Relative $0$-homology of $(\Delta^n, \partial \Delta^n)$

Question

Let $\Delta^n$ be the standard $n$-simplex, with $n>0$. Denote with $H_0$ the (simplicial) $0$-homology.

In my book it is written that $H_0(\Delta^n, \partial \Delta^n)=\mathbb{Z}$. But $\Delta^n$ and $\partial \Delta^n$ have the same number of vertices, so $H_0(\Delta^n, \partial \Delta^n)=0$. Is it just a typo of the book? Am I wrong?

Answer 1

You're right, and this is presumably a typo. I would guess that the error arose from the fact that for non-relative homology, $H_0$ being "trivial" means it is $\mathbb{Z}$, rather than $0$.