Computing Fundamental Groups

Algebraic Topology · Axiom Academy

EXAMPLE Computing Fundamental Groups Step-by-step computation of π₁ for contractible spaces, spheres, and the figure-eight Excellent work! You've completed this example. Here's what we learned: Contractible spaces: Euclidean space ℝⁿ is contractible, meaning all loops can be continuously shrunk to a point, so π₁(ℝⁿ) = 0 (the trivial group). Higher spheres: For n ≥ 2, the n-sphere Sⁿ is simply connected with π₁(Sⁿ) = 0 because any loop can be contracted using the extra dimension. Non-simply connected spaces: The figure-eight space (wedge sum of two circles) has fundamental group π₁(S¹ ∨ S¹) ≅ ℤ * ℤ, the free group on two generators, representing independent loops around each circle. van Kampen's theorem: This powerful tool lets us compute fundamental groups of spaces by decomposing them into simpler pieces and analyzing how they're glued together. These computations form the foundation for understanding more complex topological spaces!

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