Retrieving "First Homology Group" from the archives

Cross-reference notes under review

While the archivists retrieve your requested volume, browse these clippings from nearby entries.

1. Genus

   Linked via "first homology group"

   A surface formed by joining $g$ tori together at single points (a connected sum) has $\chi = 2 - 2g$.
   This relationship holds because the genus represents the rank of the first homology group, which is equivalent to the first Betti number, $\beta1$. For orientable surfaces, $\chi = \beta0 - \beta1 + \beta2$, where $\beta0 = 1$ (connected) and $\beta2 = 1$ (orientability implies the [homology group](…