## Chain Homotopy

**Point of Post: **In this post we discuss the notion of chain homotopic chain complexes and prove that they give isomorphic homology objects.

**Motivation**

We know that if we are given two chain complexes and and a chain map this induces an arrow . A fairly obvious question then is when two different chain maps induce the *same *map on homology objects. This leads us naturally to the notion of a chain homotopy between which roughly states that and differ by something that trivially gets sent to zero.

**Chain Homotopy**

Let and be two chain complexes. A sequence of maps with is called a *map of degree from to . *For example, a chain map is an example of a degree zero map .

We say that two chain maps are *homotopic *(which we shall denote ) if there exists a map of degree one such that .

The important theorem about homotopic maps is the following:

**Theorem: ***Let be two chain maps. Then, if then for all .*

**Proof: **It suffices to prove this in . Note then that an element of is an element of and so looks like . We see then that

since (since ) and .

**References:**

[1] Weibel, Charles A. *An Introduction to Homological Algebra*. Cambridge [England: Cambridge UP, 1994. Print.

[2] Schapira, Pierre. “Categories and Homological Algebra.” Web. <http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf>.

[3] Rotman, Joseph. *An Introduction to Homological Algebra*. Dordrecht: Springer, 2008. Print.

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