Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1710.07315

Dimension functions for spherical fibrations

Cihan Okay·Ergun Yalcin
View original source ↗Read source document ↗Export metadata ↓

Abstract

Given a spherical fibration $ξ$ over the classifying space $BG$ of a finite group we define a dimension function for the $m-$fold fiber join of $ξ$ where $m$ is some large positive integer. We show that the dimension functions satisfy the Borel-Smith conditions when $m$ is large enough. As an application we prove that there exists no spherical fibration over the classifying space of $\text{Qd}(p)= (\mathbb{Z}/p)^2\rtimes\text{SL}_2(\mathbb{Z}/p)$ with $p-$effective Euler class, generalizing the result of Özgün Ünlü about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in a future paper as a corollary of a previously announced program on homotopy group actions due to Jesper Grodal.

Explore related subjects

math.AT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cihan Okay, Ergun Yalcin. 2018-06-29. Dimension functions for spherical fibrations. https://doi.org/10.2140/agt.2018.18.3907

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Persistent Simple-homotopy invariants via discrete Morse theory

Persistent homology records the evolution of homological features along a filtration, but does not retain finer information related to simple-homotopy theory. In this paper, we develop two approaches to capturing such information for filtered simplicial complexes. We first introduce the Morse complexity profile, which records the minimal number of critical simplices at each filtration level. We study its invariance and stability properties and develop computable approximations using several discrete Morse matchings. We then introduce a persistent version of Whitehead torsion and show that it is invariant under both levelwise homotopy equivalence and interleaving equivalence of filtrations.

math.AT↗
arXiv2026

Signed GLMY Homology of Signed Graphs via Double Covers

We define a signed GLMY chain complex over $\mathbb{R}$ for signed digraphs using sheet-labelled regular paths. The complex is naturally isomorphic to the deck anti-invariant subcomplex of the ordinary GLMY complex on the signed double cover. The double-cover realization yields switching invariance and recovers ordinary GLMY homology for switching-balanced signings. Bidirected completion gives an orientation-independent homology theory for signed graphs. For a signed graph, the zero-dimensional homology identifies with the kernel of the signed Laplacian and has dimension equal to the number of balanced connected components. Signed GLMY homology is functorial under signed weak morphisms, which combine vertex maps with switching functions and allow compatible arrow contractions. For signed digraphs, the all-positive reduction retains the orientation sensitivity of ordinary GLMY homology, while explicit computations show additional sensitivity to the arrow signs. For a fixed digraph with five vertices and nine arrows, we classify all 512 arrow signings and obtain exactly four signed Betti vectors. Precisely 16 signings have nonzero second signed GLMY homology.

math.AT↗
arXiv2026

Gabriel Spectrum of Persistence Categories

We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space $X$, we prove that the Gabriel spectrum of ${\mathbf{Sh}}(X)$ is homeomorphic to ${\mathrm{Sk}}({\mathrm{Sob}}(X))$, where ${\mathrm{Sob}}(X)$ denotes the sobrification of $X$ and ${\mathrm{Sk}}$ indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if $X$ is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ${\mathbb R}^n$ is homeomorphic to the space of ideals of ${\mathbb R}^n$ with a natural topology.

math.AT↗