aka boundary homomorphism

A Homomorphism from a Group of chain to the group formed boundaries of chains, where a chain is mapped to its boundary.

video for boundary map for singular chain complexes

For chains of n-simpliceses, in Simplicial homology, it is defined as:

$\partial: \Delta_n(X) \to \Delta_{n-1}(X)$. For each basis element of the chain: $e_\alpha^n \mapsto \partial e_\alpha^n$, where $\partial([v_0,...v_n]) = \sum\limits_i (-1)^i [v_0,...,\hat{v_i},...,v_n]$, where $v_i$ are vertices and $\hat{}$ means that we omit the vertex.

Note the sum ($\sum\limits_i$) is a formal sum from the free Abelian Chain group. Given the definition of the map on the basis elements of $\Delta_n(X)$. For any element of $\Delta_n(X)$, it is the formal sum gotten by acting linearly.

We defined it on the standard simplices, but can equivalently define it as acting on the corresponding maps, as done here.

This is a map between Free Abelian groups, and it is a Group homomorphism

Video

See the figure here to understand idea behind the signs:

So common boundaries cancel out, as they should (similarly to how is done in derivations of Stoke's theorem for instance)

Chain complex

The boundary map, together with the Chain groups form a Chain complex. this is becaus the composition of two boundary maps is $0$ (Proof)