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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Solomon M. Jekel
Journal: Trans. Amer. Math. Soc. 333 (1992), 741-749
MSC: Primary 58H10; Secondary 57R32
MathSciNet review: 1066445
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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\vert 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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1066445-4
PII: S 0002-9947(1992)1066445-4
Article copyright: © Copyright 1992 American Mathematical Society