![Contractibility figure](https://upload.wikimedia.org/wikipedia/commons/1/1e/Contractibility_figure.png)
In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
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 ...
''X'' is contractible if the
identity map
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, un ...
on ''X'' is null-homotopic, i.e. if it 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 some constant map.
Intuitively, a contractible space is one that can be continuously shrunk to a point within that space.
Properties
A contractible space is precisely one with the
homotopy type
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 ...
of a point. It follows that all the
homotopy groups of a contractible space are
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since
singular homology
In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
is a homotopy invariant, the
reduced homology groups of a contractible space are all trivial.
For a topological space ''X'' the following are all equivalent:
*''X'' is contractible (i.e. the identity map is null-homotopic).
*''X'' is homotopy equivalent to a one-point space.
*''X''
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 deformat ...
s onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.)
*For any space ''Y'', any two maps ''f'',''g'': ''Y'' → ''X'' are homotopic.
*For any space ''Y'', any map ''f'': ''Y'' → ''X'' is null-homotopic.
The
cone
A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex.
A cone is formed by a set of line segments, half-lines, or lines con ...
on a space ''X'' is always contractible. Therefore any space can be embedded in a contractible one (which also illustrates that subspaces of contractible spaces need not be contractible).
Furthermore, ''X'' is contractible
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
there exists a
retraction from the cone of ''X'' to ''X''.
Every contractible space is
path connected
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
and
simply connected. Moreover, since all the higher homotopy groups vanish, every contractible space is
''n''-connected for all ''n'' ≥ 0.
Locally contractible spaces
A topological space ''X'' is locally contractible at a point ''x'' if for every
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
''U'' of ''x'' there is a neighborhood ''V'' of ''x'' contained in ''U'' such that the inclusion of ''V'' is nulhomotopic in ''U''. A space is locally contractible if it is locally contractible at every point. This definition is occasionally referred to as the "geometric topologist's locally contractible," though is the most common usage of the term. In
Hatcher's standard Algebraic Topology text, this definition is referred to as "weakly locally contractible," though that term has other uses.
If every point has a
local base In topology and related areas of mathematics, the neighbourhood system, complete system of neighbourhoods, or neighbourhood filter \mathcal(x) for a point x in a topological space is the collection of all neighbourhoods of x.
Definitions
Neighbour ...
of contractible neighborhoods, then we say that ''X'' is strongly locally contractible. Contractible spaces are not necessarily locally contractible nor vice versa. For example, the
comb space
In mathematics, particularly topology, a comb space is a particular subspace of \R^2 that resembles a comb. The comb space has properties that serve as a number of counterexamples. The topologist's sine curve has similar properties to the comb spa ...
is contractible but not locally contractible (if it were, it would be locally connected which it is not). Locally contractible spaces are locally ''n''-connected for all ''n'' ≥ 0. In particular, they are
locally simply connected In mathematics, a locally simply connected space is a topological space that admits a basis of simply connected sets. Every locally simply connected space is also locally path-connected and locally connected.
The circle is an example of a locally ...
,
locally path connected, and
locally connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness ...
. The circle is (strongly) locally contractible but not contractible.
Strong local contractibility is a strictly stronger property than local contractibility; the counterexamples are sophisticated, the first being given by
Borsuk and
Mazurkiewicz in their paper ''Sur les rétractes absolus indécomposables'', C.R.. Acad. Sci. Paris 199 (1934), 110-112).
There is some disagreement about which definition is the "standard" definition of local contractibility; the first definition is more commonly used in geometric topology, especially historically, whereas the second definition fits better with the typical usage of the term "local" with respect to topological properties. Care should always be taken regarding the definitions when interpreting results about these properties.
Examples and counterexamples
*Any
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
is contractible, as is any
star domain
In geometry, a Set (mathematics), set S in the Euclidean space \R^n is called a star domain (or star-convex set, star-shaped set or radially convex set) if there exists an s_0 \in S such that for all s \in S, the line segment from s_0 to s lie ...
on a Euclidean space.
*The
Whitehead manifold
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to \R^3. discovered this puzzling object while he was trying to prove the Poincaré conjecture, correcting an error in an earlier paper where ...
is contractible.
*
Spheres
The Synchronized Position Hold Engage and Reorient Experimental Satellite (SPHERES) are a series of miniaturized satellites developed by MIT's Space Systems Laboratory for NASA and US Military, to be used as a low-risk, extensible test bed for the ...
of any finite dimension are not contractible.
*The
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
in an infinite-dimensional
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
is contractible.
*The
house with two rooms
House with two rooms or Bing's house is a particular contractible, 2-dimensional simplicial complex that is not collapsible. The name was given by R. H. Bing.Bing, R. H., ''Some Aspects of the Topology of 3-Manifolds Related to the Poincaré C ...
is a standard example of a space which is contractible, but not intuitively so.
*The
Dunce hat is contractible, but not
collapsible.
*The cone on a
Hawaiian earring
In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n = 1, 2, 3, \ldots endowed with the subspace topology:
...
is contractible (since it is a cone), but not locally contractible or even locally simply connected.
*All
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s and
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 cl ...
es are ''locally'' contractible, but in general not contractible.
* The
Warsaw circle
Shape theory is a branch of topology that provides a more global view of the topological spaces than homotopy theory. The two coincide on compacta dominated homotopically by finite polyhedra. Shape theory associates with the Čech homology theory ...
is obtained by "closing up" the
topologist's sine curve
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the functio ...
by an arc connecting (0,−1) and (1,sin(1)). It is a one-dimensional continuum whose
homotopy groups are all trivial, but it is not contractible.
See also
*
References
{{DEFAULTSORT:Contractible Space
Topology
Homotopy theory
Properties of topological spaces