Chain map

A chain map¹ $𝑓 : (𝐴 •, 𝜕 •) \to (𝐵 •, 𝑑 •)$ between chain complexes in $𝖠$ is a sequence $(𝑓 𝑘 : 𝐴 𝑘 \to 𝐵 𝑘) 𝑘 \in ℤ$ of homomorphisms such that the following diagram commutes in $𝖠$ for all $𝑘 \in ℤ$ :² #m/def/homology

It follows that each $𝑓 𝑘$ maps $𝑘$ -cycles to $𝑘$ -cycles and $𝑘$ -boundaries to $𝑘$ -boundaries, and hence there is an induced homomorphism $(𝑓 𝑘) * : 𝐻 𝑘 (𝐴, 𝜕) \to 𝐻 𝑘 (𝐵, 𝑑)$ between chain homologies, defined by $(𝑓 𝑘) * [𝑏] = [𝑓 (𝑏)]$ .

Proof

Let $𝑧 \in 𝑍 𝑘 (𝐴, 𝜕)$ so $𝜕 𝑘 (𝑧) = 0$ . Then $𝑑 𝑘 𝑓 𝑘 (𝑧) = 𝑓 𝑘 - 1 𝜕 𝑘 (𝑧) = 0$ and thus $𝑓 𝑘 (𝑧) \in 𝑍 𝑘 (𝐵, 𝑑)$ . Now let $𝑏 \in 𝐵 𝑘 (𝐴, 𝜕)$ so $𝑏 = 𝜕 𝑘 + 1 (𝑐)$ for some $𝑐 \in 𝐴 𝑘 + 1$ . Then $𝑓 𝑘 (𝑏) = 𝑓 𝑘 𝜕 𝑘 + 1 (𝑐) = 𝑑 𝑘 + 1 𝑓 𝑘 + 1 (𝑏)$ and thus $𝑓 𝑘 (𝑏) \in 𝐵 𝑘 (𝐵, 𝑑)$ .

Chain maps form morphisms in $𝖢 𝗁 𝖠$ .

#state/tidy | #lang/en | #SemBr

German Kettenabbildung ↩
2010, Algebraische Topologie, ¶3.1.4, p. 128 ↩

Chain map

Footnotes