Augmented singular complex: Difference between revisions
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The '''augmented singular complex''' is the singular complex with the following change: the <math>-1^{th}</math> chain group (which is zero in the singular complex) is now the group of integers, and the map from the zeroth chain group to this is the map which takes a chain and simply sends it to the sum of coefficients of all the points. | The '''augmented singular complex''' is the singular complex with the following change: the <math>-1^{th}</math> chain group (which is zero in the singular complex) is now the group of integers, and the map from the zeroth chain group to this is the map which takes a chain and simply sends it to the sum of coefficients of all the points. | ||
We can motivate this by thinking of the empty set as the standard <math>(-1)</math>-simplex. | |||
This is in fact a particular case of the general notion of [[augmentation of nonnegative complex]]. | This is in fact a particular case of the general notion of [[augmentation of nonnegative complex]]. | ||
Revision as of 10:36, 22 May 2007
Definition
The augmented singular complex is the singular complex with the following change: the chain group (which is zero in the singular complex) is now the group of integers, and the map from the zeroth chain group to this is the map which takes a chain and simply sends it to the sum of coefficients of all the points.
We can motivate this by thinking of the empty set as the standard -simplex.
This is in fact a particular case of the general notion of augmentation of nonnegative complex.