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Bad Representations and Homotopy of Character Varieties

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Pith's one-line read For a connected reductive complex group $G$ and a rank $r$ free group, the paper proves that in a stable range the $k$th homotopy group of the smooth locus of the character variety is $\pi_k(G)^r \times \pi_{k-1}(PG)$, where $PG$ is $G$…

desk verdict A substantial paper with correct-looking central results, but the strict codimension bound in Lemma 4.13/Theorem 4.14 is false as written; fixable, but the advertised stable ranges need weakening. read the letter →

arxiv 1908.02915 v4 pith:YO6BARJE submitted 2019-08-08 math.AG math.ATmath.RT

Clément Guérin , Sean Lawton , Daniel Ramras This is my paper · ORCID
classification math.AGmath.ATmath.RT MSC 14B0514L2455Q0514D2014L3055U10
keywords charactervarietyfreegroupBorel-deSiebenthalsubgrouphomotopygroupssingularlocusreductiveGITquotientCI-group
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that the geometry of $G$-character varieties of free groups is controlled by a short list of subgroups of $G$: Levi subgroups of parabolics and Borel-de Siebenthal subgroups. Away from the singular strata, the homotopy groups of the smooth locus are as simple as possible, matching a product of $r$ copies of the homotopy of $G$ with one shifted copy of the homotopy of $PG=G/Z(G)$. This settles open conjectures about the second homotopy group, about which singularities are topological, and about which groups admit no noncentral elements commuting with an irreducible representation.

What carries the argument

The engine is a quantitative bound on the codimension of the bad locus. A bad representation must normalize either a Levi subgroup of a proper parabolic subgroup or a Borel-de Siebenthal subgroup, meaning a proper reductive subgroup of maximal rank whose root system has the same rank as the ambient group; there are only finitely many conjugacy classes. The paper classifies these subgroups for each simple Lie type and tabulates their codimensions, giving the bound $\mathrm{codim}(\mathrm{Hom}_{\mathrm{bad}}(F_r,G))>2(r-1)\min\mathrm{Rank}(G')$. Locally, the slice theorem identifies a neighborhood of $[\rho]$ with $H^1(F_r;\mathfrak{g}_{\mathrm{Ad}\,\rho})/\!/\mathrm{Stab}(\rho)$, and the principal $PG$-bundle $\mathrm{Hom}(F_r,G)_{\mathrm{good}}\to X_r(G)_{\mathrm{good}}$ turns the codimension bound into a connectedness range for maps into the good locus, after which the homotopy groups are read off from the long exact sequence of that bundle.

What would settle it

Compute the dimension of the image of $G\times\mathrm{Hom}(F_2,H)\to\mathrm{Hom}(F_2,G)$ for $G=SL_3(\mathbb{C})$ and $H$ the normalizer of a maximal torus: a generic pair in $H$ has centralizer exactly the torus, giving a 10-dimensional image and codimension 6, equal to $(r-1)\mathrm{codim}(H)$, so the strict inequality in Lemma 4.13 fails. Alternatively, evaluate the $B_r$ row of Table 3: the codimension of $D_k+B_{r-k}$ in $B_r$ should be $2k(2r-2k+1)$, not the printed $2k(2r-2k-1)$.

Watch

Extended reading notes

Core claim

The central theorem is that for $r>2$ and $1\le k\le C_{\mathrm{pasbon}}-2$, where $C_{\mathrm{pasbon}}$ is the real codimension of the union of bad and reducible representations, the good locus satisfies $\pi_k(X_r(G)_{\mathrm{good}})\cong\pi_k(G)^r\times\pi_{k-1}(PG)$. In particular $\pi_2(X_r(G)_{\mathrm{good}})\cong\pi_1(PG)$, confirming the expected generalization of the known $\pi_2$ computation for general and special linear groups. Under mild rank or rank-parameter hypotheses the good locus is the smooth locus, so the formula applies to the smooth part of the moduli space. The same circle of ideas proves that the singular locus is exactly the union of the reducible and bad loci, that every algebraic singularity is a topological singularity, and that a connected reductive group has no bad subgroups precisely when its derived subgroup is a product of special linear groups.

Load-bearing premise

The homotopy range is carried by the strict codimension bound for the bad-and-reducible locus, which is assembled from the tabulated codimensions of maximal parabolic and Borel-de Siebenthal subalgebras and from a strict inequality in Lemma 4.13 whose proof only supports a non-strict version.

Editorial extensions

If this is right

  • For $1\le k\le C_{\mathrm{pasbon}}-2$, the good locus has homotopy $\pi_k(G)^r\times\pi_{k-1}(PG)$, and in particular $\pi_2(X_r(G))\cong\pi_1(PG)$ whenever the smooth locus is the good locus.
  • When $r>3$ (or $r>2$ with no rank-one simple factors), the singular locus of the character variety is precisely the union of the reducible and bad loci, and every such singular point is a topological singularity.
  • In the stable range, the third and fourth homotopy groups of the good locus are $\mathbb{Z}^{sr}$ and $(\mathbb{Z}_2)^{rt}\times\mathbb{Z}^s$, where $s$ is the number of simple factors and $t$ counts factors of types $A_1$, $B_1$, or $C_n$ with $n>1$.
  • For the classical groups, the formula yields periodic homotopy in the stable range, reproducing Bott-periodicity-style shifts in $r$ and in the rank of $G$.
  • A connected reductive group has no bad subgroups if and only if its derived subgroup is a product of special linear groups, so the classical centralizer lemma for irreducible representations holds only for those groups.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial extension: the stable-range formula suggests that the smooth locus is, up to high-dimensional homotopy, a bundle over $BPG$ with fibre $G^r$, so the rational cohomology of the smooth locus should be computable from the rational cohomology of $G^r$ and $BPG$ in that range.
  • Editorial extension: the same Borel-de Siebenthal classification should control the singular locus and stable homotopy of character varieties of surface groups, since the one-relation presentation changes the cohomological local model but not the subgroup classification.
  • Editorial extension: the main qualitative conclusion—that the smooth locus is stably a homotopy product of the group and a loop factor—is insensitive to the exact value of the codimension constant, so small corrections to the codimension tables would shift the range of the theorem without destroying it.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the GIT quotient X_r(G) of Hom(F_r,G) by conjugation for a connected reductive complex algebraic group G. It gives a new codimension bound for the locus of bad (irreducible but not good) representations using Borel–de Siebenthal subgroups, uses this to identify the algebraic and topological singular loci of X_r(G), and then computes higher homotopy groups of the good locus in a range depending on r. The headline results are Theorem 6.10, asserting that for 1 ≤ k ≤ Cpasbon − 2 one has π_k(X_r(G)_good) ≅ π_k(G)^r × π_{k−1}(PG), and Theorem 8.3, classifying CI-groups as exactly those whose derived subgroup is a product of special linear groups. These resolve conjectures of Florentino–Lawton, Florentino–Lawton–Ramras, and Sikora.

Significance. If the main theorems are correct, this is a substantial advance: it settles long-standing conjectures about singular loci, proves the conjectured isomorphism π_2(X_r(G)) ≅ π_1(PG), and gives a clean characterization of CI-groups. The use of Borel–de Siebenthal subgroups to control bad representations is elegant and connects representation theory of free groups with root-system classification. The paper is also genuinely theorem-driven rather than example-driven, and the tables in Appendix A provide useful data. However, the current proof has load-bearing issues in the strictness of the codimension bound and in the tabulated B_r codimension, so the results cannot be accepted as written.

major comments (3)
  1. [§4, Lemma 4.13 and Theorem 4.14] Lemma 4.13 as stated is false. The statement uses codim_C(H) for both the subgroup H and the image of φ_H, and the proof via Hardt's theorem only yields the non-strict inequality codim_C(im φ_H) ≥ (r − 1) codim_C(H), not the strict inequality claimed. The bound 'each fiber has dimension at least dim H' gives an upper bound on the image dimension, and therefore a non-strict codimension lower bound. The strict version fails concretely: for G = SO(5,C), S = SO(4,C) a maximal BdS subgroup of codimension 4, and r = 3, the set of conjugates of representations with Zariski-dense image in S is contained in Hom_bad and has complex codimension (r − 1)·4 = 8, equal to 2(r − 1)Rank(DG). This contradicts the strict form of Theorem 4.14. The non-strict version may still be true and may suffice for the main π_2 conjecture, but Theorem 6.5, Remark 6.13, Corollary 6.11, and the stable ranges in Table 1 and Example 6.16 currently rely on the strict form and must be re-derived with the corrected bound.
  2. [Appendix A, Table 3, B_r row] The codimension formula for BdS subalgebras of B_r is misprinted. For s = D_k + B_{r−k}, the correct codimension in B_r is 2k(2r − 2k + 1), not 2k(2r − 2k − 1). For example, when r = 2 and k = 2, the printed formula gives −4, whereas the actual codimension of so(4,C) in so(5,C) is 4. The minimum column 2r is consistent with the corrected formula, but Theorem 4.14's proof cites this table for the required codimension bound, so the table must be corrected and the surrounding argument re-checked.
  3. [§4, proof of Theorem 4.14] The reduction to 'maximal proper parabolic or maximal proper BdS' subgroups is not justified. Proposition 4.9 shows that a bad representation normalizes some Levi subgroup of a parabolic or some BdS subgroup, but it does not show that it normalizes a maximal such subgroup. The covering argument only needs the finite family of all such subgroups, and the minimum codimension is attained on maximal ones, but the text's 'we may assume without loss of generality' step is logically incomplete. This should be rewritten so that the finite set S is taken to be all relevant subgroups, with the minimum computed over the maximal ones.
minor comments (4)
  1. [§4, Lemma 4.13] The notation should distinguish between the subgroup H and the image of φ_H; as written, codim_C(H) appears on both sides with different meanings.
  2. [§5.2, proof of Theorem 5.12] The final paragraph 'Finally, if r = 2...' is outside the theorem's hypotheses, which only cover r > 3 or r > 2 with rank conditions. Either remove this paragraph or state and prove a separate result for r = 2. The claim that ∑_{n≥1} dim u_n > 2 is also not sufficient for Lemma 5.8, and for A_2 the sum equals 2.
  3. [Appendix A, Table 3] The table header contains the typo 'BbS' and should read 'BdS'.
  4. [Example 6.17] The notation 'Z11!/32' is ambiguous and should be typeset as Z_{11!/32} or with braces around the order.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central theorems are derived from external structure theorems and prior published lemmas whose assumptions do not contain the target results.

full rationale

Walking the derivation chain, Theorem 6.10 is obtained from the principal PG-bundle Hom_good -> X_good via the long exact homotopy sequence, using a transversality bound (Lemma 6.6), nullhomotopy of adjoint-orbit inclusions quoted from prior work (Lemma 6.7, [FLR17, Lemma 4.4]), and a splitting argument (Proposition 7.1, Lemma 7.2 from Miyata). The stable range is controlled by Cpasbon, which is defined as a real codimension and then lower-bounded by Theorem 4.14 using Richardson's theorem, Luna's slice theorem, and the Tits/Borel-de Siebenthal classification; no parameter is fitted and no occurrence of the target isomorphism is assumed. Theorem 8.3 is derived from Lemma 8.1 (a centralizer computation for BdS subgroups), Corollary 8.2, and Lemma 8.4, not from Sikora's conjecture. Self-citations to [FL12] and [FLR17] appear as prior published theorems with stated assumptions that do not include the conjectures being proved; for instance, [FLR17, Lemma 4.4] is a concrete nullhomotopy statement, and [FLR17, Theorems 7.4 and 7.8] are singular-locus reductions. These are independent evidence rather than circular inputs. The skeptic's concern about Lemma 4.13 yielding only a non-strict inequality and the apparent B_r table sign error is a correctness or stable-range sharpness issue, not a reduction of the paper's central claims to their own assumptions.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new entities or fitted constants. It relies on standard classification and GIT theorems; the main input that a reader cannot check quickly is the correctness of the codimension tables.

assumptions (6)
  • standard math Borel-de Siebenthal classification of maximal rank reductive subgroups and the tabulated codimensions (Tables 2 and 3)
    Invoked in the proof of Theorem 4.14 to bound the codimension of normalizers of bad subgroups; the table values are the quantitative content of the bound.
  • standard math Richardson's theorem on singularities of semisimple character varieties and its extension in [FLR17, Theorem 7.4]
    Used in the proof of Theorem 5.10 for the r>2 and rank at least 2 case.
  • standard math Luna's slice theorem and the local model H^1(F_r; g_{Ad rho}) // Stab(rho)
    Used in Lemma 5.4 and the singularity proofs, providing the local description of points in the character variety.
  • standard math Hardt's semialgebraic triviality theorem
    Used in Lemma 4.13 to bound the codimension of images of conjugation maps.
  • domain assumption Connected reductive complex affine algebraic group G and free group rank r assumptions
    The whole paper restricts to this setting, and the results explicitly require r>2 or r>3 depending on the rank condition on simple factors.
  • standard math The computation Z_G(S)/Z(G) ~= Lambda_G / Lambda_S for BdS subgroups (Lemma 8.1)
    Used in Corollary 8.2 and Theorem 8.3 to show that BdS subgroups are bad and to classify CI groups.

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Pith. "Pith review of Bad Representations and Homotopy of Character Varieties." pith.science (2026). https://pith.science/paper/YO6BARJE

@misc{pith2026190802915,
  author       = {Pith},
  title        = {Pith review of: Bad Representations and Homotopy of Character Varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YO6BARJE}},
  note         = {Machine review of arXiv:1908.02915}
}
read the original abstract

Let G be a connected reductive complex affine algebraic group, and let X denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in X, extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino-Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel-de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of X, proving conjectures of Florentino-Lawton-Ramras. Lastly, using the earlier analysis of Borel-de Siebenthal subgroups, we prove a conjecture of Sikora about centralizers of irreducible representations in Lie groups.

Figures

Figures reproduced from arXiv: 1908.02915 by the authors.

Figure 1
Figure 1. Venn Diagram of Representations. 4. Bounding the codimension of bad representations To prove our main theorems, we need to bound the codimension of the bad locus. To do so, we need to study the Lie algebras of reductive C-groups. One can find the following terms, facts and notation in any standard text covering Lie algebras like [OV90], or [FH91] [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. BdS Subalgebras of g2. In the second diagram in [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗

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