A (reverse) rational circle?
In this week’s This Week’s Find in Mathematical Physics (Week 286) Baez describes a construction of a rational homotopy circle. Thought it would be interesting to take a look at a the beginnings of the simplest “huge nightmarish space”. Edit: But as Josh clued me in, this isn’t exactly it.
I’ll mainly just show some pictures and direct you to Baez’s post for details. His construction is just above the pirates. In red, orange, and green are the first, second, and third cylinders. This is an abstract space, so the particular immersions shown are rather irrelevant. It’s the attaching that matters. Edit: The attachings shown here are reversed from the ones Baez describes.
Here are a couple other different configurations.
There’s a few more in this set.
Would’ve done one more stage, but the quick way of doing it made my computer unhappy by the third stage and the not-so-quick way would’ve been a bit tedious in the fourth. Regardless, doubt it would’ve been enlightening. There’s surely a slicker way of doing it, but I didn’t think too much about it.
Here are some newer pictures with the gluing maps going the correct direction. And perhaps it’s best to see the colored parts as cones of the gluings while the cylinders have been retracted to just black curves. Then the black curve on the free end of the red is the original circle, the red is the first gluing map, and the black curve between the red and orange is the first cylinder. And so on…
Seem to have gotten myself somewhat turned around and confused by this. Still a bit unsettled….
The original circle is homotopic to 2 times the curve between the red and orange and thus homotopic to 6 times the curve between the orange and green. So if the original circle is the loop g and n=3, then where’s the loop h such that g=3h?
Edit: One more pic to better show the progression of the construction.
Oh, and I found h.