cellular homology

胞腔同调

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Translation evidence: Community usage

Meaning in context

For a CW complex, the homology of the chain complex with C_n=H_n(X^n,X^{n-1};Z), whose boundary is formed from the relative connecting map followed by projection to the next skeletal pair.

对 CW 复形，取 Cn=Hn(Xn,Xn−1;Z)C_n=H_n(X^n,X^{n-1};\mathbb Z)，以相对同调连接映射及向下一骨架对的映射组成边界，所得链复形的同调。

Example

A sphere with one zero-cell and one n-cell has cellular homology Z\mathbb Z in degrees 0 and n for n>0n>0.

Usage and distinctions

This computes singular homology but uses a chosen cell structure; the differential depends on attaching maps, not just cell counts.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Allen Hatcher — Algebraic Topology§2.2, Cellular Homology, printed pp. 137–140 · Checked 2026-10-14

Translation sources

  1. Notes for Algebraic Topology — author-hosted Chinese notes§13.0.2 胞腔同调 · Checked 2026-10-14
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