\equiv\left[\frac{m-1}2\right] \mod2$, we obtain $1-\left[\frac{m-1}2\right]+L\equiv0\mod2$ which is equivalent to the desired congruence. $\qedsymbol$

3.2.F (Cf. 2.6.B)   Any M-curve is of type I.

Proof. By 3.2.C, in the case of M-curve the orbit space $S/c$ is homeomorphic to a sphere with holes. In particular, it is orientable. By 3.2.D, this implies that $F$ separates $S$. $\qedsymbol$

Now consider the simplest prohibition involving the placement of the real part of the flexible curve in the projective plane.

3.2.G   The real part of a flexible curve is one-sided if and only if the degree is odd.

Proof. The proof of 3.2.G coincides basically with the proof of the same statement for algebraic curves. One has to consider a real projective line transversal to the flexible curve and calculate the intersection number of the complexification of this line and the lfexible curve. On one hand, it is equal to the degree of the flexible curve. On the other hand, the intersection points in $\mathbb{C}P^2\smallsetminus \mathbb{R}P^2$ give rise to an even contribution to the intersection number. $\qedsymbol$

Rokhlin's complex orientation formula also comes from topology. The proof presented in Section 2.7 works for a flexible curve.

At this point I want to break a textbook style exposition. Escaping a detailed exposition of prohibitions, I switch to a survey.

In the next two sections, the current state of prohibitions on the topology of a flexible curve of a given degree is outlined. (Recall that all formulations of this sort are automatically valid for real projective algebraic plane curves of the same degree.) After the survey a light outline of some proofs is proposed. It is included just to convey a general impression, rather than for more serious purposes. For complete proofs, see the surveys [Wil-78], [Rok-78], [Arn-79], [Kha-78], [Kha-86], [Vir-86] and the papers cited there.