Problem 5 - Algebraic Topology

Question

Show that every homomorphism \(\pi_{1}\left(S^{1}\right) \rightarrow \pi_{1}\left(S^{1}\right)\) can be realized as the induced homomorphism \(\varphi_{*}\) of a map \(\varphi: S^{1} \rightarrow S^{1}\).

Step-by-Step Solution

Answer

Every homomorphism \( \pi_1(S^1) \to \pi_1(S^1) \) is induced by a map \( S^1 \to S^1 \) that wraps the circle \( k \) times.

Step 1: Understand the Fundamental Group of \( S^1 \)

The fundamental group \( \pi_1(S^1) \) is isomorphic to \( \mathbb{Z} \), the group of integers under addition. This means that any element in \( \pi_1(S^1) \) can be represented as an integer, which corresponds to the class of loops that wind around the circle a certain number of times.

Step 2: Consider the Homomorphism on \( \mathbb{Z} \)

Given a homomorphism \( f: \pi_1(S^1) \rightarrow \pi_1(S^1) \), it induces a homomorphism \( f: \mathbb{Z} \rightarrow \mathbb{Z} \). Homomorphisms from \( \mathbb{Z} \) to \( \mathbb{Z} \) are entirely determined by \( f(1) = k \), where \( k \) is an integer, since \( f(n) = n \cdot k \) for any integer \( n \).

Step 3: Consider Maps \( \varphi: S^1 \rightarrow S^1 \)

A continuous map \( \varphi: S^1 \rightarrow S^1 \) can be described by how it affects a generator of \( \pi_1(S^1) \). Specifically, such maps correspond to wrapping the circle around itself \( k \) times. Thus, the map \( \varphi \), which winds the circle \( k \) times around itself, induces the homomorphism on \( \pi_1(S^1) \) that is multiplication by \( k \).

Step 4: Show that Every Homomorphism Corresponds to a Map

Since any homomorphism \( f : \mathbb{Z} \rightarrow \mathbb{Z} \) is of the form \( n \mapsto n \cdot k \), construct the map \( \varphi: S^1 \rightarrow S^1 \) such that it wraps \( S^1 \) around itself \( k \) times. The induced homomorphism on \( \pi_1(S^1) \), \( \varphi_* \), will precisely be \( n \mapsto n \cdot k \). Thus, every possible homomorphism between the fundamental groups is induced by some continuous map.

Key Concepts

Homomorphism

In mathematics, a **homomorphism** is a structural-preserving map between two algebraic structures. These structures often include groups, rings, or vector spaces. In the context of fundamental groups, which are algebraic structures used to study topological spaces, a homomorphism takes the form of a function between the fundamental groups of two spaces that respects the operation of these groups.
This means that if you have a function \( f \) that is a homomorphism from one fundamental group \( \pi_1(X) \) to another \( \pi_1(Y) \), it satisfies the condition \( f(a \cdot b) = f(a) \cdot f(b) \) for all elements \( a \) and \( b \) from \( \pi_1(X) \). Essentially, the function keeps the "combining" operation of loops intact.

• A homomorphism must map the identity loop of the first group to the identity loop of the second group.
• It translates loops in a consistent manner, respecting the core properties of the algebraic structures it connects.

When discussing the homomorphism between the fundamental groups of \( S^1 \), or the 1-dimensional sphere, we can simplify this even further since both groups are isomorphic to \( \mathbb{Z} \). This means understanding any homomorphism \( f : \pi_1(S^1) \rightarrow \pi_1(S^1) \) is equivalent to understanding a homomorphism from \( \mathbb{Z} \) to \( \mathbb{Z} \). In this case, it's easy, since such a homomorphism is simply multiplication by an integer \( k \).

Continuous Map

A **continuous map** is a function between two topological spaces that preserves the notion of closeness. For a map \( \varphi : S^1 \rightarrow S^1 \), this means that points close together on the first circle (domain) remain close together on the second circle (codomain) after mapping.
If you're visualizing continuous maps from one circle to another, think of these maps as stretching or compressing the circle, possibly wrapping it multiple times around the second circle based on how many times \( k \) the first goes around. Specifically, these wraps or winds define how a generator of the fundamental group \( \pi_1(S^1) \) is mapped to the other.

• Every continuous map maintains topological features.
• For circle mappings, they define the degree of how many times they wrap around them.
• Such maps naturally lead to induced homomorphisms, reflecting these winding properties to the algebraic level.

Understanding this concept is crucial when considering how mappings between spaces affect underlying group operations, creating a direct link between geometry and algebra.

Induced Homomorphism

An **induced homomorphism** is the result of applying a continuous map between topological spaces to their fundamental groups. Simply put, when you have a continuous map \( \varphi : X \rightarrow Y \), it not only gives a path from one space to another but also creates a specific transformation among their fundamental groups.
This induced homomorphism \( \varphi_* : \pi_1(X) \rightarrow \pi_1(Y) \) formally translates the path structure of \( X \) into the path structure of \( Y \). Given a generator of the fundamental group \( \pi_1(S^1) \), if the continuous map wraps the circle \( k \) times, then the induced homomorphism will be multiplication by \( k \). This effectively means if the map transforms loops on \( S^1 \) into loops wrapping \( k \) times on the other circle, the homomorphism reflects this by multiplying integers.

• Induced homomorphisms ensure structural integrity from topological mappings to group transformations.
• Such mappings illustrate how two seemingly distinct areas of mathematics are connected.
• They offer a way to understand complex geometric transformations using algebraic operations.

Through this linkage, one can see the interplay between algebra and topology, where continuous maps dictate not only how spaces transform but also how the algebraic representations of those spaces (in the form of fundamental groups) are related.