Homotopy
of two continuous mappings
A formalization of the intuitive idea of deformability of one mapping into another. More exactly, two mappings and
are called homotopic (denoted by
) if there exists a family of continuous mappings
, continuously depending on a parameter
, such that
,
(the choice of the segment
is here only for the sake of technical convenience; it is clear that any other segment of the real axis could have been taken just as well). This family (which is known as a homotopy connecting
with
) is a path in the space
of all continuous mappings
connecting the point
with the point
. Thus, a homotopy of mappings is a specialization to the space of mappings of the general concept of "being connected by a continuous path" . Thus, in particular, the homotopy relation is an equivalence relation, whose equivalence classes (homotopy classes) are the path-connected components of
. In order to give an exact meaning to the above-said one must specify the meaning of the expression "the mappings ft continuously depend on t" . The most natural procedure is to introduce a topology (or at least a pseudo-topology, cf. also Topological structure (topology)) in
. However, the traditional procedure is different — it is assumed, by definition, that the
continuously depend on
if the function
is continuous with respect to the totality of variables, i.e. if the mapping
defined by the formula
is continuous (this mapping is in fact often referred to as a homotopy connecting
with
).
This description of a homotopy is sometimes qualified as free, in distinction from "relative homotopyrelative" or "bound homotopybound" homotopies, which arise upon fixing a class of continuous mappings
, by imposing the requirement
for any
. Thus, given a subspace
it is possible to consider relative homotopies on
, distinguished by the fact that
on
for all
. One says in this case that the mapping
is homotopic with the mapping
relative to
; this is written as
.
Another type of "relative" homotopies is generated by selecting subspaces and
in
and
and by considering only mappings
that satisfy the condition
. Such mappings are called mappings of the pair
into the pair
(written as
), while the corresponding homotopies (i.e. the homotopies for which
for all
) are known as homotopies of pair mappings. Triplets
(with the condition
or without it), quadruplets, etc., rather than pairs, can also be considered. Thus, one can consider homotopy of pair mappings relative to a third subspace, etc. Essentially different types of "relative" homotopies may also be considered.
The task of establishing homotopy ( "relative" or not) of two given mappings is equivalent to the task of extending to
a continuous mapping into
given on
(and, in a homotopy problem
, on
). In this sense, the problem of homotopy is a special case of the problem of extension. In a wide class of individual cases, however (viz. for the so-called cofibrations, cf. Cofibration), the possibility of extending to
a continuous mapping
given on a subspace
depends only on its homotopy class. This close connection between the problem of homotopy and the problem of extension is the reason why they are considered together under the heading of so-called homotopy theory. See Homotopy type.
Comments
The context in which the term "free homotopy" is used differs slightly in the West; one uses it in the context of pointed spaces (a pointed space is a pair with
a space and
a point of
). Here then a "free" homotopy is a homotopy between arbitrary mappings, whereas for an "ordinary" homotopy one considers mappings
satisfying
and the mappings in the homotopies are to satisfy this condition too.
References
[a1] | G.W. Whitehead, "Elements of homotopy theory" , Springer (1978) pp. 228 |
Homotopy. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homotopy&oldid=32168