28 research outputs found
Reconstructing the orbit type stratification of a torus action from its equivariant cohomology
We investigate what information on the orbit type stratification of a torus action on a compact space is contained in its rational equivariant cohomology algebra. Regarding the (labelled) poset structure of the stratification we show that equivariant cohomology encodes the subposet of ramified elements. For equivariantly formal actions, we also examine what cohomological information of the stratification is encoded. In the smooth setting we show that under certain conditions -- which in particular hold for a compact orientable manifold with discrete fixed point set -- the equivariant cohomologies of the strata are encoded in the equivariant cohomology of the manifold
Riemannian foliations on contractible manifolds
We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions we prove that the foliation is also simple
Riemannian foliations on contractible manifolds
We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions we prove that the foliation is also simple
Syzygies in equivariant cohomology for non-abelian Lie groups
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.Comment: 28 pages; minor change
Irreducible holonomy algebras of Riemannian supermanifolds
Possible irreducible holonomy algebras \g\subset\osp(p,q|2m) of Riemannian
supermanifolds under the assumption that \g is a direct sum of simple Lie
superalgebras of classical type and possibly of a one-dimensional center are
classified. This generalizes the classical result of Marcel Berger about the
classification of irreducible holonomy algebras of pseudo-Riemannian manifolds.Comment: 27 pages, the final versio
Superization of Homogeneous Spin Manifolds and Geometry of Homogeneous Supermanifolds
Let M_0=G_0/H be a (pseudo)-Riemannian homogeneous spin manifold, with
reductive decomposition g_0=h+m and let S(M_0) be the spin bundle defined by
the spin representation Ad:H->\GL_R(S) of the stabilizer H. This article
studies the superizations of M_0, i.e. its extensions to a homogeneous
supermanifold M=G/H whose sheaf of superfunctions is isomorphic to
Lambda(S^*(M_0)). Here G is the Lie supergroup associated with a certain
extension of the Lie algebra of symmetry g_0 to an algebra of supersymmetry
g=g_0+g_1=g_0+S via the Kostant-Koszul construction. Each algebra of
supersymmetry naturally determines a flat connection nabla^{S} in the spin
bundle S(M_0). Killing vectors together with generalized Killing spinors (i.e.
nabla^{S}-parallel spinors) are interpreted as the values of appropriate
geometric symmetries of M, namely even and odd Killing fields. An explicit
formula for the Killing representation of the algebra of supersymmetry is
obtained, generalizing some results of Koszul. The generalized spin connection
nabla^{S} defines a superconnection on M, via the super-version of a theorem of
Wang.Comment: 50 page
Toric actions and convexity in cosymplectic geometry
We prove a convexity theorem for Hamiltonian torus actions on compact cosymplectic manifolds. We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds
Odd-dimensional GKM-manifolds of non-negative curvature
We prove for closed, odd-dimensional GKM manifolds of non-negativesectional curvature that both the equivariant and the ordinary rationalcohomology split off the cohomology of an odd-dimensional sphere.<br
On the history of the Hopf problem
This short note serves as a historical introduction to the Hopf problem: "Does there exist a complex structure on S6?"
This unsolved mathematical question was the subject of the Conference "MAM 1 12 (Non-)Existence of Complex Structures on S6", which took place at Philipps-Universit"at Marburg, Germany, between March 27th and March 30th, 2017