topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...

, 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 * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

formed by taking a solid

triangle A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

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 advant ...

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 Scotus. D ...

, 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 that ...

2-complex (such as the dunce hat) after taking the Cartesian product 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 In mathematics, the Zeeman conjecture or Zeeman's collapsibility conjecture asks whether given a finite contractible 2-dimensional CW complex K, the space K\times ,1/math> is collapsible. The conjecture, due to Christopher Zeeman, implies the Po ...

and was shown by Zeeman to imply the

Poincaré conjecture In the mathematics, mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dim ...

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 branch of mathematics, 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 deforma ...

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

CW-complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...

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

homotopic In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

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

homotopy equivalent In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...

to the disc. By contrast, it is not collapsible because it does not have a

free face Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to proc ...

References

* {{cite journal , last=Zeeman , first=E. C. , title=On the dunce hat , journal=Topology , volume=2 , issue=4 , date=1964 , pages=341–358 , doi=10.1016/0040-9383(63)90014-4 , doi-access=free Topological spaces Algebraic topology

Name

Properties

See also

References