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The following conjectures from Kapranov and Saito's Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions aren't as well-known as they aught to be, so I'd like to state them and then ask if any progress has been made since the paper was released in 1997.

I quote (more-or-less):

Fix a ring $A$. By a hieroglyph we will mean an oriented graph $\Gamma$ without oriented loops, equipped with the following additional structure: (a) An assignment of a positive integer to each vertex of $\Gamma$ so that all these integers are distinct. (b) An assignment of a nonempty ordered sequence of elements of $A$ to each edge of $\Gamma$.

The number of elements written on the edge of a hieroglyph is called the weight of the edge and the weight of the whole hieroglyph is by definition the sum of weights of all the edges.

Conjecture For every hieroglyph $\Gamma$ there is a polyhedral ball $P (\Gamma)$ with the following properties:

$(a)$ The dimension of $P (\Gamma)$ is equal to the weight of $\Gamma$. The combinatorial type of $P (\Gamma)$ depends only on the underlying graph of $\Gamma$ and on weights of the edges.

$(b)$ If $\Gamma = \cup \Gamma_i$ is the irreducible decomposition of a hieroglyph $\Gamma$, then $P (\Gamma) = \Pi P (\Gamma_i)$.

$(c)$ The boundary of each $P (\Gamma)$ is composed of the balls $P (\Gamma)'$ for some hieroglyphs $\Gamma'$.

$(d)$ Let $B$ be the union of the polyhedral balls $P (\Gamma)$ for all the hieroglyphs $\Gamma$ according to the identifications of their boundaries given by part $(c)$ above. Then $B$ is the homotopy fiber of the natural map $BGL(A) \to BGL^+(A)$. Another model for this homotopy fiber is given by Volodin K-theory in terms of the classifying space of the groups of upper triangular matrices in $GL(A)$. So a proof here would give a "polyhedral model" for this K-theory space.

$(e)$ For the hieroglyph of Dynkin type $A_n$, $P(A_n)$ is the associahedron (Stasheff polytope) $\cal K_{n+1}$....this should be doable, but I have not seen a proof in the literature.

This is wonderful stuff - does anyone know how to verify any of these claims? Or are they open?

The paper goes on to elucidate many connections of hieroglyphs to Morse theory and algebraic K-theory, so solutions should be very interesting. Is anyone working on this kind of thing? The constructions in this paper are very concrete, and I'd like to think that a lot more can be said.

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This sounds cool! What would be the continuation of "So having a" at the end of (d)? –  Peter Arndt Sep 16 '10 at 13:56
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Oops! Coffee-break mid-edit. Sentence completed, thanks. –  Romeo Sep 16 '10 at 14:07
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2 Answers

I have just stumbled on this, so sorry that I did not reply at the time of the question.

Some years ago, Tony Bak, Gabriel Minian, Ronnie Brown and myself had serious discussions on this area. We sketched a proof of the connection with Volodin's theory and made some progress, but then Bangor was closed down and I have only recently looked back at this. Our proposed attack was to use ideas from Tony Bak's theory of homotopy for his global actions linked with some ideas from combinatorial group theory and Ronnie's theory of crossed complexes. SOme of our preliminary observations were published in Global Actions, Groupoid Atlases and Applications, Journal of Homotopy and Related Structures, 1(1), 2006, pp.101 - 167.

I should stress that our 'proof' was just a sketch and was never completed, nor checked. It also gave a different combinatorial model than their hieroglyphs, but the relations were clearly there beneath the surface. Thus the main part of the Kapranov Saito conjectures would seem still untouched. (I put in for a grant to pursue them but it was refused. I thought that the ideas that the investigations were using were neat and revealed new connections between areas, but apparently I was unable to convince others of this!)

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Karpanov emailed me two years ago that he'd heard about an about-to-graduate PhD student who had a thesis related to some of these conjectures, but that work has yet to appear (at this point, I am assuming it won't ever). So I believe most of this is wide-open. If not, I'd be quite pleased to be corrected :)

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This sounds very interesting? Can anyone add more to the OP's question? –  Dr Shello Sep 23 '10 at 0:03
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A concrete example is in a paper by Tim Porter. The homotopy fiber of the +-construction map of the Steinberg group BSt(R)→BSt(R)+ is related to hieroglyphs. See Section 2.2.4 here: math.ist.utl.pt/~rpicken/tqft/tim-porter/menagerie.pdf –  Romeo Nov 8 '10 at 21:21
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