Problem 5 - Algebraic Topology

Question

Construct a connected graph \(X\) and maps \(f, g: X \rightarrow X\) such that \(f g=\mathbb{1}\) but \(f\) and \(g\) do not induce isomorphisms on \(\pi_{1} .\) [Note that \(f_{*} g_{*}=\mathbb{1}\) implies that \(f_{*}\) is surjective and \(g_{*}\) is injective.

Step-by-Step Solution

Answer

Construct \(X=S^1 \vee S^1\), \(f\) adds a loop, \(g\) collapses a loop. \(fg=\mathbb{1}\), but no isomorphism on \(\pi_1(X)\).

Step 1: Understand the Problem

We are tasked with constructing a connected graph \(X\) and maps \(f, g: X \rightarrow X\) such that the composition \(fg\) is the identity map \(\mathbb{1}\). However, neither \(f\) nor \(g\) should induce isomorphisms on the fundamental group \(\pi_{1}(X)\).

Step 2: Construct the Graph

Consider the graph \(X\) to be a wedge of two circles, \(S^1 \vee S^1\). This graph is connected and has a nontrivial fundamental group, specifically \(\pi_1(X) \cong \mathbb{Z} * \mathbb{Z}\), which is the free group on two generators.

Step 3: Define the Maps

Define the map \(g: X \rightarrow X\) such that it collapses one of the circles. For example, map both circles to one of them, making \(g\) effectively retrace one circle completely and collapse the other to a point. The map \(f: X \rightarrow X\) can then be defined as the inclusion map that adds the circle back from the collapse.

Step 4: Verify the Composition

Check the composition \(fg\): The map \(g\) collapses one circle to a point, and \(f\) maps it back, together behaving as an identity on one of the circles while being trivial on the collapsed circle, resulting in \(fg = \mathbb{1}\).

Step 5: Check the Fundamental Group Maps

For the induced maps on the fundamental group: \(g_*\) would be injective as it maps both original generators onto a single generator, not losing information of the non-collapsed circle. \(f_*\) would be surjective since it maps the generated group back onto both generators. However, since \(\pi_1(f)\) and \(\pi_1(g)\) do not preserve the entire structure of \(\pi_1(X) = \mathbb{Z} * \mathbb{Z}\), these maps do not induce isomorphisms.

Key Concepts

Connected Graph

A connected graph is one where any two vertices are connected directly or indirectly by a path. Unlike a disconnected graph, which has isolated parts, a connected graph ensures full traversal with no breaks. This property is crucial when discussing topics like the fundamental group, as it guarantees that there is a continuous path connecting every part of the graph.

• In any connected graph, there is at least one path between any two vertices, ensuring the graph is a single piece.
• This idea is critical in topology and algebraic topology, especially when constructing spaces that are easy to study in terms of their fundamental group.

For example, a pair of circles joined at a single point form a simple connected graph. Such a design is especially useful when studying maps and operations affecting the graph, like the ones seen in the exercise above, where operations on a connected graph must consider all paths available.

Graph Map

A graph map refers to a function between graphs that respects their vertex and edge structures. Typically, this involves a function that maps the vertices of one graph to another while preserving connections (edges) between these vertices.

• Graph maps are a simple way to represent transformations or operations on graphs, including collapses or extensions.
• In topology, they can alter graph properties in a controlled manner, as seen with the maps in the exercise where circles are collapsed or extended.

In the context of the exercise, the maps \(f\) and \(g\) alter the basic structure of a wedge sum graph. Each map serves a specific role: \(g\) collapses part of the graph, reducing complexity, while \(f\) reintroduces it, ensuring simplicity and manageability in the problem setting.

Wedge Sum

A wedge sum is a particular way of joining multiple spaces by identifying a single point in each, usually denoted as \(S^1 \vee S^1\) for two circles. It's a central concept in algebraic topology because it creates a new topological space with altered properties.

• A wedge sum of circles has a fundamental group that is the free product of the fundamental groups of each constituent circle.
• This operation is useful when exploring complex topological spaces since it allows combining simpler spaces with known properties.

The exercise demonstrates this by using the wedge sum of two circles. The resulting topological space has intriguing properties, such as a fundamental group \(\pi_1(X) \cong \mathbb{Z} * \mathbb{Z}\), representing two independent paths around the circles. This property allows for constructing maps like \(f\) and \(g\) to specifically explore non-isomorphic transformations of the fundamental group.