On based-free actions of compact lie groups on the hilbert cube

It is proved that a based-free action α of a given compact Lie groupG on the Hilbert cubeQ is equivalent to the standard based-free action σ if and only if the orbit spaceQ ₀/α of the free partQ ₀=Q* is aQ-manifold having the proper homotopy type of the orbit spaceQ ₀/σ. The existence of an equivariant retraction (Q ₀, σ)→(Q ₀, α) is established. It is proved that for any TikhonovG-spaceX the family of all equivariant mapsX→ conG separates the points and the closed sets inX. Translated fromMatematicheskie Zametki, Vol. 65, No. 2, pp. 163–174, February, 1999.     

A combinatorial principle and endomorphism rings I: Onp-groups

Two lines of research are involved here. One is a combinatorial principle, proved in ZFC for many cardinals (e.g., any λ = λ^ℵ ₀) enabling us to prove things which have been proven using the diamond or for strong limit cardinals of uncountable cofinality. The other direction is building abelian groups with few endomorphisms and/or prescribed rings of endomorphisms. We prove that for a ringR, whose additive group is thep-adic completion of a freep-adic module,R is isomorphic to the endomorphism ring of some separable abelianp-groupG divided by the ideal of small endomorphisms, withG of power λ for any λ = λ^ℵ ₀≧|R|. Dedicated to the memory of Abraham Robinson on the tenth anniversary of his death The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research.     

Singular del Pezzo surfaces that are equivariant compactifications

We determine which singular del Pezzo surfaces are equivariant compactifications of \mathbbG_\texta² \mathbb{G}_{\text{a}}^2 , to assist with proofs of Manin’s conjecture for such surfaces. Additionally, we give an example of a singular quartic del Pezzo surface that is an equivariant compactification of \mathbbG_\texta {\mathbb{G}_{\text{a}}} ⋊ \mathbbG_\textm {\mathbb{G}_{\text{m}}} . Bibliography: 32 titles.     

Joint ergodicity and mixing

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ₁σ₂,...,σ, ofG, the sequence ((1/N)Σ _n=0 ^N−1 σ ₁ ⁿ f ¹·σ ₂ ⁿ f ²· ··· · σ _s ⁿ f ^sconverges inL ²(G) for everyf ₁,f ₂,…,f _s∈L ^∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π _i=1 ^s ∫ _G f ₁ d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.     

On shadowing property for inverse limit spaces

We study here the G-shadowing property of the shift map σ on the inverse limit space X _f, generated by an equivariant self-map f on a metric G-space X.      

Antiperfect morse stratification

For an equivariant Morse stratification that contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincaré series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal G-bundle over a connected, closed, nonorientable surface Σ is locally equivariantly \mathbbQ{\mathbb{Q}}-antiperfect when G = U(2), SU(2), U(3), SU(3); we propose that the Morse stratification is actually equivariantly \mathbbQ{\mathbb{Q}}-antiperfect in these cases. Our proposal yields formulas of Poincaré series P_t^G(Hom(p₁(S),G);\mathbbQ){P_t^G({\rm Hom}(\pi_1(\Sigma),G);\mathbb{Q})} when G = U(2), SU(2), U(3), SU(3). Our U(2), SU(2) formulas agree with formulas proved by T. Baird, who also verified our conjectural U(3) formula.     

Gersten conjecture for equivariant K-theory and applications

For a reductive group scheme G over a regular semi-local ring A, we prove the Gersten conjecture for the equivariant K-theory. As a consequence, we show that if F is the field of fractions of A, then K^G₀(A) @ K^G₀(F){K^G_0(A) \cong K^G_0(F)}, generalizing the analogous result for a dvr by Serre (Inst Hautes études Sci Publ Math 34:37–52, 1968). We also show the rigidity for the K-theory with finite coefficients of a Henselian local ring in the equivariant setting. We use this rigidity theorem to compute the equivariant K-theory of algebraically closed fields.     

The relation between the Baum-Connes Conjecture and the Trace Conjecture

We prove a version of the L ²-index Theorem of Atiyah, which uses the universal center-valued trace instead of the standard trace. We construct for G-equivariant K-homology an equivariant Chern character, which is an isomorphism and lives over the ring ℤ⊂λ ^G ⊂ℚ obtained from the integers by inverting the orders of all finite subgroups of G. We use these two results to show that the Baum-Connes Conjecture implies the modified Trace Conjecture, which says that the image of the standard trace K ₀(C ^* _r (G))→ℝ takes values in λ ^G . The original Trace Conjecture predicted that its image lies in the additive subgroup of ℝ generated by the inverses of all the orders of the finite subgroups of G, and has been disproved by Roy [15]. Oblatum 10-IV-2001 & 18-X-2001?Published online: 15 April 2002     

The degree of maps of free G-manifolds and the average

Suppose that G is a compact Lie group, M and N are orientable, free G-manifolds and f : M → N is an equivariant map. We show that the degree of f satisfies a formula involving data given by the classifying maps of the orbit spaces M/G and N/G. In particular, if the generator of the top dimensional cohomology of M/G with integer coefficients is in the image of the cohomology map induced by the classifying map for M, then the degree is one. The condition that the map be equivariant can be relaxed: it is enough to require that it be “nearly equivariant”, up to a positive constant. We will also discuss the G-average construction and show that the requirement that the map be equivariant can be replaced by a somewhat weaker condition involving the average of the map. These results are applied to maps into real, complex and quaternionic Stiefel manifolds. In particular, we show that a nearly equivariant map of a complex or quaternionic Stiefel manifold into itself has degree one. Dedicated to the memory of Jean Leray     

Equivariant fixed-point indices of iterated maps

We describe an equivariant version (for actions of a finite group G) of Dold’s index theory, [10], for iterated maps. Equivariant Dold indices are defined, in general, for a G-map U → X defined on an open G-subset of a G-ANR X (and satisfying a suitable compactness condition). A local index for isolated fixed-points is introduced, and the theorem of Shub and Sullivan on the vanishing of all but finitely many Dold indices for a continuously differentiable map is extended to the equivariant case. Homotopy Dold indices, arising from the equivariant Reidemeister trace, are also considered.      

The non-commutative Weil algebra

For any compact Lie group G, together with an invariant inner product on its Lie algebra ?, we define the non-commutative Weil algebra ? _G as a tensor product of the universal enveloping algebra U(?) and the Clifford algebra Cl(?). Just like the usual Weil algebra W _G =S(?^*)⊗∧?^*, ? _G carries the structure of an acyclic, locally free G-differential algebra and can be used to define equivariant cohomology ℋ _G (B) for any G-differential algebra B. We construct an explicit isomorphism ?: W _G →? _G of the two Weil algebras as G-differential spaces, and prove that their multiplication maps are G-chain homotopic. This implies that the map in cohomology H _G (B)→ℋ _G (B) induced by ? is a ring isomorphism. For the trivial G-differential algebra B=ℝ, this reduces to the Duflo isomorphism S(?) ^G ≅U(?) ^G between the ring of invariant polynomials and the ring of Casimir elements. Oblatum 13-III-1999 & 27-V-1999 / Published online: 22 September 1999     

Equivariant frame fields on spheres with complementary equivariant complex structures

LetG be a finite group and letM be a unitary representation space ofG. We consider the existence problem of equivariant frame fields on the unit sphereS(M) whose orthogonal complements in the tangent bundleT(S(M)) admitG-equivariant complex structures. Under mild fixed point conditions we give a complete solution for this problem whenG is either ℤ/2ℤ or a finite group of odd order. This article was processed by the author using theLaT_EX style filecljourl from Springer-Verlag.     

Endomorphism algebras of transitive permutation modules for p-groups

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = End_kG(k_H ↑^G), showing that it is a split extension of a nilpotent ideal by the group algebra kN_G(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of k_H ↑^G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic. Received: 3 November 2008     

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002     

A generalization of de Bruijn graphs and classification of endomorphisms of Cuntz algebras by graph invariants

The classification problem of endomorphisms of the Cuntz algebra O_N\mathcal{O}_{N} is solved by using graph theory. We introduce permutative de Bruijn graphs as generalizations of de Bruijn graphs. Branching laws for a permutative endomorphism ρ of O_N\mathcal{O}_{N} are computed by using the permutative de Bruijn graph associated with ρ. According to this correspondence between endomorphisms and graphs, we classify permutative endomorphisms of O_N\mathcal{O}_{N} by graph invariants concretely.     

Real loci of based loop groups

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S ¹ on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat’s convexity theorem. Namely, the images of Ω(G) and Ω(G) ^τ (fixed point set of τ) under the T × S ¹ moment map on Ω(G) are equal. The space Ω(G) ^τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in \mathbbZ₂ {\mathbb{Z}_2} of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [HHP].     

Mackey systems of monoidal categories and G-spectra

We construct an equivariant infinite loop space machine denned on certain class of monoidal O _G -categories which have built-in Mackey structure. Applications include the equivariant infinite delooping of the classifying space BF(G) for stable spherical G-fibrations and also the construction of an infinite loop G-space E(X, G) with ₀ ^HE (X, G) naturally isomorphic to the equivariant Whitehead groups Wh _H (X) of given G-space X.Dedicated to Professor Shôrô Araki on his sixtieth birthday     

Commutative rings whose zero-divisor graph is a proper refinement of a star graph

A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G _c ^* be the subgraph of G induced on the vertex set V (G)\ {c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G _c ^* has at least two connected components. We prove that the diameter of the induced graph G _c ^* is two if Z(R)² ≠ {0}, Z(R)³ = {0} and G _c ^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)* \ {c} such that the ideal [N(α) ∩ N(β)]∪ {0} is generated by only one element of Z(R)*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs K _n with some end vertices adjacent to a single vertex of K _n .     

The chromatic number of the product of two ℵ1-chromatic graphs can be countable

We prove (in ZFC) that for every infinite cardinal ϰ there are two graphsG ₀,G ₁ with χ(G ₀)=χ(G ₁)=ϰ⁺ and χ(G ₀×G ₁)=ϰ. We also prove a result from the other direction. If χ(G ₀)≧≧ℵ₀ and χ(G ₁)=k<ω, then χ(G ₀×G ₁)=k.     

Mappings of group shifts

A group shift is a proper closed shift-invariant subgroup ofG ^ℤ ₂ whereG is a finite group. We consider a class of group shifts in whichG is a finite field and show that mixing is a necessary and sufficient condition on such a group shift for all codes from it into another group shift to be affine and all codes from another group shift into it to be affine. As a corollary, it will follow forG=ℤ _p that two mixing group shifts are topologically conjugate if and only if they are equal.