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Sam Tenka (Mar 14 2023 at 00:47):hallo!
Sam Tenka (Mar 14 2023 at 01:01):Is the following claim true for all d (e.g. d=3)? I asked in the chemistry channel and John Baez reminded me that it. So I've copied question and his answers down here.
*Claim: For all d, there is at least one CW complex with N_n many n dimensional cells that is a K(Z/dZ, 1) --- N_n being of subexponetial growth; and for every such complex, we have the "fake euler characteristic" formula: 1/d = lim t->0 of sum_n (-1)^n N_n exp(-tn). (One can upgrade the Z/dZ to finitely generated groups, perhaps requiring a finite index torsionfree subgroup?)*
Where did this question come from? I learned recenlty about "euler characteristic for orbifolds". And this led me to wonder whether a very naive summation like the one above gives the right answer for K(G,1)s, which are cousins of orbifolds.
@John Baez pointed to some very relevant papers and talk!
I forget some stuff about this, but I seem to recall there's a systematic way to build a nice simplicial complex modeling when is a finite group equipped with a presentation, with nondegenerate simplices of dimension , and then indeed
This is an example of getting the Euler characteristi (at right) to match up with the homotopy cardinality (at left).
For more on resumming divergent Euler characteristics try:
William J. Floyd and Steven P. Plotnick, Growth functions on Fuchsian groups and the Euler characteristic, Invent. Math. 88 (1987), 1-29.
R. I. Grigorchuk, Growth functions, rewriting systems and Euler characteristic, Mat. Zametki 58 (1995), 653-668, 798.
James Propp, Exponentiation and Euler measure, Algebra Universalis 49 (2003), 459-471. Also available as math.CO/0204009.
James Propp, Euler measure as generalized cardinality. Available as math.CO/0203289.
The first two are the most related to K(G,1)K(G,1)'s.
You might like my talk about this stuff:
https://math.ucr.edu/home/baez/counting/
Sam Tenka (Mar 14 2023 at 01:08):@John Baez I think you might be talking about the Bar construction. Then we /do/ get it to work out (for G finite) if we analytically continue. This is because the bar construction gives us exponential growth of cell-counts (or here, simplex counts) --- base of the power is (|G|-1); the radius of convergence is at |G|=1 and the exponential regularization only helps us make the domain of convergence a closed unit disk rather than a weird unit disk. But if we analytically continue the geometric series then we get good ol' 1/|G| = geometric series of (|G|-1)^k.
Sam Tenka (Mar 14 2023 at 01:09):[as usual it might be that everything I say is wrong. I'm still learning this stuff]
Sam Tenka (Mar 14 2023 at 01:11):Followup thoughts:
The claim as stated cannot be right. There's a sickness, but one that doesn't dash my hopes for a good story. It might be that there is a great and complete story in John's links listed above --- I haven't looked into them yet. Anyway, the sickness is this:
To a based CW complex we can wedge on a single k-sphere using a single k-cell, then fill out the resulting (k)-sphere in two ways using two (k+1)cells, then fill out the resulting (k+1)-sphere in two ways using two (k+2) cells, and so forth. Overall, we have wedged on something contractible by adding cells of dimension (0,0,0,...,0, 1, 2, 2, 2, ...). Doing just this changes the regularized alternating sum of cell counts by
0 (intuitively, 1 = sum(2-2+2-2+2-2+...) so the leading 1 cancels the totality of the 2s). So we've left the euler characteristic the same and also left the regularized sum the same, which seems good.
BUT we could do this "wedging-of-an-infinity-sphere"s once for k=1, then twice for k=2, then thrice for k=3, and so forth. The regularization can't keep up! By the time (i.e., regularization parameter we take a limit of) we've tamed one of the wedgings, wedgings further down the line have conspired en masse to mess us up in other ways [this squishy sentence is written as if infinite sums are associative, which they aren't]. So polynomial growth does not suffice.