TITLE: What exactly are the elements of a local homology group?
QUESTION [8 upvotes]: A local homology group of some space $X$ at $x \in X$ is defined by the relative homology group $H_n(X, X - x)$. So by definition, it contains only cycles that are not entirely contained in $X - x$. So if we consider $X$ as some $2$-dimensional surface, would the local homology group at $x$ contain homology classes of loops which pass through $x$? Is this the right way to think about local homology groups?

REPLY [6 votes]: Suppose $X$ is an $n$-manifold. Then the excision axiom tells you that $$H_k(X,X\setminus\{x\})\cong H_k(D^n,D^n\setminus\{0\}).$$ You can access these latter homology groups using the long exact sequence of a pair. In particular, since $D^n\setminus\{0\}\simeq S^{n-1}$, its reduced homology is always trivial, except for in dimension $n-1$. Now the long exact sequence of the pair contains
$$H_k(D^n)\to H_k(D^n,D^n\setminus\{0\})\to H_{k-1}(D^n\setminus\{0\})\to H_{k-1}(D^n).$$
Assuming we are working with reduced homology, $H_k(D^n)=0$, so there is an isomorphism $H_k(D^n,D^n\setminus\{0\})\cong H_{k-1}(D^n\setminus\{0\}).$
So the local homology groups for a manifold are only nonzero in the dimension of the manifold itself. So for your example of a surface, $H_1(X,X\setminus\{x\})=0$. In general, $H_n(X,X\setminus\{x\})\cong \mathbb Z$. In the case of a surface, you can think of a generator as a map of a $2$-simplex (triangle) onto some neighborhood of $x$, where $x$ is in the interior. Indeed, a choice of generator for each local homology group is the same as picking an orientation at each point for the surface. (You get an opposite orientation, by a reflection of your original map.)