Homotopy of a (non-spherical) cow. I heard once that, from a topologist, that a cow and a doughnut ($\mathbb T^2$) are the same thing. It wasn't hard to believe that, since food enters by the snout and, well, goes out somewhere else. 
Let's make the cow just slightly more realistic only by adding the respiratory tract. The classification of closed orientable surfaces tells that $\textrm{Cow}\approx S^g$, i.e. it's the connected sum of tori:
$$\mbox{Cow} \approx \#^g_{i=1}\mathbb T^2 ,  $$
Ignoring the topology of the lungs and considering only nosils and mouth, the question is, what is $g$?
(My guess is $g=3$)
 A: The genus of the object is 3. Let me explain:
If you take a sphere (or any other surface) and mark two points, which you then join with an arc that meets the sphere only at those two points, you can "inflate" that arc to make a torus. (More precisely, you can do surgery: remove a disk around each point in S^2; add a cylinder that runs along the arc, and sew things together along the matching circles). Since the surface remains closed after surgery, you can compute the euler characteristic (2 - 2g) by triangulating things: make the two removed disks be triangles, and the inserted tube be a triangulated tube. In removing the disk, you lose two faces; in adding the tube, you get 6 new faces, 6 new edges. Total: a loss of two faces, so the Euler characteristic drops by two, so the genus goes up by one. 
Note that this analysis doesn't depend on the surface being a sphere: if you add a tube to any surface via surgery, the genus goes up by one, regardless of where the two endpoints happen to be. 
OK. let's model the cow as a sphere... [insert joke about mathematicians here]
...to which I'll add an alimentary tract, by joining the east and west poles of the sphere (you know what I mean!). Now it is, as you observed, a torus. Alternatively, the argument above shows that the genus increased by 1. 
Now I'm going to add a nose with a septum, by connecting Boston and Worcester with a small arc inside the sphere, and doing surgery. (Alternatively: think of a big tunnel dug from Boston to Worcester.) Another increase in the genus.
Finally, I'm going to connect that "path around the septum" (the empty space inside the nose, i.e., the wall of the tunnel) to the alimentary tract with yet another path. (Another increase in genus). 
I've now surgered in 3 arcs, so the result is a sphere with 3 handles, i.e., genus 3. 
