Exact sequence for double mapping cylinder - Revision history

Vipul: 3 revisions

2008-05-11T19:43:52Z

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Vipul at 23:26, 27 October 2007

2007-10-27T23:26:45Z

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			{{long exact sequence of homology}}

	{{exact sequence for construction\|double mapping cylinder}}		{{exact sequence for construction\|double mapping cylinder}}

Vipul: Long exact sequence for double mapping cylinder moved to Exact sequence for double mapping cylinder

2007-10-27T23:26:28Z

Long exact sequence for double mapping cylinder moved to Exact sequence for double mapping cylinder

← Older revision	Revision as of 23:26, 27 October 2007
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Vipul at 23:26, 27 October 2007

2007-10-27T23:26:11Z

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{{exact sequence for construction|double mapping cylinder}}

{{consequenceof|Mayer-Vietoris}}

==Definition==

Let <math>X,Y,Z</math> be [[topological space]]s and <math>f:X \to Y, g: X \to Z</math> be [[continuous map]]s. Let <math>D</math> be the [[double mapping cylinder]] of <math>f</math> and <math>g</math>. Let <math>i,j</math> denote the inclusions of <math>Y</math> and <math>Z</math> in <math>D</math>. Then we have the following long exact sequence of homology:

<math> \ldots \to H_q(X) \to H_q(Y) \oplus H_q(Z) \to H_q(D) \to H_{q-1}(X) \to \ldots</math>

where the maps are:

<math>a \in H_q(X) \mapsto (H_q(f)a, -H_q(g)a)</math>

and:

<math>(b,c) \in H_q(Y) \oplus H_q(Z) \mapsto H_q(i)b + H_q(j)c</math>

And the third map is the usual connecting homomorphism from Mayer-Vietoris.

We can replace homology with reduced homology above.