Mapping Spheres into Spaces

Algebraic Topology · Axiom Academy

INTRO Mapping Spheres into Spaces Discover how continuous maps from spheres reveal the hidden shape of spaces Start by exploring what happens when we map a circle into different spaces. Move the slider to trace out a path. Step 2: Different Maps, Same Idea? Now we have two different maps from S¹ into the same space. Can we continuously deform one into the other? Step 3: Grouping Maps into Classes Select different maps to see which ones can be deformed into each other. Maps in the same homotopy class are equivalent. Step 4: Beyond Circles to Spheres The same idea works for higher-dimensional spheres! Explore maps from S² (a 2-sphere) into S³ (a 3-sphere). We just explored these: loops in a space with a basepoint. Now we map entire 2-dimensional spheres (like Earth's surface) into a space. This generalizes to spheres of any dimension! Step 5: The Importance of Basepoints Why do we insist on basepoints and based homotopies? By fixing a basepoint x₀ in X and requiring all maps to send the basepoint of Sⁿ to x₀, we can define a group operation : we can "concatenate" two maps by connecting them at the basepoint. For loops (n=1), we can compose two loops by traveling along the first, then the second, both starting and ending at x₀. This gives π₁(X, x₀) the structure of a group . Based homotopies ensure that when we deform maps, the basepoint stays fixed. This is crucial for the group structure to be well-defined. We're turning topology (continuous shapes) into algebra (groups)!

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