Homotopy Type

Let $X$ be a topological space.

Definition. The homotopy type of $X$ is its equivalence class under homotopy equivalence. That is, two spaces $X$ and $Y$ have the same homotopy type if and only if $X \simeq Y$: there exist continuous maps $f: X \to Y$ and $g: Y \to X$ with $g \circ f \simeq \mathrm{id}_X$ and $f \circ g \simeq \mathrm{id}_Y$.

Proposition. If $X$ and $Y$ have the same homotopy type, then they share all homotopy-invariant algebraic data: homotopy groups, singular homology groups, singular cohomology ring, and Euler characteristic (when defined).

Proof sketch. Each of these invariants is a functor from the homotopy category $\mathsf{hTop}$ (or from pointed spaces in the case of homotopy groups) to an algebraic category; homotopy equivalences become isomorphisms in $\mathsf{hTop}$, hence are sent to isomorphisms by any such functor. $\square$

Proposition. Every compact polyhedron (finite simplicial complex) has the homotopy type of a finite CW complex, and conversely.

Examples.

• $\{*\}$, $D^n$, $\mathbb{R}^n$, and any convex subset of $\mathbb{R}^n$ all represent the contractible homotopy type.
• $S^1$, the annulus $\{z \in \mathbb{C} : 1 \leq |z| \leq 2\}$, the cylinder $S^1 \times [0,1]$, and the Möbius band all represent the same homotopy type.
• $S^2$ and $T^2$ have different homotopy types: $\pi_1(S^2) = 0$ while $\pi_1(T^2) \cong \mathbb{Z} \times \mathbb{Z}$.

Remark. In higher category theory, homotopy types are identified with $\infty$-groupoids – $\infty$-categories in which every morphism at every level is invertible. This identification is the content of the homotopy hypothesis (Grothendieck), proven in several models of higher categories.