41 problems

Fried's conjecture. The function $\zeta_{g,\rho}(s)$ is well-defined and nonvanishing at $s=0$, and

Let $K\subset{\mathbb S}^3$ be a hyperbolic knot, let $\gamma$ be a simple closed curve on $\partial\nu(K)$, and let $\rho_0$ be the discrete faithful representation associated wit…

Let $\pi: M \rightarrow S$ be a smooth fibration, with total space $M$, and let $F\to M$ be a flat complex vector bundle. The Wagoner conjecture. Reidemeister–Franz torsion and Ray…

Let $M$ be a 3-manifold with torus boundary, let $\rho$ range over representations into $\operatorname{SL}_2(\mathbb{C})$, and let…

Let $M$ be an oriented compact $3$-manifold with torus boundary, and let $X^\mathrm{irr}(M)$ be the character variety of irreducible $\mathrm{SL}_{2}(\mathbb{C})$-representations.…

Lück's homotopy-invariance conjecture. $L^2$-Reidemeister torsion is a homotopy invariant.

Let $F$ be a unitary flat vector bundle on a closed Riemannian manifold $X$. Ray and Singer's analytic torsion is the torsion invariant constructed from the spectrum of the Laplaci…

Torsion equality conjecture. The $L_2$-analytic and $L_2$-Reidemeister torsions of $M$ are equal.

Let $M$ be a $3$-manifold, let $g$ be any non-negative integer, and consider the irreducible representations of the fundamental group of $M$ together with their adjoint Reidemeiste…

Let $\mathring{D}$ be an essential open diagram, let $\omega(\mathring{D})$ denote its Ohtsuki–Takata invariant, and let the holonomy representation determine the associated adjoin…

Let $M$ be a cusped hyperbolic $3$-manifold with an essential ideal triangulation detecting its geometric representation. Its 1-loop invariant is an element of the invariant trace…

Let $M$ be a closed $3$-manifold, let $\varphi$ be a smooth transitive Anosov flow on $M$, and let $\rho$ be an acyclic unitary representation of $\pi_1(M)$. Define the twisted Rue…

Higher Whitehead torsion and Reidemeister trace compatibility conjecture. The following diagram commutes up to natural homotopy:

Let $M$ be a smooth closed manifold, let $\mathcal{H}(M)$ be its stable $h$-cobordism space, and suppose the higher Reidemeister trace constructions define a map … Let…

A complete non-compact hyperbolic three-manifold is a hyperbolic three-manifold with complete finite-volume hyperbolic structure and noncompact underlying space. The 1-loop invaria…

Let $K$ be a hyperbolic knot. For a parameter $u$, let $\rho_u\colon\pi_1(S^3\setminus K)\to\operatorname{SL}(2;\mathbb{C})$ be the irreducible representation specified by the meri…

Let $K$ be a hyperbolic knot. Let $\operatorname{Vol}(K)$ be its hyperbolic volume, let $\operatorname{CS}^{\mathrm{SO}(3)}(K)$ be its $\mathrm{SO}(3)$ Chern--Simons invariant, and…

Let $\mathscr{H}\subset S^3$ be a hyperbolic knot. Let $\operatorname{cv}(\mathscr{H})$ be its complex volume, and let $T(\mathscr{H})$ be the adjoint cohomological Reidemeister to…

Let $\mathfrak{g}$ be the Lie algebra of a semisimple complex Lie group $G$, let $M$ be a connected closed oriented $3$-manifold, and let $R^{\rm irr}_G(M)$ be the set of conjugacy…

The 1-loop invariant conjecture. The invariant agrees with the torsion up to sign:

Let $M$ be a cusped hyperbolic 3-manifold, let $\mathcal{T}$ be an ideal triangulation, and choose an epimorphism $\alpha:\pi_1(M)\twoheadrightarrow\mathbb{Z}$, equivalently an inf…

Let $M$ be a cusped hyperbolic 3-manifold, let $z$ be an algebraic solution of Thurston's equations, let $\rho_z$ be its holonomy representation, and let $\gamma$ be an essential s…

DGG 1-loop invariant conjecture. The 1-loop invariant is equal to the adjoint torsion:

Let $K$ be a knot in $S^3$, let $X_K$ be its exterior, and let $\phi\in H^1(X_K;\mathbb{Z})$ be a generator. A pair $(\rho,\phi)$ is assumed to be admissible, and…

Let $M$ be a compact oriented $3$-manifold with torus boundary, and let $\mathcal{X}(M)$ be the character variety of irreducible representations…