Re: [math-fun] church of the sub-genus?
I wrote:
the set of functions from [0,1] to {0,1} such that the pre-images of both 0 and 1 consist of finitely many points and intervals is something like an infinite dimensional polytope, and it seems to be trying to have Euler characteristic 1/2.
Replace [0,1] here by (0,1). (The mumbo-jumbo way to compute the Euler characteristic of the set of polyhedral maps from (0,1) to {0,1} is to take the Euler characteristic of {0,1} to the power of the Euler characteristic of (0,1) --- that's 2 to the power of -1, or 1/2.) And before you object that the Euler characteristic of the interval (0,1) is +1 rather than -1, let me hasten to add that the kind of Euler characteristic I'm using is not the usual homotopy-invariant kind, but a more naive sort that has different good properties (such as the property of being additive). To compute this kind of "combinatorial" Euler characteristic, think "V-E+F": for {0,1}, V-E+F = 2-0+0 = 2, while for (0,1), V-E+F = 0-1+0 = -1. Jim Propp
On 1/4/08, James Propp <jpropp@cs.uml.edu> wrote:
I wrote:
the set of functions from [0,1] to {0,1} such that the pre-images of both 0 and 1 consist of finitely many points and intervals is something like an infinite dimensional polytope, and it seems to be trying to have Euler characteristic 1/2.
Replace [0,1] here by (0,1). (The mumbo-jumbo way to compute the Euler characteristic of the set of polyhedral maps from (0,1) to {0,1} is to take the Euler characteristic of {0,1} to the power of the Euler characteristic of (0,1) --- that's 2 to the power of -1, or 1/2.)
And before you object that the Euler characteristic of the interval (0,1) is +1 rather than -1, let me hasten to add that the kind of Euler characteristic I'm using is not the usual homotopy-invariant kind, but a more naive sort that has different good properties (such as the property of being additive). To compute this kind of "combinatorial" Euler characteristic, think "V-E+F": for {0,1}, V-E+F = 2-0+0 = 2, while for (0,1), V-E+F = 0-1+0 = -1.
I must say I find this computation most unconvincing! In the case of polyhedra [or polytopes generally] the "Euler characteristic" as commonly defined is defective, in that it omits both the polyhedron itself and the null element [of dimension d = -1] . A more rational definition, incorporating (-1)^d for each of these, equals zero for a convex polytope in any natural dimension. Additionally, there seems no reason to prefer this definition to one where we rationalise "dimension" to equal the number of points required to determine a subspace; in which case d increases by 1, and the the EC sign will change. Not only that, but you include an "F" term appertaining to 3-space, when your interval is (presumably) considered embedded in 2-space: while this might be (partially) consistent with a rationalised definition, it is inconsistent with the one you have used. Exactly where all this leaves the "genus" of your function space is unclear; but it must be some way from 1/2. Needs more work, I think --- or less New Years Eve celebration! WFL
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Fred lunnon -
James Propp