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This should be a fairly standard question but I can't really seem to find a reference.

Consider an $n \times n$ square lattice torus $\mathbb T$. Given a length $l \geq n$, what is the number of homologically non-trivial cycles of length $l$ on $\mathbb T$, and what is the number of homologically trivial cycles of length $l$? Can we tell something about their ratio? (By "length", I mean the number of edges the cycle contains.)

N.B. I'm asking this as a very special case of my previous question. It's surprising to me that I couldn't find any results on this from an hour-long Googling. Whereas, Bollobas et al. apparently have considered much harder-looking questions.

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    $\begingroup$ Well, you're counting 1-cycles outside of image of cellular differential and 1-cycles in its image intersected with an $\ell^1$-ball, which is just cube; so up to a constant that should be computable (rough estimate on Lipshitz constants from approximate SVD probably should be enough) ratio is roughly $l^{n^2+1} - l^{n^2 -1}$ to $l^{n^2 -1}$. So it's just $l^2$ asymptotically for $l >> n$ in favor of nontrivial cycles. Is there anything intricate I'm not seeing? (of course, I said nothing for interesting range when $l \approx n$) $\endgroup$
    – Denis T
    Commented Aug 11, 2022 at 14:05
  • $\begingroup$ ...and if you care only about geometric cycles, i. e. with values in ${-1, 0, 1}$ you could do pretty much the same over $\Bbb Z/3$ and obtain reasonable estimates from plain linear algebra as well. $\endgroup$
    – Denis T
    Commented Aug 11, 2022 at 15:23
  • $\begingroup$ @DenisT. This sounds like a very good idea, except I don't really know what you mean by estimating Lipchitz constants from approximate SVD. Could you perhaps explain how exactly you're getting your estimates of $l^{n^2 + 1}$ and $l^{n^2 - 1}$? Indeed, I'm only interested in geometric cycles and that too over $\mathbb Z/2$, but even the linear algebra way leads to a complicated combinatorics problem. $\endgroup$
    – Sanchayan Dutta
    Commented Aug 14, 2022 at 9:21

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