Fibrant extensions of free G-spaces

We show that any equivariant fibrant extension of a compact free G-space is also free. This result allows us to prove that the orbit space of any equivariant fibrant compact space E is also fibrant, provided that E has only one orbit type.     

Linearization of proper group actions

We prove that if G is a locally compact group acting properly (in the sense of R. Palais) on a space X that is metrizable by a G-invariant metric, then X can be embedded equivariantly into a normed linear G-space E endowed with a linear isometric G-action which is proper on the complement E?{0}. If, in addition, G is a Lie group then E?{0} is a G-equivariant absolute extensor. One can make this equivariant embedding even closed, but in this case the non-proper part of the linearizing G-space E may be an entire subspace instead of {0}.     

Equivariant RO(G)-graded bordism theories

Let G be a finite group. The RO(G)-graded bordism theories of Pulikowski [7] and Kosniowski [3] are studied. Representing equivariant Thom spectra are constructed, and the relevant transversality results proved. New methods for splitting away from the order of G are described, and behavior in the presence of a gap hypothesis is examined.     

On isometrically universal spaces, mappings, and actions of groups

In this paper we first consider some well-known classes of separable metric spaces which are isometrically ω-saturated (see [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559]) and, therefore, contain isometrically universal spaces. We put some problems concerning such spaces most of which are related with the properties of the isometrically universal Urysohn space. Furthermore, using the defined notions of isometrically universal mappings and G-spaces (which are analogies of the notion of isometrically universal spaces) we introduce the notions of an isometrically ω-saturated class of mappings and an isometrically ω-saturated class of G-spaces (in which there are “many” isometrically universal elements). We prove that all results of Sections 6.1 and 7.1 of [S.D. Iliadis, Universal Spaces and Mappings, North-Holland Mathematics Studies, vol. 198, Elsevier, 2005, xvi+559] can be reformulated for isometrically ω-saturated classes of spaces and G-spaces, respectively. In particular, we prove that if D and R are isometrically ω-saturated classes of spaces, then the class of all mappings with the domain in D and range in R is an isometrically ω-saturated class of mappings and, therefore, in this class there are isometrically universal elements. As a corollary of this result we have that since the class of all mappings is isometrically ω-saturated, in this class there are isometrically universal mappings. Similarly, if G is an arbitrary separable metric group and P is an isometrically ω-saturated class of spaces, then the class of all G-spaces (X,F), where X is an element of P, is an isometrically ω-saturated class of G-spaces and, therefore, in this class there are isometrically universal elements. In particular, for any separable metric group G, in the class of all G-spaces there are isometrically universal G-spaces. We also pose some problems concerning isometrically universal mappings and G-spaces some of which concern the Urysohn space.     

Universal G-spaces for proper actions of locally compact groups

We establish the existence of universal G-spaces for proper actions of locally compact groups on Tychonoff spaces. A typical result sounds as follows: for each infinite cardinal number τ every locally compact, non-compact, σ-compact group G of weight w(G)?τ, can act properly on Rτ?{0} such that Rτ?{0} contains a G-homeomorphic copy of every Tychonoff proper G-space of weight ?τ. The metric cones Cone(G/H) with H⊂G a compact subgroup such that G/H is a manifold, are the main building blocks in our approach. As a byproduct we prove that the cardinality of the set of all conjugacy classes of such subgroups H⊂G does not exceed the weight of G.     

Lifting paths on quotient spaces

Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X/G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn-Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g∈G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the first named author in a number of ways.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

Whitney preserving maps on finite graphs

For a Whitney preserving map f:X→G we show the following: (a) If X is arcwise connected and G is a graph which is not a simple closed curve, then f is a homeomorphism; (b) If X is locally connected and G is a simple closed curve, then X is homeomorphic to either the unit interval [0,1], or the unit circle S¹. As a consequence of these results, we characterize all Whitney preserving maps between finite graphs. We also show that every hereditarily weakly confluent Whitney preserving map between locally connected continua is a homeomorphism.     

Examples of cell-like decompositions of the infinite-dimensional manifolds σ and Σ

Denote by σ the subspace of Hilbert space {(x_i)?l₂:x_i=0 for all but finitely many i}. Examples of cell-like decompositions of σ are constructed that have decomposition spaces that are not homeomorphic to σ. At one extreme is a cell-like decomposition G of σ produced using ghastly finite dimensional examples such that the decomposition space σ?G contains no embedded 2-cell but (σ?G)×$\text{R}$ is homeomorphic to σ. At the other extreme is a cell-like decomposition G of σ satisfying: (a) the nondegeneracy set N_G={g?G:g≠point} consists of countably many arcs (necessarily tame); (b) the nondegeneracy set N_G is a closed subset of the decomposition space σ?G; (c) each map f:B²→σ?G of a 2-cell into σ?G can be approximated arbitrarily closely by an embedding; (d) σ?G is not homeomorphic to σ but (σ?G)×$\text{R}$ is homeomorphic to σ. The fact that both conditions (a) and (b) can be satisfied (and have (d) hold) is directly attributable to σ’s incompleteness as a topological space.     

G-compactifications of pseudocompact G-spaces

Examples of a pseudocompact (even countably compact) G-space which is not G-Tychonoff and of a locally compact pseudocompact (even countably compact) G-Tychonoff space X with βGX≠βX are constructed.     

Fréchet-Urysohn fans in free topological groups

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fréchet-Urysohn fan S_ω in a topological group G admitting a functorial embedding [0,1]⊂G. The latter means that each autohomeomorphism of [0,1] extends to a continuous homomorphism of G. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space X containing a topological copy of the space Q of rationals give topological groups containing S_ω.     

On the continuity of left translations in the LUC-compactification

Consider the continuity of left translations in the LUC-compactification GLUC of a locally compact group G. For every X⊆G, let κ(X) be the minimal cardinality of a compact covering of X in G. Let U(G) be the points in GLUC that are not in the closure of any X⊆G with κ(X)<κ(G). We show that the points at which no left translation in U(G) is continuous are dense in U(G). This result is a generalization of a theorem by van Douwen concerning discrete groups. We obtain a new proof for the fact that the topological center of GLUC?G is empty.     

Embedding EnG in Euclidean space

It is shown that if G is an arbitrary upper semicontinuous decomposition of Eⁿ for which π(N_G embeds in S^m for some m?3, then the decomposition space $\text{E}{}^{\mathrm{n}}\text{G}$ embeds as a closed subset of E^n+m+1. The proof consists of constructing a cell-like upper semicontinuous decomposition G? of E^n+m+1 which intersects Eⁿ to yield precisely G and using Edwards' Cell-Like Approximation Theorem to show that G? is shrinkable. As an immediate corollary, $\text{E}{}^{\mathrm{n}}\text{G}$ embeds in E^n+2k+2 whenever G is an arbitrary k-dimensional upper semicontinuous decomposition of Eⁿ. This is an improvement of (n?1)-dimensions over the corresponding dimension theoretic result and examples due to Daverman show that this result is sharp in case n is odd and off by no more than one dimension in case n is even.     

Integral closure of invariant ideals, toroidal resolution, and equivariant vector bundles

We establish a connection between equivariant integrally closed ideal sheaves on a G-fibration Y over a G-spherical variety X with an affine fiber V and equivariant vector bundles on the universal toroidal resolution of X. As an application, we reduce the study of invariant integrally closed ideals of V×X to that of some smaller variety in the case of X=M_n,m. Moreover, we present an affirmative answer to a problem raised by Michel Brion [Comment. Math. Helv. 66 (1991) 237-262] for two special infinite series.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Left and right uniform structures on functionally balanced groups

Let G be a Hausdorff topological group. It is shown that there is a class C of subspaces of G, containing all (but not only) precompact subsets of G, for which the following result holds:Suppose that for every real-valued discontinuous function on G there is a set A∈C such that the restriction mapping f|A has no continuous extension to G; then the following are equivalent:

(i)

the left and right uniform structures of G are equivalent,

(ii)

every left uniformly continuous bounded real-valued function on G is right uniformly continuous,

(iii)

for every countable set A⊂G and every neighborhood V of the unit e of G, there is a neighborhood U of e in G such that AU⊂VA.

As a consequence, it is proved that items (i), (ii) and (iii) are equivalent for every inframetrizable group. These results generalize earlier ones established by Itzkowitz, Rothman, Strassberg and Wu, by Milnes and by Pestov for locally compact groups, by Protasov for almost metrizable groups, and by Troallic for groups that are quasi-k-spaces.     

Equivariant homology and cohomology of groups

We provide and study an equivariant theory of group (co)homology of a group G with coefficients in a Γ-equivariant G-module A, when a separate group Γ acts on G and A, generalizing the classical Eilenberg-MacLane (co)homology theory of groups. Relationship with equivariant cohomology of topological spaces is established and application to algebraic K-theory is given.     

Hausdorff compactifications and lebesgue sets

Let X be a completely regular Hausdorff space and let H be a subset of C¹(X) which separates points and closed sets. By embedding X into a cube whose factors are indexed by H, a Hausdorff compactification e_HX of X is obtained. Given two subsets F and G of C¹(X) which separate points from closed sets, in the present paper we obtain a necessary and sufficient condition for the equivalence of e_FX and e_GX. The condition is expressed in terms of the space X and the sets F and G alone, herewith solving a question raised by Chandler.     

Parametric bing and Krasinkiewicz maps

It is shown that if is a perfect map between metrizable spaces and Y is a C-space, then the function space C(X,I) with the source limitation topology contains a dense Gδ-subset of maps g such that every restriction map gy=g|f⁻¹(y), y∈Y, satisfies the following condition: all fibers of gy are hereditarily indecomposable and any continuum in f⁻¹(y) either contains a component of a fiber of gy or is contained in a fiber of gy.