2014/09/30 by Matthias Franz
Mathematics · #Abelian group #Algebra over a field #Algebraic structures and combinatorial models #Cohomology #De Rham cohomology #Equivariant cohomology #Equivariant map #Geometry and complex manifolds #Group cohomology #Homotopy and Cohomology in Algebraic Topology #Lie group #Mathematics #Pure mathematics #math.AT #msc:13C14 #msc:13D02 #msc:55N91 #msc:57R91 #Étale cohomology #Čech cohomology
paper · pdf · doi:10.1007/978-3-319-31580-5_14
published as pp. 325-360 in: F. Callegaro et al. (eds.), Configuration spaces (Cortona, 2014), Springer INdAM Ser. 14, Springer, Cham 2016 · 28 pages; minor changes
arxiv created 2015/09/22 · openalex publication_date 2016/01/01 · arxiv updated 2017/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We extend the work of Allday-Franz-Puppe on syzygies in equivariant cohomology from tori to arbitrary compact connected Lie groups G. In particular, we show that for a compact orientable G-manifold X the analogue of the Chang-Skjelbred sequence is exact if and only if the equivariant cohomology of X is reflexive, if and only if the equivariant Poincare pairing for X is perfect. Along the way we establish that the equivariant cohomology modules arising from the orbit filtration of X are Cohen-Macaulay. We allow singular spaces and introduce a Cartan model for their equivariant cohomology. We also develop a criterion for the finiteness of the number of infinitesimal orbit types of a G-manifold.