https://en.formulasearchengine.com/api.php?action=feedcontributions&user=84.226.22.141&feedformat=atom formulasearchengine - User contributions [en] 2021-09-24T14:22:47Z User contributions MediaWiki 1.37.0-alpha https://en.formulasearchengine.com/index.php?title=Shelling_(topology)&diff=268908 Shelling (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 &lt;math&gt;\Delta&lt;/math&gt; be a finite or countably infinite simplicial complex. An ordering &lt;math&gt;C_1,C_2,\ldots&lt;/math&gt; of the maximal simplices of &lt;math&gt;\Delta&lt;/math&gt; is a '''shelling''' if the complex &lt;math&gt;B_k:=\left(\bigcup_{i=1}^{k-1}C_i\right)\cap C_k&lt;/math&gt; is pure and &lt;math&gt;(\dim C_k-1)&lt;/math&gt;-dimensional for all &lt;math&gt;k=2,3,\ldots&lt;/math&gt;. That is, the &quot;new&quot; simplex &lt;math&gt;C_k&lt;/math&gt; meets the previous simplices along some union &lt;math&gt;B_k&lt;/math&gt; of top-dimensional simplices of the boundary of &lt;math&gt;C_k&lt;/math&gt;. If &lt;math&gt;B_k&lt;/math&gt; is the entire boundary of &lt;math&gt;C_k&lt;/math&gt; then &lt;math&gt;C_k&lt;/math&gt; is called '''spanning'''.<br /> <br /> For &lt;math&gt;\Delta&lt;/math&gt; not necessarily countable, one can define a shelling as a well-ordering of the maximal simplices of &lt;math&gt;\Delta&lt;/math&gt; 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.&lt;ref&gt;{{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 /> }}&lt;/ref&gt;<br /> <br /> * There is an unshellable [[triangulation]] of the [[tetrahedron]].&lt;ref&gt;{{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 /> }}&lt;/ref&gt;<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> 84.226.22.141 https://en.formulasearchengine.com/index.php?title=Deformation_retract&diff=241377 Deformation retract 2014-02-28T12:25:26Z <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''',&lt;ref&gt;{{cite journal|title=Sur les rétractes|author=K. Borsuk|journal=Fund. Math.|volume=17|year=1931|pages=2–20}}&lt;/ref&gt; 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 /> :&lt;math&gt;r:X \to A&lt;/math&gt;<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 /> :&lt;math&gt;\iota : A \hookrightarrow X&lt;/math&gt;<br /> <br /> the [[Inclusion map|inclusion]], a retraction is a continuous map ''r'' such that <br /> <br /> :&lt;math&gt;r \circ \iota = id_A,&lt;/math&gt;<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 &lt;math&gt;r:X \to A&lt;/math&gt; is a retraction, then the composition &lt;math&gt;\iota \circ r&lt;/math&gt; is an [[idempotent]] continuous map from ''X'' to ''X''. Conversely, given any idempotent continuous map &lt;math&gt;s:X\to X&lt;/math&gt;, 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&lt;sup&gt;n&lt;/sup&gt;'' as well as the [[Hilbert cube]] ''I&lt;sup&gt;ω&lt;/sup&gt;'' are absolute retracts.<br /> <br /> === Neighborhood retract ===<br /> If there exists an [[open set]] ''U'' such that <br /> <br /> :&lt;math&gt;A \subset U \subset X&lt;/math&gt;<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&lt;sup&gt;n&lt;/sup&gt;'' is an absolute neighborhood retract.<br /> <br /> === Deformation retract and strong deformation retract===<br /> A continuous map <br /> <br /> :&lt;math&gt;F:X \times [0, 1] \to X \, &lt;/math&gt;<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 /> :&lt;math&gt; F(x,0) = x, \; F(x,1) \in A ,\quad \mbox{and} \quad F(a,1) = a \mbox{ for every } a \in A .&lt;/math&gt;<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 /> :&lt;math&gt;F(a,t) = a\,&lt;/math&gt;<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&lt;sup&gt;n&lt;/sup&gt;'' is a strong deformation retract of '''R'''&lt;sup&gt;''n''+1&lt;/sup&gt;\{0}; as strong deformation retraction one can choose the map <br /> :&lt;math&gt;F(x,t)=\left((1-t)+{t\over \|x\|}\right) x.&lt;/math&gt;<br /> <br /> ===Neighborhood deformation retract===<br /> A closed subspace ''A'' is a '''neighborhood deformation retract''' of ''X'' if there exists a continuous map &lt;math&gt;u:X \rightarrow I&lt;/math&gt; (where &lt;math&gt;I=[0,1]&lt;/math&gt;) such that &lt;math&gt;A = u^{-1} (0)&lt;/math&gt; and a homotopy<br /> &lt;math&gt;H:X\times I\rightarrow X&lt;/math&gt; such that &lt;math&gt;H(x,0) = x&lt;/math&gt; for all &lt;math&gt;x \in X&lt;/math&gt;, &lt;math&gt;H(a,t) = a&lt;/math&gt; for all<br /> &lt;math&gt;(a,t) \in A\times I&lt;/math&gt;, and &lt;math&gt;h(x,1) \in A&lt;/math&gt; for all &lt;math&gt;x \in u^{-1} [ 0 , 1)&lt;/math&gt;.&lt;ref name='steenrod'&gt;{{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}}&lt;/ref&gt;<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.&lt;ref name='hatcher'&gt;{{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}}&lt;/ref&gt;<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