https://en.formulasearchengine.com/api.php?action=feedcontributions&user=84.226.22.141&feedformat=atomformulasearchengine - User contributions [en]2021-09-24T14:22:47ZUser contributionsMediaWiki 1.37.0-alphahttps://en.formulasearchengine.com/index.php?title=Shelling_(topology)&diff=268908Shelling (topology)2014-02-28T15:49:38Z<p>84.226.22.141: /* Definition */ Explanatory sentence</p>
<hr />
<div>In [[mathematics]], a '''shelling''' of a [[simplicial complex]] is a way of gluing it together from its maximal simplices (simplices that are not a face of another simplex) in a well-behaved way. A complex admitting a shelling is called '''shellable'''.<br />
<br />
==Definition==<br />
A ''d''-dimensional simplicial complex is called '''pure''' if its maximal simplices all have dimension ''d''. Let <math>\Delta</math> be a finite or countably infinite simplicial complex. An ordering <math>C_1,C_2,\ldots</math> of the maximal simplices of <math>\Delta</math> is a '''shelling''' if the complex <math>B_k:=\left(\bigcup_{i=1}^{k-1}C_i\right)\cap C_k</math> is pure and <math>(\dim C_k-1)</math>-dimensional for all <math>k=2,3,\ldots</math>. That is, the "new" simplex <math>C_k</math> meets the previous simplices along some union <math>B_k</math> of top-dimensional simplices of the boundary of <math>C_k</math>. If <math>B_k</math> is the entire boundary of <math>C_k</math> then <math>C_k</math> is called '''spanning'''.<br />
<br />
For <math>\Delta</math> not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of <math>\Delta</math> having analogous properties.<br />
<br />
==Properties==<br />
* A shellable complex is [[homotopy|homotopy equivalent]] to a [[wedge sum]] of [[n-sphere|spheres]], one for each spanning simplex and of corresponding dimension.<br />
* A shellable complex may admit many different shellings, but the number of spanning simplices, and their dimensions, do not depend on the choice of shelling. This follows from the previous property.<br />
<br />
==Examples==<br />
* Every [[Coxeter complex]], and more generally every [[building (mathematics)|building]], is shellable.<ref>{{Cite journal<br />
| issn = 0001-8708<br />
| volume = 52<br />
| issue = 3<br />
| pages = 173–212<br />
| last = Björner<br />
| first = Anders<br />
| title = Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings<br />
| journal = Advances in Mathematics<br />
| date = June 1984<br />
| doi = 10.1016/0001-8708(84)90021-5<br />
}}</ref><br />
<br />
* There is an unshellable [[triangulation]] of the [[tetrahedron]].<ref>{{Cite journal<br />
| issn = 1088-9485<br />
| volume = 64<br />
| issue = 3<br />
| pages = 90–91<br />
| last = Rudin<br />
| first = M.E.<br />
| title = An unshellable triangulation of a tetrahedron<br />
| journal = Bull. Am. Math. Soc.<br />
| date = 1958-02-14<br />
}}</ref><br />
<br />
==References==<br />
* {{cite book |author=Dmitry Kozlov |title=Combinatorial Algebraic Topology |publisher=Springer |location=Berlin |year=2008 |isbn=978-3-540-71961-8 |oclc= |doi=}}<br />
{{reflist}}<br />
\<br />
<br />
[[Category:Topology]]<br />
[[Category:Algebraic topology]]<br />
<br />
<br />
{{topology-stub}}</div>84.226.22.141https://en.formulasearchengine.com/index.php?title=Deformation_retract&diff=241377Deformation retract2014-02-28T12:25:26Z<p>84.226.22.141: /* Deformation retract and strong deformation retract */</p>
<hr />
<div>{{redirect|Retract|other meanings including concepts in group theory and category theory|Retraction (disambiguation)}}<br />
In [[topology]], a branch of mathematics, a '''retraction''',<ref>{{cite journal|title=Sur les rétractes|author=K. Borsuk|journal=Fund. Math.|volume=17|year=1931|pages=2–20}}</ref> is a continuous mapping from the entire [[space (mathematics)|space]] into a [[Subspace topology|subspace]] which preserves the position of all points in that subspace. A '''deformation retraction''' is a [[function (mathematics)|map]] which captures the idea of ''[[continuous function|continuously]] shrinking'' a space into a subspace.<br />
<br />
== Definitions ==<br />
=== Retract ===<br />
Let ''X'' be a [[topological space]] and ''A'' a [[subspace (topology)|subspace]] of ''X''. Then a continuous map <br />
<br />
:<math>r:X \to A</math><br />
<br />
is a '''retraction''' if the [[function (mathematics)#Restrictions and extensions|restriction]] of ''r'' to ''A'' is the [[Identity function|identity map]] on ''A''; that is, ''r''(''a'') = ''a'' for all ''a'' in ''A''. Equivalently, denoting by <br />
<br />
:<math>\iota : A \hookrightarrow X</math><br />
<br />
the [[Inclusion map|inclusion]], a retraction is a continuous map ''r'' such that <br />
<br />
:<math>r \circ \iota = id_A,</math><br />
<br />
that is, the composition of ''r'' with the inclusion is the identity of ''A''. Note that, by definition, a retraction maps ''X'' [[onto]] ''A''. A subspace ''A'' is called a '''retract''' of ''X'' if such a retraction exists. For instance, any space retracts to a point in the obvious way (the constant map yields a retraction). If ''X'' is [[Hausdorff space|Hausdorff]], then ''A'' must be closed.<br />
<br />
If <math>r:X \to A</math> is a retraction, then the composition <math>\iota \circ r</math> is an [[idempotent]] continuous map from ''X'' to ''X''. Conversely, given any idempotent continuous map <math>s:X\to X</math>, we obtain a retraction onto the image of ''s'' by restricting the codomain.<br />
<br />
A space ''X'' is known as an '''absolute retract''' if for every [[normal space]] ''Y'' that contains ''X'' as a closed subspace, ''X'' is a retract of ''Y''. The [[unit cube]] ''I<sup>n</sup>'' as well as the [[Hilbert cube]] ''I<sup>ω</sup>'' are absolute retracts.<br />
<br />
=== Neighborhood retract ===<br />
If there exists an [[open set]] ''U'' such that <br />
<br />
:<math>A \subset U \subset X</math><br />
<br />
and ''A'' is a retract of ''U'', then ''A'' is called a '''neighborhood retract''' of ''X''.<br />
<br />
A space ''X'' is an '''absolute neighborhood retract''' (or '''ANR''') if for every normal space ''Y'' that embeds ''X'' as a closed subset, ''X'' is a neighborhood retract of ''Y''. The ''n''-sphere ''S<sup>n</sup>'' is an absolute neighborhood retract.<br />
<br />
=== Deformation retract and strong deformation retract===<br />
A continuous map <br />
<br />
:<math>F:X \times [0, 1] \to X \, </math><br />
<br />
is a ''deformation retraction'' of a space ''X'' onto a subspace ''A'' if, for every ''x'' in ''X'' and ''a'' in ''A'',<br />
<br />
:<math> F(x,0) = x, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a \mbox{ for every } a \in A .</math><br />
<br />
In other words, a deformation retraction is a [[homotopy]] 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 [[homotopy equivalence]].<br />
<br />
A retract need not be a deformation retract. For instance, having a single point as a deformation retract would imply a space is path connected (in fact, it would imply contractibility of the space).<br />
<br />
''Note:'' An equivalent definition of deformation retraction is the following. A continuous map ''r'': ''X'' → ''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. <br />
<br />
If, in the definition of a deformation retraction, we add the requirement that <br />
<br />
:<math>F(a,t) = a\,</math><br />
<br />
for all ''t'' in [0, 1] 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 [[Allen Hatcher]], take this as the definition of deformation retraction.)<br />
<br />
As an example, the [[n-sphere|''n''-sphere]] ''S<sup>n</sup>'' is a strong deformation retract of '''R'''<sup>''n''+1</sup>\{0}; as strong deformation retraction one can choose the map <br />
:<math>F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.</math><br />
<br />
===Neighborhood deformation retract===<br />
A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map <math>u:X \rightarrow I</math> (where <math>I=[0,1]</math>) such that <math>A = u^{-1} (0)</math> and a homotopy<br />
<math>H:X\times I\rightarrow X</math> such that <math>H(x,0) = x</math> for all <math>x \in X</math>, <math>H(a,t) = a</math> for all<br />
<math>(a,t) \in A\times I</math>, and <math>h(x,1) \in A</math> for all <math>x \in u^{-1} [ 0 , 1)</math>.<ref name='steenrod'>{{cite journal | journal= Michigan Math. J. | last1=Steenrod | first1=N. E. | title=A convenient category of topological spaces | volume=14 | issue=2 | year=1967 | pages=133–152}}</ref><br />
<br />
==Properties==<br />
Deformation retraction is a particular case of homotopy equivalence. In fact, two spaces are [[homotopy equivalent]] [[if and only if]] they are both deformation retracts of a single larger space. <br />
<br />
Any topological space which deformation retracts to a point is [[contractible space|contractible]] and vice versa. However, there exist contractible spaces which do not strongly deformation retract to a point.<ref name='hatcher'>{{Citation | last1=Hatcher | first1=Allen | title=Algebraic topology | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | publisher=[[Cambridge University Press]] | isbn=978-0-521-79540-1 | year=2002}}</ref><br />
<br />
==Notes==<br />
{{Reflist}}<br />
<br />
==External links==<br />
* {{PlanetMath attribution|id=6255|title=Neighborhood retract}}<br />
<br />
[[Category:Topology]]</div>84.226.22.141