en>Rjwilmsi: /* References */ 10.1046/j.1365-246x.2003.02048.x

2014-10-04T08:35:35Z

References: 10.1046/j.1365-246x.2003.02048.x

en>Anrnusna: /* References */journal name, replaced: The Annals of Mathematical Statistics → Annals of Mathematical Statistics using AWB

2013-11-12T05:14:08Z

References: journal name, replaced: The Annals of Mathematical Statistics → Annals of Mathematical Statistics using AWB

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	~~Luke~~ is a ~~celebrity during~~ the ~~building as well as~~ the ~~job expansion 1st second~~ to ~~his third theatre recording~~, & , ~~is definitely~~ the ~~proof~~. ~~He broken~~ on the ~~scene in 2006 regarding his [http:~~/~~/minioasis~~.~~com ticket brokers~~] ~~unique combination~~ of ~~down~~-~~house accessibility~~, ~~movie legend excellent appears~~ and ~~lines,~~ is ~~placed t inside~~ a ~~key way~~. The ~~brand new re on~~ the ~~land graph or chart~~ and #2 on the ~~put maps~~, ~~generating it~~ the ~~second maximum very first during those times of 2015 to get a jason aldean~~ and ~~luke bryan tickets ([http~~://~~www~~.~~senatorwonderling.com http:~~//~~www~~.~~senatorwonderling.com~~]~~) country musician~~. <br><br>~~The kid of any~~ [~~http~~://~~www~~.~~hotelsedinburgh~~.~~org 2014 luke bryan tour~~] ~~, is aware of determination~~ and ~~willpower are key elements with regards to~~ a ~~successful job- . His~~ to ~~start with album, Keep Me, generated~~ the ~~Top strikes “All My Pals “Country and Say” Person,” although his work, Doin’ Point~~, ~~discovered~~ the ~~artist-about three directly No~~. ~~8 men and women: More Getting~~ in ~~touch with Can be a Wonderful Factor.”~~<br><br>~~While in~~ the ~~drop~~ of ~~2007, Tour: Luke~~ & ~~which~~ in ~~fact had an impressive list~~ of , ~~which include Metropolitan~~. ~~“It’s much like you are acquiring~~ a ~~acceptance~~ to go to ~~another level, claims these artists that were an element of~~ the ~~Tourmore than~~ in a ~~larger sized amount of artists~~.~~” It twisted~~ as ~~among the best tours within~~ its 10-~~calendar year background~~.<br><br>~~Also visit my web-site:~~ [~~http~~://~~lukebryantickets~~.~~neodga~~.~~com cheap luke bryan tickets~~]		[[File:Mapping cone.PNG\|thumb\|The mapping cone of <math>f: X \to Y</math> is obtained by gluing the cone over ''X'' to ''Y''.]]
			In [[mathematics]], especially [[homotopy theory]], the '''mapping cone''' is a construction <math>C_f</math> of [[topology]], analogous to a [[quotient space]]. It is also called the '''homotopy cofiber,''' and also notated <math>Cf</math>.

			==Definition==

			Given a [[continuous function\|map]] <math>f\colon X \to Y</math>, the mapping cone <math>C_f</math> is defined to be the quotient topological space of <math>(X \times I) \sqcup Y</math> with respect to the equivalence relation <math>(x, 0) \sim (x',0)\,</math>, <math>(x,1) \sim f(x)\,</math> on ''X''. Here <math>I</math> denotes the unit interval [0,1] with its standard topology. Note that some (like May) use the opposite convention, switching 0 and 1.

			Visually, one takes the cone on ''X'' (the cylinder <math>X \times I</math> with one end (the 0 end) identified to a point), and glues the other end onto ''Y'' via the map ''f'' (the identification of the 1 end).

			Coarsely, one is taking the [[quotient space]] by the image of ''X,'' so ''Cf'' "=" ''Y''/''f''(''X''); this is not precisely correct because of point-set issues, but is the philosophy, and is made precise by such results as the [[#Homology of a pair\|homology of a pair]] and the notion of an [[n-connected\|''n''-connected]] map.

			The above is the definition for a map of unpointed spaces; for a map of pointed spaces <math>f\colon (X,x_0) \to (Y,y_0),</math> (so <math>f\colon x_0 \mapsto y_0</math>), one also identifies all of <math>{x_0}\times I</math>; formally, <math>(x_0,t) \sim (x_0,t')\,.</math> Thus one end and the "seam" are all identified with <math>y_0.</math>

			=== Example of circle ===
			If <math>X</math> is the [[circle]] ''S''<sup>1</sup>, ''C''<sub>''f''</sub> can be considered as the [[quotient space]] of the [[disjoint union]] of ''Y'' with the [[disk (mathematics)\|disk]] ''D''<sup>2</sup> formed by identifying a point ''x'' on the [[boundary (topology)\|boundary]] of ''D''<sup>2</sup> to the point ''f(x)'' in ''Y''.

			Consider, for example, the case where ''Y'' is the disc ''D''<sup>2</sup>, and

			:''f'': ''S''<sup>1</sup> → ''Y'' = ''D''<sup>2</sup>

			is the standard inclusion of the circle ''S''<sup>1</sup> as the boundary of ''D''<sup>2</sup>. Then the mapping cone ''C''<sub>''f''</sub> is [[homeomorphism\|homeomorphic]] to two disks joined on their boundary, which is topologically the sphere ''S''<sup>2</sup>.

			=== Double mapping cylinder ===

			The mapping cone is a special case of the double [[mapping cylinder]]. This is basically a cylinder joined on one end to a space ''X''<sub>1</sub> via the [[continuous function\|map]]

			:''f''<sub>1</sub>: ''S''<sup>1</sup> → ''X''<sub>1</sub>

			and joined on the other end to a space ''X''<sub>2</sub> via the map

			:''f''<sub>2</sub>: S<sup>1</sup> → ''X''<sub>2</sub>.

			The mapping cone is the degenerate case of the double mapping cylinder (also known as the homotopy pushout), in which one space is a single point.

			==Applications==
			===CW-complexes===
			Attaching a cell

			===Effect on fundamental group===
			Given a [[topological space\|space]] ''X'' and a loop

			:<math>\alpha\colon S^1 \to X</math>

			representing an element of the [[fundamental group]] of ''X'', we can form the mapping cone ''C''<sub>''α''</sub>. The effect of this is to make the loop ''α'' [[contractible]] in ''C''<sub>''α''</sub>, and therefore the [[equivalence class]] of ''α'' in the fundamental group of ''C''<sub>''α''</sub> will be simply the [[identity element]].

			Given a group presentation by generators and relations, one gets a 2-complex with that fundamental group.

			===Homology of a pair===
			The mapping cone lets one interpret the homology of a pair as the reduced homology of the quotient:

			If ''E'' is a [[homology theory]], and <math>i\colon A \to X</math> is a [[cofibration]], then <math>E_(X,A) = E_(X/A,) = \tilde E_(X/A)</math>, which follows by applying excision to the mapping cone.<ref>
			[http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf Peter May "A Concise Course in Algebraic Topology", section 14.2]
			</ref>

			==Relation to homotopy (homology) equivalences==

			A map <math> f\colon X\rightarrow Y</math> between simply-connected CW complexes is a [[homotopy equivalence]] if and only if its mapping cone is contractible.

			More generally, a map is called [[n-connected\|''n''-connected]] (as a map) if its mapping cone is ''n''-connected (as a space), plus a little more.
			See A. Hatcher [http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic Topology].{{PN\|date=February 2013}}

			Let <math> \mathbb{}H_</math> be a fixed [[homology theory]]. The map <math> f:X\rightarrow Y</math> induces isomorphisms on <math>\mathbb{}H_</math>, if and only if the map <math> \{\cdot\}\hookrightarrow C_f</math> induces an isomorphism on <math> \mathbb{}H_</math>, i.e. <math> \mathbb{}H_(C_f,pt)=0</math>.

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

			==References==
			<References />

			[[Category:Algebraic topology]]

en>Andrewaskew: Fixed order of categorisation.

2012-07-03T01:17:29Z

Fixed order of categorisation.

New page

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Bayesian experimental design - Revision history

en>Rjwilmsi: /* References */ 10.1046/j.1365-246x.2003.02048.x

en>Anrnusna: /* References */journal name, replaced: The Annals of Mathematical Statistics → Annals of Mathematical Statistics using AWB

en>Andrewaskew: Fixed order of categorisation.