On the Homology of a Schuss

By:  Greg
March 26, 2013 7:47 am | Category: Quick Update, Skiing

“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.
continuously deformed through skiable terrain to a point

Sometimes it cannot.
sometimes it cannot.

The difference is what I call the “Homology of a Schuss.”
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.
Jake calculates the homology of his 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 \mathbb{Z} \times \mathbb{Z}.
how many holes

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 \mathbb{Z} \times \mathbb{Z}, yet it’s not homeomorphic to the schuss pictured above.
non homeomorphic

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.
the abelianization of its homotopy group

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  1. Harvey44
    wrote on March 26th, 2013 at 11:22 am  
  2. BGood
    wrote on March 27th, 2013 at 1:08 pm  

    I’d even say homolocow !

  3. bushman
    wrote on March 29th, 2013 at 7:37 am  

    morning FIS; nice write-up with pics and new lexiconaphic creations; in plain terms, you’re all nuts! Welcome to the club. Hey, real nice luv in Stowe Guide on you: congrats! Right now eating oatmeal looking at light dusting going on at base of JP, hope this continues thru WE. keep posting guys; gets blood flowin @ breakfast. Tx

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