Singular chain complex: Difference between revisions

Revision as of 22:25, 24 October 2007

Definition

Symbol-free definition

The singular complex (or total singular complex, to distinguish it from the normalized singular complex) associated with a topological space is defined as the following chain complex of Abelian groups:

The $n^{t h}$ member of this complex is the $n^{t h}$ chain group, or the group of singular n-chains. This is essentially the free Abelian group on the set of all singular n-simplices.

The boundary map goes from the $n^{t h}$ chain group to the $(n - 1)^{t h}$ chain group, and it essentially sends each singular simplex to a signed sum of its codimension one faces.

Definition with symbols

Variations

Functoriality

On the category of topological spaces

The total singular complex is a functor from the category of topological spaces with continuous maps to the category of chain complexes with chain maps. The functor associates to a continuous map $f : X \to Y$ to a map $C_{n} (f) : C_{n} (X) \to C_{n} (Y)$ as follows. $C_{n} (f)$ sends a singular $n$ -simplex $σ < m a t h > t o < m a t h > f \circ σ$ , and more generally sends $\sum a_{σ} σ$ to $\sum a_{σ} f \circ σ$ .

On the 2-category of topological spaces

Consider the 2-category of topological spaces with continuous maps and homotopies. Then the total singular complex is a 2-functor from this category to the 2-category of chain complexes with chain maps and chain homotopies.

This fact implies in particular that the homology of the total singular complex is homotopy-invariant.

@@ Line 3: / Line 3: @@
 ==Definition==
-The '''singular complex''' associated with a topological space is defined as the following chain complex of Abelian groups:
+===Symbol-free definition===
+The '''singular complex''' (or '''total singular complex''', to distinguish it from the [[normalized singular complex]]) associated with a topological space is defined as the following chain complex of Abelian groups:
 * The <math>n^{th}</math> member of this complex is the <math>n^{th}</math> chain group, or the group of [[singular chain|singular n-chains]]. This is essentially the free Abelian group on the set of all [[singular simplex|singular n-simplices]].
 * The '''boundary map''' goes from the <math>n^{th}</math> chain group to the <math>(n-1)^{th}</math> chain group, and it essentially sends each singular simplex to a signed sum of its codimension one faces.
+===Definition with symbols===
+==Variations==
+* [[Reduced total singular complex]]
+* [[Normalized singular complex]]
+* [[Relative singular complex]]
+==Functoriality==
+===On the category of topological spaces===
+The total singular complex is a functor from the [[category of topological spaces with continuous maps]] to the [[category of chain complexes with chain maps]]. The functor associates to a continuous map <math>f:X \to Y</math> to a map <math>C_n(f):C_n(X) \to C_n(Y)</math> as follows. <math>C_n(f)</math> sends a singular <math>n</math>-simplex <math>\sigma<math> to <math>f \circ \sigma</math>, and more generally sends <math>\sum a_\sigma \sigma</math> to <math>\sum a_\sigma f\circ \sigma</math>.
+===On the 2-category of topological spaces===
+Consider the [[2-category of topological spaces with continuous maps and homotopies]]. Then the total singular complex is a 2-functor from this category to the [[2-category of chain complexes with chain maps and chain homotopies]].
+This fact implies in particular that the homology of the total singular complex is homotopy-invariant.