5.2 Digression: Topology of Closed Two-Dimensional Submanifolds of $\mathbb{R}P^3$

For brevity, we shall refer to closed two-dimensional submanifolds of $\mathbb{R}P^3$ as topological spatial surfaces, or simply surfaces when there is no danger of confusion.

Since the homology group $H_2(\mathbb{R}P^3; \mathbb{Z}_2)$ is $\mathbb{Z}_2$, a connected surface can be situated in $\mathbb{R}P^3$ in two ways: zero-homologous, and realizing the nontrivial homology class.

In the first case it divides the projective space into two domains being the boundary for both domains. Hence, the surface divides its tubular neighborhood, i. e. it is two-sided.

In the second case the complement of the surface in the projective space is connected. (If it was not connected, the surface would bound and thereby realize the zero homology class.) Moreover, it is one-sided.

The latter can be proved in many ways. For example, if the surface was two-sided and its complement was connected, there would exist a nontrivial infinite cyclic covering of $\mathbb{R}P^3$, which would contradict the fact that $\pi_1(\mathbb{R}P^3)=\mathbb{Z}_2.$ The infinite cyclic covering could be constructed by gluing an infinite sequence of copies of $\mathbb{R}P^3$ cut along the surface: each copy has to be glued along one of the sides of the cut to the other side of the cut in the next copy.

Another proof: take a projective plane, make it transversal to the surface, and consider the curve which is their intersection. Its homology class in $\mathbb{R}P^2$ is the image of the nontrivial element of $H_2(\mathbb{R}P^3; \mathbb{Z}_2)$ under the inverse Hopf homomorphism $in^!:H_2(\mathbb{R}P^3; \mathbb{Z}_2)\to H_1(\mathbb{R}P^2; \mathbb{Z}_2)$. This is an isomorphism, as one can see taking the same construction in the case when the surface is another projective plane. Thus the intersection is a one-sided curve in $\mathbb{R}P^2$. Hence the normal fibration of the original surface in $\mathbb{R}P^3$ is not trivial. This means that the surface is one-sided.

A connected surface two-sidedly embedded in $\mathbb{R}P^3$ is orientable, since it bounds a part of the ambient space which is orientable. Therefore, such a surface is homeomorphic to sphere or to sphere with handles. There is no restriction to the number of handles: one can take an embedded sphere bounding a small ball, and adjoin to it any number of handles.

A one-sidedly embedded surface is nonorientable. Indeed, its normal bundle is nonorientable, while the restriction of the tangent bundle of $\mathbb{R}P^3$ to the surface is orientable (since $\mathbb{R}P^3$ is). The restriction of the tangent bundle of $\mathbb{R}P^3$ to the surface is the Whitney sum of the normal and tangent bundles of the surface. Therefore it cannot happen that only one of these three bundles is not orientable.

Contrary to the case of two-sided surfaces, in the case of one-sided surfaces there is an additional restriction on their topological types.

5.2.A   The Euler characteristic of a connected surface one-sidedly embedded to $\mathbb{R}P^3$ is odd.

In particular, it is impossible to embed a Klein bottle to $\mathbb{R}P^3$. (The Euler characteristic of a connected surface two-sidedly embedded into $\mathbb{R}P^3$ is even, but it follows from orientability: the Euler characteristic of any closed oriented surface is even.) By topological classification of closed surfaces, a nonorientable connected surface with odd Euler characteristic is homeomorphic to the projective plane or to the projective plane with handles. Any surface of this sort can be embedded into $\mathbb{R}P^3$: for the projective plane $\mathbb{R}P^3$ is the native ambient space, and one can adjoin to it in $\mathbb{R}P^3$ any number of handles. We denote a sphere with $g$ handles by $S_g$ and a projective plane with $g$ handles by $P_g$.

Proof. [Proof of 5.2.A] Let $S$ be a connected surface one-sidedly embedded into $\mathbb{R}P^3$. By a small shift it can be made transversal to the projective plane $\mathbb{R}P^2$ standardly embedded into $\mathbb{R}P^3$. Since both surfaces are embedded one-sidedly, they realize the same homology class in $\mathbb{R}P^3$. Therefore their union bounds in $\mathbb{R}P^3$: one can color the complement $\mathbb{R}P^3\smallsetminus (S\cup\mathbb{R}P^2)$ into two colors in such a way that the components adjacent from the different sides to the same (two-dimensional) piece of $S\cup\mathbb{R}P^2$ would be of different colors. It is a kind of checkerboard coloring.

Consider the disjoint sum $Q$ of the closures of those components of $\mathbb{R}P^3\smallsetminus (S\cup\mathbb{R}P^2)$ which are colored with the same color. It is a compact 3-manifold. It is oriented since each of the components inherits orientation from $\mathbb{R}P^3$. The boundary of this 3-manifold is composed of pieces of $S$ and $\mathbb{R}P^2$. It can be thought of as the result of cutting both surfaces along their intersection curve and regluing. The intersection curve is replaced by its two copies, while the rest part of $S$ and $\mathbb{R}P^2$ does not change. Since the intersection curve consists of circles, its Euler characteristic is zero. Therefore $\chi(\partial Q)=\chi(S)+\chi(\mathbb{R}P^2)=\chi(S)+1$. On the other hand, $\chi(\partial Q)$ is even since $\partial Q$ is a closed oriented surface ( $\partial Q$ inherits orientation from $Q$). Thus $\chi(S)$ is odd. $\qedsymbol$

A one-sided connected surface in $\mathbb{R}P^3$ contains a loop which is not contractible in $\mathbb{R}P^3$. Such a loop can be detected in the following way: Consider the intersection of the surface with any one-sided transversal surface (e. g., $\mathbb{R}P^2$ or a surface obtained from the original one by a small shift). The homology class of the intersection curve is the self-intersection of the nonzero element of $H_2(\mathbb{R}P^3 ; \mathbb{Z}_2)$. Since the self-intersection is the nonzero element of $H_1(\mathbb{R}P^3 ; \mathbb{Z}_2)$, the intersection curve contains a component noncontractible in $\mathbb{R}P^3$.

A two-sided connected surface in $\mathbb{R}P^3$ can contain no loops noncontractible in $\mathbb{R}P^3$ (this happens, for instance, if the surface lies in an affine part of $\mathbb{R}P^3$). Of course, if a surface contains a loop noncontractible in $\mathbb{R}P^3$, it is not contractible in $\mathbb{R}P^3$ itself. Moreover, then it meets any one-sided surface, since the noncontractible loop realizes the nonzero element of $H_1(\mathbb{R}P^3 ; \mathbb{Z}_2)$ and this element has nonzero intersection number with the homology class realized by a one-sided surface.

If any loop on a connected surface $S$ embedded in $\mathbb{R}P^3$ is contractible in $\mathbb{R}P^3$ (which means that the embedding homomorphism $\pi_1(S)\to\pi_1(\mathbb{R}P^3)$ is trivial), then there is no obstruction to contract the embedding, i. e., to construct a homotopy between the embedding $S\to \mathbb{R}P^3$ and a constant map. One can take a cell decomposition of $S$, contract the 1-skeleton (extending the homotopy to the whole $S$), and then contract the map of the 2-cell, which is possible, since $\pi_2(\mathbb{R}P^3)=0$. A surface of this sort is called contractible (in $\mathbb{R}P^3$).

It may happen, however, that there is no isotopy relating the embedding of a contractible surface with a map to an affine part of $\mathbb{R}P^3$. The simplest example of a contractible torus which cannot be moved by an isotopy to an affine part of $\mathbb{R}P^3$ is shown in Figure 32.

Figure:

As it was stated above, the complement $\mathbb{R}P^3\smallsetminus S$ of a connected surface $S$ two-sidedly embedded in $\mathbb{R}P^3$ consists of two connected components. If $S$ is not contractible in $\mathbb{R}P^3$ then both of them are not contractible, since a loop on $S$ noncontractible in $\mathbb{R}P^3$ can be pushed to each of the components. They may be positioned in $\mathbb{R}P^3$ in the same way.

The simplest example of this situation is provided by a one-sheeted hyperboloid. It is homeomorphic to torus and its complement consists of two solid tori. So, this is a Heegaard decomposition of $\mathbb{R}P^3$. There exists an isotopy of $\mathbb{R}P^3$ made of projective transformation exchanging the components. - $(^{\arabic{endnotectr}})$endnoteiii One-sheeted hyperboloid can be presented as a parabolic hyperboloid defined by equation $xy=z$. It is invariant under rotation by $\pi$ around the axes $OX$ (i. e. transformation $(x,y,z)\mapsto(x,-y,-z)$), which exchanges the components of the complement.

A connected surface decomposing $\mathbb{R}P^3$ into two handlebodies is called a Heegaard surface. Heegaard surfaces are the most unknotted surfaces among two-sided noncontractible connected surfaces. They may be thought of as unknotted noncontractible surfaces.

If a connected surface $S$ is contractible in $\mathbb{R}P^3$, then the components $C_1$ and $C_2$ can be distinguished in the following way: for one of them, say $C_1$, the inclusion homomorphism $\pi_1(C_1)\to\pi_1(\mathbb{R}P^3)$ is trivial, while for the other one the inclusion homomorphism $\pi_1(C_2)\to\pi_1(\mathbb{R}P^3)$ is surjective. This follows from the van Kampen theorem. The component with trivial homomorphism is called the interior of the surface. It is contractible in $\mathbb{R}P^3$ (in the same sense as the surface is).

A contractible connected surface $S$ in $\mathbb{R}P^3$ is said to be unknotted, if it is contained in some ball $B$ embedded into $\mathbb{R}P^3$ and divides this ball into a ball with handles (which is the interior of $S$) and a ball with handles with an open ball deleted. Any two unknotted contractible surfaces of the same genus are ambiently isotopic in $\mathbb{R}P^3$. Indeed, first the balls containing them can be identified by an ambient isotopy (see, e. g., Hirsch [Hir-76], Section 8.3), then it follows from uniqueness of Heegaard decomposition of sphere that there is an orientation preserving homeomorphism of the ball mapping one of the surfaces to the other. Any orientation preserving homeomorphism of a 3-ball is isotopic to the identity.

At most one component of a (closed) surface embedded in $\mathbb{R}P^3$ may be one-sided. Indeed, a one-sided closed surface cannot be zero-homologous in $\mathbb{R}P^3$ and the self-intersection of its homology class (which is the only nontrivial element of $H_2(\mathbb{R}P^3 ; \mathbb{Z}_2)$) is the nonzero element of $H_1(\mathbb{R}P^3 ; \mathbb{Z}_2)$. Therefore any two one-sided surfaces in $\mathbb{R}P^3$ intersect.

Moreover, if an embedded surface has a one-sided component, then all other components are contractible. The contractible components are naturally ordered: a contractible component of a surface can contain other contractible component in its interior and this gives rise to a partial order in the set of contractible components. If the interior of contractible surface $A$ contains a surface $B$, then one says that $A$ envelopes $B$.

The connected components of a surface embedded in $\mathbb{R}P^3$ divide $\mathbb{R}P^3$ into connected regions. Let us construct a graph of adjacency of these regions: assign a vertex to each of the regions and connect two regions with an edge if the corresponding regions are adjacent to the same connected two-sided component of the surface. Since the projective space is connected and its fundamental group is finite, the graph is contractible, i. e., it is a tree. It is called region tree of the surface.

Consider now a (closed) surface without one-sided components. It may contain several noncontractible components. They decompose the projective space into connected domains, each of which is not contractible in $\mathbb{R}P^3$. Let us construct a graph of adjacency of these domains: assign a vertex to each of the domains and connect two vertices with an edge if the corresponding domains are adjacent. Edges of the graph correspond to noncontractible components of the surface. For the same reasons as above, this graph is contractible, i. e. it is a tree. This tree is called the domain tree of the surface.

Contractible components of the surface are distributed in the domains. Contractible components which are contained in different domains cannot envelope one another. Contractible components of the surface which lie in the same domain are partially ordered by enveloping. They divide the domain into regions. Each domain contains only one region which is not contractible in $\mathbb{R}P^3$. If the domain does not coincide with the whole $\mathbb{R}P^3$ (i.e., the surface does contain noncontractible components), then this region can be characterized also as the only region which is adjacent to all the noncontractible components of the surface comprising the boundary of the domain. Indeed, contractible components of the surface cannot separate noncontractible ones.

The region tree of a surface contains a subtree isomorphic to the domain tree, since one can assign to each domain the unique noncontractible region contained in the domain and two domains are adjacent iff the noncontractible regions contained in them are adjacent. The complement of the noncontractible domains tree is a union of adjacency trees for contractible subdomains contained in each of the domains.

Let us summarize what can be said about topology of a spatial surface in the terms described above.

If a surface is one-sided (i. e., contains a one-sided component), then it is a disjoint sum of a projective plane with handles and several (maybe none) spheres with handles. Thus, it is homeomorphic to

$$P_g\amalg S_{g_1}\amalg \dots S_{g_k},$$

where $\amalg$ denotes disjoint sum.

All two-sided components are contractible and ordered by enveloping. The order is easy to incorporate into the notation of the topological type above. Namely, place notations for components enveloped by a component $A$ immediately after $A$ inside brackets $\langle \rangle$. For example,

$$P_0\amalg S_1\amalg S_1\amalg S_0\langle S_1\amalg S_0\amalg S_2\langle S_1\amalg S_0 \rangle\amalg S_2\langle S_1\amalg S_0 \rangle\rangle$$

denotes a surface consisting of a projective plane, two tori, which do not envelope any other component, a sphere, which envelopes a torus and a sphere without components inside them and a two spheres with two handles each of which envelopes empty sphere and torus. To make the notations shorter, let us agree to skip index 0, i. e. denote projective plane $P_0$ by $P$ and sphere $S_0$ by $S$, and denote the disjoint sum of $k$ fragments identical to each other by $k$ followed by the notation of the fragment. These agreements shorten the notation above to

$$P\amalg 2S_1\amalg S\langle S_1\amalg S\amalg 2S_2\langle S_1\amalg S \rangle\rangle.$$

If a surface is two-sided (i. e. does not contain a one-sided component), then it is a disjoint sum $S_{g_1}\amalg \dots S_{g_k},$ of spheres with handles. To distinguish in notations the components noncontractible in $\mathbb{R}P^3$, we equip the corresponding symbols with upper index 1. Although we do not make any difference between two components of the complement of noncontractible connected surface (and there are cases when they cannot be distinguished), in notations we proceed as if one of the components is interior: the symbols denoting components of the surface which lie in one of the components of the complement of the noncontractible component $A$ are placed immediately after the notation of $A$ inside braces $\{ \}$. Our choice is the matter of convenience. It correspond to the well-known fact that usually, to describe a tree, one introduces a partial order on the set of its vertices.

In these notations,

$$S_1\amalg S\langle3S\rangle\amalg S_1^1\{S_3\amalg 2S_2^1\{3S\amalg S_1\}\}$$

denotes a two-sided surface containing three noncontractible components. One of them is a torus, two others are spheres with two handles. The torus bounds a domain containing a contractible empty torus and a sphere enveloping three empty spheres. There is a domain bounded by all three noncontractible components. It contains a contractible empty sphere with three handles. Each of the noncontractible spheres with two handles bounds a domain containing empty contractible torus and three empty spheres.

This notation system is similar to notations used above to described isotopy types of curves in the projective plane. However, there is a fundamental difference: the notations for curves describe the isotopy type of a curve completely, while the notations for surfaces are far from being complete in this sense. Although topological type of the surface is described, knotting and linking of handles are completely ignored. In the case when there is no handle, the notation above does provide a complete description of isotopy type.