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 then we identify the line segment with and with in the
orientation indicated by the order given of the vertices. (See Figure 1-6.) Show that
the dunce cap is contractible.
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
Proof of the Qustion. Suppose is a map from to itself. The cone for is obtained by pinching the top of the mapping cylinder to a point. As is the cylinder with the bottom pasted to by the map , is with pasted to by the map . So the dunce cap is just defined as
which is homotopic to the identity by a linear homotopy (note that we make the choice of for an easy definition of the homotopy)
So the dunce cap is homotopic to which is contractible.