Homotopy Groups

Entry conditions

Use homotopy groups only when you have pointed spaces (a chosen basepoint).

Definitions

• The fundamental group ${\pi }_1(X,x_0)$ is the group of loops based at $x_0$, modulo homotopy.
• Higher homotopy groups ${\pi }_n(X,x_0)$ use maps from the $n$-sphere $S^n$ into $X$ that send a basepoint to $x_0$.

Vocabulary (plain language)

• Loop: a path that starts and ends at the basepoint.
• Basepoint: a chosen reference point in the space.

Symbols used

• ${\pi }_1$: fundamental group.
• ${\pi }_n$: $n$-th homotopy group.

Intuition

Homotopy groups measure the kinds of holes in a space, organized by dimension.

Worked example

The circle $S^1$ has ${\pi }_1(S^1)\cong \mathbb{Z}$, reflecting how loops can wrap around it.

How to recognize the structure

• You can specify a basepoint.
• You can define loops or sphere-maps with homotopy equivalence classes.

Common mistakes

• Forgetting to fix a basepoint.