The Fundamental Group of the Circle

Topology · Axiom Academy

LESSON The Fundamental Group of the Circle Unit 8 - Introduction to Algebraic Topology: π₁(S¹) = ℤ One of the most fundamental results in algebraic topology is computing the fundamental group of the circle . This result bridges geometry and algebra in a beautiful way. That is, what are all the homotopy classes of loops based at a point on the circle? The circle is the set of all points in the plane at distance 1 from the origin. A loop on the circle starts at a basepoint, travels around the circle (possibly multiple times, in either direction), and returns to where it started. The animation above shows a loop on the circle. Notice how the loop can wind around the circle multiple times. This winding number will be key to our answer. The key insight is that loops on a circle are classified by how many times they wind around the circle. Try the different buttons above to see loops with different winding numbers. Notice that two loops with the same winding number can be continuously deformed into each other (they're homotopic!). Step 3: The Group Structure on Loops The fundamental group isn't just a set—it has a group structure . We can compose loops by doing one after the other. Given two loops and , their composition is the loop that first traverses , then traverses . The animation shows composing loops with winding numbers 1 and 2. The resulting loop has winding number 3 = 1 + 2. We're now ready to state the main result:

This is the written version of the interactive lesson above. See the full Topology course.