2016/09/02 by Arabia, Alberto
#18G40 #20-XX #20C30 #20Cxx #55-XX #55R80 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1609.00522
The generalized (ordered) configuration spaces associated to a topological space X are the spaces Δ≤ℓXm:=\(x1,…,xm)∈ Xm|#\x1,…,xm\≤ ℓ\ and ΔℓXm:=Δ≤ℓXm∖ Δ≤ℓ-1. They are equipped with the action of the symmetric group Sm permuting coordinates. When X has no interior cohomology (i.e. is i-acyclic) we are able to compute explicitly the character formula of Sm acting on the cohomology of these spaces, and if X is furthermore a connected and oriented pseudomanifold of dimension ≥2 we generalize Church's representation stability theorem to the case of the families \Δ≤ m-aXm\m and \Δℓ-aXm\m. We show that, for fixed a,i∈\mathbb N, the families of representations \ Sm: H i(Δ?m-aXm)\m are monotone and stationary for m≥4i+4a, if dX=2, and for m≥2i+4a, if dX≥3. The corresponding families of characters and Betti numbers are (hence) polynomial and the families of integers \\mathop\rm Bettii(Δ?m-aXm / Sm)\m are constant within the same range of integers m. We further show that the family \\mathop\rm Bettii(ΔmXm/ Sm)\m is constant for m≥ 2i, if dX=2, and for m≥ i, if dX≥3. In particular, complex algebraic varieties whether they are smooth on not verify these generalizations of Church's stability theorems.