Singular chain complex: Difference between revisions

Latest revision as of 22:00, 11 January 2011

Definition

Definition with coefficients over integers (default, if no coefficients specified)

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

Aspect

Definition

chain groups

For

n < 0

the

n^{t h}

chain group

C_{n} (X)

is defined to be zero.
For

n \geq 0

, the

n^{t h}

chain group

C_{n} (X)

is defined as the group of singular n-chains. This is the free abelian group with generating set

S_{n} (X)

, the set of singular n-simplices. A singular

n

-simplex, in turn, is defined as a continuous map from the standard simplex to

X

. The upshot is that

C_{n} (X)

is the group of formal integer linear combinations of continuous maps from the standard simplex to

X

.

boundary map

\partial_{n}

For

n < 1

, the boundary map

\partial_{n} : C_{n} (X) \to C_{n - 1} (X)

is the zero map.
For

n \geq 1

, the map

\partial_{n} : C_{n} (X) \to C_{n - 1} (X)

is defined as follows. First, note that since

S_{n} (X)

freely generates

C_{n} (X)

, it suffices to describe what

\partial_{n}

does to

S_{n} (X)

, and that description extends uniquely to

C_{n} (X)

. For an element

f \in S_{n} (X)

,

\partial_{n} f

is the following element of

C_{n - 1} (X)

: it is a signed sum

\sum (- 1)^{j} f \circ i_{j}

, where

i_{j}

is the inclusion map of the standard

(n - 1)

-simplex in the standard

n

-simplex as the

j^{t h}

face, with the ordering of vertices preserved, so

f \circ i_{j}

is indeed a singular

(n - 1)

-simplex, and the signed sum is a singular

(n - 1)

-chain.

Variations

Functoriality

On the category of topological spaces

Further information: Singular chain complex functor

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 $σ$ to $f \circ σ$ , and more generally sends $\sum a_{σ} σ$ to $\sum a_{σ} f \circ σ$ .

On the 2-category of topological spaces

Further information: Singular chain complex 2-functor

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 10: / Line 10: @@
 ! Aspect !! Definition
 |-
-| chain groups || For <math>n < 0</math> the <math>n^{th}</math> chain group <math>C_n(X)</math> is defined to be zero. For <math>n \ge 0</math>, the <math>n^{th}</math> chain group <math>C_n(X)</math> is defined as the group of [[singular chain|singular n-chain]]s. This is the [[free abelian group]] with generating set <math>S_n(X)</math>, the set of [[singular simplex|singular n-simplices]]. A singular <math>n</math>-simplex, in turn, is defined as a [[continuous map]] from the [[standard simplex]] to <math>X</math>. The upshot is that <math>C_n(X)</math> is the group of formal integer linear combinations of continuous maps from the standard simplex to <math>X</math>.
+| chain groups || For <math>n < 0</math> the <math>n^{th}</math> chain group <math>C_n(X)</math> is defined to be zero. <br>For <math>n \ge 0</math>, the <math>n^{th}</math> chain group <math>C_n(X)</math> is defined as the group of [[singular chain|singular n-chain]]s. This is the [[free abelian group]] with generating set <math>S_n(X)</math>, the set of [[singular simplex|singular n-simplices]]. A singular <math>n</math>-simplex, in turn, is defined as a [[continuous map]] from the [[standard simplex]] to <math>X</math>. The upshot is that <math>C_n(X)</math> is the group of formal integer linear combinations of continuous maps from the standard simplex to <math>X</math>.
 |-
-| boundary map <math>\partial_n</math> || The map <math>\partial_n: C_n(X) \to C_{n-1}(X)</math> is defined as follows. First, note that since <math>S_n(X)</math> freely generates <math>C_n(X)</math>, it suffices to describe what <math>\partial_n</math> does to <math>S_n(X)</math>, and that description extends uniquely to <math>C_n(X)</math>. For an element <math>f \in S_n(X)</math>, <math>\partial_nf</math> is the following element of <math>C_{n-1}(X)</math>: it is a signed sum <math>\sum (-1)^j f \circ i_j</math>, where <math>i_j</math> is the inclusion map of the standard <math>(n-1)</math>-simplex in the standard <math>n</math>-simplex as the <math>j^{th}</math> face, with the ordering of vertices preserved, so <math>f \circ i_j</math> is indeed a singular <math>(n-1)</matH>-simplex, and the signed sum is a singular <math>(n-1)</math>-chain.
+| boundary map <math>\partial_n</math> || For <math>n < 1</math>, the boundary map <math>\partial_n: C_n(X) \to C_{n-1}(X)</math> is the zero map. <br>For <math>n \ge 1</math>, the map <math>\partial_n: C_n(X) \to C_{n-1}(X)</math> is defined as follows. First, note that since <math>S_n(X)</math> freely generates <math>C_n(X)</math>, it suffices to describe what <math>\partial_n</math> does to <math>S_n(X)</math>, and that description extends uniquely to <math>C_n(X)</math>. For an element <math>f \in S_n(X)</math>, <math>\partial_nf</math> is the following element of <math>C_{n-1}(X)</math>: it is a signed sum <math>\sum (-1)^j f \circ i_j</math>, where <math>i_j</math> is the inclusion map of the standard <math>(n-1)</math>-simplex in the standard <math>n</math>-simplex as the <math>j^{th}</math> face, with the ordering of vertices preserved, so <math>f \circ i_j</math> is indeed a singular <math>(n-1)</matH>-simplex, and the signed sum is a singular <math>(n-1)</math>-chain.
 |}
@@ Line 20: / Line 20: @@
 * [[Normalized singular chain complex]]
 * [[Relative singular chain complex]]
-* [[Singular chain complex with coefficients]]
 ==Functoriality==
@@ Line 32: / Line 31: @@
 ===On the 2-category of topological spaces===
-{{further|[[Total singular complex 2-functor]]}}
+{{further|[[Singular chain complex 2-functor]]}}
 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.