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Contracting the Dunce Hat

Contracting the Dunce Hat. Daniel Rajchwald George Francis John Dalbec IlliMath 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.

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Contracting the Dunce Hat

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  1. Contracting the Dunce Hat Daniel Rajchwald George Francis John Dalbec IlliMath 2010

  2. 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

  3. 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 facet 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.”

  4. 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.

  5. 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.

  6. 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.

  7. 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

  8. 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

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

  10. 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.

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