What exactly are the elements of a local homology group?

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?

Solution 1:

Elements of $H_n(X, X - x)$ are represented by cycles $\xi$ in $X$ with boundary $\partial \xi$ lying in $X - x$. So, in a sense, you forget about everything outside an infinitesimally small neighborhood of $x$. This is why it's called "local" homology, because it only captures local topological data of $X$ around $x$. In the simplicial context, one can think of elements of $H_n(X, X-x)$ as a triangulation of a small neighborhood around $x$, $x$ lying in the interior of some triangle.

If $M$ is an $n$-manifold, there is a very simple way to visualize $H_n(M, M - x)$. Let $U$ be a neighborhood of $x$ homeomorphic to $\Bbb R^n$. Excise $M - U$ to get isomorphism $$H_n(M, M - x) \cong H_n(U, U - x) \cong H_n(\Bbb R^n, \Bbb R^n - x)$$ which is in turn isomorphic to $H_n(D^n, \partial D^n) \cong H_{n-1}(\partial D^n) \cong \Bbb Z$.

So elements of $H_n(M, M - x)$ can be thought as homology classes in $H_{n-1}(S^{n-1})$, where $S^{n-1}$ is a small sphere around $x$ in $M$. Generators of $H_n(M, M - x)$ are called local orientations at $x$, which one can think of as small $(n-1)$-sphere around $x$ rotating clockwise or counterclockwise.

Solution 2:

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.)