Jonginn Yun

Homotopy

Two maps $f,g:X\to Y$ are homotopic if one can be continuously deformed into the other:

$$H:X\times[0,1]\to Y,\qquad H(-,0)=f,\quad H(-,1)=g.$$

Homotopy keeps track of qualitative shape. It ignores metric details but remembers obstructions such as holes.

Fundamental Group

At a base point $x_0\in X$, loops based at $x_0$ can be multiplied by concatenation. After quotienting by homotopy of loops, one obtains the fundamental group

$$\pi_1(X,x_0).$$

This group detects one-dimensional holes. For example,

$$\pi_1(S^1)\cong \mathbb Z,$$

because a loop around the circle has an integer winding number.

Covering Spaces

A covering map $p:\widetilde X\to X$ is a map such that every point of $X$ has a neighborhood $U$ whose preimage is a disjoint union of copies of $U$.

Covering spaces turn questions about loops into questions about lifting paths. A loop in $X$ may lift to a path in $\widetilde X$ whose endpoint records winding information.

Why This Matters Later

Principal bundles and gauge theory often contain global information that cannot be seen in a single coordinate patch. Homotopy and covering spaces provide the first algebraic language for such global obstructions.