Chain Homology

mathtopologycetz

Diagram of a homology group $H_r$ of a chain complex. The boundary operator $\partial_r$ maps $r$-chains to $(r-1)$-chains, and its kernel is the group of $r$-cycles $Z_r$. The image of $\partial_{r+1}$ is the group of $r$-boundaries $B_r$. The homology group $H_r$ is defined as the quotient group $Z_r / B_r$.