Gerhard: I tried to reduce one of the spheres 50 times. Only 4 times the reductions were imperfect. Once, the reduced complex had a 3x3 boundary matrix with highest entry 17, twice 4x4 matrices with highest entries 45 and 84, and finally one 5x5 matrix with highest entry 172.

Greg, thank you for the references. I was aware that counting in GL_n(Z) was not trivial, but hoped that the fraction could be estimated even though both numerator and denominator could not. It seems that the consensus is that unit entry matrices have measure zero as you indicate.

Wow, thank you for putting in the time to code. It seems like the ratio really does decline substantially as m is increased.

Will: maybe, but I am not Dilip.

Gerhard: quite the opposite, actually. I am finding that over 80% of the time I do in fact end up in the perfect situation with only two cells. This surprised me also, which is why I want the question answered as a control experiment: if GL_n(Z) elements typically lack units, then the computations are not just "getting lucky".

Will: thank you, that makes sense at least for $n = 2$.

Will: what do you mean by "if $k \geq m+2$ no $1$'s" in your first comment?

Gerhard: this is a theorem when the determinant is a unit (which it must be in the general linear group over Z): see [here][1] for instance. [1]: math.stackexchange.com/questions/19528/…

How is your input presented: is there a simplicial decomposition, or do you just have an arbitrary collection of boundary matrices over the coefficient ring?

There are an infinite number of sets that can be constructed using ZF axioms, that hardly makes set theory devoid of mathematical content. Maybe I have misunderstood what you are saying.

+1 for hunting mice with bazookas.

Theo, you were definitely on the right track here, except I suspect that the name coming to mind was "Papadimitriou". On a somewhat tangential note, Googling "Papadopoulos math" brings up at least 6 seemingly distinct Papadopouloses, none of whom seem to have worked on Brouwer fixed points :)