Problem 9 - Algebraic Topology

Question

Show that if \(f: S^{n} \rightarrow S^{n}\) has degree \(d\) then \(f^{*}: H^{n}\left(S^{n} ; G\right) \rightarrow H^{n}\left(S^{n} ; G\right)\) is multiplication by \(d .\)

Step-by-Step Solution

Answer

The map generates the homomorphism by multiplication with the degree \(d\).

Step 1: Understand the Mapping Degree

The degree of a map \(f: S^n \rightarrow S^n\) is an integer \(d\) that represents how many times the domain space wraps around the range space under the mapping \(f\). For maps between spheres, this degree can be understood as the number of times the map covers the sphere. The degree is a fundamental topological invariant.

Step 2: Review Homology and Cohomology

The homology group \(H_n(S^n; G)\) for a sphere \(S^n\) is isomorphic to the group \(G\), and the same holds for the cohomology group \(H^n(S^n; G)\). These groups capture the topological structure of the sphere at dimension \(n\).

Step 3: Examine Induced Homomorphism in Cohomology

Given a map \(f: S^n \rightarrow S^n\), it induces a map \(f^{*}: H^n(S^n; G) \rightarrow H^n(S^n; G)\) in cohomology, which is a homomorphism. This induced map reflects how \(f\) transforms the topology of the space with respect to the chosen coefficients \(G\).

Step 4: Relate Degree to Induced Map

The degree \(d\) of the map \(f\) directly affects the induced homomorphism on cohomology. For cohomology class \([a] \in H^n(S^n; G)\), the induced map is given by \(f^*([a]) = d[a]\), meaning it multiplies each cohomology class by the degree \(d\). This results from properties of the orientation of the mappings where degree counts the signed sum of preimages.

Key Concepts

Cohomology

Cohomology is an algebraic tool used in topology to study and classify spaces. It is a type of homology theory that helps to analyze the properties of spaces using algebraic objects known as cohomology groups. These groups, denoted as \(H^n(X; G)\), associate algebraic structures with the topological space \(X\) at a particular dimension \(n\) and coefficients \(G\).
Cohomology groups differ from homology groups by capturing more detailed information about a space's structure. While homology deals with voids and holes, cohomology captures mappings and provides a means to study these using functions and algebra. It builds on the concept of duality, offering insights into how different parts of a space can be "glued" together through these algebraic structures.
Understanding cohomology can be a bit abstract, but think of it as a tool that helps measure how different parts of a space can interact or map between one another. This interaction is especially significant when exploring spheres and maps between them, as in the provided exercise about the degree of a map.

Induced homomorphism

In topology, when you have a continuous map between two spaces, it often leads to a transformation at the level of homology or cohomology groups. This transformation is known as an induced homomorphism. Specifically, for a map \(f: X \rightarrow Y\), the induced map, denoted \(f^*\), acts between the cohomology groups of \(X\) and \(Y\): \(f^*: H^n(Y; G) \rightarrow H^n(X; G)\).
Think of an induced homomorphism as a way to transport topological information from one space to another via the map \(f\). It helps to understand how features of a map affect the structure of the spaces it connects, effectively measuring the impact of the map at the level of cohomology.
In the context of mapping spheres, an induced homomorphism reflects how the degree of the mapping (i.e., how one sphere wraps around another) impacts the corresponding cohomology classes. It is an essential concept in understanding the relationship between a map's geometric properties and algebraic topology.

Topological invariant

Topological invariants are properties of a space that remain unchanged under homeomorphisms, meaning they are preserved through continuous deformations such as stretching and bending. These properties can include characteristics like genus, Euler characteristic, and degrees of maps, among others.
A topological invariant serves as a powerful tool to distinguish between different topological spaces. If two spaces share the same set of invariants, they might be homeomorphic, suggesting that they are essentially the same shape in a topological sense. However, differing invariants provide a clear distinction that no such continuous transformation can exist between them.
For instance, the degree of a map between two spheres, as mentioned in the exercise, is a topological invariant. It provides insight into how one sphere can be mapped onto another, reflecting how the map affects the space's structure. This degree reflects not only the map's essential geometry but also its signature in topological terms, as can be observed in its effect on cohomology groups through induced homomorphisms.