Transcript 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.
• (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.