FF-Dugundji spaces, FF-Milutin spaces and absolute FF-valued retracts

For every functional functor $F:Comp\to Comp$ in the category $Comp$ of compact Hausdorff spaces we define the notions of $F$-Dugundji and $F$-Milutin spaces, generalizing the classical notions of a Dugundji and Milutin spaces. We prove that the class of $F$-Dugundji spaces coincides with the class of absolute $F$-valued retracts. Next, we show that for a monomorphic continuous functor $F:Comp\to Comp$ admitting tensor products each Dugundji compact is an absolute $F$-valued retract if and only if the doubleton $\{0,1\}$ is an absolute $F$-valued retract if and only if some points $a\in F(\{0\})\subset F(\{0,1\})$ and $b\in F(\{1\})\subset F(\{0,1\})$ can be linked by a continuous path in $F(\{0,1\})$. We prove that for the functor $Lip_k$ of $k$-Lipschitz functionals with $k<2$, each absolute $Lip_k$-valued retract is openly generated. On the other hand the one-point compactification of any uncountable discrete space is not openly generated but is an absolute $Lip_3$-valued retract. More generally, each hereditarily paracompact scattered compact space $X$ of finite scattered height $n$ is an absolute $Lip_k$-valued retract for $k=2^{n+2}-1$.Comment: 12 page

(Metrically) quarter-stratifiable spaces and their applications in the theory of separately continuous functions

We introduce and study (metrically) quarter-stratifiable spaces and then apply them to generalize Rudin and Kuratowski-Montgomery theorems about the Baire and Borel complexity of separately continuous functions.Comment: 19 page

Topologies on groups determined by sequences: Answers to several questions of I.Protasov and E.Zelenyuk

We answer several questions of I.Protasov and E.Zelenyuk concerning topologies on groups determined by T-sequences. A special attention is paid to studying the operation of supremum of two group topologies.Comment: 7 page

A quantitative generalization of Prodanov-Stoyanov Theorem on minimal Abelian topological groups

A topological group $X$ is defined to have $compact$ $exponent$ if for some number $n\in\mathbb N$ the set $\{x^n:x\in X\}$ has compact closure in $X$. Any such number $n$ will be called a compact exponent of $X$. Our principal result states that a complete Abelian topological group $X$ has compact exponent (equal to $n\in\mathbb N$) if and only if for any injective continuous homomorphism $f:X\to Y$ to a topological group $Y$ and every $y\in \bar{f(X)}$ there exists a positive number $k$ (equal to $n$) such that $y^k\in f(X)$. This result has many interesting implications: (1) an Abelian topological group is compact if and only if it is complete in each weaker Hausdorff group topology; (2) each minimal Abelian topological group is precompact (this is the famous Prodanov-Stoyanov Theorem); (3) a topological group $X$ is complete and has compact exponent if and only if it is closed in each Hausdorff paratopological group containing $X$ as a topoloical subgroup (this confirms an old conjecture of Banakh and Ravsky).Comment: 14 page

A functional representation of the capacity multiplication monad

Functional representations of the capacity monad based on the max and min operations were considered in \cite{Ra1} and \cite{Ny1}. Nykyforchyn considered in \cite{Ny2} some alternative monad structure for the possibility capacity functor based on the max and usual multiplication operations. We show that such capacity monad (which we call the capacity multiplication monad) has a functional representation, i.e. the space of capacities on a compactum $X$ can be naturally embedded (with preserving of the monad structure) in some space of functionals on $C(X,I)$. We also describe this space of functionals in terms of properties of functionals

On the sequential closure of the set of continuous functions in the space of separately continuous functions

For separable metrizable spaces $X,Y$ and a metrizable topological group $Z$ by $S(X\times Y,Z)$ we denote the space of all separately continuous functions $f:X\times Y\to Z$ endowed with the topology of layer-wise uniform convergence, generated by the subbase consisting of the sets $[K_X\times K_Y,U]=\{f\in S(X\times Y,Z):f(K_X\times K_Y)\subset U\}$, where $U$ is an open subset of $Z$ and $K_X\subset X$, $K_Y\subset Y$ are compact sets one of which is a singleton. We prove that every separately continuous function $f:X\times Y\to Z$ with zero-dimensional image $f(X\times Y)$ is a limit of a sequence of jointly continuous functions in the topology of layer-wise uniform convergence.Comment: 5 page

Geometric structures on moment-angle manifolds

The moment-angle complex Z_K is cell complex with a torus action constructed from a finite simplicial complex K. When this construction is applied to a triangulated sphere K or, in particular, to the boundary of a simplicial polytope, the result is a manifold. Moment-angle manifolds and complexes are central objects in toric topology, and currently are gaining much interest in homotopy theory, complex and symplectic geometry. The geometric aspects of the theory of moment-angle complexes are the main theme of this survey. We review constructions of non-Kahler complex-analytic structures on moment-angle manifolds corresponding to polytopes and complete simplicial fans, and describe invariants of these structures, such as the Hodge numbers and Dolbeault cohomology rings. Symplectic and Lagrangian aspects of the theory are also of considerable interest. Moment-angle manifolds appear as level sets for quadratic Hamiltonians of torus actions, and can be used to construct new families of Hamiltonian-minimal Lagrangian submanifolds in a complex space, complex projective space or toric varieties.Comment: 60 page

On the cohomology of quotients of moment-angle complexes

We describe the cohomology of the quotient Z_K/H of a moment-angle complex Z_K by a freely acting subtorus H in T^m by establishing a ring isomorphism of H*(Z_K/H,R) with an appropriate Tor-algebra of the face ring R[K], with coefficients in an arbitrary commutative ring R with unit. This result was stated in [BP02, 7.37] for a field R, but the argument was not sufficiently detailed in the case of nontrivial H and finite characteristic. We prove the collapse of the corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with respect to `strongly homotopy multiplicative' maps in the category DASH, following Munkholm [Mu74]. Our collapse result does not follow from the general results of Gugenheim-May and Munkholm.Comment: 3 page

Cohomology of face rings, and torus actions

In this survey article we present several new developments of `toric topology' concerning the cohomology of face rings (also known as Stanley-Reisner algebras). We prove that the integral cohomology algebra of the moment-angle complex Z_K (equivalently, of the complement U(K) of the coordinate subspace arrangement) determined by a simplicial complex K is isomorphic to the Tor-algebra of the face ring of K. Then we analyse Massey products and formality of this algebra by using a generalisation of Hochster's theorem. We also review several related combinatorial results and problems.Comment: 28 pages, more minor changes, to be published in the LMS Lecture Note

The structure of infinite 2-groups with a unique 2-element subgroup

We prove that each infinite 2-group with a unique 2-element subgroup is isomorphic either to the quasicyclic 2-group or to the infinite group of generalized quaternions.Comment: 4 page