In
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, the path space fibration over a
based space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
is a
fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics.
Fibrations are used, for example, in postnikov-systems or obstruction theory.
In this article, all map ...
of the form
:
where
*
is the
path space of ''X''; i.e.,
equipped with the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and ...
.
*
is the fiber of
over the base point of ''X''; thus it is the
loop space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology ...
of ''X''.
The space
consists of all maps from ''I'' to ''X'' that may not preserve the base points; it is called the free path space of ''X'' and the fibration
given by, say,
, is called the free path space fibration.
The path space fibration can be understood to be dual to the
mapping cone Mapping cone may refer to one of the following two different but related concepts in mathematics:
* Mapping cone (topology)
* Mapping cone (homological algebra)
In homological algebra, the mapping cone is a construction on a map of chain complexe ...
. The reduced fibration is called the mapping fiber or, equivalently, the
homotopy fiber In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction ...
.
Mapping path space
If
is any map, then the mapping path space
of
is the pullback of the fibration
along
. (A mapping path space satisfies the universal property that is dual to that of a mapping cylinder, which is a push-out. Because of this, a mapping path space is also called a mapping cocylinder.)
Since a fibration pulls back to a fibration, if ''Y'' is based, one has the fibration
:
where
and
is the
homotopy fiber In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction ...
, the pullback of the fibration
along
.
Note also
is the composition
:
where the first map
sends ''x'' to
; here
denotes the constant path with value
. Clearly,
is a
homotopy equivalence
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 ...
; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.
If
is a fibration to begin with, then the map
is a
fiber-homotopy equivalence In algebraic topology, a fiber-homotopy equivalence is a map over a space ''B'' that has homotopy inverse over ''B'' (that is we require a homotopy be a map over ''B'' for each time ''t''.) It is a relative analog of a homotopy equivalence
In t ...
and, consequently, the fibers of
over the path-component of the base point are homotopy equivalent to the homotopy fiber
of
.
Moore's path space
By definition, a path in a space ''X'' is a map from the
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, ...
''I'' to ''X''. Again by definition, the product of two paths
such that
is the path
given by:
:
.
This product, in general, fails to be
associative
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
on the nose:
, as seen directly. One solution to this failure is to pass to
homotopy
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 ...
classes: one has