# Algebraic Topology (math.AT)

• Recent developments have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for describing morphisms of homotopy operadic algebras. We provide new elementary examples which clearly resemble such connection, based on spectral graph theory. These observations are important for bringing new ideas from non-commutative probability into TDA and stochastic topology, and in the opposite direction.
• The space of polynomial differential equations of a fixed degree with a center singularity has many irreducible components. We prove that pull back differential equations form an irreducible component of such a space. The method used in this article is inspired by Ilyashenko and Movasati s method. The main concepts are the Picard Lefschetz theory of a polynomial in two variables with complex coefficients, the Dynkin diagram of the polynomial and the iterated integral.
• With $G = \mathbb{Z}/p$, $p$ prime, we calculate the ordinary $G$-cohomology (with Burnside ring coefficients) of $\mathbb{C}P_G^\infty = B_G U(1)$, the complex projective space, a model for the classifying space for $G$-equivariant complex line bundles. The $RO(G)$-graded ordinary cohomology was calculated by Gaunce Lewis, but here we extend to a larger grading in order to capture a more natural set of generators, including the Euler class of the canonical bundle, as well as a significantly simpler set of relations.
• We construct a canonical basis of two-cycles, on a $K3$ surface, in which the intersection form takes the canonical form $2E_8(-1) \oplus 3H$. The basic elements are realized by formal sums of smooth submanifolds.
• We associate invariants such as permutation cycles and local cycles at infinity with $2-$standard consecutive structures (refer Definition $23$) to a line arrangement (refer Definition $2$) which has global cyclicity (refer Definition $19$) over fields with $1-ad$ structure (refer Definition $1$) to describe the gonality structures (refer Definition $8$) in Theorem $11$ when there exists a local permutation chart where the intersections points corresponding to simple transpositions satisfy One Sided Property $11$. While generalizing the features of a finite set of linear inequalities in two variables we compute modified simplicial homology groups of line arrangements over arbitrary fields in Theorem $2$ and ask Question $1$ about associating spaces which describe these invariant homology groups. In this article we could associate regions (refer Definition $5$) describing these invariants over fields with $1-ad$ structure. We construct a graph of isomorphism of classes of line arrangements over fields with $1-ad$ structure using the associated invariants and Elementary Collineation Transformations (ECT) in Theorem $13$ and in Note $11$. Here we prove a representation Theorem $13$ where we represent each isomorphism class with a given set of distinct slopes. We also prove a formulation of Polygonal Jordan Curve Theorem $3$ over fields with $1-ad$ structure and an isomorphism Theorem $14$ for those line arrangement collineation maps which preserve nook points and central pairs for quadrilateral substructures (refer Definition $30$). At the end of the article we ask some open questions on line-folds (refer Definition $31$).
• We investigate under which assumptions a subclass of flat quasi-coherent sheaves on a quasi-compact and semi-separated scheme allows to "mock" both the homotopy category of projective modules and the homotopy category of totally acyclic complexes of projectives. Our methods are based on module theoretic properties of the subclass of flat modules involved as well as their behaviour with respect to Zariski localizations. As a consequence we get that, for such schemes, the derived category of flats is equivalent to the derived category of very flats and the equivalence restricts to the full subcategories of F-totally acyclic complexes. Furthermore, the equivalences are derived from a Quillen equivalence between the corresponding models.
• Aug 22 2017 math.AT arXiv:1708.05871v1
We introduce notions of \it upper chernrank and \it even cup length of a finite connected CW-complex and prove that \it upper chernrank is a homotopy invariant. It turns out that determination of \it upper chernrank of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$-dimensional complex manifold we obtain an upper bound of its even cup length. For a finite connected even dimensional CW-complex with its \it upper chernrank equal to its dimension, we provide a method of computing its even cup length. Finally, we compute \it upper chernrank of many interesting spaces.
• A multigraph is a nonsimple graph which is permitted to have multiple edges, that is, edges that have the same end nodes. We introduce the concept of spanning simplicial complexes $\Delta_s(\mathcal{G})$ of multigraphs $\mathcal{G}$, which provides a generalization of spanning simplicial complexes of associated simple graphs. We give first the characterization of all spanning trees of a uni-cyclic multigraph $\mathcal{U}_{n,m}^r$ with $n$ edges including $r$ multiple edges within and outside the cycle of length $m$. Then, we determine the facet ideal $I_\mathcal{F}(\Delta_s(\mathcal{U}_{n,m}^r))$ of spanning simplicial complex $\Delta_s(\mathcal{U}_{n,m}^r)$ and its primary decomposition. The Euler characteristic is a well-known topological and homotopic invariant to classify surfaces. Finally, we device a formula for Euler characteristic of spanning simplicial complex $\Delta_s(\mathcal{U}_{n,m}^r)$.
• The Lusternik-Schnirelmann category $cat(X)$ is a homotopy invariant which is a numerical bound on the number of critical points of a smooth function on a manifold. In this paper we calculate the Lusternik-Schnirelmann category of the configuration space of $2$ distinct points in Complex Projective $n-$space for all $n\geq 1$.