Everything about Homotopy totally explained
In
topology, two
continuous functions from one
topological space to another are called
homotopic (
Greek homos = identical and
topos = place) if one can be "continuously deformed" into the other, such a deformation being called a
homotopy between the two functions. An outstanding use of homotopy is the definition of
homotopy groups and
cohomotopy groups, important
invariants in
algebraic topology.
In practice, there are technical difficulties in using homotopies with certain pathological spaces. Consequently most algebraic topologists work with
compactly generated spaces,
CW complexes, or
spectra.
Formal definition
Formally, a homotopy between two
continuous functions
f and
g from a
topological space
X to a topological space
Y is defined to be a
continuous function
H:
X × [0,1] →
Y from the
product of the space
X with the
unit interval [0,1]
to
Y such that, for all points
x in
X,
H(
x,0)=
f(
x)
and
H(
x,1)=
g(
x).
If we think of the second
parameter of
H as "time", then
H describes a "continuous deformation" of
f into
g: at time 0 we've the function
f, at time 1 we've the function
g.
Properties
Continuous functions
f and
g (both from topological space
X to
Y) are said to be homotopic iff there's a homotopy
H taking
f to
g as described above.
Being homotopic is an
equivalence relation on the set of all continuous functions from
X to
Y.
This homotopy relation is compatible with
function composition in the following sense: if
f1,
g1:
X →
Y are homotopic, and
f2,
g2:
Y →
Z are homotopic, then their compositions
f2 o
f1 and
g2 o
g1:
X →
Z are homotopic as well.
Homotopy equivalence and null-homotopy
Given two spaces
X and
Y, we say they're
homotopy equivalent or of the same
homotopy type if there exist continuous
maps
f:
X →
Y and
g:
Y →
X such that
g o
f is homotopic to the
identity map id
X and
f o
g is homotopic to id
Y.
The maps
f and
g are called
homotopy equivalences in this case. Clearly, every
homeomorphism is a homotopy equivalence, but the converse isn't true: for example, a solid disk isn't homeomorphic to a single point, although the disk and the point are homotopy equivalent.
Intuitively, two spaces
X and
Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and
R2 - is homotopy equivalent to the
unit circle S1. Those spaces that are homotopy equivalent to a point are called
contractible.
A function
f is said to be
null-homotopic if it's
homotopic to a constant function. (The homotopy from
f to a constant function is then sometimes called a
null-homotopy.) For example, it's simple to show that a map from the
circle S1 is null-homotopic precisely when it can be extended to a map of the disc
D2.
It follows from these definitions that a space
X is contractible if and only if the identity map from
X to itself—which is always a homotopy equivalence—is null-homotopic.
Homotopy invariance
Homotopy equivalence is important because in
algebraic topology many concepts are
homotopy invariant, that is, they respect the relation of homotopy equivalence. For example, if
X and
Y are homotopy equivalent spaces, then:
An example of an algebraic invariant of topological spaces which isn't homotopy-invariant is
compactly supported homology (which is, roughly speaking, the homology of the
compactification, and compactification isn't homotopy-invariant).
Homotopy category
The idea of homotopy can be turned into a formal category of
category theory. The
homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces
X and
Y are isomorphic in this category if and only if they're homotopy-equivalent. Then a
functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
For example, homology groups are a
functorial homotopy invariant: this means that if
f and
g from
X to
Y are homotopic, then the
group homomorphisms induced by
f and
g on the level of
homology groups are the same: H
n(
f) = H
n(
g) : H
n(
X) → H
n(
Y) for all
n. Likewise, if
X and
Y are in addition
path-connected, then the group homomorphisms induced by
f and
g on the level of
homotopy groups are also the same: π
n(
f) = π
n(
g) : π
n(
X) → π
n(
Y).
Relative homotopy
In order to define the
fundamental group, one needs the notion of
homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if
f and
g are continuous maps from
X to
Y and
K is a
subset of
X, then we say that
f and
g are homotopic relative to
K if there exists a homotopy
H:
X × [0,1] →
Y between
f and
g such that
H(
k,
t) =
f(
k) =
g(
k) for all
k∈
K and
t∈[0,1]. Also, if
g is a
retract from
X to
K and
f is the identity map, this is known as a strong
deformation retract of
X to
K.
Timelike homotopy
On a
Lorentzian manifold, certain curves are distinguished as
timelike. A
timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No
closed timelike curve (CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be
multiply connected by timelike curves. A manifold such as the
3-sphere can be
simply connected (by any type of curve), and yet be multiply timelike connected.
Homotopy extension property
Another useful property involving homotopy is the
homotopy extension property,
which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with
cofibrations.
Isotopy
In case the two given continuous functions
f and
g from the topological space
X to the topological space
Y are
homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of
isotopy, which is a homotopy,
H, in the notation used before, such that for each fixed
t,
H(
x,
t) gives a homeomorphism.
Requiring that two homeomorphisms be isotopic really is a stronger requirement than that they be homotopic. For example, the map of the
unit disc in
R2 defined by
f(
x,
y) = (−
x, −
y) is equivalent to a 180-degree
rotation around the origin, and so the identity map and
f are isotopic because they can be connected by rotations. However, the map on the interval [−1,1] in
R defined by
f(
x) = −
x is
not isotopic to the identity. Loosely speaking, any homotopy from
f to the identity would have to exchange the endpoints, which would mean that they'd have to 'pass through' each other. Moreover,
f has changed the orientation of the interval, hence it can't be isotopic to the identity.
In
geometric topology—for example in
knot theory—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots,
K1 and
K2, in three-
dimensional space. The intuitive idea of
deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism,
h, such that
h moves
K1 to
K2. An
ambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold.
Further Information
Get more info on 'Homotopy'.
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