Problem 10 - A First Course in Algebraic Topology

Question

(j) A subset \(\Sigma\) of a space is a simple closed curve if it is homeomorphic to \(\mathrm{S}^{1}\). Let \(\mathrm{p}: \mathrm{S}^{2} \rightarrow \mathrm{R} \mathrm{P}^{2}\) be the canonical projection of the sphere onto the projective plane. Prove that if \(\Sigma\) is a simple closed curve in \(\mathbf{R} \mathrm{P}^{2}\) then \(\mathrm{p}^{-1}(\Sigma)\) is either a simple closed curve in \(\mathrm{S}^{2}\) or is a union of two disjoint simple closed curves. (Ilint: Consider \(\Sigma\) as the image of a closed path in \(R P^{2}\).

Step-by-Step Solution

Answer

\( p^{-1}(\Sigma) \) is either one simple closed curve or two disjoint simple closed curves in \( S^2 \).

Step 1: Understanding the Definitions and Setup

First, recall that a simple closed curve is a subset of a topological space homeomorphic to the circle \( S^1 \). The projection \( p : S^2 \rightarrow \mathrm{RP}^2 \) is a canonical map that identifies antipodal points on the sphere. \( \mathrm{RP}^2 \), the real projective plane, is obtained by identifying each point \( x \) with \( -x \) on \( S^2 \). The goal is to determine the preimage \( p^{-1}(\Sigma) \) for a simple closed curve \( \Sigma \) in \( \mathrm{RP}^2 \).

Step 2: Consider the Preimage of \( \Sigma \) under \( p \)

If \( \Sigma \subset \mathrm{RP}^2 \) is a simple closed curve (homeomorphic to \( S^1 \)), then considered as the image of a path in \( \mathrm{RP}^2 \), we want to determine the structure of \( p^{-1}(\Sigma) \). Consider two cases based on the nature of \( \Sigma \): \( \Sigma \) may or may not pass through an antipodal point-equivalence in \( S^2 \).

Step 3: Case 1 - \( \Sigma \) Does Not Pass Through Antipodal Equivalence

If \( \Sigma \) does not pass through any points in \( \mathrm{RP}^2 \) that stem from antipodal points on \( S^2 \), then \( p^{-1}(\Sigma) \) will be homeomorphic to \( \Sigma \) itself. Therefore, in this case, \( p^{-1}(\Sigma) \) will be a simple closed curve in \( S^2 \) since the map is otherwise locally a homeomorphism.

Step 4: Case 2 - \( \Sigma \) Passes Through an Antipodal Point

If \( \Sigma \) passes through an antipodal equivalence in \( \mathrm{RP}^2 \), then the points on \( \Sigma \) that are images of antipodal pairs in \( S^2 \) will result in \( p^{-1}(\Sigma) \) being a union of two disjoint simple closed curves. This is because the path is forced to 'split' at each antipodal equivalence, resulting in one path in \( S^2 \) for each half of the antipodal point.

Step 5: Conclusion of the Proof

Combining these cases, we see that \( p^{-1}(\Sigma) \) is either a simple closed curve in \( S^2 \) itself if \( \Sigma \) does not interact with any antipodal equivalence, or it is a union of two disjoint simple closed curves otherwise. This matches the required conditions of the problem statement.

Key Concepts

Simple Closed Curve

A *simple closed curve* is an essential concept in algebraic topology. It refers to a curve in a space that forms a loop without any self-intersections. Moreover, it is homeomorphic to the unit circle, denoted as \( \mathrm{S}^1 \). Homeomorphic means that there is a continuous transformation that can morph one shape into the other without tearing or gluing.

A simple closed curve encloses an area within a space and its characteristics remain consistent even if it undergoes continuous deformations. A simple closed curve in a topological space brings a sense of circularity and boundedness.

Here are some properties of a simple closed curve:

• It has no endpoints, forming a complete loop.
• All points on the curve are homeomorphic to those on a circle.
• The curve may twist or turn, but it never intersects itself.

These traits make the concept integral when discussing shapes and spaces within algebraic topology.

Real Projective Plane

The *Real Projective Plane* \( \mathrm{RP}^2 \) is a fascinating geometric construct. It can be thought of as a two-dimensional surface where every point \( x \) is identified with its antipodal point \( -x \). This characteristic merges opposite points from a three-dimensional sphere, \( \mathrm{S}^2 \), forming a unique surface without boundaries.

Visualizing the real projective plane may pose a challenge because it transcends our normal three-dimensional perception. Some properties and implications of \( \mathrm{RP}^2 \) include:

• It cannot be embedded in three-dimensional Euclidean space without self-intersection.
• It serves as a key example in topology due to its non-orientability.
• Models such as the Boy’s surface attempt to visualize \( \mathrm{RP}^2 \).

This construct helps in understanding how spaces behave when certain symmetric conditions, like antipodal point identification, are applied. Such understanding is crucial in proving theorems involving transformations of surfaces.

Homeomorphic

In algebraic topology, the term *homeomorphic* is used to describe when two spaces are topologically equivalent. Being homeomorphic means there exists a bijective (one-to-one) mapping between the spaces that is continuous, with its inverse also being continuous. This transformation allows one shape to be continuously deformed into another without any cuts or joins.

Homeomorphisms preserve the intrinsic topological structure of a shape, such as:

• Connectedness - if a space is connected, its homeomorphic counterpart must also be connected.
• Compactness - the property of being "compact" is preserved.
• Dimension - the dimension of a shape does not increase or decrease under homeomorphism.

This profound concept helps mathematicians conclude that shapes perceived different in Euclidean sense might be identical in a topological space, allowing simplified solutions to complex problems by shifting perspectives.

Canonical Projection

The concept of *canonical projection* is central to understanding how certain types of equivalences are established in topology. Canonical projection refers to a specific type of mapping where elements of a set are grouped (or identified) based on some equivalence relation. In our context, it is the mapping \( p: \mathrm{S}^2 \rightarrow \mathrm{RP}^2 \) that identifies points on a sphere which are opposite each other.

Key features of canonical projections include:

• Canonical projection maps points in \( \mathrm{S}^2 \) such that \( x \equiv -x \) are identical in \( \mathrm{RP}^2 \).
• It significantly reduces dimension; a complex structure is simplified into a more manageable form, \( \mathrm{RP}^2 \).
• The concept is vital in various proofs where dimensions are reduced by identifying equivalent points.

Such techniques allow insights into properties of non-Euclidean spaces, giving rise to broader understanding of geometric constructs and their relationships in higher dimensions.