Homotopy Groups Calculator
Compute $\pi_k(X)$ for common topological spaces
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Theory & Concepts
What is $\pi_k(X)$?
The $k$-th homotopy group, $\pi_k(X)$, measures the number of distinct ways you can map a $k$-dimensional sphere ($S^k$) into a space $X$, up to continuous deformation.
- $\pi_1(X)$: The Fundamental Group. It measures 1D loops. If you can shrink a loop to a point, it’s trivial ($0$).
- $\pi_k(X)$ for $k \ge 2$: Higher homotopy groups. These are always Abelian groups. They detect higher-dimensional “holes” or voids.
Common Questions
Why is $\pi_3(S^2) = \mathbb{Z}$?
Intuitively, a 3D sphere seems “too big” to wrap around a 2D sphere. However, the Hopf Fibration is a map that wraps $S^3$ around $S^2$ in a way that links non-trivially. It cannot be “slipped off” or deformed to a point.
What is the “Figure-8” space?
It is the Wedge Sum $S^1 \vee S^1$: two circles glued at a single point. Its fundamental group is the “Free Group on 2 generators”, meaning loops can wind around circle A and circle B in any order (AB $\neq$ BA).