Fundamental Group of the Plane

Topology · Axiom Academy

EXAMPLE Fundamental Group of the Plane Unit 8 - Introduction to Algebraic Topology - π₁(ℝ²) = 0 Example 1: Proving ℝ² is Simply Connected Show that every loop in the plane can be continuously shrunk to a point A space is simply connected if it is path-connected and every loop can be continuously contracted to a point. Equivalently, . To prove that , we need to show that for any continuous loop , there exists a homotopy that shrinks this loop to a point. Let be any continuous loop in with . Define the homotopy by the straight-line homotopy: This continuously deforms the loop by scaling it toward the basepoint. Check that this is indeed a valid homotopy: Since is convex, every point on the straight line from to lies in . Therefore, is continuous, and every loop is null-homotopic. Why does the convexity of ℝ² matter in this proof? Example 2: Straight-Line Homotopy in Action Visualize how a specific loop contracts to a point Consider a circular loop centered at the origin with radius 1: Using our straight-line homotopy formula : As varies from 0 to 1, we get concentric circles of decreasing radius. Interactive: The animation shows the loop at different stages of the homotopy At each stage , we get a loop: At : Single point (the origin) The loop has been continuously contracted! What would happen if we tried this homotopy on a loop in ℝ² \ (0,0) (plane with origin removed) where the loop winds around the origin? Example 3: Generalization to Convex Sets

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