In
mathematics, a configuration space is a construction closely related to
state space
A state space is the set of all possible configurations of a system. It is a useful abstraction for reasoning about the behavior of a given system and is widely used in the fields of artificial intelligence and game theory.
For instance, the to ...
s or
phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point in a high-dimensional space. In mathematics, they are used to describe assignments of a collection of points to positions in 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 po ...
. More specifically, configuration spaces in mathematics are particular examples of
configuration spaces in physics in the particular case of several non-colliding particles.
Definition
For a topological space
, the ''n''
th (ordered) configuration space of X is the set of ''n''-
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
s of pairwise distinct points in
:
:
This space is generally endowed with the subspace topology from the inclusion of
into
. It is also sometimes denoted
,
, or
.
There is a natural
action
Action may refer to:
* Action (narrative), a literary mode
* Action fiction, a type of genre fiction
* Action game, a genre of video game
Film
* Action film, a genre of film
* ''Action'' (1921 film), a film by John Ford
* ''Action'' (1980 fil ...
of the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on the points in
given by
:
This action gives rise to the
th unordered configuration space of ,
:
which is the
orbit space of that action. The intuition is that this action "forgets the names of the points". The unordered configuration space is sometimes denoted
,
, or
. The collection of unordered configuration spaces over all
is the
Ran space, and comes with a natural topology.
Alternative formulations
For a topological space
and a finite set
, the configuration space of with particles labeled by is
:
For
, define
. Then the
th configuration space of ''X'' is
, and is denoted simply
.
Examples
* The space of ordered configuration of two points in
is
homeomorphic to the product of the Euclidean 3-space with a circle, i.e.
.
*More generally, the configuration space of two points in
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 sphere
.
*The configuration space of
points in
is the classifying space of the
th
braid group
A braid (also referred to as a plait) is a complex structure or pattern formed by interlacing two or more strands of flexible material such as textile yarns, wire, or hair.
The simplest and most common version is a flat, solid, three-strande ...
(see
below).
Connection to braid groups
The -strand braid group on a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
topological space is
:
the
fundamental group of the
th unordered configuration space of . The -strand pure braid group on is
:
The first studied braid groups were the Artin braid groups
. While the above definition is not the one that
Emil Artin gave,
Adolf Hurwitz implicitly defined the Artin braid groups as fundamental groups of configuration spaces of the complex plane considerably before Artin's definition (in 1891).
It follows from this definition and the fact that
and
are
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
s of type
, that the unordered configuration space of the plane
is a
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
for the Artin braid group, and
is a classifying space for the pure Artin braid group, when both are considered as
discrete group
In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
s.
Configuration spaces of manifolds
If the original space
is a
manifold, its ordered configuration spaces are open subspaces of the powers of
and are thus themselves manifolds. The configuration space of distinct unordered points is also a manifold, while the configuration space of ''not necessarily distinct'' unordered points is instead an
orbifold
In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
D ...
.
A configuration space is a type of
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
or (fine)
moduli space. In particular, there is a universal bundle
which is a sub-bundle of the trivial bundle
, and which has the property that the fiber over each point
is the ''n'' element subset of
classified by ''p''.
Homotopy invariance
The homotopy type of configuration spaces is not
homotopy invariant. For example, the spaces
are not homotopy equivalent for any two distinct values of
:
is empty for
,
is not connected for
,
is an
Eilenberg–MacLane space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane space Saunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
of type
, and
is
simply connected for
.
It used to be an open question whether there were examples of ''compact'' manifolds which were homotopy equivalent but had non-homotopy equivalent configuration spaces: such an example was found only in 2005 by Riccardo Longoni and Paolo Salvatore. Their example are two three-dimensional
lens spaces, and the configuration spaces of at least two points in them. That these configuration spaces are not homotopy equivalent was detected by
Massey product
In algebraic topology, the Massey product is a cohomology operation of higher order introduced in , which generalizes the cup product. The Massey product was created by William S. Massey, an American algebraic topologist.
Massey triple product
Le ...
s in their respective universal covers. Homotopy invariance for configuration spaces of
simply connected closed manifolds remains open in general, and has been proved to hold over the base field
. Real homotopy invariance of simply connected compact
manifolds with simply connected boundary of dimension at least 4 was also proved.
Configuration spaces of graphs
Some results are particular to configuration spaces of
graphs. This problem can be related to robotics and motion planning: one can imagine placing several robots on tracks and trying to navigate them to different positions without collision. The tracks correspond to (the edges of) a graph, the robots correspond to particles, and successful navigation corresponds to a path in the configuration space of that graph.
For any graph
,
is an Eilenberg–MacLane space of type
and
strong deformation retracts to a
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 ...
of dimension
, where
is the number of vertices of
degree at least 3.
Moreover,
and
deformation retract to
non-positively curved cubical complexes of dimension at most
.
Configuration spaces of mechanical linkages
One also defines the configuration space of a mechanical linkage with the graph
its underlying geometry. Such a graph is commonly assumed to be constructed as concatenation of rigid rods and hinges. The configuration space of such a linkage is defined as the totality of all its admissible positions in the Euclidean space equipped with a proper metric. The configuration space of a generic linkage is a smooth manifold, for example, for the trivial planar linkage made of
rigid rods connected with revolute joints, the configuration space is the n-torus
.
The simplest singularity point in such configuration spaces is a product of a cone on a homogeneous quadratic hypersurface by a Euclidean space. Such a singularity point emerges for linkages which can be divided into two sub-linkages such that their respective endpoints trace-paths intersect in a non-transverse manner, for example linkage which can be aligned (i.e. completely be folded into a line).
See also
*
Configuration space (physics)
*
State space (physics)
In physics, a state space is an abstract space in which different "positions" represent, not literal locations, but rather states of some physical system. This makes it a type of phase space.
Specifically, in quantum mechanics a state space is a ...
References
{{DEFAULTSORT:Configuration Space
Manifolds
Topology
Algebraic topology