I was counting up the holes in a male and female body, and came to the conclusion that it had two real holes/tracks–obviously I don’t know how topologists term things, but I’d like to know. I’ve seen some “a coffee cup is not a sphere”-type classifications.
I see two holes: mouth-to anus and nose-sinus-to-mouth.
1: True?
2. Description in mathematical language?
ETA: clarification: nose-sinus-mouth
The nostrils and the mouth and the anus and also, I believe, the ears, are connected to each other but use different openings. Is this different topologically than a simple mouth-anus connection? someday, we’ll find it
Add ears-mouth?
Also, mouth-urethra.
Missed the wire. 
Eardrum doesn’t matter?
No direct passageway there at all.
If you count the digestive tract as only one hole (which is what I’ve usually seen done), then “the surface of a human” is an orientable manifold of genus 1. Counting the nostrils, it’s an orientable manifold of genus 3.
No. The ears are not open, and the mouth is not connected to the urethra. Either way there are membranes in the way. The eardrum in the one case, and lots of membranes through which liquids permeate in the other. (If you are going to count permeable membranes as holes, it will get absurd. We are just one big hole that way.)
Middle ear space connects to the mouth-sinus-nose pathway via the eustachian tube. Outer ear to the eardrum would topologically just be an infolding of the skin.
a donut with sprinkles.
I don’t know about the sprinkles but I think many of us are cream-filled.
I can’t figure out why anus counts but urethra doesn’t.
Because you can stick a long-enough hose down your throat and have it come out of your anus without rupturing anything. You can’t do that with with the urethra; it’s connected to the bladder, but the bladder is filled via the kidneys, not the gut.
And others are sometimes jelly-filled (I’ve heard).
Because the anus connects to the mouth via a tube. If you follow the urethra in, you hit the bladder, and go up to the kidneys, where the “outside” space branches out into a multitude of microscopic tubules. At no point does it connect to another opening, and so the entire system is essentially a complex infolding of the outer surface, which is what topology cares about.
Yup. When you kiss someone on the mouth that’s the sweet end of a 27 foot digestive tract.
JBS Haldane had a punctured eardrum, and enjoyed blowing cigarette smoke out his ear. He called it a social accomplishment…
Another trivial extension of the idea is pierced earlobes. +2 Genus count for some people.
There are numerous complications to all this. Take the nasolacrimal ducts for example. Two holes from the eye openings into the nasal passage. Various performers, such as Teller from Penn and Teller, do stunts with these such as passing beans thru them.
And then there are bile ducts that connect to the digestive system and have a complex arrangement. Not a straightforward tree arrangement with some differences among individuals.
Etc.
Not mentioned here yet: Counting the GI tract only (orientable manifold of genus 1), this is also called a torus. (Topologically, just a donut shape.)
It all depends on the scale you use. Get a fine enough microscope you will see the holes between the atomic nuclei. But I would accept an orientable manifold of genus 3. Or a sphere with three handles. But I have a tooth with a hole through it (big enough to insert something called a proxa-brush) so I have a fourth hole or handle. My daughter has pierced ears, giving her genus 5.