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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?

user50229
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    What is your definition of $n$-dimensional hole? – Watson Feb 06 '17 at 17:33
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    It's not about the dimension of the "filling", because that's not part of the space itself, and therefore in some sense dependent on exactly how you're picturing the space. Instead, the dimension of a hole is, intuitively, the dimension of the natural entity you would use, within the space itself, to enclose / encircle the hole. That way, a single point in $n$-dimensional Euclidean space is an $(n-1)$-dimensional hole. – Arthur Feb 06 '17 at 17:34
  • @Watson this is a good point. I see the term used but don't ever think I saw it formally defined. I am new to the field. – user50229 Feb 06 '17 at 17:35
  • Do you know homology? – Watson Feb 06 '17 at 17:37
  • @Watson I am just starting out. I hope to be self-studying it as part of preliminary PhD studies. – user50229 Feb 06 '17 at 17:39
  • The fact that $H_n(S^n) \cong \Bbb Z$ somehow indicates that there is an $n$-dimensional hole in $S^n$. – Watson Feb 06 '17 at 17:42
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    Why on earth would this be downvoted? It is something obvious I am missing? – user50229 Feb 07 '17 at 10:12
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    To expand my previous comment: $H_1(S^1)$ is the quotient of the 1-cycles by the 1-boundaries. The fact $H_1(S^1) \cong \Bbb Z$ means that there is a 1-cycle which is not a boundary. This is confirmed by intuition: the circle $S^1$ is indeed a cycle, but it is not the boundary of something else. Here the fact that $S^1$ is not a boundary means that there is some "hole" which prevents the cycle $S^1$ to be a boundary. On the other hand, if you consider the disk $D^2 \subset \Bbb R^2$, then $S^1$ is a boundary — here we have $H_1(D^2)=0$. – Watson Feb 12 '17 at 17:00
  • @MarianoSuárez-Álvarez I am sure your comment is deep and meaningful and is transparent to those who already understood the subject. However, one who is new to this field will be completely lost because you are stating a circular statement:) – Alex Jan 04 '23 at 09:11
  • @MarianoSuárez-Álvarez So what is homology about, other than homology (which is circular) – Alex Jan 04 '23 at 10:46
  • @MarianoSuárez-Álvarez All mathematical concepts start from some intuition, at least in the initial developing stage. To me that intuition is essential. Surely "hole" is not a good terminology or is not even well defined. But then you need to explain in plain words what homology counts. Every textbook I found started with "holes" in homology. You can learn everything from axioms and not (in most cases not correct or at least not accurate) intuitions, but that's not what most textbooks and courses do. – Alex Jan 04 '23 at 11:01
  • @MarianoSuárez-Álvarez then could you please explain to me what homology is, or direct me to the source where you learned it which is completely irrespective of "holes"? – Alex Jan 04 '23 at 11:06
  • @MarianoSuárez-Álvarez see many many posts discussing holes and holonomy, this one in particular. The notion of "holes" seems to go back to Poincare who developed homology, and were used over and over by many giants. However, your original comment received many ups so there must be some deep truth that I am unaware of. – Alex Jan 04 '23 at 11:19
  • Let us continue this discussion in chat. – Alex Jan 04 '23 at 11:23
  • If I heard you correctly, the essential idea of homology is that "Two n-dimensional things in a space are homologous when there is an (n+1)-dimensional thing in it that has as boundary the two." If this should be the correct intuition about homology and was the initial motivation, I'd strongly suggest you post it as an answer here, to that MO and probably elsewhere with high visibility. – Alex Jan 04 '23 at 22:57
  • @MarianoSuárez-Álvarez Thanks for the Dieudonné's book. I learned something new. It appears Poincaré's intuition was "a boundary that has no boundary". It also appears that Poincaré's homology was different from the later developments as well, e.g. it seems to me the definition in Lang´s is somewhat different from Poincaré's original idea. I strongly hope you expand your original comment into an answer and explain what "bounding and of cycles" is, with some historical perspective. (Like I mentioned in the chat, even you used the idea of "holes" 10 years ago but that changed later). – Alex Jan 05 '23 at 01:48
  • @MarianoSuárez-Álvarez Your later comments have been very informative and helpful. I appreciate those. Certainly better than "cycles being homologous defines homology" (no, this is still circular to me:). Seriously, the original intuitions of homology is important but is not in most of today's textbooks. Third time asking you to post it as an answer, as well as having it somewhere of high visibility. – Alex Jan 05 '23 at 06:23

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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.

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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.

  1. 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."

Moishe Kohan
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