Problem 15 - Algebraic Topology

Question

Given a space \(X\) with basepoint \(x_{0} \in X,\) we may construct a CW complex \(L(X)\) having a single 0 -cell, a 1 -cell for each loop in \(X\) based at \(x_{0},\) and a 2 -cell for each map of a standard triangle \(T\) into \(X\) taking the three vertices to the basepoint. Such a 2-cell is attached to the three 1 -cells that are the loops obtained by restricting the map to the three edges of \(T .\) Show that \(\pi_{1}(L(X))\) is isomorphic to \(\pi_{1}\left(X, x_{0}\right)\) via an isomorphism induced by a natural map \(L(X) \rightarrow X\).

Step-by-Step Solution

Answer

The groups \(\pi_1(L(X))\) and \(\pi_1(X, x_0)\) are isomorphic via the map \(L(X) \rightarrow X\).

Step 1: Understand the Components of CW Complex

The CW complex given, \(L(X)\), is constructed with one 0-cell, many 1-cells, and several 2-cells. The 0-cell corresponds to the basepoint \(x_0\). Each 1-cell represents a loop in \(X\) based at \(x_0\). The 2-cells correspond to maps from a standard triangle \(T\) into \(X\). Each of these maps takes the vertices of \(T\) to the basepoint \(x_0\) and is attached along 1-cells that represent loops from the map's edge restrictions.

Step 2: Define Loop Map and Fundamental Group

The map from \(L(X)\) to \(X\) takes the unique 0-cell of \(L(X)\) to the basepoint \(x_0\), each 1-cell representing a loop to the corresponding loop in \(X\), and the 2-cells ensure the relations among the loops are consistent with \(\pi_1(X, x_0)\). The fundamental group \(\pi_1(L(X))\) is the set of equivalence classes of loops based at the 0-cell of \(L(X)\), the same as equivalence classes of loops based at \(x_0\) in \(X\).

Step 3: Establish the Isomorphism

Each 1-cell loop in \(L(X)\) exactly corresponds to an element of the fundamental group \(\pi_1(X, x_0)\). Since triangles (2-cells) represent the homotopy relations in \(X\), and they are preserved in \(L(X)\) upon gluing, the presentation of \(\pi_1(L(X))\) mimics that of \(\pi_1(X, x_0)\). The natural map from \(L(X)\) to \(X\) is a continuous function inducing an isomorphism \(\pi_1(L(X))\) to \(\pi_1(X, x_0)\), ensuring that for each path in \(L(X)\) there corresponds exactly one loop in \(X\) modulo homotopy.

Key Concepts

CW Complex

A CW complex is a topological space constructed by gluing together cells of varying dimensions. It provides a flexible way to build spaces that are important in algebraic topology. In the exercise, the space \(L(X)\) is a CW complex with the following structure:

• 0-Cell: This is a point, which in our context, is the basepoint \(x_0\).
• 1-Cells: These are line segments representing loops in the space \(X\) that start and end at the basepoint \(x_0\).
• 2-Cells: These are shapes like triangles that are mapped into \(X\) and glued along the segments to ensure the structure of relations between loops holds.

Each cell plays a critical role in constructing the space by adding layers of structure and complexity.
The 0-cell provides a starting point, while 1-cells capture the loop-like paths essential for understanding \(X\)'s fundamental group. The 2-cells further define how these loops relate via homotopy, completing the topological picture of \(L(X)\).

Fundamental Group

The fundamental group, denoted as \(\pi_1\), is a way to study the shape of a space through loops. Each loop is a path that starts and ends at a fixed point, known as the basepoint. In the problem, we're looking at the fundamental group \(\pi_1(L(X))\) and its relation to \(\pi_1(X, x_0)\).

• In \(L(X)\), each 1-cell represents a loop from the basepoint around the complex, embodying an element of the fundamental group.
• The fundamental group is about finding all possible loops in this space that can be continuously changed into one another, a process known as homotopy.

Importantly, the exercise demonstrates that the fundamental group of the CW complex \(L(X)\) is isomorphic to that of the original space \(X\). This means there is a one-to-one correspondence between loops in both spaces, preserving the loop structure through an isomorphism, created by mapping from \(L(X)\) to \(X\). This underlies the idea that the fundamental structure of \(X\) in terms of loops is fully captured by the structure of \(L(X)\).

Homotopy

Homotopy is a concept that describes when two shapes, functions, or spaces can be transformed into each other through continuous deformations. This idea is essential in understanding how loops in our spaces relate to each other.

• Continuous Transformation: Homotopy considers two paths equivalent if one can be smoothly morphed into the other without breaking. In our problem, this relates to how the 2-cells dictate path changes.
• Triangles and Loops: The 2-cells in a CW complex, particularly in \(L(X)\), facilitate homotopy by defining relationships between loops. These triangles map into \(X\), anchoring their vertices to the basepoint and aligning their edges with loop paths.
• Preservation of Structure: The whole setup ensures that the homotopy relations (like contractions of loops) present in \(X\) are reflected in \(L(X)\), thanks to the attachment of 2-cells that represent these transformations.

By understanding homotopy, we see how spaces are connected deeply, beyond just the loops themselves, ensuring that the spaces \(X\) and \(L(X)\) mirror each other in terms of topological properties. This is key in proving that the respective fundamental groups are isomorphic, as the relations between loops, dictated by homotopy, are preserved.