scholarly journals On representation varieties of 3–manifold groups

2017 ◽  
Vol 21 (4) ◽  
pp. 1931-1968 ◽  
Author(s):  
Michael Kapovich ◽  
John Millson
Keyword(s):  
Representation Varieties

scholarly journals TOPOLOGY OF THE REPRESENTATION VARIETIES WITH BOREL MOLD FOR UNSTABLE CASES

2011 ◽  
Vol 91 (1) ◽  
pp. 55-87 ◽  
Author(s):  
KAZUNORI NAKAMOTO ◽  
TAKESHI TORII
Keyword(s):  
Configuration Space ◽  
Cohomology Group ◽  
Cohomology Ring ◽  
Picard Group ◽  
Homotopy Equivalent ◽  
Representation Variety ◽  
Stable Case ◽  
Representation Varieties

AbstractIn this paper we show that, in the stable case, when m≥2n−1, the cohomology ring H*(Repn(m)B) of the representation variety with Borel mold Repn(m)B and $H^{\ast }(F_{n}(\mathbb {C}^m)) \otimes H^{\ast }(\mathrm {Flag}(\mathbb { C}^n)) \otimes \Lambda (s_{1}, \ldots , s_{n-1})$ are isomorphic as algebras. Here the degree of si is 2m−3 when 1≤i<n. In the unstable cases, when m≤2n−2, we also calculate the cohomology group H*(Repn(m)B) when n=3,4 . In the most exotic case, when m=2 , Rep n (2)B is homotopy equivalent to Fn (ℂ2)×PGL n (ℂ) , where Fn (ℂ2) is the configuration space of n distinct points in ℂ2. We regard Rep n (2)B as a scheme over ℤ, and show that the Picard group Pic (Rep n (2)B) of Rep n (2)B is isomorphic to ℤ/nℤ. We give an explicit generator of the Picard group.

scholarly journals Homology TQFT's and the Alexander–Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

2003 ◽  
Vol 55 (4) ◽  
pp. 766-821 ◽  
Author(s):  
Thomas Kerler
Keyword(s):  
Moduli Spaces ◽  
Hopf Algebras ◽  
Large Family ◽  
Quantum Invariants ◽  
Topological Quantum ◽  
Irreducible Components ◽  
Skein Theory ◽  
Higher Rank ◽  
The One ◽  
Representation Varieties

AbstractWe develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology ofU(1)-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra= ℤ/2 n ⋊ Λ* ℝ2on the other side. We find that both TQFT's are SL(2; ℝ)-equivariant functors and, as such, are isomorphic. The SL(2; ℝ)-action in the Hennings construction comes from the natural action onand in the case of the Frohman–Nicas theory from the Hard–Lefschetz decomposition of theU(1)-moduli spaces given that they are naturally Kähler. The irreducible components of this TQFT, corresponding to simple representations of SL(2; ℤ) and Sp(2g; ℤ), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg–Witten theories, Casson type theories for homology circlesà laDonaldson, higher rank gauge theories following Frohman and Nicas, and the ℤ=pℤ reductions of Reshetikhin.Turaev theories over the cyclotomic integers ℤ[ζp]. We also conjecture that the Hennings TQFT for quantum-sl2is the product of the Reshetikhin–Turaev TQFT and such a homological TQFT.

scholarly journals The Algebraic de Rham Cohomology of Representation Varieties

2018 ◽  
Vol 70 (3) ◽  
pp. 702-720
Author(s):  
Eugene Z. Xia
Keyword(s):  
De Rham Cohomology ◽  
Natural Family ◽  
De Rham ◽  
The One ◽  
Representation Varieties ◽  
Algebraic De Rham Cohomology ◽  
Natural Families

AbstractThe SL(2, ℂ)-representation varieties of punctured surfaces form natural families parameterized by monodromies at the punctures. In this paper, we compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauß-Manin connection on the natural family of the smooth SL(2, ℂ)-representation varieties of the one-holed torus.

scholarly journals Representation varieties detect essential surfaces

2018 ◽  
Vol 25 (3) ◽  
pp. 803-817
Author(s):  
Stefan Friedl ◽  
Takahiro Kitayama ◽  
Matthias Nagel
Keyword(s):  
Essential Surfaces ◽  
Representation Varieties

scholarly journals Non-abelian cohomology jump loci from an analytic viewpoint

2014 ◽  
Vol 16 (04) ◽  
pp. 1350025 ◽  
Author(s):  
Alexandru Dimca ◽  
Ştefan Papadima
Keyword(s):  
Minimal Model ◽  
Nilpotent Groups ◽  
Exponential Map ◽  
Flat Connections ◽  
Differential Graded Algebra ◽  
Irreducible Components ◽  
Rank One ◽  
Positive Weights ◽  
Projective Manifolds ◽  
Representation Varieties

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

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