On the homotopy fixed point problem for free loop spaces and other function complexes

Let G be a finite group, let X and Y be finite G-complexes, and suppose that for each K G, Y ^K is dim(X ^K)-connected and simple. G acts on the function complex F(X, Y) by conjugation of maps. We give a complete analysis of the homotopy fixed point set of the space F(X, Y). As a corollary, we are able to analyze at a prime p, the homotopy fixed point set of the circle action on X, where X denotes the free loop space of X, and X is a simply connected finite complex.Supported in part by NSF DMS 86-02430.To A. Grothendieck on the occasion of his sixtieth birthday     

Transitive Actions on Lorentz Manifolds with Noncompact Stabilizer

If a topological group G acts on a topological space X, then we say that the action is orbit nonproper provided that, for some xX, the orbit map ggx:GX is nonproper. We consider the problem of classifying the connected, simply connected real Lie groups G admitting a locally faithful, orbit nonproper, isometric action on a connected Lorentz manifold. In an earlier paper, we found three collections of groups such that G admits such an action iff G is in one of the three collections. In another paper, we effectively described the first collection. In this paper, we show that the second collection contains a small, effectively described collection of groups, and, aside from those, it is contained in the union of the first and third collections. Finally, in a third paper, we effectively describe the third collection, thus solving the stated problem.     

Surjectivity for Hamiltonian loop group spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,), and assume that the moment map :XL ^* is proper. We consider the function ||²:X, and use a version of Morse theory to show that the inclusion map j:^-1(0)X induces a surjection j ^*:H _G ^*(X)H _G ^*(^-1(0)), in analogy with Kirwans surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.     

The involution on the equivariant Whitehead group

For a finite group G we define an involution on the equivariant Whitehead group given by reversing the direction of an equivariant h-cobordism. It turns out that the involution is not compatible with the splitting of the equivariant Whitehead group into a direct sum of algebraic Whitehead groups, certain correction terms involving the transfer maps of the normal sphere bundles of the various fixed point sets come in. However, if the group has odd order, these transfer maps all vanish. We prove a duality formula for a G-homotopy equivalence (f f): (M; M) (N, N) relating the equivariant Whitehead torsion of f and (f,f).     

Separately <Emphasis Type="Italic">F</Emphasis>σ-measurable functions are close to functions of the first Baire class

We prove that a Borel separately F -measurable function f: X×YR on the product of Polish spaces is a function of the first Baire class on the complement X×Y\M of a certain projectively meager set MX×Y.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 4, pp. 573–576, April, 2004.     

Equivariant Surgery Theory: Deleting-Inserting Theorems of Fixed Point Manifolds on Spheres and Disks

The paper gives a tool to delete and insert fixed point manifolds for smooth actions of finite Oliver groups on spheres and disks. A similar result was already given in a joint article with E. Laitinen and K. Pawaowski for those of finite nonsolvable groups on spheres. It is useful in classifying smooth actions on spheres from the view point of fixed point data. The methods employed in the present paper are equivariant surgery and equivariant connected sum associated with elements in the Burnside ring. The idea of killing surgery obstructions is as follows: Let G be a finite group not of prime power order, C a contractible, finiteG CW complex, and an element in a K theoretic group arising as an obstruction class of geometric object f. It often holds that (1-[C])^m becomes trivial for large integers m where [C] is the element represented byC in the Burnside ring (G). One expects that the algebraic object (1 - [C])^m is realizable as the obstruction class of G connected sum of f's related to (1 - [C])^m. Since it is true for the case here, we can kill the obstruction by taking G connected sum off's     

HigherG-signatures for Lipschitz manifolds

Using the Teleman signature operator and Kasparov'sKK-theory, we prove a strong De Rham theorem and a higherG-signature theorem for Lipschitz manifolds. These give in particular a substitute for the usualG-signature theorem that applies to certain nonsmooth actions on topological manifolds. Then we present a number of applications. Among the most striking are a proof that nonlinear similarities preserve renormalized Atiyah-Bott numbers, and a proof that under suitable gap, local flatness, and simple connectivity hypotheses, a compact (topological)G-manifoldM is determined up to finite ambiguity by its isovariant homotopy type and by the classes of the equivariant signature operators on all the fixed sets . These could also be proved using joint work of Cappell, Shaneson, and the second author on topological characteristic classes.Partially supported by NSF Grants DMS-87-00551 and DMS-90-02642 (J.R.) and by NSF Grants, a Sloan Foundation Fellowship, and a Presidential Young Investigator award (S.W.).     

Radical Classes of Lattice-Ordered Groups vs. Classes of Compact Spaces

For a given class T of compact Hausdorff spaces, let Y(T) denote the class of -groups G such that for each gG, the Yosida space Y(g) of g belongs to T. Conversely, if R is a class of ;-groups, then T(R) stands for the class of all spaces which are homeomorphic to a Y(g) for some gGR. The correspondences TY(T) and RT(R) are examined with regard to several closure properties of classes. Several sections are devoted to radical classes of -groups whose Yosida spaces are zero-dimensional. There is a thorough discussion of hyper-projectable -groups, followed by presentations on Y(e.d.), where e.d. denotes the class of compact extremally disconnected spaces, and, for each regular uncountable cardinal , the class Y(disc), where disc stands for the class of all compact -disconnected spaces. Sample results follow. Every strongly projectable -group lies in Y(e.d.). The -group G lies in Y(e.d.) if and only if for each gG Y(g) is zero-dimensional and the Boolean algebra of components of g, comp(g), is complete. Corresponding results hold for Y(disc). Finally, there is a discussion of Y(F), with F standing for the class of compact F-spaces. It is shown that an Archimedean -group G is in Y(F) if and only if, for each pair of disjoint countably generated polars P and Q, G=P +Q .     

Lebesgue–Cech Dimensionality and Baire Classification of Vector-Valued Separately Continuous Mappings

For a metrizable space X with finite Lebesgue–Cech dimensionality, a topological space Y, and a topological vector space Z, we consider mappings f: X × Y Z continuous in the first variable and belonging to the Baire class in the second variable for all values of the first variable from a certain set everywhere dense in X. We prove that every mapping of this type belongs to the Baire class + 1.     

Isolated points in duals of certain locally compact groups

Summary LetG be a separable locally compact group with dual space. consists of all equivalence classes of irreducible unitary representations ofG, and is endowed with the Fell-topology. We study the topological properties in of the square-integrable representations ofG. [ is square-integrable provided there is a coordinate functiong((g)v, v),gG, for which is inL ²(G) w.r.t. left Haar measure onG.]SupposeG contains an open normal subgroupN of the formeKN ⁿ e whereK is compact. (All groups with a compact invariant neighborhood of the identity, [IN] groups, satisfy this condition.) In this case we show that if is square-integrable then {} is an open point of.Finally, our techniques are used to prove this result for arbitrary (non connected) nilpotent Lie groups.     

On the positive definiteness of n ↦ epnα for α close to 2

For >2, let Q ⁺() be the infimum of those q>0 for which the function n e^pn is positive definite on N ₀ for every pq. We shall prove that Q ⁺()0 as 2.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Bott-Chern currents,excess normal bundles and the Chern character

LetY andY be two complex submanifolds of a complex manifoldX, and assume thatU=YY is also a submanifold ofX. Let Ñ be the excess normal bundle toU inX. We attach certain Bott-Chern currents to holomorphic Hermitian vector bundles onY, Y and to resolutions of the corresponding sheaves by holomorphic Hermitian complexes overX. We show that these currens verify natural functorial properties. Our results extend earlier results of Bismut-Gillet-Soulé to the case where Ñ is nonzero.     

On σ-discrete borel mappings via quasi-metrics

Let X and Y be metrizable spaces. We show that, for a mapping f : X Y, there exists a quasi-metric X inducing the topology of X such that f regarded as a mapping from (X, max{, ^–1}) to Y is continuous if and only if f in the original topology of X is a -discrete map of Borel class 1. Further, we prove that, for every -discrete mapping f: X Y of Borel class + 1, there exists a compatible quasi-metric on X such that f : (X, max{, ^–1}) Y is of Borel class . We also investigate a more general situation when the range of the mapping under consideration is not necessarily metrizable. In passing, we obtain some results related to the behaviour of absolutely Borel sets and absolutely analytic spaces with respect to compatible quasi-metrics.     

Estimation of the Multivariate Normal Precision Matrix under the Entropy Loss

Let X ₁, , X _n (n > p) be a random sample from multivariate normal distribution N _p (, ), where R ^p and is a positive definite matrix, both and being unknown. We consider the problem of estimating the precision matrix ^–1. In this paper it is shown that for the entropy loss, the best lower-triangular affine equivariant minimax estimator of ^–1 is inadmissible and an improved estimator is explicitly constructed. Note that our improved estimator is obtained from the class of lower-triangular scale equivariant estimators.     

On the domain of attraction of an operator between supremum and sum

Summary Between the operations which produce partial maxima and partial sums of a sequenceY ₁,Y ₂, ..., lies the inductive operation:X _n =X _n-1(X _n-1+Y _n ),n1, for 0<<1. If theY _n are independent random variables with common distributionF, we show that the limiting behavior of normed sequences formed from {X _n ,n1}, is, for 0<<1, parallel to the extreme value case =0. ForFD() we give a full proof of the convergence, whereas forFD()D(), we only succeeded in proving tightness of the involved sequence. The processX _n is interesting for some applied probability models.     

A Result on Localization of Equilibria

In this paper, we deal with a general equilibrium problem where a bimap F: A×B X×Y2 ^Z is involved. This problem contains the scalar equilibrium problem as a very special case. The general equilibrium is considered via the properties of the map G: B2 ^A naturally associated to the problem. The main result shows that, to have solutions on every convex subsets B ₁ of B, localized via a map T: B2 ^A , a necessary and sufficient condition is the KKM property of the map G with respect to T. The assumptions require that T satisfies a regularity condition with respect to G, and it is proved that this condition is quite sharp, providing a suitable counterexample.     

The Higher K-Theory of Real Curves

If X is a smooth curve defined over the real numbers , we show that K _n (X) is the sum of a divisible group and a finite elementary Abelian 2-group when n 2. We determine the torsion subgroup of K _n (X), which is a finite sum of copies of and 2, only depending on the topological invariants of X() and X(), and show that (for n 2) these torsion subgroups are periodic of order 8.     

p-Group actions on homology spheres

When an arbitraryp-groupG acts on a _n -homologyn-sphereX, it is proved here that the dimension functionn:S(G)(S(G) is the set of subgroups ofG), defined byn(H)=dimX ^H, (dim here is cohomological dimension) is realised by a real representation ofG, and that there is an equivariant map fromX to the sphere of this representation. A converse is also established.     

Ends of maps,II

Versions of the finiteness obstruction and simple homotopy theory within overX are developed. This provides a setting for obstructions to the map analogs of the end ands-cobordism theorems for manifolds. These are applied to study equivariant mapping cylinder neighborhoods in topological group actions, triangulations of locally triangulable spaces, and block bundle structures on approximate fibrations.