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Chapter 2: Problem 4

Construct a surjective map \(S^{n} \rightarrow S^{n}\) of degree zero, for each \(n \geq 1\).

Short Answer

Expert verified
Map \( f(x) = -x \) works; it's surjective and has degree zero.

Step by step solution

01

Understanding the Concept of Surjective Maps and Degree

A surjective map (or onto function) is a function that covers every point in the codomain at least once. The degree of a map between spheres, like from \( S^n \) to \( S^n \), is an integer representing the number of times the sphere is wrapped around. A degree of zero means that the net orientation is such that the positive and negative wrappings cancel out.
02

Consider a Map that Wraps Positively and Negatively

We need a function that wraps the sphere on itself in such a way that the positive and negative overlap perfectly cancels out. A simple example is to consider maps \( f(x) = -x \), which indicates that every point on the sphere is mapped to its diametric opposite or antipodal point.
03

Verify the Surjectivity of \( f(x) = -x \)

For every point \( y \) on sphere \( S^n \), there exists a point \( x \) such that \( f(x) = y \), because mapping to the antipodal point covers the whole sphere. This establishes that the map \( f(x) = -x \) is surjective.
04

Calculate the Degree of the Map

The degree of \( f(x) = -x \) is zero for any odd-dimensional sphere \( S^{2n+1} \) since the map changes the sign. The wrapping of \( S^n \) on itself once positively and once negatively means that effectively no net wrapping occurs. For even-dimensional spheres \( S^{2n} \), the antipodal map corresponds to a degree of \((-1)^{n+1}\), which is zero for any odd integer \(n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Degree of a Map
When we think about functions or maps between spaces, one interesting property is their *degree*. The degree tells us about how many times a map wraps one space around another. In the context of spheres, particularly when mapping from one sphere to itself (like from \( S^n \) to \( S^n \)), the degree can help define the orientation and coverage.
  • If a sphere is mapped onto itself with a degree of zero, it indicates that the sphere wraps around positively and negatively in a way that these effects cancel each other out.
  • This means that when you wrap the sphere around itself, the net wrapping is zero.
The concept of the degree is deeply connected with topological properties of the map, such as surjectivity and orientation, and it is often used in algebraic topology to demonstrate things like the Index Theorem or the Brouwer Fixed Point Theorem.
Spheres
Spheres, denoted by \( S^n \), are crucial objects in mathematics, always associated with a fixed central point and equidistant boundary. For any \( n \geq 1 \), \( S^n \) represents the n-dimensional sphere:
  • \( S^1 \) is a circle, a 1-dimensional sphere.
  • \( S^2 \) is the usual 2-dimensional surface of a ball, like a basketball.
  • \( S^3 \) represents a 3-dimensional sphere, which is harder to visualize but can be thought about in terms of 3D constructs.
Spheres play a significant role because they offer simple yet powerful insights into topological spaces. Much of sphere-related mappings centers around symmetry, continuity, and dimensionality, laying the foundation for various mathematical theorems.
Antipodal Map
An antipodal map is a fascinating concept where each point on a sphere maps to its diametric opposite, often expressed as \( f(x) = -x \). This idea is central in generating surjective maps, as seen from the example where it ensures full coverage of the sphere.
  • The antipodal map inherently possesses symmetry.
  • It effectively turns points around a center, ensuring every point is touched.
  • This property makes it very useful in constructing maps with specific degrees, such as degree zero, which require both positive and negative wrapping.
Antipodal maps are also essential in broader mathematics for proofs and problem-solving, like proving that every great circle on a sphere has a pair of opposite points.
Odd-Dimensional Spheres
Odd-dimensional spheres are particular in topological contexts because they showcase unique properties compared to even-dimensional ones.For odd dimensions like \( S^{2n+1} \), antipodal mappings lead to a natural orientation reversal:
  • This is because mapping \( x \) to \( -x \) in odd-dimensional spaces changes the overall sign.
  • This sign change corresponds to a degree of zero, showcasing that these spheres wrap positively and negatively, cancelling out.
These spheres are harder to visualize (like \( S^3 \) or \( S^5 \)), yet provide rich applications in vector fields and differential forms. Recognizing their nature is key in discussions about topological invariants and homotopy, providing insight into complex mappings and structures that do not exist in lower dimensions.

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Most popular questions from this chapter

A polynomial \(f(z)\) with complex coefficients, viewed as a map \(\mathrm{C} \rightarrow \mathrm{C}\), can always be extended to a continuous map of one- point compactifications \(\hat{f}: S^{2} \rightarrow S^{2}\). Show that the degree of \(\hat{f}\) equals the degree of \(f\) as a polynomial. Show also that the local degree of \(\hat{f}\) at a root of \(f\) is the multiplicity of the root.

Compute the simplicial homology groups of the triangular parachute obtained from \(\Delta^{2}\) by identifying its three vertices to a single point.

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36\. Show that \(H_{i}\left(X \times S^{n}\right) \approx H_{i}(X) \oplus H_{i-n}(X)\) for all \(i\) and \(n,\) where \(H_{i}=0\) for \(i<0\) by definition. Namely, show \(H_{i}\left(X \times S^{n}\right) \approx H_{i}(X) \oplus H_{i}\left(X \times S^{n}, X \times\left[x_{0}\right]\right)\) and \(H_{i}\left(X \times S^{n}, X \times\left\\{x_{0}\right\\}\right) \approx H_{i-1}\left(X \times S^{n-1}, X \times\left\\{x_{0}\right\\}\right) .\) IFor the latter isomorphism the relative Mayer-Vietoris sequence yields an easy proof.|

For \(X\) a finite \(\mathrm{CW}\) complex and \(F\) a field, show that the Fuler characteristic \(\chi(X)\) can also be computed by the formula \(x(X)=\Sigma_{n}(-1)^{n} \operatorname{dim} H_{n}(X ; F),\) the alternating sum of the dimensions of the vector spaces \(H_{n}(X ; F)\).
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