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On the edge of the stable range
Algebraic Topology · Submitted 31 Aug 2016
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range.
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The abstract, sentence by sentence
We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the "edge" in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range, a description of the last unstable group up to a single ambiguity, and a lower bound on the rank of the penultimate unstable group. We give similar applications to the general linear groups of the integers and of the field of order 2, this time recovering the known stablility range. The results can also be applied to general linear groups of arbitrary principal ideal domains, symmetric groups, and braid groups. Our methods require us to use field coefficients throughout.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. | contribution 0.75 | 0.97 | 0.83 | asserted 1.00 |
| s1 | (These last two unstable groups are the "edge" in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range, a description of the last unstable group up to a single ambiguity, and a lower bound on the rank of the penultimate unstable group. | result 0.58 | 0.95 | 0.21 | asserted 0.99 |
| s2 | We give similar applications to the general linear groups of the integers and of the field of order 2, this time recovering the known stablility range. | result 0.54 | 0.92 | 0.05 | asserted 1.00 |
| s3 | The results can also be applied to general linear groups of arbitrary principal ideal domains, symmetric groups, and braid groups. | contribution 0.59 | 0.79 | 0.05 | asserted 0.70 |
| s4 | Our methods require us to use field coefficients throughout. | limitation 0.90 | 0.81 | 0.02 | asserted 1.00 |
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The paper
Original abstract
We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These last two unstable groups are the "edge" in our title.) Applying our results to automorphism groups of free groups yields a new proof of homological stability with an improved stable range, a description of the last unstable group up to a single ambiguity, and a lower bound on the rank of the penultimate unstable group. We give similar applications to the general linear groups of the integers and of the field of order 2, this time recovering the known stablility range. The results can also be applied to general linear groups of arbitrary principal ideal domains, symmetric groups, and braid groups. Our methods require us to use field coefficients throughout.
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How the literature uses it
6 arXiv papers cite this in 13 sentences
Papers citing it · 7 sentences without a confident label
Compares with it 4 papers
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This makes it clear that <ref> is analogous to [ref] .
Stability in the high-dimensional cohomology of congruence subgroups · Homological vanishing for the Steinberg module of a field · 0.98
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See also Hepworth [ref] for a related result on certain families of groups.
Higher order representation stability and ordered configuration spaces of manifolds · Stability for configuration spaces · 0.89
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A related result regarding the relative groups $H_d(G_σ^2d, G_σ^2d-1;)$ has recently been given by Hepworth [ref] .
Cellular $E_k$-algebras · A generic homological stability result · 0.86
Cites it as background 1 paper
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The question remained for a while: what happens at the `edge' [ref] of this region, for $^n(_2n(_2);_2)$?
The third Milgram-Priddy class lifts · Introduction · 0.90
What it cites
Cites as background 1 paper
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Homological stability is known to hold for many families of groups, including symmetric groups [ref] , general linear groups [ref] , mapping class groups of surfaces and 3-manifolds [ref] , diffeomorphism groups of highly connected manifolds [ref] , and automorphism groups of free groups [ref] .
Homological stability for moduli spaces of high dimensional manifolds. I · Introduction · 0.95
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