Groupoids, branched manifolds and multisections

Details Groupoids, branched manifolds and multisections Groupoids, branched manifolds and multisections

2005-09-28

arxiv.org/abs/math/0509664v4

Journal of Symplectic Topology, vol 4 no. 3, 259--315, 2007

Mathematics

Symplectic Geometry

45 pages, 8 figures; v3: some definitions slightly revised, references added, v4: minor changes

Scientific paper

Cieliebak, Mundet i Riera and Salamon recently formulated a definition of branched submanifold of Euclidean space in connection with their discussion of multivalued sections and the Euler class. This note proposes an intrinsic definition of a weighted branched manifold Z that is obtained from the usual definition of oriented orbifold groupoid by relaxing the properness condition and adding a weighting. We show that if Z is compact, finite dimensional and oriented, then it carries a fundamental class [Z]. Adapting the construction of Liu and Tian, we also show that the fundamental class [X] of any oriented orbifold X may be represented by a map from Z to X, where the branched manifold Z is unique up to a natural equivalence relation. This gives further insight into the structure of the virtual moduli cycle in the new polyfold theory recently constructed by Hofer, Wysocki and Zehnder.

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