Category: Algebraic Topology

The “Dunce Cap” Space Is Contractible

Here is the exercise 6 on P. 50 in the book Topology and Geometry by Glen Bredon. I put it here because I found the drawing of this cap very lovely. Indeed I like that most of the pictures in this book are lovely sketches.

Question. The “dunce cap” space is the quotient of a triangle (and interior) obtained by identifying all three edges in an inconsistent manner. That is, if the vertices of the triangle are p, q, r then we identify the line segment (p, q) with (q, r) and with (p, r) in the
orientation indicated by the order given of the vertices. (See Figure 1-6.) Show that
the dunce cap is contractible.the dunce cap

Following the development in the book, I will use the following theorem that the homotopy type of a mapping cylinder or cone
depends only on the homotopy class of the map [Theorem 14.18, Topology and Geometry by Glen Bredon]. The idea is to identify the dunce cap as a mapping cone.

Theorem 14.18. If f_0\simeq f_1:X\to Y \text{ then } M_{f_0 } \simeq M_{f_1}\text { rel } X+Y \text{ and }C_{f_0}\simeq C_{f_1}\text{ rel } Y+\mathrm{vertex}.

Proof of the Qustion. Suppose f: S^1\to S^1 is a map from S^1 to itself. The cone C_f=M_f/S^1 \times\{1\} for f is obtained by pinching the top of the mapping cylinder to a point. As M_f  is the cylinder S_1\times [0,1] with the bottom pasted to S_1 by the map f, C_f is D_2 with \partial D_2 pasted to S_1 by the map f. So the dunce cap is just C_f \text{ with } f: S^1\to S^1 defined as

f(e^{2\pi i t})= \begin{cases} e^{2\pi i (3t)}, 0\leq t\leq 2/3\\ e^{2\pi i(2- 3t)}, 2/3\leq t\leq 1. \end{cases}

which is homotopic to the identity by a linear homotopy (note that we make the choice of f for an easy definition of the homotopy)

H(e^{2\pi i t},s)= \begin{cases} e^{2\pi i (3t(1-s)+st}, 0\leq t\leq 2/3\\ e^{2\pi i[(2- 3t)(1-s)+st]}, 2/3\leq t\leq 1. \end{cases}.

So the dunce cap is homotopic to C_{id}\simeq D^2 which is contractible.