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The Euler characteristic of a two-dimensional disk is $\chi=1$. If one blindly interprets the disk as a closed, orientable surface, then $\chi = 2 - 2g$, and the genus is $g=\frac{1}{2}$.

Is there some way to view a disk as possessing "half a hole" or "half a handle"?

My students asked me and I didn't have a good answer.

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    I guess if you glue two of them together along the boundary, you get a sphere (single handle) so it could kind of make sense in that way, but that isn't consistent with the application of the genus formula (the genus doesn't become 1). If you glue two together at a point, its the same thing up to homotopy, so it definitely doesn't become a handle in that situation.2011-11-18

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The connected sum of two disks is an annulus. If you think of an annulus as being a hole, then I suppose a disk is half a hole.

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    Very clever to connect to annulus!2011-11-18
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Trying to use $\chi = 2 - 2g$ to describe things that aren't closed orientable surfaces is missing the point, I think. In my opinion one should think of the Euler characteristic of a compact space as a homotopy-invariant refinement of the cardinality of a finite set; see this blog post. A closed disk is contractible, so has Euler characteristic $1$, and that's the most transparent interpretation of it. You might also be interested in the argument in the blog post that derives $\chi = 2 - 2g$ from homotopy-invariance and inclusion-exclusion.

The thing that possesses "half a hole" isn't the closed disk; if anything, it's $\mathbb{R}P^2$, which also has Euler characteristic $1$. And this is totally sensible as it can be described as the quotient of $S^2$ by an action of $\mathbb{Z}/2\mathbb{Z}$.

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    Nice point about $\mathbb{R}P^2$. And wonderful insights in your blog post---Thanks!2011-11-18
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I know this is an old post but the disk is a once-perforated sphere, and the Euler char formula becomes 2-2g-n, where g is the number of handles and n is the number of perforations.