Problem 2 - Algebraic Topology

Question

Given a map \(f: S^{2 n} \rightarrow S^{2 n}\), show that there is some point \(x \in S^{2 n}\) with either \(f(x)=x\) or \(f(x)=-x .\) Deduce that every map \(\mathbb{R P}^{2 n} \rightarrow \mathbb{R} P^{2 n}\) has a fixed point. Construct maps \(\mathrm{RP}^{2 n-1} \rightarrow \mathrm{RP}^{2 n-1}\) without fixed points from linear transformations \(\mathbb{R}^{2 n} \rightarrow \mathbb{R}^{2 n}\) without eigenvectors.

Step-by-Step Solution

Answer

Use Borsuk-Ulam theorem to find a point \( x \in S^{2n} \) where \( f(x) = x \) or \( f(x) = -x \). Every map \( \mathbb{R}P^{2n} \rightarrow \mathbb{R}P^{2n} \) has a fixed point. Construct maps without fixed points using complex rotations without real eigenvectors.

Step 1: Understand Spherical Symmetry

Given the map \( f: S^{2n} \rightarrow S^{2n} \), where \( S^{2n} \) denotes the \( 2n \)-dimensional sphere, we need to find a point \( x \in S^{2n} \) such that \( f(x) = x \) or \( f(x) = -x \). The symmetry of the sphere suggests that if we try to associate antipodal points, one of the mapping criteria might hold.

Step 2: Applying the Borsuk-Ulam Theorem

To solve this problem, we can use the Borsuk-Ulam theorem, which states that for any continuous function \( g: S^n \rightarrow \mathbb{R}^n \), there is a point \( x \in S^n \) such that \( g(x) = g(-x) \). Here, redefine \( f \) to consider both possibilities \( f(x) = x \) or \( f(x) = -x \) by defining a map \( g(x) = f(x) - x \).

Step 3: Mapping Function Analysis

Consider the continuous function \( g(x) = f(x) - x \) on \( S^{2n} \). By Borsuk-Ulam theorem, there exists a point \( x \in S^{2n} \) such that \( g(x) = g(-x) = 0 \). Thus \( f(x) - x = 0 \) or \( f(x) - (-x) = 0 \), giving \( f(x) = x \) or \( f(x) = -x \).

Step 4: Deduction for Projective Spaces

Deduce that every continuous map \( \mathbb{R}P^{2n} \rightarrow \mathbb{R}P^{2n} \) must have a fixed point because such a map can be represented by a lift to \( S^{2n} \) conforming to our previous discussion, forcing a point to map to itself or to its antipodal point on \( S^{2n} \).

Step 5: Constructing Maps with No Fixed Points

For constructing maps \( \mathrm{RP}^{2n-1} \rightarrow \mathrm{RP}^{2n-1} \) without fixed points, consider linear transformations in \( \mathbb{R}^{2n} \) which do not have eigenvectors in \( \mathbb{R} \). A complex rotation transformation, which would lack real eigenvectors, can be used in this context to demonstrate such mappings. These maps rotate points and avoid fixed points in \( \mathrm{RP}^{2n-1} \).

Key Concepts

Fixed Points

In mathematics, a fixed point is a point that remains unchanged under a given map or function. This means that if you apply the function to the point, you get the same point back. Formally, for a function \( f \), a point \( x \) is a fixed point if \( f(x) = x \). In the context of the exercise, we are interested in maps on spheres and projective spaces. You can think of this similar to finding a location on a spinning globe that returns to the same spot after one complete rotation. Some important properties of fixed points include:

• Fixed points provide stability in dynamic systems.
• They are critical in many mathematical fields like game theory, computer science, and economics for solutions that remain constant over time.

You should also be aware that, beyond exact fixed points, sometimes points map to their opposite or antipodal points (like on a sphere where every point has a direct opposite point). This concept is where the Borsuk-Ulam theorem gives us many interesting results.

Symmetric Maps

Symmetric maps take into account the structure of the spaces they act upon. For example, if you consider a sphere, symmetry involves considering how each point on the sphere behaves relative to its opposite or antipodal point. Symmetric maps are thus often involved in understanding behaviors such as reflection and inversion. Key points include:

• A symmetric map \( f \) on a sphere \( S^n \) may carry a point \( x \) to its opposite, \( -x \).
• Symmetric maps are particularly important in higher-dimensional spaces where intuitive geometric symmetry is not immediately visible but is critical for analysis.

In our context, the symmetry of the sphere helps us use the Borsuk-Ulam theorem to ensure certain conditions are met, namely, either exact fixed points or antipodal mappings.

Projective Spaces

Projective spaces are a fascinating concept in geometry. They are a way of looking at spaces where you "remove" the notion of parallel lines by introducing points at infinity. This is important because it allows for a consistent behavior under projection and perspective changes. Some defining features:

• Each line passing through the origin in \( \mathbb{R}^{n+1} \) corresponds to a point in \( \mathbb{R}P^n \).
• Projective spaces are essential in many areas, such as algebraic geometry and projective geometry, for handling degenerate cases smoothly.

The property that every map \( \mathbb{R}P^{2n} \to \mathbb{R}P^{2n} \) must have a fixed point ties back to how mappings lift to spheres, emphasizing the rich geometry in higher dimensions.

Linear Transformations

Linear transformations describe how vectors in one space can be systematically changed into vectors in another, preserving the operations of vector addition and scalar multiplication. Key points include:

• They are often represented by matrices which makes computations involving them systematic and swift.
• Eigenvectors and eigenvalues are central concepts, helping understand how transformations act along specific directions in space.

In the original exercise, when discussing maps with no fixed points, we refer to linear transformations in \( \mathbb{R}^{2n} \) that lack real eigenvectors. This usually involves complex transformations, such as rotations, which ensure vector directions are consistently shifted. This guides the formation of continuous maps on \( \mathrm{RP}^{2n-1} \) with no fixed points.