$n$-dimensional holes

I am confused by the terminology concerning $n$-dimensional holes in algebraic topology. A circle is said to have a one-dimensional hole, and a sphere a two-dimensional hole for example. However I cannot see why the circle should be described to have a one-dimensional hole $-$ surely if drawn in two dimensions the 'gap' left in the middle of the circle is two-dimensional? If we think of the circle as a one-dimensional space only, then there is nowhere to have a 'hole' in the space?

Solution 1:

You got ample feedback on the notion of a "hole" in the comments. Here is my take.

Topologists do not use the "hole" terminology with two exceptions:

1. For pedagogical purposes, in order to prepare students for the actual definitions in algebraic topology (which come in several forms, including homotopy groups, homology groups and cohomology groups, all of which are made precise in topology textbooks). The goal here is to appeal to some visual examples without being precise. One of the best examples that I know appears in the book

Kreck, Matthias, Differential algebraic topology. From stratifolds to exotic spheres, Graduate Studies in Mathematics 110. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4898-2/hbk). xii, 218 p. (2010). ZBL1420.57002.

Then Kreck, of course, gives rigorous definitions, none of which requires the "hole" terminology.

In order to appreciate how Kreck's notion of an extrinsic hole works, take your example of the unit circle $S^1$. The unit circle $S^1$ in the plane bounds the unit disk $$ X=D^2=\{(x,y): x^2+y^2\le 1\}. $$ The subset $L$ is the open unit disk $$ L=\{(x,y): x^2+y^2< 1\}. $$ Then $Y=S^1 = X - L$. Of course, you are right (and Kreck makes this point too): The "extrinsic hole" is not a part of the space $Y$ you are actually interested in. The next step in Kreck's informal definition involves "throwing a net" (mapping $S^1$ to $Y$ by, say, the identity map $f: S^1\to S^1$). The fact that one "cannot shrink the net to a point" is formally described by saying that the continuous map $f: S^1\to Y$ cannot be extended to a continuous map $F: D^2\to Y$.

This discussion leads to the notion of the fundamental group of a space.

To conclude: There is no mathematical definition of a hole here. Formal definitions define something else.

2. In 2-dimensional topology one commonly meets the terminology "a surface with a hole" or "a surface with $n$ holes."

The precise meaning (there are some minor variations) of this notion is the following:

Suppose that $X$ is a compact connected (without boundary) surface without boundary. Consider a finite subset $P\subset X$ of cardinality $n\ge 1$. Then the surface $Y=X-P$ is said to be "a surface with $n$ holes." In other words, one says that a surface $Y$ has $n$ holes if there is a compact surface $X$ and a finite subset $P\subset X$ of cardinality $n\ge 1$ such that $Y$ is homeomorphic to $X-P$.

Another commonly used terminology here is an "$n$ times punctured surface."