The Fundamental Group

Loops up to homotopy: the first invariant that detects holes.

The idea

The fundamental group $\pi_{1}(X, x_{0})$ is the first algebraic invariant of a space: it attaches to $X$ a group, and what it detects is holes. Invariants such as compactness and connectedness answer yes or no, which separates few spaces; a group carries far more information, and two spaces can be told apart by showing their groups are not isomorphic.

A loop detects a hole by being unable to shrink past it, and that is what the definition records.

Definition (Fundamental group).

Fix a basepoint $x_{0} \in X$. A loop at $x_{0}$ is a continuous path $f \colon [0,1] \to X$ with $f(0) = f(1) = x_{0}$, and two loops are counted as the same when one can be deformed into the other with the basepoint held fixed. The fundamental group $\pi_{1}(X, x_{0})$ is the set of these homotopy classes of loops, with the operation of running one loop and then the other, $[f][g] = [f \cdot g]$.

Under this operation the classes do form a group. Concatenation is associative up to homotopy, which suffices because homotopic loops were identified at the outset; the class of the constant loop is the identity; and running a loop backwards gives its inverse.

A trivial $\pi_{1}$ says every loop shrinks to a point. A nontrivial one says some loop cannot, and records how it wraps.

Ways to work on it

• Walkthrough. Loops, homotopy, and the group law; trivial _1 versus winding numbers.
• Practice. Name the fundamental group of a standard space.
• Hardest. Use basepoint independence and homotopy invariance to compute a fundamental group.

Not sure where to start? Take the ten-question placement test.