Contracting the Dunce Hat

Daniel RajchwaldGeorge Francis

John DalbecIlliMath 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.

IlliDunce

• IlliDunce 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 IlliDunce 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 IlliMath 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.