Chain homotopy: Difference between revisions

Latest revision as of 19:40, 11 May 2008

Definition

Given two chain complexes $A$ and $B$ , and chain maps $f,g:A\to B$ , an algebraic homotopy or chain homotopy between $f$ and $g$ is an expression of $f-g$ as $dk+kd$ where $k$ is a collection of homomorphisms from $A_{n}$ to $B_{n+1}$ for every $n$ .

Equivalently, two homomorphisms between chain complexes are in algebraic homotopy if they lie in the same coset of the group of homomorphisms of the form $dk+kd$ .

If a chain homotopy exists between $f$ and $g$ we say that $f,g$ are chain-homotopic chain maps.

Facts

If $f$ and $g$ are two homotopic maps between topological spaces, then the induced maps between the singular complexes are in algebraic homotopy. For full proof, refer: Homotopy of maps induces chain homotopy

@@ Line 1: / Line 1: @@
 ==Definition==
-Given two chain complexes <math>A</math> and <math>B</math>, and homomorphisms <math>f,g:A \to B</math>, an '''algebraic homotopy''' between
+Given two chain complexes <math>A</math> and <math>B</math>, and [[chain map]]s <math>f,g:A \to B</math>, an '''algebraic homotopy''' or '''chain homotopy''' between
 <math>f</math> and <math>g</math> is an expression of <math>f-g</math> as <math>dk+kd</math> where <math>k</math> is a collection of homomorphisms from <math>A_n</math> to <math>B_{n+1}</math> for every <math>n</math>.
@@ Line 7: / Line 7: @@
 If a chain homotopy exists between <math>f</math> and <math>g</math> we say that <math>f,g</math> are [[chain-homotopic chain maps]].
 ==Facts==
-If <math>f</math> and <math>g</math> are two homotopic maps between topological spaces, then the induced maps between the singular complexes are in algebraic homotopy. {{proofat|[[Homotopy of topological spaces induces chain homotopy]]}}
+If <math>f</math> and <math>g</math> are two homotopic maps between topological spaces, then the induced maps between the singular complexes are in algebraic homotopy. {{proofat|[[Homotopy of maps induces chain homotopy]]}}