scholarly journals Étale homological stability and arithmetic statistics

2018 ◽  
Vol 69 (3) ◽  
pp. 951-974
Author(s):  
Benson Farb ◽  
Jesse Wolfson
Keyword(s):  
Homological Stability ◽  
Arithmetic Statistics

scholarly journals Homological stability for spaces of embedded surfaces

2017 ◽  
Vol 21 (3) ◽  
pp. 1387-1467 ◽  
Author(s):  
Federico Cantero ◽  
Oscar Randal-Williams
Keyword(s):  
Homological Stability ◽  
Embedded Surfaces

scholarly journals Homological stability of series of groups

2010 ◽  
Vol 246 (1) ◽  
pp. 31-47 ◽  
Author(s):  
Tim Cochran ◽  
Shelly Harvey
Keyword(s):  
Homological Stability

scholarly journals Homological Stability for Spaces of Commuting Elements in Lie Groups

2020 ◽  
Author(s):  
Daniel A Ramras ◽  
Mentor Stafa
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Stable Range ◽  
Rational Homology ◽  
Infinite Dimensional ◽  
Fixed Group ◽  
Character Varieties ◽  
Representation Stability ◽  
Equivariant Homology ◽  
Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

scholarly journals Stability of the Cohomology of the Space of Complex Irreducible Polynomials in Several Variables

2019 ◽  
Author(s):  
Weiyan Chen
Keyword(s):  
Finite Fields ◽  
Irreducible Polynomials ◽  
Compactly Supported ◽  
Several Variables ◽  
Homological Stability

Abstract We prove that the space of complex irreducible polynomials of degree $d$ in $n$ variables satisfies two forms of homological stability: first, its cohomology stabilizes as $d\to \infty $, and second, its compactly supported cohomology stabilizes as $n\to \infty $. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.

scholarly journals A NOTE ON RATIONAL HOMOLOGICAL STABILITY OF AUTOMORPHISMS OF MANIFOLDS

2020 ◽  
Vol 71 (3) ◽  
pp. 1069-1079
Author(s):  
Manuel Krannich
Keyword(s):  
Invariant Theory ◽  
Characteristic Classes ◽  
Classical Invariant Theory ◽  
Homological Stability ◽  
Classical Invariant

Abstract By work of Berglund and Madsen, the rings of rational characteristic classes of fibrations and smooth block bundles with fibre $D^{2n}\sharp (S^n\times S^n)^{\sharp g}$, relative to the boundary, are for $2n\ge 6$ independent of $g$ in degrees $*\le (g-6)/2$. In this note, we explain how this range can be improved to $*\le g-2$ using cohomological vanishing results due to Borel and the classical invariant theory. This implies that the analogous ring for smooth bundles is independent of $g$ in the same range, provided the degree is small compared to the dimension.

scholarly journals Arithmetic statistics of modular symbols

2018 ◽  
Vol 212 (3) ◽  
pp. 997-1053 ◽  
Author(s):  
Yiannis N. Petridis ◽  
Morten S. Risager
Keyword(s):  
Modular Symbols ◽  
Arithmetic Statistics

scholarly journals Homological stability for topological chiral homology of completions

2016 ◽  
Vol 292 ◽  
pp. 755-827 ◽  
Author(s):  
Alexander Kupers ◽  
Jeremy Miller
Keyword(s):  
Homological Stability
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