173.55.64.130: removing opinion and false information

2013-11-26T05:20:14Z

removing opinion and false information

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			In [[mathematics]], specifically [[algebraic topology]], the '''mapping cylinder''' of a [[function (mathematics)\|function]] <math>f</math> between [[topological space]]s <math>X</math> and <math>Y</math> is the [[quotient space\|quotient]]
			:<math>M_f = (([0,1]\times X) \amalg Y)\,/\,\sim</math>
			where the union is disjoint, and ∼ is the equivalence relation generated by
			:<math>(0,x)\sim f(x)\quad\text{for each }x\in X.</math>
			That is, the mapping cylinder <math>M_f</math> is obtained by gluing one end of <math>X\times[0,1]</math> to <math>Y</math> via the map <math>f</math>. Notice that the "top" of the cylinder <math>\{1\}\times X</math> is homeomorphic to <math>X</math>, while the "bottom" is the space <math>Y</math>.

			See <ref>Algebraic Topology by Allen Hatcher. Page 2</ref> for more details.

			==Basic properties==
			The bottom ''Y'' is a [[deformation retract]] of <math>M_f</math>.
			The projection <math>M_f \to Y</math> splits (via <math>Y \ni y \mapsto y \in Y \subset M_f</math>), and a deformation retraction <math> R </math> is given by:

			:<math> R: M_f \times I \rightarrow M_f</math>
			:<math>([t,x],s) \mapsto [s\cdot t,x]</math>

			(where points in <math> Y </math> stay fixed, which is well-defined, because <math>[0,x]=[s\cdot 0,x]</math> for all <math>s</math>).

			The map <math>f:X \to Y</math> is a homotopy equivalence if and only if the "top" <math>\{1\}\times X</math> is a strong deformation retract of <math>M_f</math>. A proof can be found in.<ref>Algebraic Topology by Allen Hatcher. Corollary 0.16</ref> An explicit formula for the strong deformation retraction is produced in.<ref>A Short Note on Mapping Cylinders by A. Aguado</ref>

			==Interpretation==
			The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent [[cofibration]], in the following sense:

			Given a map <math>f\colon X \to Y</math>, the mapping cylinder is a space <math>M_f</math>, ''together with'' a cofibration <math>\tilde f\colon X \to M_f</math> and a surjective [[homotopy equivalence]] <math>M_f \to Y</math> (indeed, ''Y'' is a [[deformation retract]] of <math>M_f</math>), such that the composition <math>X \to M_f \to Y</math> equals ''f''.
			[[Image:Mapping cylinder.png\|right]]

			Thus the space ''Y'' gets replaced with a homotopy equivalent space <math>M_f</math>, and the map ''f'' with a lifted map <math>\tilde f</math>. Equivalently, the diagram
			:<math>f\colon X \to Y</math>
			gets replaced with a diagram
			:<math>\tilde f\colon X \to M_f</math>
			together with a homotopy equivalence between them.

			The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.

			Note that pointwise, a [[cofibration]] is a closed [[Injective function\|inclusion]].

			==Applications==
			Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be [[injective]].

			Consequently, theorems or techniques (such as [[homology (mathematics)\|homology]], [[cohomology]] or [[homotopy theory]]) which are only dependent on the homotopy class of spaces and maps involved may be applied to <math>f\colon X\rightarrow Y</math> with the assumption that <math>X \subset Y</math> and that <math>f</math> is actually the inclusion of a subspace.

			Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of <math>X</math> to points of <math>Y,</math> and hence of embedding <math>X</math> within <math>Y,</math> despite the fact that the function need not be one-to-one.

			===Categorical application and interpretation===
			One can use the mapping cylinder to construct [[homotopy limits]]{{Citation needed\|date=June 2010}}: given a diagram, replace the maps by cofibrations (using the mapping cylinder) and then take the ordinary pointwise limit (one must take a bit more care, but mapping cylinders are a component).

			Conversely, the mapping cylinder is the [[homotopy pushout]] of the diagram where <math>f\colon X \to Y</math> and <math>\text{id}_X\colon X \to X</math>.

			===Mapping telescope===
			Given a sequence of maps

			:<math>X_1 \to_{f_1} X_2 \to_{f_2} X_3 \to\cdots</math>

			the mapping telescope is the homotopical [[direct limit]]. If the maps are all already cofibrations (such as for the orthogonal groups <math>O(n) \subset O(n+1)</math>), then the direct limit is the union, but in general one must use the mapping telescope. The mapping telescope is a sequence of mapping cylinders, joined end-to-end. The picture of the construction looks like a stack of increasingly large cylinders, like a telescope.

			Formally, one defines it as
			:<math>\Bigl(\coprod_i [0,1] \times X_i\Bigr) / ((0,x_i) \sim (1,f(x_i)))</math>

			==See also==
			* [[Mapping cylinder (homological algebra)]]

			== References ==
			{{Reflist}}
			* {{cite journal \|ref=agu \|last=Aguado \|first=A. \|title=A Short Note on Mapping Cylinders \|year=2012 \|url=http://arxiv.org/abs/1206.1277}}
			* {{cite book \|ref=hatc \|last=Hatcher \|first=A. \|title=Algebraic Topology \|publisher=Cambridge University Press \|year=2002\|isbn=0-521-79540-0 \|url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}
			* {{cite book \|ref=may \|last=May \|first=J.P \|title=A Concise Course in Algebraic Topology \|publisher=The University of Chicago Press \|year=1999\|isbn=978-0-2265-1183-2 \|url=http://www.math.uchicago.edu/~may/CONCISE}}

			{{DEFAULTSORT:Mapping Cylinder}}
			[[Category:Algebraic topology]]

en>Stevefal at 21:28, 24 August 2012

2012-08-24T21:28:42Z

New page

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173.55.64.130: removing opinion and false information

en>Stevefal at 21:28, 24 August 2012