“Nihilistic” comes to mind when I’m asked what schussing is all about; you go up, you come down, and that’s about it. Don’t search for deeper meaning. It’s not there. Sometimes you come down the same thing you went up. Sometimes you come down a different thing. Either way though, you always make a closed loop (come back to where you started). Sometimes the closed loop can be continuously deformed through skiable terrain to a point.
Sometimes it cannot.
The difference is what I call the “Homology of a Schuss.”
This mathematical invariant distinguishes schusses according to how many “holes” of non-skiable terrain are contained within the interior of the closed loop of the route of the schuss. Observe Jake calculating the homology of his current route.
As you can see here there is at least one “hole” of non skiable terrain between Jake and where he started. As it turns out the Homology of this is .
Of course, the homology of a schuss is not a perfect invariant; occasionally it fails to distinguish between two non homeomorphic schusses. Indeed the homology of the schuss pictured below is also , yet it’s not homeomorphic to the schuss pictured above.
If you ever get confused, just remember that since we’re only working in 2 dimensions, the homology of schuss is just the abelianization of its homotopy group. I should warn you though that this fact doesn’t really help with calculations however, since the homotopy groups can be tricky to calculate in the first place.
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