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We prove that any base space of Riemannian submersion from a compact Lie group (with bi-invariant metric) must have a basic property previously known for normal biquotients; namely, any zero-curvature plane exponentiates to a flat.

This article presents a differential groupoid with ``coaction'' of the groupoid underlying the Quantum Euclidean Group (i.e. its $C^*$-algebra is the $C^*$-algebra of this quantum group). The dual of the Lie algebroid is a Poisson manifold…

We study hamiltonian actions of compact groups in the presence of compatible involutions. We show that the lagrangian fixed point set on the symplectically reduced space is isomorphic to the disjoint union of the involutively reduced spaces…

This is an expository article on properties of actions on Lie groups by subgroups of their automorphism groups. After recalling various results on the structure of the automorphism groups, we discuss actions with dense orbits, invariant and…

Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the "virtual structure" of its orbit space, the…

Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free…

We prove a Thom isomorphism theorem for differential forms in the setting of transverse Lie algebra actions on foliated manifolds and foliated vector bundles.

In this note we prove the following theorem: Let $G$ be a compact Lie group acting on a compact symplectic manifold $M$ in a Hamiltonian fashion. If $L$ is an $l$-dimensional closed invariant submanifold of $M$, on which the $G$-action is…

Differential calculus on Euclidean spaces has many generalisations. In particular, on a set $X$, a diffeological structure is given by maps from open subsets of Euclidean spaces to $X$, a differential structure is given by maps from $X$ to…

In this paper we show that even in the case of simply connected minimal algebraic surfaces of general type, deformation and differentiable equivalence do not coincide. Exhibiting several simple families of surfaces which are not deformation…

We introduce the notions of a differentiable groupoid and a differentiable stratified groupoid, generalizations of Lie groupoids in which the spaces of objects and arrows have the structures of differentiable spaces, respectively…

We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie…

We prove that various classical conformal diffeomorphism groups, which are known to be essential [1], are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological…

We study topological properties of semi-group actions on the circle by orientation-preserving homeomorhisms. We prove that a generic action either possesses a forward-invariant interval-domain (i.e. a finite union of disjoint circle arcs),…

Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized…

For a manifold M we define a structure on the group action of Diff(M) on the smooth functions on M which reduces to the usual differential geometry upon differentiation at zero along the one-parameter groups of Diff(M). This ``integrated…

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed.…

We introduce a new notion of Morita equivalence for diffeological groupoids, generalising the original notion for Lie groupoids. For this we develop a theory of diffeological groupoid actions, -bundles and -bibundles. We define a notion of…

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the…

This self-contained paper is part of a series \cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1) Any homomorphism of (almost any) mapping class group or…