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.      

Free <Emphasis Type="Italic">G</Emphasis>-spaces and maximal equivariant compactifications

For a compact Lie group G we prove that every free (resp., semifree) G-space admits a based-free (resp., semifree) G-compactification. Examples of finite- and infinite-dimensional G-spaces are presented that do not admit a free or based-free G-compactification. We give several characterizations of the maximal G-compactification β_GX that are further applied to prove the formula (β_GX)/H=β_G/H(X/H) for arbitrary closed normal subgroup H⊂G. Mathematics Subject Classification (2000) 54H15, 54D35     

On Well-posed Mutually Nearest and Mutually Furthest Point Problems in Banach Spaces

Let G be a non-empty closed(resp．bounded closed)boundedly relatively weakly compact subset in a strictly convex Kadec Banach space X．Let K(X)denote the space of all non-empty compact convex subsets of X endowed with the Hausdorff distance．Moreover,let KG(X)denote the closure of the set {A∈K(x)：A∩G=0}．We prove that the set of all A∈KG(X)(resp．A∈K(X)),such that the minimization (resp．maximization)problem min(A,G)(resp．max(A,G))is well posed,contains a dense Gδ-subset of KG(X)(resp．K(X))．thus extending the recent results due to Blasi,Myjak and Papini and Li．     

On p-rank representations

The p-rank of an algebraic curve X over an algebraically closed field k of characteristic p>0 is the dimension of the vector space H¹(X_et,F_p). We study the representations of finite subgroups GAut(X) induced on H¹(X_et,F_p)k, and obtain two main results.First, the sum of the nonprojective direct summands of the representation, i.e., its core, is determined explicitly by local data given by the fixed point structure of the group acting on the curve. As a corollary, we derive a congruence formula for the p-rank.Secondly, the multiplicities of the projective direct summands of quotient curves, i.e., their Borne invariants, are calculated in terms of the Borne invariants of the original curve and ramification data. In particular, this is a generalization of both Nakajima's equivariant Deuring–Shafarevich formula and a previous result of Borne in the case of free actions.     

On equivariant Kasparov theory and Spanier-Whitehead duality

Suppose thatG is a second countable compact Lie group and thatA andB are commutativeG-C*-algebras. Then the Kasparov groupKK _* ^G (A, B) is a bifunctor onG-spaces. It is computed here in terms of equivariant stable homotopy theory. This result is a consequence of a more general study of equivariant Spanier-Whitehead duality and uses in an essential way the extension of the Kasparov machinery to the setting of -G-C*-algebras. As a consequence, we show that if (X, x ₀) is a based separable compact metricG-ENR (such as a smooth compactG-manifold) and (Y, y ₀) is a based countableG-CW-complex then there is a natural isomorphism

Equivariant K-Theory of Compact Connected Lie Groups Dedicated to Daniel Quillen on the Occasion of his Sixtieth Birthday

We compute the equivariant K-theory K _G ^* (G)for a compact connected Lie group Gsuch that ₁ (G)is torsion free (where Gacts on itself by conjugation). We prove that K _G ^* (G)is isomorphic to the algebra of Grothendieck differentials on the representation ring. We also study a special example of a compact connected Lie group Gwith ₁ (G)torsion, namely PSU(3), and compute the corresponding equivariant K-theory.     

Existence of finitely dominatedCW-complexes withG 1(X) = π1(X) and non-vanishing finiteness obstruction

We show for a finite abelian groupG and any element in the image of the Swan homomorphism sw: that it can be realized as the finiteness obstruction of a finitely dominated connectedCW-complexX with fundamental group π₁(X) =G such that π₁(X) is equal to the subgroupG ₁(X) defined by Gottlieb. This is motivated by the observation that anyH-spaceX satisfies π₁(X) =G ₁(X) and still the problem is open whether any finitely dominatedH-space is up to homotopy finite.     

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.     

Equivariant sheaf cohomdlogy

In this paper we study Grothendieck's equivariant sheaf cohomology H(X,G;G) for non-discrete topological groups G and G-sheavesG on a G-Space X. For compact groups and locally compact, totally disconnected groups we obtain detailed results relating H(X,G;-) to H(X;-)^G and H(X/G;-). Furthermore we point out the connection between H(X,G;-) and Borel's equivariant cohomology H_G(X;-).     

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.     

Shape Morphisms to Transitive G-Spaces

The following problem plays an important role in shape theory: find conditions that guarantee that a shape morphism F:X Y of a topological space X to a topological space Y is generated by a continuous mapping f:X Y. In the present paper, we study this problem in equivariant shape theory and give a solution for shape-equivariant morphisms to transitive G-spaces, where G is a compact group with countable base. As a corollary, we prove a sufficient condition for equivariant shapes of a G-space X to be equal to the group G itself. We also prove some statements concerning equivariant bundles that play the key role in the proof of the main results and are of interest on their own.     

Stable equivariant abelianization, its properties, and applications

Let G be a finite group. For a based G-space X and a Mackey functor M, a topological Mackey functor is constructed, which will be called the stable equivariant abelianization of X with coefficients in M. When X is a based G-CW complex, is shown to be an infinite loop space in the sense of G-spaces. This gives a version of the RO(G)-graded equivariant Dold-Thom theorem. Applying a variant of Elmendorf's construction, we get a model for the Eilenberg-Mac Lane spectrum HM. The proof uses a structure theorem for Mackey functors and our previous results.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

The Fractional Chromatic Number Of The Categorical Product Of Graphs

We prove that the identity

On Finite Groups with a Given Number of Centralizers

For a finite group G, let Cent(G) denote the set of centralizers of single elements of G and #Cent(G) = |Cent(G)|. G is called an n-centralizer group if #Cent(G) = n, and a primitive n-centralizer group if #Cent(G) = #Cent(G/Z(G)) = n. In this paper, we compute #Cent(G) for some finite groups G and prove that, for any positive integer n 2, 3, there exists a finite group G with #Cent(G) = n, which is a question raised by Belcastro and Sherman [2]. We investigate the structure of finite groups G with #Cent(G) = 6 and prove that, if G is a primitive 6-centralizer group, then G/Z(G) A₄, the alternating group on four letters. Also, we prove that, if G/Z(G) A₄, then #Cent(G) = 6 or 8, and construct a group G with G/Z(G) A₄ and #Cent(G) = 8.This research was in part supported by a grant from IPM.2000 Mathematics Subject Classification: 20D99, 20E07     

On the negative cyclic homology of <Emphasis Type="Italic">shc</Emphasis>-algebras

Let be a field of characteristic and S ¹ the unit circle. We prove that the shc-structure on a cochain algebra (A,d _A ) induces an associative product on the negative cyclic homology HC _*⁻ A. When the cochain algebra (A,d _A ) is the algebra of normalized cochains of the simply connected topological space X with coefficients in , then HC _*⁻ A is isomorphic as a graded algebra to the S ¹-equivariant cohomology algebra of LX, the free loop space of X. We use the notion of shc-formality introduced in Topology 41, 85–106 (2002) to compute the S ¹-equivariant cohomology algebras of the free loop space of the complex projective space when n + 1 = 0 [p] and of the even spheres S ²ⁿ when p = 2.      

On the <Emphasis Type="Italic">G</Emphasis>-compactifications of the rational numbers

We show that for any sufficiently homogeneous metrizable compactum X there is a Polish group G acting continuously on the space of rational numbers such that X is its unique G-compactification. This allows us to answer Problem 995 in the ‘Open Problems in Topology II’ book in the negative: there is a one-dimensional Polish group G acting transitively on for which the Hilbert cube is its unique G-completion.      

On representation theorem of <Emphasis Type="Italic">G</Emphasis>-expectations and paths of <Emphasis Type="Italic">G</Emphasis>-Brownian motion

We give a very simple and elementary proof of the existence of a weakly compact family of probability measures {Pθ ： θ∈θ} representing an important sublinear expectation- G-expectation E[·]. We also give a concrete approximation of a bounded continuous function X（ω） by an increasing sequence of cylinder functions Lip（Ω） in order to prove that Cb（Ω） belongs to the completion of Lip（Ω） under the natural norm E[｜·｜].     

The Atiyah-Segal completion theorem for C*-algebras

We generalize the Atiyah-Segal completion theorem to C ^*-algebras as follows. Let A be a C ^*-algebra with a continuous action of the compact Lie group G. If K _* ^G (A) is finitely generated as an R(G)-module, or under other suitable restrictions, then the I(G)-adic completion K _* ^G (A) is isomorphic to RK _*([A C(EG)]^G), where RK _* is representable K-theory for - C ^*-algebras and EG is a classifying space for G. As a corollary, we show that if and are homotopic actions of G, and if K _*(C ^* (G,A,)) and K _*(C ^* (G,A,)) are finitely generated, then K _*(C ^*(G,A,))K_*(C ^* (G,A,)). We give examples to show that this isomorphism fails without the completions. However, we prove that this isomorphism does hold without the completions if the homotopy is required to be norm continuous.This work was partially supported by an NSF Graduate Fellowship and by an NSF Postdoctoral Fellowship.