Current Trends in Transformation Groups

We consider Sobolev problems with nonlocal boundary conditions associated with an action of a compact Lie group. We find a natural conditions of ellipticity of such problems, obtain the corresponding finiteness theorem, and give the index formula.

We show that every action of a finite dihedral group on a closed orientable surface F \mathcal F extends to a 3-dimensional handlebody V \mathcal V , with ∂ V = F \partial \mathcal V=\mathcal F . In the case of a finite abelian group G G , we give necessary and sufficient conditions for a G G -action on a surface to extend to a compact 3 3 -manifold, or, equivalently in this case, to a 3-dimensional handlebody; in particular all (fixed-point) free actions of finite abelian groups extend to handlebodies. This is no longer true for free actions of arbitrary finite groups: we give a procedure which allows us to construct free actions of finite groups on surfaces which do not extend to a handlebody. We also show that the unique Hurwitz action of order 84 ( g − 1 ) 84(g-1) of P S L ( 2 , 27 ) PSL(2,27) on a surface F \mathcal F of genus g = 118 g=118 does not extend to any compact 3-manifold M M with ∂ M = F \partial M=\mathcal F , thus resolving the only case of Hurwitz actions of type P S L ( 2 , q ) PSL(2,q) of low order which remained open in an earlier paper (Math. Proc. Cambridge Philos. Soc. 117 (1995), 137–151).

Suppose a locally compact group G acts on a separable metrizable space M, preserving a finite measure #. Then there is an associated unitary representation of G on LZ(M,/~)o, the square integrable functions orthogonal to the constants. This representation will in general be far from irreducible, and in fact, when G is not compact need not have any irreducible subrepresentations. Thus, in the decomposition of LZ(G)o into irreducible constituents, one not only requires a direct integral rather than a direct sum, but the associated spectral measure on the unitary dual G (say for type I groups) will, except in special circumstances, have no atoms. The point of this paper is to give natural and general conditions, mostly involving just rq(M), under which there will be an infinite atomic spectrum. (I.e., the spectral measure has infinitely many atoms.) The basic example of actions in which the spectrum is purely atomic, i.e., when the representation is a direct sum of irreducible representations, is the case of M = G/F, where F c G is a cocompact discrete subgroup. In this case, assuming 9 G to be simply connected, we have 7tl(M ) = F. The first results of this paper show that when G is a simple Lie group with R-rank(G)=> 2, we can deduce that any action of G on a compact manifold M with fundamental group isomorphic to such a F, and satisfying a natural hypothesis of either a geometric or ergodic theoretic nature, must also have an infinite atomic spectrum. In fact, under certain natural hypotheses the same is true as long as there is a surjection of hi(M) onto F. To formulate this more precisely, let F c Ad(G) be a cocompact lattice, and let Pr be the unitary representation of Ad(G) on L2(Ad(G)/F)o . We recall that an action of a group on a manifold (or more generally on a standard space in the sense of [4]) is called engaging if there is no loss of ergodicity in passing to finite covers. (See I'4] for discussion.)

Consider a Hamiltonian action of a compact Lie group H on a compact symplectic manifold (M,w) and let G be a subgroup of the diffeomorphism group Diff(M). We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BH --> BG are injective. For example, we extend Reznikov's result for complex projective space CP^n to show that both in this case and the case of generalized flag manifolds the natural map H_*(BSU(n+1)) --> H_*(BG) is injective, where G denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if lambda is a Hamiltonian circle action that contracts in G = Ham(M,w) then there is an associated nonzero element in pi_3(G) that deloops to a nonzero element of H_4(BG). This result (as well as many others) extends to c-symplectic manifolds (M,a), ie, 2n-manifolds with a class a in H^2(M) such that a^n is nonzero. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

Abstract

In this paper, actions of finite groups on aspherical spaces are studied. A complete homotopy classification of free actions of finite groups on aspherical spaces is obtained, including some results in nonfree actions. For example, a relation between cohomologies of finite groups and the lattice structure of their subgroups is obtained by this method; this relation is formulated in terms of spectral sequences.

Polynomials for blending parametric curves in Lie groups are defined. Properties of these polynomials are proved. Blending parametric curves in Lie groups with these polynomials is considered. Then application of the proposed technique to construction of spline curves on smooth manifolds is presented. As an example, construction of spherical spline curves using the proposed approach is depicted.

In this paper, we introduce a way to define a free cyclic group action on a homotopy sphere and examine the stable parallelizability of its orbit spaces. J. Ewing et al [3] answered the stable parallelizability problem for the classical lens space, that is, the orbit space of the standard sphere under a linear cyclic group action. Let Wi, wZ9 . . . , n+i be positive rational numbers. A polynomial /(£i> £25 • • • 5 £n+i) is called a weighted homogeneous polynomial of type (wi, w^ . . . , z#B+i) if it can be expressed as a linear combination of monomials \2 y n+l Zl %2 « e 8 Zn+l

Classification of free actions of finite groups on the 3-torus

Let G be a solvable Lie group. Let $\chi$ be a character on G with values in $\mathbb R$. Let $\mathfrak B$ be a Banach space on which G is made to act by bounded operators, with the equality $\Vert gb\Vert=\chi(g)\Vert b\Vert$ satisfied for all $g\in G$ and $b\in\mathfrak B$. A criterion is given on the pair $(G,\chi)$ for the cohomology group $H^1(G,\mathfrak B)$ to vanish. Using this criterion, we classify up to Cr conjugacy all the volume-preserving locally free Cr actions of G on closed manifolds with dimension $\dim(G)+1$, for G belonging to a wide class of solvable Lie groups.

This book introduces a new context for global homotopy theory, i.e., equivariant homotopy theory with universal symmetries. Many important equivariant theories naturally exist not just for a particular group, but in a uniform way for all groups in a specific class. Prominent examples are equivariant stable homotopy, equivariant $K$-theory or equivariant bordism. Global equivariant homotopy theory studies such uniform phenomena, i.e., the adjective `global' refers to simultaneous and compatible actions of all compact Lie groups. We give a self-contained treatment of unstable and stable global homotopy theory, modeled by orthogonal spaces respectively orthogonal spectra under global equivalences. Specific topics include the global stable homotopy category, operations on equivariant homotopy groups, global model structures, and ultra-commutative multiplications. The book includes many explicit examples and detailed calculations.

In this paper, we study the regularized mean curvature flow starting from\ninvariant hypersurfaces in a Hilbert space equipped with an isometric almost\nfree Hilbert Lie group action whose orbits are minimal regularizable\nsubmanifolds, where "almost free" means that the stabilizers of the group\naction are finite. First we obtain the evolution equations for some geometric\nquantities along the regularized mean curvature flow. Next, by using the\nevolution equations, we prove a horizontally strongly convexity preservability\ntheorem for the regularized mean curvature flow. From this theorem, we derive\nthe strongly convexity preservability theorem for the mean curvature flow\nstarting from compact Riemannian suborbifolds in the orbit space (which is a\nRiemannian orbifold) of the Hilbert Lie group action.\n

Essential results about actions of compact Lie groups on connected manifolds are generalized to proper actions of arbitrary groups on connected cohomology manifolds. Slices are replaced by certain fiber bundle structures on orbit neighborhoods. The group dimension is shown to be effectively finite. The orbits of maximal dimension form a dense open connected subset. If some orbit has codimension at most $2$, then the group is effectively a Lie group.

Slice theorem and orbit type stratification in infinite dimensions

Mensch, streckh deine Vernunfft hieher, diese dinge zu begreiffen! Johann Keppler (1571–1630) There is nothing that nuclear spins will not do for you, as long as you treat them as human beings. Erwin Louis Hahn (1949) Ein maschinelles ‘agencement’ ist den Schichten zugewandt, reinen Intensitäten, die sie zirkulieren läßt um die Selektion der ‘Konsistenzebene’ zu sichern, und der sich die Subjekte zuordnen, welchen sie einen Namen nur als Spur einer Intensität läßt. Gilles Deleuze und Félix Guattari (1988) The sense we make of the world is governed by our conceptual means. More specifically, sense is made of nature by projecting a conceptual structure onto observed events. This is accomplished by means of a semantic filter, which transforms raw data into semantically significant events. George L. Farre (1997) Phase coherent wavelets form a unified basis of the multichannel reconstruction analysis–synthesis filter bank, implemented by high-resolution synthetic aperture radar (SAR) imaging and clinical MRI. The construction of unitary bank filters is performed by the Kepplerian stratification technique. It allows for the cross-sectional quadrature filtering of phase histories in contiguous local frequency encoding subband multichannels relative to the rotating coordinate frame of quadrature reference. The Kepplerian quadrature detection strategy, which is based on the unitary filter bank construction takes place in symplectic affine planes of incidence immersed into the three-dimensional compact super-encoding projective space. The planes are implemented in the quadrature format of a phase-splitting network of Fourier analysis of the stratified Heisenberg nilpotent Lie group G. The action of the basic Lie group G of quantum physics under its sub-Riemannian geometry admits a matrix coordinatization by transvections. In terms of projective geometry, the longitudinal dilations act as a spectral transform on the transvections of the G