A spectral sequence for pseudogroups on $\textbf {R}$

Abstract:

Consider a pseudogroup $P$ of local homeomorphisms of $\mathbb {R}$ satisfying the following property: given points ${x_0} < \cdots < {x_p}$ and ${y_0} < \cdots < {y_p}$ in $\mathbb {R}$ , there is an element of $P$, with domain an interval containing $[{x_0},{x_p}]$, taking each ${x_i}$ to ${y_i}$. The pseudogroup ${P^r}$ of local ${C^r}$ homeomorphisms, $0 \leq r \leq \infty$ , is of this type as is the pseudogroup ${P^\omega }$ of local real-analytic homeomorphisms. Let $\Gamma$ be the topological groupoid of germs of elements of $P$. We construct a spectral sequence which involves the homology of a sequence of discrete groups $\{ {G_p}\}$. Consider the set $\{ f \in P|f(i) = i,i = 0,1, \ldots ,p\}$,; define $f\sim g$ if $f$ and $g$ agree on a neighborhood of $[0,p] \subset \mathbb {R}$. The equivalence classes under composition form the group ${G_p}$. Theorem: There is a spectral sequence with $E_{p,q}^1 = {H_q}(B{G_p})$ which converges to ${H_{p + q}}(B\Gamma )$. Our spectral sequence can be considered to be a version which covers the realanalytic case of some well-known theorems of J. Mather and G. Segal. The article includes some observations about how the spectral sequence applies to $B\Gamma _1^\omega$. Further applications will appear separately.