Algebraic Topology

New submissions

• New submissions
• Cross-lists
• Replacements

New submissions for Tue, 14 May 24

[1]  arXiv:2405.07372 [pdf, ps, other]

Title: Spaces of non-resultant systems of real bounded multiplicity determined by a toric variety

Authors: Andrzej Kozlowski, Kohhei Yamaguchi

Comments: arXiv admin note: text overlap with arXiv:2105.14601, arXiv:2009.04255

Subjects: Algebraic Topology (math.AT); Algebraic Geometry (math.AG)

For any field $\Bbb F$ and positive integers $m,n,d$ with $(m,n)\not= (1,1)$, Farb and Wolfson defined the certain affine varieties ${\rm Poly}^{d,m}_n(\Bbb F)$ as generalizations of spaces first studied by Arnold, Vassiliev, Segal and others. As a natural generalization of this, for each fan $\Sigma$ and $r$-tuple $D=(d_1,\cdots ,d_r)$ of positive integers, the current authors defined spaces ${\rm Poly}^{D,\Sigma}_n(\Bbb F)$, where $r$ is the number of one dimensional cones in $\Sigma$. These spaces can also be regarded as generalizations of the space ${\rm Hol}^*_D(S^2,X_\Sigma)$ of based rational curves from the Riemann sphere $S^2$ to the toric variety $X_\Sigma$ of degree $D$, where $X_\Sigma$ denotes the toric variety (over $\Bbb C$) corresponding to the fan $\Sigma$. In this paper, we define spaces ${\rm Q}^{D,\Sigma}_n(\Bbb F)$ ($\Bbb F=\Bbb R$ or $\Bbb C$) which are real analogues of ${\rm Poly}^{D,\Sigma}_n(\Bbb F)$ and which can be viewed as a generalizations of spaces considered by Arnold, Vassiliev and others in the context of real singularity theory. We prove that homotopy stability holds for these spaces and compute the stability dimensions explicitly.

Cross-lists for Tue, 14 May 24

[2]  arXiv:2405.06912 (cross-list from math.CT) [pdf, ps, other]

Title: A simple model for twisted arrow $\infty$-categories

Authors: Takeshi Torii

Comments: 16 pages

Subjects: Category Theory (math.CT); Algebraic Topology (math.AT)

Twisted arrow $\infty$-categories of $(\infty,1)$-categories were introduced by Lurie, and they have various applications in higher category theory. Abell\'{a}n Garc\'{i}a and Stern gave a generalization to twisted arrow $\infty$-categories of $(\infty,2)$-categories. In this paper we introduce another simple model for twisted arrow $\infty$-categories of $(\infty,2)$-categories.

[3]  arXiv:2405.07566 (cross-list from math.AC) [pdf, other]

Title: Homological stability for general linear groups over Dedekind domains

Authors: Oscar Randal-Williams

Comments: 18 pages

Subjects: Commutative Algebra (math.AC); Algebraic Topology (math.AT); K-Theory and Homology (math.KT)

We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only stabilisation by the free module of rank 1. We show the same kind of stability holds for Clausen and Jansen's reductive Borel--Serre spaces.

[4]  arXiv:2405.07936 (cross-list from math.AG) [pdf, ps, other]

Title: A Perspective on the Foundations of Derived Analytic Geometry

Authors: Oren Ben-Bassat, Jack Kelly, Kobi Kremnizer

Comments: Comments and feedback very welcome; 274 pages

Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Complex Variables (math.CV); Number Theory (math.NT)

We show how one can do algebraic geometry with respect to the category of simplicial objects in an exact category. As a biproduct, we get a theory of derived analytic geometry.

Replacements for Tue, 14 May 24

[5]  arXiv:2210.12784 (replaced) [pdf, other]

Title: On the top-dimensional cohomology of arithmetic Chevalley groups

Authors: Benjamin Brück, Yuri Santos Rego, Robin J. Sroka

Comments: 8 pages; v2: small changes according to referee's suggestions; to appear in Proc. Amer. Math. Soc

Subjects: Algebraic Topology (math.AT); Group Theory (math.GR); Number Theory (math.NT)

[6]  arXiv:2305.11447 (replaced) [pdf, ps, other]

Title: A note on Samelson product in $Sp(n)$

Authors: Sajjad Mohammadi

Comments: 12pages. arXiv admin note: text overlap with arXiv:2204.08867

Subjects: Algebraic Topology (math.AT)

[7]  arXiv:2112.02707 (replaced) [pdf, ps, other]

Title: On representation categories of $A_\infty$-algebras and $A_\infty$-coalgebras

Authors: Abhishek Banerjee, Anita Naolekar

Comments: Paper significantly expanded with two new sections

Subjects: Category Theory (math.CT); Algebraic Topology (math.AT)

[8]  arXiv:2208.01778 (replaced) [pdf, other]

Title: On Good $2$-Query Locally Testable Codes from Sheaves on High Dimensional Expanders

Authors: Uriya A. First, Tali Kaufman

Comments: Sections 1-8 are subsumed and improved by arxiv:2403.19388. Other sections may be subsumed by future works of the authors. Comments are welcome. No changes from last version

Subjects: Combinatorics (math.CO); Computational Complexity (cs.CC); Algebraic Topology (math.AT)

[9]  arXiv:2211.00762 (replaced) [pdf, ps, other]

Title: $\infty$-Dold-Kan correspondence via representation theory

Authors: Chiara Sava

Comments: Exposition improved, expanded with new (counter)examples. arXiv admin note: text overlap with arXiv:1409.5003, arXiv:1512.06267 by other authors

Subjects: Representation Theory (math.RT); Algebraic Topology (math.AT); Category Theory (math.CT)

[10]  arXiv:2307.15148 (replaced) [pdf, ps, other]

Title: Isotropic and numerical equivalence for Chow groups and Morava K-theories

Authors: Alexander Vishik

Comments: to appear in Inventiones Mathematicae

Subjects: Algebraic Geometry (math.AG); Algebraic Topology (math.AT); K-Theory and Homology (math.KT)

[11]  arXiv:2404.19710 (replaced) [pdf, other]

Title: A rank decomposition for the topological classification of neural representations

Authors: Kosio Beshkov, Gaute T. Einevoll

Subjects: Machine Learning (cs.LG); Algebraic Topology (math.AT); Neurons and Cognition (q-bio.NC)

• New submissions
• Cross-lists
• Replacements