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Contracting the Dunce Hat
Daniel RajchwaldGeorge Francis
John DalbecIlliMath 2010
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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
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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.”
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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.
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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.
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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.
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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
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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
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Further Goals
• Document Tsurgua’s and Dalbec’s work as a stepping stone towards new/more generalized results
• Publish a paper
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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.