Deformation Retract
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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of mathematics, a retraction is a
continuous mapping In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
from 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 ...
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 deformation retraction is a mapping that captures the idea of ''continuously shrinking'' a space into a subspace. An absolute neighborhood retract (ANR) is a particularly
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
type of topological space. For example, every
topological manifold In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
is an ANR. Every ANR has 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 very simple topological space, 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 ...
.


Definitions


Retract

Let ''X'' be a topological space and ''A'' a subspace of ''X''. Then a continuous map :r\colon X \to A is a retraction if the
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...
of ''r'' to ''A'' is 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 ''A''; that is, r(a) = a for all ''a'' in ''A''. Equivalently, denoting by :\iota\colon A \hookrightarrow X the
inclusion Inclusion or Include may refer to: Sociology * Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society. ** Inclusion (disability rights), promotion of people with disabiliti ...
, a retraction is a continuous map ''r'' such that :r \circ \iota = \operatorname_A, that is, the composition of ''r'' with the inclusion is the identity of ''A''. Note that, by definition, a retraction maps ''X''
onto In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element of ...
''A''. A subspace ''A'' is called a retract of ''X'' if such a retraction exists. For instance, any non-empty space retracts to a point in the obvious way (the constant map yields a retraction). If ''X'' is Hausdorff, then ''A'' must be a closed subset of ''X''. If r: X \to A is a retraction, then the composition ι∘''r'' is an
idempotent Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
continuous map from ''X'' to ''X''. Conversely, given any idempotent continuous map s: X \to X, we obtain a retraction onto the image of ''s'' by restricting the
codomain In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either th ...
.


Deformation retract and strong deformation retract

A continuous map :F\colon X \times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\to X is a ''deformation retraction'' of a space ''X'' onto a subspace ''A'' if, for every ''x'' in ''X'' and ''a'' in ''A'', : F(x,0) = x, \quad F(x,1) \in A ,\quad \mbox \quad F(a,1) = a. In other words, a deformation retraction is a
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 defor ...
between a retraction and the identity map on ''X''. The subspace ''A'' is called a deformation retract of ''X''. A deformation retraction is a special case of 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 defor ...
. A retract need not be a deformation retract. For instance, having a single point as a deformation retract of a space ''X'' would imply that ''X'' is path connected (and in fact that ''X'' is
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
). ''Note:'' An equivalent definition of deformation retraction is the following. A continuous map r: X \to A is a deformation retraction if it is a retraction and its composition with the inclusion is homotopic to the identity map on ''X''. In this formulation, a deformation retraction carries with it a homotopy between the identity map on ''X'' and itself. If, in the definition of a deformation retraction, we add the requirement that :F(a,t) = a for all ''t'' in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
and ''a'' in ''A'', then ''F'' is called a strong deformation retraction. In other words, a strong deformation retraction leaves points in ''A'' fixed throughout the homotopy. (Some authors, such as
Hatcher Hatcher is a surname. Notable people with the surname include: *Allen Hatcher (born 1944), U.S. mathematician * Anna Granville Hatcher (1905–1978), U.S. linguist *Edwin Starr (born Charles Edwin Hatcher, 1942–2003), U.S. soul singer * Chris Hat ...
, take this as the definition of deformation retraction.) As an example, the ''n''-sphere ''S^'' is a strong deformation retract of \reals^ \backslash \; as strong deformation retraction one can choose the map :F(x,t)=\left((1-t)+\right) x.


Cofibration and neighborhood deformation retract

A map ''f'': ''A'' → ''X'' of topological spaces is a (
Hurewicz Witold Hurewicz (June 29, 1904 – September 6, 1956) was a Polish mathematician. Early life and education Witold Hurewicz was born in Łódź, at the time one of the main Polish industrial hubs with economy focused on the textile industry. His ...
)
cofibration In mathematics, in particular homotopy theory, a continuous mapping :i: A \to X, where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
if it has the
homotopy extension property In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual ...
for maps to any space. This is one of the central concepts of homotopy theory. A cofibration ''f'' is always injective, in fact a
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
to its image. If ''X'' is Hausdorff (or a
compactly generated In mathematics, compactly generated can refer to: * Compactly generated group, a topological group which is algebraically generated by one of its compact subsets *Compactly generated space In topology, a compactly generated space is a topological s ...
weak Hausdorff space In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed. In particular, every Hausdorff space is weak Hausdorff. As ...
), then the image of a cofibration ''f'' is closed in ''X''. Among all closed inclusions, cofibrations can be characterized as follows. The inclusion of a closed subspace ''A'' in a space ''X'' is a cofibration if and only if ''A'' is a neighborhood deformation retract of ''X'', meaning that there is a continuous map u: X \rightarrow
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> with A = u^\!\left(0\right) and a homotopy H: X \times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\rightarrow X such that H(x,0) = x for all x \in X, H(a,t) = a for all a \in A and t \in
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
and H\left(x,1\right) \in A if u(x) < 1. For example, the inclusion of a subcomplex in a CW complex is a cofibration.


Properties

* One basic property of a retract ''A'' of ''X'' (with retraction r: X \to A) is that every continuous map f: A \rightarrow Y has at least one extension g: X \rightarrow Y, namely g = f \circ r. * Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are homotopy equivalent
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 b ...
they are both homeomorphic to deformation retracts of a single larger space. * Any topological space that deformation retracts to a point is contractible and vice versa. However, there exist contractible spaces that do not strongly deformation retract to a point.


No-retraction theorem

The
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
of the ''n''-dimensional ball, that is, the (''n''−1)-sphere, is not a retract of the ball. (See .)


Absolute neighborhood retract (ANR)

A closed subset X of a topological space Y is called a neighborhood retract of Y if X is a retract of some open subset of Y that contains X. Let \mathcal be a class of topological spaces, closed under homeomorphisms and passage to closed subsets. Following Borsuk (starting in 1931), a space ''X'' is called an absolute retract for the class \mathcal, written \operatorname \left(\mathcal\right), if ''X'' is in \mathcal and whenever ''X'' is a closed subset of a space Y in \mathcal, X is a retract of Y. A space X is an absolute neighborhood retract for the class \mathcal, written \operatorname \left(\mathcal\right), if X is in \mathcal and whenever X is a closed subset of a space Y in \mathcal, X is a neighborhood retract of Y. Various classes \mathcal such as
normal space In topology and related branches of mathematics, a normal space is a topological space ''X'' that satisfies Axiom T4: every two disjoint closed sets of ''X'' have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. T ...
s have been considered in this definition, but the class \mathcal of
metrizable space In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
s has been found to give the most satisfactory theory. For that reason, the notations AR and ANR by themselves are used in this article to mean \operatorname \left(\right) and \operatorname \left(\right). A metrizable space is an AR if and only if it is contractible and an ANR. By Dugundji, every locally convex metrizable
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
V is an AR; more generally, every nonempty
convex subset In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex ...
of such a vector space V is an AR. For example, any
normed vector space In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length ...
(
complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
or not) is an AR. More concretely, Euclidean space \reals^, the
unit cube A unit cube, more formally a cube of side 1, is a cube whose sides are 1 unit long.. See in particulap. 671. The volume of a 3-dimensional unit cube is 1 cubic unit, and its total surface area is 6 square units.. Unit hypercube The term '' ...
I^,and the
Hilbert cube In mathematics, the Hilbert cube, named after David Hilbert, is a topological space that provides an instructive example of some ideas in topology. Furthermore, many interesting topological spaces can be embedded in the Hilbert cube; that is, ...
I^ are ARs. ANRs form a remarkable class of "
well-behaved In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition, it is sometimes called well-behaved. Th ...
" topological spaces. Among their properties are: *Every open subset of an ANR is an ANR. *By
Hanner Hanner may refer to: People * Bob Hanner (1945–2019), American businessman and politician * Dave Hanner (1930–2008), American football player, coach and scout * Flint Hanner (1898–1973), American mulit-sport track and field athlete and coac ...
, a metrizable space that has an
open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alph ...
by ANRs is an ANR. (That is, being an ANR is a
local property In mathematics, a mathematical object is said to satisfy a property locally, if the property is satisfied on some limited, immediate portions of the object (e.g., on some ''sufficiently small'' or ''arbitrarily small'' neighborhoods of points). P ...
for metrizable spaces.) It follows that every topological manifold is an ANR. For example, the sphere ''S^'' is an ANR but not an AR (because it is not contractible). In infinite dimensions, Hanner's theorem implies that every Hilbert cube manifold as well as the (rather different, for example not locally compact)
Hilbert manifold In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbert space. The concept of a Hilbert manifold ...
s and
Banach manifold In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more involved and formal definition is given below) ...
s are ANRs. *Every locally finite
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 ...
is an ANR. An arbitrary CW complex need not be metrizable, but every CW complex has the homotopy type of an ANR (which is metrizable, by definition). *Every ANR ''X'' is locally contractible in the sense that for every open neighborhood U of a point x in X, there is an open neighborhood V of x contained in U such that the inclusion V \hookrightarrow U is homotopic to a
constant map In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic propertie ...
. A finite-dimensional metrizable space is an ANR if and only if it is locally contractible in this sense. For example, the Cantor set is a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
subset of the real line that is not an ANR, since it is not even
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 ...
. *Counterexamples: Borsuk found a compact subset of \reals^ that is an ANR but not strictly locally contractible. (A space is strictly locally contractible if every open neighborhood U of each point x contains a contractible open neighborhood of x.) Borsuk also found a compact subset of the Hilbert cube that is locally contractible (as defined above) but not an ANR. *Every ANR has the homotopy type of a CW complex, by Whitehead and
Milnor John Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Univ ...
. Moreover, a locally compact ANR has the homotopy type of a locally finite CW complex; and, by West, a compact ANR has the homotopy type of a finite CW complex. In this sense, ANRs avoid all the homotopy-theoretic pathologies of arbitrary topological spaces. For example, the
Whitehead theorem In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result w ...
holds for ANRs: a map of ANRs that induces an isomorphism on homotopy groups (for all choices of base point) is a homotopy equivalence. Since ANRs include topological manifolds, Hilbert cube manifolds, Banach manifolds, and so on, these results apply to a large class of spaces. *Many mapping spaces are ANRs. In particular, let ''Y'' be an ANR with a closed subspace ''A'' that is an ANR, and let ''X'' be any compact metrizable space with a closed subspace ''B''. Then the space \left(Y, A\right)^ of maps of
pairs Concentration, also known as Memory, Shinkei-suijaku (Japanese meaning "nervous breakdown"), Matching Pairs, Match Match, Match Up, Pelmanism, Pexeso or simply Pairs, is a card game in which all of the cards are laid face down on a surface and tw ...
\left(X, B\right) \rightarrow \left(Y, A\right) (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 ...
on the
mapping space In mathematics, the category of compactly generated weak Hausdorff spaces CGWH is one of typically used categories in algebraic topology as a substitute for the category of topological spaces, as the latter lacks some of the pleasant properties one ...
) is an ANR. It follows, for example, that 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 topolo ...
of any CW complex has the homotopy type of a CW complex. *By Cauty, a metrizable space X is an ANR if and only if every open subset of X has the homotopy type of a CW complex. *By Cauty, there is a metric linear space ''V'' (meaning a topological vector space with a translation-invariant metric) that is not an AR. One can take V to be separable and an
F-space In functional analysis, an F-space is a vector space X over the real or complex numbers together with a metric d : X \times X \to \R such that # Scalar multiplication in X is continuous with respect to d and the standard metric on \R or \Complex ...
(that is, a complete metric linear space).Cauty (1994), Fund. Math. 146: 85–99. (By Dugundji's theorem above, V cannot be locally convex.) Since V is contractible and not an AR, it is also not an ANR. By Cauty's theorem above, V has an open subset U that is not homotopy equivalent to a CW complex. Thus there is a metrizable space U that is strictly locally contractible but is not homotopy equivalent to a CW complex. It is not known whether a compact (or locally compact) metrizable space that is strictly locally contractible must be an ANR.


Notes


References

* * * * * * * * * * * *


External links

* {{PlanetMath attribution, id=6255, title=Neighborhood retract Topology