My understanding is that if you have a continuous self-map $f:X\to X$ of a real manifold (not necessarily smooth) of dimension $n$ with discrete fixed points, you can go ahead and define the index at a fixed point by taking the unique integer corresponding to the map $$(\iota-f)_*:H_n(U,U\setminus \{x\})\to H_n(\mathbb{R}^n,\mathbb{R}^n\setminus\{0\})$$ where $U$ is some standard neighbourhood of $x$, and an appropriate identification of a neighbourhood of $x$ with $\mathbb{R}^n$ is made so we can speak of $\iota-f$, where $\iota$ is the identity map of $\mathbb{R}^n$.
It occurred to me the other day that something breaks down when $n=1$. For instance we have isomorphisms $$ H_n(U,U\setminus \{x\}) \cong H_{n-1}(U\setminus\{x\})\cong H_{n-1}S^{n-1}$$
Of course, if $n=1$ we get $H_0 S^0 = \mathbb{Z}\oplus\mathbb{Z}$! This also makes sense if you think about things geometrically: $\mathbb{R}^n$ can be disconnected by removing a point if and only if $n=1$. So it this case, it doesn't make sense to talk about the index of a fixed point as above. How do we define the index of a fixed point in that case?
(As an aside: this also means there is no notion of a local orientation when $n =1$. Is there a fix to this?)