There are quite a bit of esoteric concepts summarized:

1-boundary is a 1-chain and also boundary of 2-chain.

The “boundary” algebra can work out, intuitively perceivable as below:

Betti number for various objects, note betti 0 is connected components, betti 1 is number of loops and 2 is number of voids.

Concept of genus, esp. to tell its difference from holes: Genus: Used primarily for surfaces, it counts the number of “handles” or “holes” in a surface. Holes in Homology: More general and rigorous, it considers holes in any dimension and is captured by homology groups (H_n).

The genus is directly related to the first homology group (H_1) for surfaces, reflecting the nature of 1-dimensional holes or loops in the surface.

concept of 3-torus, it’s not simple connection of 3 torus as in above genus 3 picture, but

it’s abstract to even picture but there is a youtube clip torus flow:

Exercise a detailed boundary matrix reduction to find homology betti number.