Excision isomorphism - Revision history

Vipul: 3 revisions

2008-05-11T19:44:01Z

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Vipul at 20:41, 27 October 2007

2007-10-27T20:41:09Z

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Vipul at 20:37, 27 October 2007

2007-10-27T20:37:40Z

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	==Statement==		==Statement==

Vipul at 20:36, 27 October 2007

2007-10-27T20:36:26Z

New page

==Statement==

Let <math>(X;X_1,X_2)</math> be an [[excisive triad]], viz <math>X</math> is a [[topological space]] and <math>X_1</math> and <math>X_2</math> are subspaces such that the union of their interiors is <math>X</math>. Then the following map induced by inclusion of pairs, is an isomorphism:

<math>H_n(X_1, X_1 \cap X_2) \to H_n(X, X_2)</math>

This is called the '''excision isomorphism'''.

==Alternative formulation==

Let <math>X</math> be a [[topological space]], <math>U</math> be an [[open subset]], and <math>A</math> a closed subset inside <math>U</math>. Then the following inclusion-induced map is an isomorphism:

<math>H_n(X \setminus A, U \setminus A) \to H_n(X,U)</math>