A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

Regularity of the Donaldson Geometric Flow

We prove a regularity theorem for the solutions of the Donaldson geometric flow equation on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The minimal initial conditions lay in the Besov space $B^{1,p}_{2}(M, {\varLambda }^{2})$ B 2 1 , p ( M , Λ 2 ) for p > 4. The Donaldson geometric flow was introduced by Simon Donaldson in Donaldson (Asian J. Math.3, 1–16 1999). For a detailed exposition see Krom and Salamon (J. Symplectic Geom.17, 381–417 2019).

A pair of homotopy-theoretic version of TQFT’s induced by a Brown functor

The purpose of this paper is to study some obstruction classes induced by a construction of a homotopy-theoretic version of projective TQFT (projective HTQFT for short). A projective HTQFT is given by a symmetric monoidal projective functor whose domain is the cospan category of pointed finite CW-spaces instead of a cobordism category. We construct a pair of projective HTQFT’s starting from a [Formula: see text]-valued Brown functor where [Formula: see text] is the category of bicommutative Hopf algebras over a field [Formula: see text] : the cospanical path-integral and the spanical path-integral of the Brown functor. They induce obstruction classes by an analogue of the second cohomology class associated with projective representations. In this paper, we derive some formulae of those obstruction classes. We apply the formulae to prove that the dimension reduction of the cospanical and spanical path-integrals are lifted to HTQFT’s. In another application, we reproduce the Dijkgraaf–Witten TQFT and the Turaev–Viro TQFT from an ordinary [Formula: see text]-valued homology theory.

AN APPLICATION OF WALL-CROSSING TO NOETHER–LEFSCHETZ LOCI

Consider a smooth projective 3-fold $X$ satisfying the Bogomolov–Gieseker conjecture of Bayer–Macrì–Toda (such as ${\mathbb{P}}^3$, the quintic 3-fold or an abelian 3-fold). Let $L$ be a line bundle supported on a very positive surface in $X$. If $c_1(L)$ is a primitive cohomology class, then we show it has very negative square.

Block Extensions, Local Categories and Basic Morita Equivalences

Let $(\mathcal{K},\mathcal{O},k)$ be a $p$-modular system where $p$ is a prime and $k$ algebraically closed, let $b$ be a $G$-invariant block of the normal subgroup $H$ of a finite group $G$, having defect pointed group $Q_\delta$ in $H$ and $P_\gamma$ in $G$ and consider the block extension $b\mathcal{O}G$. One may attach to $b$ an extended local category $\mathcal{E}_{(b,H,G)}$, a group extension $L$ of $Z(Q)$ by $N_G(Q_\delta )/C_H(Q)$ having $P$ as a Sylow $p$-subgroup, and a cohomology class $[\alpha ]\in H^2(N_G(Q_\delta )/QC_H(Q),k^\times )$. We prove that these objects are invariant under the $G/H$-graded basic Morita equivalences. Along the way, we give alternative proofs of the results of Külshammer and Puig (1990), and Puig and Zhou (2012) on extensions of nilpotent blocks. We also deduce by our methods a result of Zhou (2016) on $p^{\prime}$-extensions of inertial blocks.

The Bundle of Simultaneously Diagonalizable N-tuples of Matrices

In this paper, a review of the simultaneous diagonalization of n-tuples of matrices for its applications in sciences is presented. For example, in quantum mechanics, position and momentum operators do not have a shared base that can represent the states of the system because they not commute, which is why switching operators form a key element of quantum physics since they define quantities that are compatible, that is, defined simultaneously. We are going to study this kind of family of linear operators using geometric constructions such as the principal bundles and associating them with a cohomology class measuring the deviation of the local product structure from the global product structure.

Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

On the topological complexity of aspherical spaces

The well-known theorem of Eilenberg and Ganea [Ann. Math. 65 (1957) 517–518] expresses the Lusternik–Schnirelmann category of an Eilenberg–MacLane space [Formula: see text] as the cohomological dimension of the group [Formula: see text]. In this paper, we study a similar problem of determining algebraically the topological complexity of the Eilenberg–MacLane spaces [Formula: see text]. One of our main results states that in the case when the group [Formula: see text] is hyperbolic in the sense of Gromov, the topological complexity [Formula: see text] either equals or is by one larger than the cohomological dimension of [Formula: see text]. We approach the problem by studying essential cohomology classes, i.e. classes which can be obtained from the powers of the canonical class (as defined by Costa and Farber) via coefficient homomorphisms. We describe a spectral sequence which allows to specify a full set of obstructions for a cohomology class to be essential. In the case of a hyperbolic group, we establish a vanishing property of this spectral sequence which leads to the main result.

Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights

Let ${\boldsymbol{\Sigma }}=(\Sigma ,\mathbb{M},\mathbb{O})$ be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair $(\tau ,\omega )$ consisting of a triangulation $\tau $ of ${\boldsymbol{\Sigma }}$ and a function $\omega :\mathbb{O}\rightarrow \{1,4\}$, we define a chain complex $C_\bullet (\tau , \omega )$ with coefficients in $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$. Given ${\boldsymbol{\Sigma }}$ and $\omega $, we define a colored triangulation of ${\boldsymbol{\Sigma }_\omega }=(\Sigma ,\mathbb{M},\mathbb{O},\omega )$ to be a pair $(\tau ,\xi )$ consisting of a triangulation of ${\boldsymbol{\Sigma }}$ and a 1-cocycle in the cochain complex that is dual to $C_\bullet (\tau , \omega )$; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [25]. We define the flip graph of ${\boldsymbol{\Sigma }_\omega }$ as the graph whose vertices are the pairs $(\tau ,x)$ consisting of a triangulation $\tau $ and a cohomology class $x\in H^1(C^\bullet (\tau , \omega ))$, with an edge connecting two such pairs, $(\tau ,x)$ and $(\sigma ,z),$ if and only if there exist 1-cocycles $\xi \in x$ and $\zeta \in z$ such that $(\tau ,\xi )$ and $(\sigma ,\zeta )$ are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface $\Sigma $ is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair $(\tau ,\omega )$ over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the $2^{|\mathbb{O}|}$ skew-symmetrizable matrices that Felikson–Shapiro–Tumarkin associated in [17] to any given triangulation of ${\boldsymbol{\Sigma }}$. In the prequel [25] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.