Loops and Holes

Topology · Axiom Academy

Unit 8 - Introduction to Algebraic Topology Imagine you're holding a rubber band. You can stretch it, twist it, move it around—but there's something special about it. If you lay it flat on a table, you've created a loop . Now here's a fundamental question in topology: Can you shrink this loop down to a single point without leaving the surface? Click the canvas to shrink the loop But what if there's a hole in your surface? Imagine a table with a hole cut out in the middle, and your rubber band is wrapped around that hole. Now try to shrink the loop to a point. You can't! The hole prevents you from pulling the loop tight. The loop is "trapped" around the hole. Click to attempt shrinking the loop (it can't!) Consider a donut (mathematicians call it a torus ). It has a hole through its center. On a torus, there are different kinds of loops! This loop sits on the surface and can be shrunk to a point This loop goes through the hole and cannot be shrunk to a point This simple idea—whether loops can be shrunk to a point—is the foundation of algebraic topology . By studying loops, mathematicians can: Count the number of holes in a space Distinguish between different surfaces (a sphere vs. a torus) Understand the connectivity of complex spaces Solve problems in physics, data analysis, and robotics Interactive comparison of contractible vs. non-contractible loops

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