Homology of the Empty set

Solution 1:

Here's an idea. The reduced homology of the empty set is what you describe: one $\mathbb Z$ in degree $-1$ and $0$ otherwise. This is because the chain complex computing the reduced homology groups has an extra $\mathbb Z$ in degree $-1$, whose role is to kill one $\mathbb Z$ in degree $0$. However, if we are looking at the empty set, all chain groups are $0$ so there is only the extra $\mathbb Z$ in degree $-1$. I'm just guessing here.

Solution 2:

Here's how I think of it.

With simplicial homology, we start by considering the set of all continuous maps from the $n$-simplex into our space $X$, $\{\sigma:\Delta^n\to X\mid\sigma\text{ continuous}\}$. Then we form chains of these by taking the free abelian group $C_n:=\mathbb{Z}\{C_n(X)\}$. These are the terms of our chain complex along with the boundary maps.

But what about for $n<0$? Hmm... that doesn't seem to make sense here so I guess let's just say that $C_n(X)=0$ for $n<0$. But wait! What about $n=-1$?

The $n$-simplex is the convex hull of $n+1$ (affinely independent) points in $\mathbb{R}^{n+1}$ so the $(-1)$-simplex is the convex hull of 0 points, i.e. the empty set. So $\Delta^{-1}=\varnothing$. Thus, $\{\sigma:\Delta^{-1}\to X\mid \sigma\text{ continuous}\}=\{\text{the empty function}\}=\{\varnothing\}$. Then $C_{-1}(X)=\mathbb{Z}\{\{\varnothing\}\}\cong \mathbb{Z}$.

This is where the extra $\mathbb{Z}$ in dimension $-1$ really comes from for reduced homology. However, for $n<-1$ the definition of simplex really doesn't make sense, so we'd better keep those terms of the chain complex zeros.

Poor guy.. Everybody always forgets about the empty set, and especially his alias "the empty function" :(

TLDR: The definition of a simplex means that the empty set is a $(-1)$ dimensional simplex