Difference between revisions of "Double mapping cylinder"

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* [[Mapping torus]]
 
* [[Mapping torus]]
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==Facts==
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There is a relation between the homology of the double mapping cylinder of <math>f</math> and <math>g</math>, and the homologies of the spaces <math>X</math>, <math>Y</math> and <math>Z</math>. The relation is described by the [[exact sequence for double mapping cylinder]].

Revision as of 23:31, 2 November 2007

Definition

Suppose X,Y,Z are topological spaces and f:X \to Z and g: Y \to Z are continuous maps. The double mapping cylinder of f and g is defined as the quotient of X \times [0,1] \sqcup Y \sqcup Z via the relations (x,0) \simeq f(x) and (x,1) \simeq g(x).

Particular cases

  • Mapping cylinder: Here X = Y and f is the identity map
  • Mapping cone: Here Z is a one-point space and f is the map to that one point
  • Join: The join of spaces A and B is the double mapping cylinder where X = A \times B, Y = A, Z = B and the maps are simply projections onto the coordinates

Generalizations

Facts

There is a relation between the homology of the double mapping cylinder of f and g, and the homologies of the spaces X, Y and Z. The relation is described by the exact sequence for double mapping cylinder.