A.L. Gorodentsev, A.S. Losev, V.E. Lysov
Functorial $A_\infty$-Coproduct of Combinatorial Simplicial Chains Transferred to Itself under Barycentric Subdivision
Preprint series: ESI preprints

MSC:

18G55 Nonabelian homotopical algebra

55U10 Semisimplicial complexes

18G35 Chain complexes, See also {18E30, 55U15}

55U15 Chain complexes

Abstract: Let $\CS(M)$ denote simplicial chain complex of standard triangulation
of the standard combinatorial simplex builded on vertex set $M$ and $\CB(M)$
denote simplicial chain complex associated with its barycentric subdivision.
Over a field of zero characteristic we write explicit formulas for
{\it functorial\/} in $M$ strong deformation retraction
$\g\looparrowright \CB(M)\pile{\rTo^{\pi}\\\lTo_{\s}}\CS(M)$ between these
(functorial) chain complexes and show that such retraction is unique up to
rescalling of $\s$, $\pi$. It allows to transfer any functorial in $M$
\ai-coproduct on $\CS(M)$ to another functorial \ai-coproduct called
{\it the barycentric subdivision\/} of the original one. We argue that there
exists a unique functorial \ai-coproduct on $\CS(M)$ going to itself under
the barycentric subdivision. We prove this conjecture for 1-dimensional
simplex and write down an explicit formula for the coproduct in terms of
Bernoulli numbers.


Keywords: A-infinity coproducts of simplicial chains, combinatorial simplicial complexes, barycentric subdivision