The topology of moduli spaces of free group representations. (English) Zbl 1200.14093
Let \(G\) be a complex affine reductive group, with compact real form \(K\). As is well known, \(G\) deformation retracts to \(K\).
This paper concerns the Geometric Invariant Theory quotient \(\mathfrak{X}_r(G)\) of the Cartesian power \(G^r\) by diagonal \(G\)-conjugation, and the corresponding quotient \(\mathfrak{X}_r(K)\) of \(K^r\) by diagonal \(K\)-conjugation. The powers \(G^r\) (respectively \(K^r\)) are equivalently described as the spaces of representations of a rank \(r\) free group \(\mathbb{F}_r\) into \(G\) (respectively \(K\)). Clearly \(\mathsf{Hom}(\mathbb{F}_r,G)\approx G^r\) deformation retracts to \(\text{Hom}(\mathbb{F}_r,K)\approx K^r\).
The main result is that the quotient \(\mathfrak{X}_r(G)\) deformation retracts to the quotient \(\mathfrak{X}_r(K)\). Using this the authors show that \(SL(n,\mathbb{C})\)-character variety is homotopy-equivalent to \(S^8\) when \(n=3, r=2\), and homotopy-equivalent to \(S^6\) when \(n=2, r=3\). For \(n=2\), the main result is due to S. Bradholdt and D. Cooper [Rend. Ist. Mat. Univ. Trieste 32, Suppl. 1, 45–53 (2001; Zbl 1062.14505)]. A similar result is proved when \(\mathbb{F}_r\) is replaced by a free abelian group, but counterexamples are given for other finitely generated groups. The result also enables a complete determination of the Poincaré polynomial of \(\mathfrak{X}_r(SL(2,\mathbb{C})\).
The paper is nicely written, and achieves a pleasant balance between the commutative algebra and the topology.
This paper concerns the Geometric Invariant Theory quotient \(\mathfrak{X}_r(G)\) of the Cartesian power \(G^r\) by diagonal \(G\)-conjugation, and the corresponding quotient \(\mathfrak{X}_r(K)\) of \(K^r\) by diagonal \(K\)-conjugation. The powers \(G^r\) (respectively \(K^r\)) are equivalently described as the spaces of representations of a rank \(r\) free group \(\mathbb{F}_r\) into \(G\) (respectively \(K\)). Clearly \(\mathsf{Hom}(\mathbb{F}_r,G)\approx G^r\) deformation retracts to \(\text{Hom}(\mathbb{F}_r,K)\approx K^r\).
The main result is that the quotient \(\mathfrak{X}_r(G)\) deformation retracts to the quotient \(\mathfrak{X}_r(K)\). Using this the authors show that \(SL(n,\mathbb{C})\)-character variety is homotopy-equivalent to \(S^8\) when \(n=3, r=2\), and homotopy-equivalent to \(S^6\) when \(n=2, r=3\). For \(n=2\), the main result is due to S. Bradholdt and D. Cooper [Rend. Ist. Mat. Univ. Trieste 32, Suppl. 1, 45–53 (2001; Zbl 1062.14505)]. A similar result is proved when \(\mathbb{F}_r\) is replaced by a free abelian group, but counterexamples are given for other finitely generated groups. The result also enables a complete determination of the Poincaré polynomial of \(\mathfrak{X}_r(SL(2,\mathbb{C})\).
The paper is nicely written, and achieves a pleasant balance between the commutative algebra and the topology.
Reviewer: William Goldman (College Park)
MSC:
| 14L24 | Geometric invariant theory |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 20E05 | Free nonabelian groups |
| 14D20 | Algebraic moduli problems, moduli of vector bundles |
| 20F29 | Representations of groups as automorphism groups of algebraic systems |
| 20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |
| 57M07 | Topological methods in group theory |
Citations:
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