Contracting the Dunce Hat Daniel Rajchwald George Francis

Contracting the Dunce Hat Daniel Rajchwald George Francis John Dalbec Illi. Math 2010

Background • Dunce hat is a cell complex that is contractible but not collapsible. Significance having both of these properties is due in part to EC Zeeman. • (Zeeman Conjecture) He observed that any contractible 2 -complex (such as the dunce hat) after taking the Cartesian product with the closed unit interval seemed to be collapsible. Shown to imply Poincare Conjecture

Collapsibility • It is not collapsible because it does not have a free face. • (Wikipedia) “Let K be a simplicial complex, and suppose that s is a simplex in K. We say that s has a free face t if t is a face of s and t has no other cofaces. We call (s, t) a free pair. If we remove s and t from K, we obtain another simplicial complex, which we call an elementary collapse of K. A sequence of elementary collapses is called a collapse. A simplicial complex that has a collapse to a point is called collapsible. ”

Contractibility • The dunce hat can be deformed into the spine of a 3 -ball, showing that it is contractible, i. e. it can be continuously deformed into a point. • Definition: Two functions, f: X ->Y, g: X->Y between topological spaces X and Y are said to be homotopic if there exists a continuous function H: [0, 1] x X - > Y such that H(0, x) = f(x) and H(1, x) = g(x) for each x in X.

Contractibility (cont) • A topological space X is said to be contractible if the identity map I: X->X, I(x)=x is homotopic to a constant map g: X->X, g(x) = z for some z in X.

Illi. Dunce • Illi. Dunce RTICA is an animation used to show the contraction of the dunce hat. The contraction was discovered by John Dalbec. • George Francis translated his animation to the animation to Illi. Dunce in 2001.

The Contraction • First Phase: Move points up (map symmetric about the altitude) • Second Phase: Factor the first phase through the quotient • Third Phase: Push along the free edge towards the dunce hat’s rim • Fourth Phase: Contract the rim to the vertex

Mathematica • Mimi Tsuruga translated George Francis’s duncehat. c to Mathematica during Illi. Math 2004. • Code focused on functions “fff” and “eee. ” – “fff” maps the first stage of the homotopy – “eee” readjusts the locations of the points as the dunce hat becomes double pleated

Further Goals • Document Tsurgua’s and Dalbec’s work as a stepping stone towards new/more generalized results • Publish a paper

References • [1] E. C. Zeeman. On the dunce hat. Topology, 2(4): 341 -348, December 1963. • [2] John Dalbec. Contracting the Dunce Hat, July 2010.