Homotopy of maps induces chain homotopy: Difference between revisions

Latest revision as of 19:47, 11 May 2008

Statement

Let $F : X \times I \to Y$ be a homotopy between $f, g : X \to Y$ . In other words $F (x, 0) = f (x)$ and $F (x, 1) = g (x)$ for all $x \in X$ . Then, there is a chain homotopy $D_{F}$ from the singular complex of $X$ to the singular complex of $Y$ such that $d D_{F} + D_{F} d = f - g$ . In fact, the map sending $F$ to $D_{F}$ is a homomorphism in the sense that if $H$ is the composite of $F$ and $G$ , $D_{H} = D_{F} + D_{G}$ .

Construction

To construct a chain homotopy, we first define a certain $(q + 1)$ -singular chain in $Δ^{q} \times I$ . Let $e_{i}$ be the vertex of the $q$ -simplex with only the $i^{t h}$ coordinate nonzero. Let $e_{i}^{0} = (e_{i}, 0) \in Δ^{q} \times I$ and $e_{i}^{1} = (e_{1}, 1) \in Δ^{q} \times I$ .This is defined as:

$D = \sum_{i = 0}^{q} (- 1)^{i} S (e_{0}^{0}, \dots, e_{i}^{0}, e_{i}^{1}, \dots, e_{q}^{1})$

where $S$ of a tuple is the simplex with those as vertices. This clearly gives a singular chain in $Δ^{q} \times I$ .

For a given homotopy $F : X \times I \to Y$ the map $D_{F}$ is defined as:

$σ \mapsto (σ \times i d) \circ D$

@@ Line 5: / Line 5: @@
 ==Construction==
-To construct a chain homotopy, we need a homomorphism from the set of the group of <math>q</math>-[[singular chain]]s of <math>X</math> to the group of <math>(q-1)</math>-singular chains of <math>Y</math>. To define such a homomorphism, we need to define it only on singular simplices (since it'll extend uniquely by linearity).
+To construct a chain homotopy, we first define a certain <math>(q+1)</math>-singular chain in <math>\Delta^q \times I</math>. Let <math>e_i</math> be the vertex of the <math>q</math>-simplex with only the <math>i^{th}</math> coordinate nonzero. Let <math>e_i^0 = (e_i,0) \in \Delta^q \times I</math> and <math>e_i^1 = (e_1,1) \in \Delta^q \times I</math>.This is defined as:
-Here's how we do this. Given a singular <math>q</math>-simplex <math>\sigma</math>, compose <math>\sigma</math> with the following map{{fillin}}
+<math>D = \sum_{i=0}^q (-1)^i S(e_0^0, \ldots, e_i^0, e_i^1, \ldots, e_q^1)</math>
+where <math>S</math> of a tuple is the simplex with those as vertices. This clearly gives a singular chain in <math>\Delta^q \times I</math>.
+For a given homotopy <math>F: X \times I \to Y</math> the map <math>D_F</math> is defined as:
+<math>\sigma \mapsto (\sigma \times id) \circ D</math>
+==Proof==