topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, the dunce hat is a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

formed by taking a solid

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

and

gluing Adhesive, also known as glue, cement, mucilage, or paste, is any non-metallic substance applied to one or both surfaces of two separate items that binds them together and resists their separation. The use of adhesives offers certain advantage ...

all three sides together, with the orientation of one side reversed. Simply gluing two sides oriented in the opposite direction would yield a cone much like the

dunce cap ''Dunce'' is a mild insult in English meaning "a person who is slow at learning or stupid". The etymology given by Richard Stanyhurst is that the word is derived from the name of the Scottish scholastic theologian and philosopher John Duns Scotu ...

, but the gluing of the third side results in identifying the base of the cap with a line joining the base to the point.

Name

The name is due to E. C. Zeeman, who observed that any

contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within t ...

2-complex (such as the dunce hat) after taking the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

with the

closed unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, ...

seemed to be collapsible. This observation became known as the Zeeman conjecture and was shown by Zeeman to imply the

Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured b ...

Properties

The dunce hat is

, but not collapsible. Contractibility can be easily seen by noting that the dunce hat embeds in the 3-ball and the 3-ball

deformation retract In topology, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformation retraction is a mappi ...

s onto the dunce hat. Alternatively, note that the dunce hat is the

CW-complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...

obtained by gluing the boundary of a 2-cell onto the circle. The gluing map is

homotopic In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

to the identity map on the circle and so the complex is

homotopy equivalent In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

to the disc. By contrast, it is not collapsible because it does not have a free face. Dunce hat animated

References

{{reflist, refs= {{cite book, title=Algorithmic Topology and Classification of 3-Manifolds, volume=9, series=Algorithms and Computation in Mathematics, first=Sergei, last=Matveev, publisher=Springer, year=2007, isbn=9783540458999, pages=46–58, url=https://books.google.com/books?id=vFLgAyeVSqAC&pg=PA46, contribution=1.3.4 Zeeman's Collapsing Conjecture {{cite journal , last=Zeeman , first=E. C., author-link=Christopher Zeeman , title=On the dunce hat , journal=

Topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...

, volume=2 , issue=4 , date=1964 , pages=341–358 , doi=10.1016/0040-9383(63)90014-4 , doi-access= Topological spaces Algebraic topology

Name

Properties

See also

References