arXiv subjects

Shahryar Ghaed Sharaf

Publications and source records attributed to Shahryar Ghaed Sharaf.

5 recordsLinked to original sources

arXiv2026

A Decomposition Theorem for Topological Semi-small Maps

A continuous surjection $f: X \rightarrow Y$ is an almost covering map if there exists a nowhere dense closed set $R \subset Y$ with a finite decomposition $R = \bigsqcup_β T_β$ such that the restriction $f\vert_{f^{-1}(Y \setminus R)}$ is a covering map and, for each $β$, the restriction $f\vert_{f^{-1}(T_β)}$ is a fiber bundle. We refer to an almost covering map as a topological semi-small map if, for each $β$, the fiber of $f\vert_{f^{-1}(T_β)}$ and the space $T_β$ satisfy the same dimension condition as for algebraic semi-small maps. We first study the topological properties of almost coverings and develop the necessary tools. In the next step, we restrict our attention to topological semi-small maps and prove that, under the assumption that for each $β$ and each trivializing neighborhood $U \subset T_β$ the connecting homomorphisms in the Gysin sequence of the normal bundle of $f^{-1}(U) \subset X$ vanish in suitable degrees, the derived direct image of the constant sheaf on the domain admits a decomposition.

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arXiv2026

A Decomposition Theorem for Topological Branched Coverings

In the context of complex algebraic varieties, the decomposition theorem for semi-small maps provides a decomposition of the direct image of the constant sheaf. In this work, we develop a decomposition theorem for branched coverings of topological spaces. To achieve this, we start by constructing a decomposition theorem for unramified covering maps via an explicit gluing construction using transition functions. For a given branched covering of closed topological manifolds, we use the previous result to establish a decomposition of the direct image of the constant sheaf on the covering space. In the next step, we generalize our discussion to the case where the target space is not necessarily a topological manifold.

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arXiv2026

Topology of the orbit space of a maximal torus action on $\operatorname{Gr}(2,\mathbb{C}^{5})$

In this work, we take a topological approach and introduce a homotopy model of the orbit space of a maximal torus action on $\operatorname{Gr}(2, \mathbb{C}^{5})$. Using this homotopy model, we compute the integral homology groups and study the homotopy type of the orbit space. More precisely, we endow the homotopy model with a CW complex that contains one top cell and determine the homotopy class of the attaching map of the top cell of the orbit space.

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arXiv2025

Link Bundles of Compact Toric Varieties of Real Dimension 8

The main goal of this work is to determine the Betti numbers of the links of isolated singularities in a compact toric variety of real dimension 8, using the CW-structure of the links. Additionally, we construct the intersection spaces associated with these links. Using the duality of the Betti numbers of intersection spaces, we conclude that, similar to the case of toric varieties of real dimension 6, the Betti numbers of the links contain only one non-combinatorial invariant parameter. In the final section, we extend our discussion to arbitrary compact toric varieties and their associated link bundles. We show that for any given link $\mathcal{L}$, there exists a fiber bundle $π: \mathcal{L} \to X$ with fiber $S^{1}$, where the base space $X$ is a compact toric variety. Furthermore, using the Chern-Spanier exact sequences for sphere bundles, we show that for the fiber bundle $π:\mathcal{L}\longrightarrow X$, where $\dim_{\mathbb{R}}(X)=6$, the non-combinatorial invariant parameters appearing in the Betti numbers of $\mathcal{L}$ and $X$ are equal. In addition, we provide an algebraic description of the non-combinatorial invariant parameter of $X$ in terms of the cohomological Euler class of the fiber bundle.

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arXiv2024

Link Bundles and Intersection Spaces of Complex Toric Varieties

There exist several homology theories for singular spaces that satisfy generalized Poincaré duality, including Goresky-MacPherson's intersection homology, Cheeger's $L^2$ cohomology and the homology of intersection spaces. The intersection homology and $L^2$ cohomology of toric varieties is known. Here, we compute the rational homology of intersection spaces of complex 3-dimensional toric varieties and compare it to intersection homology. To achieve this, we analyze cell structures and topological stratifications of these varieties and determine compatible structures on their singularity links. In particular, we compute the homology of links in 3-dimensional toric varieties. We find it convenient to use the concept of a rational homology stratification. It turns out that the intersection space homology of a toric variety, contrary to its intersection homology, is not combinatorially invariant and thus retains more refined information on the defining fan.

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