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The main idea of this paper is to generate supra and infra topologies from simple undirected graphs. For this, we have introduced two new operators namely supra and infra operators which are defined on the power set of the vertex set of a graph. Moreover, we have also proved that the supra operator satisfying Kuratowski’s closure axiom will yield a topology. Further it was extended to develop the concept of connectedness and separation axioms on G-supra and G-infra spaces.

We prove that the existence of totally real immersions of manifolds is a closed property under cut-and-paste constructions along submanifolds including connected sums. We study the existence of totally real embeddings for simply connected 5-manifolds and orientable 6-manifolds and determine the diffeomorphism and homotopy types. We show that the fundamental group is not an obstruction for the existence of a totally real embedding for high-dimensional manifolds in contrast with the situation in dimension four.

We show that 𝐾(2)-locally, the smash product of the string bordism spectrum and the spectrum 𝑇₂ splits into copies of Morava 𝐸-theories. Here, 𝑇₂ is related to the Thom spectrum of the canonical bundle over Ω𝑆𝑈(4).

In this paper, we consider the weighted 𝑝-Laplacian Lichnerowicz equation Δ p , f u + c u σ = 0 on smooth metric measure spaces, where 𝑐 ≥ 0, 𝑝 > 1, and 𝜎 ≤ 𝑝 − 1 are real constants. A local gradient estimate for positive solutions to this equation is derived, and as applications, we give a corresponding Liouville property and Harnack inequality.

We prove that the 𝐶₂-equivariant Eilenberg–MacLane spectrum associated with the constant Mackey functor F _ 2 is equivalent to a Thom spectrum over Ω𝜌𝑆𝜌⁺¹.

We give a simple alternative proof for the 𝐶¹,¹–convex extension problem which has been introduced and studied by D. Azagra and C. Mudarra (2017). As an application, we obtain an easy constructive proof for the Glaeser-Whitney problem of 𝐶¹,¹ extensions on a Hilbert space. In both cases we provide explicit formulae for the extensions. For the Glaeser-Whitney problem the obtained extension is almost minimal, that is, minimal up to a multiplicative factor in the sense of Le Gruyer (2009).

This paper provides a new way to understand the equivariant slice filtration. We give a new, readily checked condition for determining when a 𝐺-spectrum is slice 𝑛-connective. In particular, we show that a 𝐺-spectrum is slice greater than or equal to 𝑛 if and only if for all subgroups 𝐻, the 𝐻-geometric fixed points are (𝑛/𝐻| − 1)-connected. We use this to determine when smashing with a virtual representation sphere 𝑆𝑉 induces an equivalence between various slice categories. Using this, we give an explicit formula for the slices for an arbitrary 𝐶𝑝-spectrum and show how a very small number of functors determine all of the slices for 𝐶𝑝𝑛 -spectra.

The colored Jones polynomial is a 𝑞-polynomial invariant of links colored by irreducible representations of a simple Lie algebra. A 𝑞-series called a tail is obtained as the limit of the 𝔰𝔩₂ colored Jones polynomials {𝐽𝑛(𝐾; 𝑞)}𝑛 for some link 𝐾, for example, an alternating link. For the 𝔰𝔩₃ colored Jones polynomials, the existence of a tail is unknown. We give two explicit formulas of the tail of the 𝔰𝔩₃ colored Jones polynomials colored by (𝑛, 0) for the (2, 2𝑚)-torus link. These two expressions of the tail provide an identity of 𝑞-series. This is a knot-theoretical generalization of the Andrews–Gordon identities for the Ramanujan false theta function.

We show that the existence of a countable dense homogeneous metric space which is connected and meager in itself is independent of 𝖹𝖥𝖢.

As an application of Behrens and Rezk’s spectral algebra model for unstable 𝜐𝑛-periodic homotopy theory, we give explicit presentations for the completed 𝐸-homology of the Bousfield-Kuhn functor on odd-dimensional spheres at chromatic level 2, and compare them to the level 1 case. The latter reflects earlier work in the literature on 𝐾-theory localizations.