The Fundamental Group

Topology · Axiom Academy

Understanding π₁(X, x₀): the first algebraic invariant of topological spaces Step 1: Loops Based at a Point Before we can define the fundamental group, we need to understand what we're counting. The fundamental group studies loops in a space—continuous paths that start and end at the same point. A loop in a topological space based at a point is a continuous function such that . We call the basepoint of the loop. Step 2: When Are Two Loops the Same? Not all loops are created equal! Two loops might look different, but if we can continuously deform one into the other (while keeping the basepoint fixed), we consider them homotopic —essentially the same from a topological perspective. Two loops and based at are homotopic (written ) if there exists a continuous function such that: for all (basepoint stays fixed) Step 3: Defining the Fundamental Group Now we can define the fundamental group! It consists of all homotopy classes of loops—that is, we group together all loops that are homotopic to each other. The fundamental group of a space with basepoint , denoted , is the set of homotopy classes of loops based at . where denotes the homotopy class of the loop . In other words, each element of is represented by a loop, but we identify loops that can be continuously deformed into each other. Step 4: The Group Operation (Loop Composition) What makes the fundamental group a group ? We need a binary operation! We can compose two loops by traversing one, then the other.

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