Answer: The Dunce Hat Is Contractible

Key Idea / Intuition

The dunce hat looks bizarre — its edge identification pattern creates no obvious retraction to a point, and the space is famously not obviously contractible by inspection. Yet it is contractible! The trick is not to try to visualize a deformation retraction directly, but instead to use the fact that the fundamental group is trivial (by van Kampen or direct calculation) and appeal to Whitehead's theorem for CW-complexes: a simply connected 2-dimensional CW-complex with trivial homology is contractible. The deeper lesson is that algebraic triviality can force geometric triviality, even when the geometry looks weird.

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Formal Proof / Solution

Step 1: Build the CW structure

The dunce hat $D$ has a CW structure:

• One 0-cell: $v$ (all three vertices of the triangle are identified to a single point).
• One 1-cell: $a$ (all three edges are identified to a single edge).
• One 2-cell: $e^2$, attached via the word $a \cdot a \cdot a^{-1}$.

So the attaching map of $e^2$ goes around the single loop $a$ with total degree $1 + 1 - 1 = 1$.

Step 2: Compute the fundamental group

The 1-skeleton is $S^1$ (one 0-cell and one 1-cell), so $\pi_1(\text{1-skeleton}) = \mathbb{Z}$, generated by $[a]$.

The 2-cell is attached by the loop $a \cdot a \cdot a^{-1} = a$, which represents the element $1 + 1 - 1 = 1$ in $\mathbb{Z}$ (i.e., the generator $[a]$ itself).

By van Kampen's theorem (or the standard formula for CW-complexes): $\pi_1(D) = \mathbb{Z} / \langle [a] \rangle = \mathbb{Z}/\mathbb{Z} = \{1\}.$

So $D$ is simply connected.

Step 3: Compute homology

The cellular chain complex is: $0 \to \mathbb{Z} \xrightarrow{\partial_2} \mathbb{Z} \xrightarrow{\partial_1} \mathbb{Z} \to 0.$

• $\partial_1(a) = v - v = 0$ (both endpoints are identified to $v$).
• $\partial_2(e^2)$: the attaching word is $aaa^{-1}$, so the degree is $1 + 1 + (-1) = 1$.

Thus $\partial_2$ is multiplication by $1$, which is an isomorphism. So: $H_0(D) = \mathbb{Z}, \quad H_1(D) = \ker \partial_1 / \operatorname{im} \partial_2 = \mathbb{Z}/\mathbb{Z} = 0, \quad H_2(D) = \ker \partial_2 = 0.$

All reduced homology groups vanish.

Step 4: Conclude contractibility

$D$ is a connected, simply connected CW-complex with all homology groups trivial. By Whitehead's theorem (or more elementarily, by the Hurewicz theorem + Whitehead):

A simply connected CW-complex with trivial reduced homology is contractible.

Therefore, the dunce hat is contractible, and $\pi_1(D) = \{1\}$.

The Surprise

The dunce hat is a compact 2-dimensional space that looks "badly" glued — intuitively it seems like it should have some interesting topology. Yet it collapses to a point. This makes it a famous example showing that:

1. Simple connectivity + vanishing homology $\Rightarrow$ contractibility (for CW-complexes).
2. A space can be contractible without admitting any obvious geometric retraction.
3. The dunce hat is not collapsible in the simplicial sense (a subtler combinatorial notion), showing contractibility $\not\Rightarrow$ collapsibility — a surprise in combinatorial topology.