Topology and its Applications最新文献

We prove in a constructive way that, given $\alpha >1$, there exists a quasisymmetrically rigid metric carpet of Hausdorff dimension >α and whose peripheral circles are all rectifiable.

This paper offers a novel homotopical characterization of strongly contextual simplicial distributions with binary outcomes, specifically those defined on the cone of a 1-dimensional space. In the sheaf-theoretic framework, such distributions correspond to non-signaling distributions on measurement scenarios where each context contains 2 measurements with binary outcomes. To establish our results, we employ a homotopical approach that includes collapsing measurement spaces and introduce categories associated with simplicial distributions that can detect strong contextuality.

We construct a multiplicative spectral sequence converging to the cohomology algebra of the diagonal complex of a bisimplicial set with coefficients in a field. The construction provides a spectral sequence converging to the cohomology algebra of the classifying space of a category internal to the category of topological spaces. By applying the machinery to a Borel construction, we explicitly determine the mod p cohomology algebra of the free loop space of the real projective space for each odd prime p. This example is emphasized as an important computational case. Moreover, we represent generators in the singular de Rham cohomology algebra of the diffeological free loop space of a non-simply connected manifold M with differential forms on the universal cover of M via Chen's iterated integral map.

In this research paper, we investigate ultra-pseudo metric spaces and prove that a completely regular topological space is homeomorphic to a subspace of a product of ultra-pseudo metric spaces.

We show the Teichmüller space of a non-orientable surface with marked points (considered as a Klein surface) can be identified with a subspace of the Teichmüller space of its orientable double cover. It is also well known that the mapping class group $\mathrm{Mod}(N_g;k)$ of a non-orientable surface can be identified with a subgroup of $\mathrm{Mod}(S_{g-1};2k)$, the mapping class group of its orientable double cover. These facts, together with the classical Nielsen realization theorem, are used to prove that every finite subgroup of $\mathrm{Mod}(N_g;k)$ can be lifted isomorphically to a subgroup of the group of diffeomorphisms $\mathrm{Diff}(N_g;k)$. In contrast, we show the projection $\mathrm{Diff}\left(N_g\right)\to \mathrm{Mod}\left(N_g\right)$ does not admit a section for large g.

In this paper, we introduce polytopal k-wedge construction and blowdown of a simple polytope and inspect the effect on the retraction sequence of a simple polytope due to the k-wedge construction and blowdown. In relation to these constructions, we introduce the k-wedge and blowdown of a quasitoric orbifold. We compare the torsions in the integral cohomologies of k-wedges and blowdowns of a quasitoric orbifold with the original one. These two constructions provide infinitely many integrally equivariantly formal quasitoric orbifolds from a given one.

We will calculate the density, the spread and related cardinal functions on lexicographic products of GO-spaces, and give their applications.

Groups with a topology that is in consistent one way or another with the algebraic structure are considered. Well-known classes of groups with a topology are topological, paratopological, semitopological, and quasitopological groups. We also study other ways of matching topology and algebraic structure. The minimum requirement in this paper is that the group is a right semitopological group (such groups are often called right topological groups). We study when a group with a topology is a topological group; research in this direction began with the work of Deane Montgomery and Robert Ellis. (Invariant) semi-neighborhoods of the diagonal are used as a means of study.

The theory of generalized metrizable spaces is an important topic of general topology. This paper is a survey of research methods and achievements on generalized metrizable properties in topological groups and weakly topological groups. We mainly study this kind of properties in topological groups, semitopological groups, paratopological groups, quasitopological groups and free topological groups, and focus on the influence of separation properties, conditions for weakly topological groups to become topological groups, cardinal invariants, weak first-countability, three-space properties and remainders in compactifications on topological groups and related structures. Finally, some unsolved problems in this field are listed for researchers.

In this paper, we introduce the notion of a semi-stratifiable frame as an extension of classical semi-stratifiability and also as the monotonization of perfect frames. We show that stratifiable frames are precisely the monotonically normal semi-stratifiable frames. Moreover, we present an insertion theorem for semi-stratifiable frames in terms of real functions and thereby obtain an insertion theorem for stratifiable frames.