Spaces of Maps into Classifying Spaces for Equivariant Crossed Complexes, II: The General Topological Group Case

The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces.     

The Reduction of Equivariant Dynamics to the Orbit Space for Compact Group Actions

Symmetry introduces degeneracies in dynamical systems, as well as in bifurcation problems. An “obvious” idea in order to remove these degeneracies is to project the dynamics onto the quotient space obtained by identifying points in phase space which lie in the same group orbits (the so-called orbit space). Unfortunately, several difficulties arise when one tries to implement this idea. First, the orbit space is not, in general a manifold. Second, how does one explicitely realize the orbit space, and how does one compute and analyze the projected dynamics? In this paper I will describe the methods which have been developped in order to answer these questions, and I will show on three examples how they apply. We shall see that, although not always suitable to treat equivariant dynamics, these methods sometimes lead to insightful reductions.     

On the Least Number of Fixed Points of an Equivariant Map

The problem on the least number of fixed points of an equivariant map of a compact polyhedron on which a finite group acts is considered. For such a map, the least number of fixed points and the least number of fixed orbits are estimated in terms of invariants of the type of Nielsen numbers. The estimates obtained are sharp. The results are similar to those of P. Wong, but their assumptions are essentially weaker. Some notations are refined. The proofs are constructive.     

The group of self homotopy equivalences of some localized aspherical complexes

By studying the group of self homotopy equivalences of the localization (at a prime p and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent, ?^m _#(X_p ) is in general different from ?^m _#(X)p. That is the case even when X = K (G, 1) is a finite complex and/or G satisfies extra finiteness or nilpotency conditions, for instance, when G is finite or virtually nilpotent. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Split Exact Sequence of Equivariant K-groups of Virtually Nilpotentt Groups

Let be an extension of a torsion free nilpotent group by a finite group G, of odd order. Then admits a cocompact proper action on ⁿ . This action determines an action of G on a nil-manifold by isometries. In this paper, the equivariant K-theory of the finite group action is studied and it is shown that the forget control map is a split monomorphism.     

Equivariant Extensions of Categorical Groups

If is a group, then the category of -graded categorical groups is equivalent to the category of categorical groups supplied with a coherent left-action from . In this paper we use this equivalence and the homotopy classification of graded categorical groups and their homomorphisms to develop a theory of extensions of categorical groups when a fixed group of operators is acting. For this kind of extensions we show a suitable Schreiers theory and a precise theorem of classification, including obstruction theory, which generalizes both known results when the group of operators is trivial (categorical group extensions theory) or when the involved categorical groups are discrete (equivariant group extensions theory).Mathematics Subject Classifications (2000) 18D10, 18B40, 20J05, 20J06.Partially supported by MTM2004-01060.     

含两组状态变量的等变分歧问题在左右等价群下的开折

基于奇点理论中光滑映射芽的左右等价关系, 讨论多参数等变分歧问题关于左右等价的开折．将这种等变分歧问题的状态变量分为两组,其中属于同一组的诸状态变量可以独立地变化,而属于另一组的诸状态变量在变化过程中依赖于前一组中的诸状态变量．应用光滑映射芽开折理论中的相关方法和技巧,得到了一个含两组状态变量的多参数等变分歧问题的开折是通用开折的充要条件．     

Lie group structures on symmetry groups of principal bundles

In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M. This is done by first considering Lie group structures on the group of vertical bundle automorphisms Gau(P). Then the full automorphism group Aut(P) is considered as an extension of the open subgroup DiffP(M) of diffeomorphisms of M preserving the equivalence class of P under pull-backs, by the gauge group Gau(P). We derive explicit conditions for the extensions of these Lie group structures, show the smoothness of some natural actions and relate our results to affine Kac-Moody algebras and groups.     

Equivariant resolution of points of indeterminacy

We prove an equivariant form of Hironaka's theorem on elimination of points of indeterminacy. Our argument uses canonical resolution of singularities and an extended version of Sumihiro's equivariant Chow lemma.

     

Fundamental groups of algebraic fiber spaces

Let be a dominant morphism, where E and B are smooth irreducible complex quasi-projective varieties. Suppose that the general fiber of $f$ is connected. We present an algebro-geometric condition under which the boundary homomorphism is well-defined, and makes the sequence exact. As an application, we calculate the fundamental group of the complement to the dual hypersurface of a smooth projective curve. Received: October 3, 2001     

Euler-Poincaré reduction in principal fibre bundles and the problem of Lagrange

We compare Euler-Poincaré reduction in principal fibre bundles, as a constrained variational problem on the connections of this fibre bundle and constraint defined by the vanishing of the curvature of the connection, with the corresponding problem of Lagrange. Under certain cohomological condition we prove the equality of the sets of critical sections of both problems with the one obtained by application of the Lagrange multiplier rule. We compute the corresponding Cartan form and characterise critical sections as the set of holonomic solutions of the Cartan equation and, in particular, under a certain regularity condition for the problem, we prove the holonomy of any solution of this equation.     

紧黎曼对称空间到Grassmann流形的等变等距极小浸入

在本文中，我们利用李群及其表示理论作为主要工具，　讨论了紧黎曼对称空间到Ｇｒａｓｓｍａｎｎ　流形的等变等距极小浸入问题．     

Equivariant embeddings of principal Z-bundles into complex vector bundles

Let π: E→X be a principal $\text{Z}$_n-bundle and p:V→X an m-dimensional complex vector bundle over, say, a connected CW-complex X. An equivariant embedding of π into p is an embedding h:E → V commuting with projections such that h(e · z)=zh(e) for all eεE and $\text{zε}\text{Z}{}_{\mathrm{n}}\text{?S}{}^1\text{?}\text{Z}$. We compute the primary obstruction $\text{cεH}{}^{\mathrm{2m}}\text{(X;}\text{Z}\text{)}$ to embedding π equivariantly into p. If dim X?2m, then c=0 if and only if π admits an equivariant embedding into p. If dim X>2m and π embeds equivariantly into p, then c=0. Other embedding criteria exist in case p is the trivial m-plane bundle ε^m. We use these criteria for a discussion of the classification of the equivalence classes of principal $\text{Z}$-bundles that admit equivariant embeddings into ε^m. Finally, we offer an example of a principal $\text{Z}$-bundle that admit an ordinary but not an equivariant embedding into ε¹.     

Disconnected Equivariant Rational Homotopy Theory

We present a variant of the disconnected equivariant rational homotopy theory to complete the result shown in [8]. For a finite group G let O(G) be the category of its canonical orbits. We prove that the category O(G)-DGA _Q of O(G)S-complete differential graded algebras over the rationals is a closed model category, where S runs over all O(G)-sets. Then, by means of the equivariant KS-minimal models, we show that the homotopy category of O(G)-DGA _Q is equivalent to the rational homotopy category of G-nilpotent disconnected simplicial sets provided G is a finite Hamiltonian group.     

Rank of the fundamental group of any component of a function space

We compute the rank of the fundamental group of any connected component of the space for and connected, nilpotent CW complexes of finite type with finite. For the component corresponding to a general homotopy class , we give a formula directly computable from the Sullivan model for . For the component of the constant map, our formula retrieves a known expression for the rank in terms of classical invariants of and . When both and are rationally elliptic spaces with positive Euler characteristic, we use our formula to determine the rank of the fundamental group of any component of explicitly in terms of the homomorphism induced by on rational cohomology.

     

Homotopy Classification of Categorical Torsors

The long-known results of Schreier on group extensions are here raised to a categorical level by giving a factor set theory for torsors under a categorical group (G,) over a small category . We show a natural bijection between the set of equivalence classes of such torsors and [B({}),B(G,)], the set of homotopy classes of continuous maps between the corresponding classifying spaces. These results are applied to algebraically interpret the set of homotopy classes of maps from a CW-complex X to a path-connected CW-complex Y with _i (Y)=0 for all i1,2.     

Configurations, braids, and homotopy groups

The main results of this article are certain connections between braid groups and the homotopy groups of the -sphere. The connections are given in terms of Brunnian braids over the disk and over the -sphere. The techniques arise from the natural structure of simplicial and -structures on fundamental groups of configuration spaces.

     

Counterexample to a problem of Geoghegan-West

Let be a Banach space and its general linear group. Let denote the operator norm and ``' the pointwise convergence topology on . Is the identity map a homotopy equivalence? The answer is negative. One of the possible counterexamples is a well-known James space ---the ``space of counterexamples in Banach spaces theory'.

     

Rational Mackey functors for compact lie groups I

A Mackey functor M is a structure analogous to the representationring functor H R(H) encoding good formal behaviour under inductionand restriction. More explicitly, M associates an abelian groupM(H) to each closed subgroup H of a fixed compact Lie groupG, and to each inclusion K H it associates a restriction map and an induction map . This paper gives an analysis of thecategory of Mackey functors M whose values are rational vectorspaces: such a Mackey functor may be specified by giving a suitablycontinuous family consisting of a Q ₀(W_G(H))-module V(H) foreach closed subgroup H with restriction maps V(K) V(K) wheneverK is normal in K and K/K is a torus (a ‘continuous Weyl-toralmodule’). We show that the category of rational Mackeyfunctors is equivalent to the category of rational continuousWeyl-toral modules. In Part II this will be used to give analgebraic analysis of the category of rational Mackey functors,showing in particular that it has homological dimension equalto the rank of the group. 1991 Mathematics Subject Classification:19A22, 20C99, 22E15, 55N91, 55P42, 55P91.     

Sur les feuilletages algébriques de Rolle

L'objet de ce travail est l'étude des feuilletages algébriques de Rolle dans . On montre que leur restriction au complémentaire d'un nombre fini de feuilles possède une structure de produit. On précise aussi la topologie de certaines de leurs feuilles. Received: June 22, 1996