10. Introduction to Homotopy Theory

Introduction to Homotopy Theory

Fundamental Group and Homotopy

Fundamental Group: The fundamental group \( \pi_1(X, x_0) \) of a topological space \( X \) with basepoint \( x_0 \) is a group that captures information about the possible ways loops in \( X \) based at \( x_0 \) can be continuously deformed to each other.

Homotopy: Two continuous maps \( f, g: X \rightarrow Y \) between topological spaces are said to be homotopic if there exists a continuous map \( H: X \times [0, 1] \rightarrow Y \) such that \( H(x, 0) = f(x) \) and \( H(x, 1) = g(x) \) for all \( x \) in \( X \). Intuitively, two maps are homotopic if one can be continuously deformed into the other.

Higher Homotopy Groups

Higher Homotopy Groups: While the fundamental group captures information about loops in a space, higher homotopy groups \( \pi_n(X) \) for \( n > 1 \) capture information about higher-dimensional analogs of loops, known as spheres or spheres of dimension \( n \). These groups measure the non-trivial ways such spheres can be continuously deformed within the space.

Applications to Topological Classification

Homotopy theory provides powerful tools for classifying topological spaces up to homotopy equivalence. For example:

• Two spaces with isomorphic fundamental groups are homotopy equivalent.
• Homotopy invariants such as homology and cohomology groups can be used to distinguish between non-homotopy equivalent spaces.
• Homotopy theory plays a fundamental role in algebraic topology, which studies topological spaces using algebraic methods.