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arxiv: 2606.12206 · v1 · pith:JEXZDKQBnew · submitted 2026-06-10 · 🧮 math.AT

Stable homology of complex braid groups

Andrea Bianchi , Filippo Callegaro , Luigi Caputi , Paolo Salvatore This is my paper

Pith reviewed 2026-06-27 07:27 UTC · model grok-4.3

classification 🧮 math.AT
keywords stable homologycomplex braid groupsArtin groupsquillenizationclassifying spacestype B(e,e,n)type D
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The pith

The stable homology of complex braid groups of types B(e,e,n) and B(2e,e,n) is computed explicitly via quillenization of their classifying spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes the stable homology of the complex braid groups in the families B(e,e,n) and B(2e,e,n) for each fixed e at least 2 as n grows. It obtains these groups by explicitly calculating a quillenization of the stable classifying spaces. This calculation covers every infinite family of complex braid groups. A sympathetic reader cares because the same computation proves the identification of stable homology for Artin groups of type D that Fuchs claimed in the 1970s.

Core claim

We compute the stable homology of complex braid groups of types B(e,e,n) and B(2e,e,n) for fixed e≥2 and increasing n by explicitly computing a quillenization of their stable classifying spaces. This accounts for the stable homology of all infinite families of complex braid groups and provides a proof of an identification of the stable homology of Artin groups of type D claimed by Fuchs in the 1970s.

What carries the argument

The quillenization of the stable classifying spaces of the braid groups, which is computed explicitly to determine the stable homology.

If this is right

  • The stable homology of Artin groups of type D is identified as previously claimed.
  • Stable homology is now known for every infinite family of complex braid groups.
  • The quillenization method supplies concrete descriptions of the homology groups in the stable range.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same quillenization technique may apply to stable homology computations for other Artin groups or reflection groups.
  • The explicit groups obtained could be compared with stable homology arising from configuration spaces or mapping class groups.
  • These results provide concrete test cases for conjectures relating braid group homology to algebraic K-theory.

Load-bearing premise

The quillenization of the stable classifying spaces of these braid groups can be explicitly computed and yields the correct stable homology groups.

What would settle it

Direct computation of homology groups of the classifying spaces for large n, followed by checking whether the groups match those predicted by the quillenization.

read the original abstract

We compute the stable homology of complex braid groups of types $B(e,e,n)$ and $B(2e,e,n)$ for fixed $e\ge2$ and increasing $n$. This accounts for the stable homology of all infinite families of complex braid groups. We achieve this by explicitly computing a quillenization of their stable classifying spaces. In particular, we provide a proof of an identification of the stable homology of Artin groups of type $D$ claimed by Fuchs in the '70s.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The manuscript computes the stable homology of complex braid groups of types B(e,e,n) and B(2e,e,n) for fixed e≥2 and increasing n. This is achieved by explicitly computing a Quillenization of their stable classifying spaces. As a special case, the work provides a proof of an identification of the stable homology of Artin groups of type D previously claimed by Fuchs in the 1970s.

Significance. If the computations are correct, the result completes the stable homology for all infinite families of complex braid groups. The explicit Quillenization approach and the resolution of Fuchs' claim constitute a concrete contribution in algebraic topology.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for noting its significance in completing the stable homology computations for all infinite families of complex braid groups, as well as resolving Fuchs' claim for type D Artin groups. The recommendation of 'uncertain' appears to stem from the absence of listed major comments; we provide the following point-by-point responses to the referee report as presented.

Circularity Check

0 steps flagged

No significant circularity; explicit computation of quillenization stands independently

full rationale

The paper's central result is an explicit computation of the quillenization of the stable classifying spaces for the indicated families of complex braid groups, which directly yields the stable homology groups. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided abstract or description. The recovery of Fuchs' type-D claim follows from the same construction rather than from any prior result by the authors. The derivation is presented as self-contained against external topological benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; cannot identify any.

pith-pipeline@v0.9.1-grok · 5600 in / 981 out tokens · 17612 ms · 2026-06-27T07:27:28.711729+00:00 · methodology

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Reference graph

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