[Paper Review] $T^3$-fibrations on compact six-manifolds

Patrick Baier|ArXiv.org|Sep 13, 2001

Geometric Analysis and Curvature Flows9 references3 citations

TL;DR

This paper constructs explicit $T^3$-fibrations over $S^3$ with singular fibers over torus knots by leveraging monodromy representations of knot groups into $\mathrm{SL}(3,\mathbb{Z})$. It shows that topological invariants of the total space can be computed algebraically from the monodromy, leading to new examples of $T^3$-fibrations on $S^3\times S^3$ and connected sums involving $S^4\times S^2$, with discriminant locus a torus knot $\mathfrak{t}(2p',3q')$. The construction generalizes to all such torus knots satisfying number-theoretic conditions on $p'$ and $q'$.

ABSTRACT

We describe a simple way of constructing torus fibrations $T^3 o X o S^3$ which degenerate canonically over a knot or link in $S^3$. We show that the topological invariants of $X$ can be computed algebraically from the monodromy representation of the fibration. We use this to obtain some new $T^3$-fibrations $S^3 imes S^3 o S^3$ and $(S^3 imes S^3)#(S^3 imes S^3)#(S^4 imes S^2) o S^3$ whose discriminant locus is a torus knot.

Motivation & Objective

• To develop a systematic method for constructing $T^3$-fibrations over $S^3$ with canonical singularities over knots or links.
• To show that topological invariants of the total space $X$ can be computed purely algebraically from the monodromy representation of the knot group on $H_1(T^3,\mathbb{Z})$.
• To construct new examples of $T^3$-fibrations on compact six-manifolds, including $S^3\times S^3$ and connected sums involving $S^4\times S^2$.
• To classify $M$-representations of torus knot groups $G(p,q) = \pi_1(S^3 \setminus \mathfrak{t}(p,q))$ into $\mathrm{SL}(3,\mathbb{Z})$ that admit canonical compactification to $T^3$-fibrations.

Proposed method

• Uses affine $T^n$-bundles with monodromy representations $\varrho: \pi_1(S^3 \setminus \mathfrak{k}) \to \mathrm{SL}(3,\mathbb{Z})$ to define fibrations over $S^3$ with singular fibers over a knot $\mathfrak{k}$.
• Applies a canonical compactification procedure to linear $T^3$-bundles over $S^3 \setminus \mathfrak{k}$, using the monodromy to extend the fibration across the knot.
• Employs the Chern class $c(f) \in H^2(B, \mathcal{L})$ to classify affine $T^n$-bundles and determines when a section exists (i.e., when $c(f) = 0$).
• Utilizes the universal cover $\widehat{B}$ and the action of $\pi_1(B)$ via $\varrho$ to construct the total space as $X = \widehat{B} \times_\varrho \mathbb{T}^3$.
• Applies representation theory of knot groups, particularly for torus knots $\mathfrak{t}(p,q)$, to classify $M$-representations into $\mathrm{SL}(3,\mathbb{Z})$.
• Reduces the computation of topological invariants (e.g., Betti numbers, Euler characteristic) to algebraic invariants of the monodromy representation $\varrho$.

Experimental results

Research questions

• RQ1 Which monodromy representations $\varrho: \pi_1(S^3 \setminus \mathfrak{k}) \to \mathrm{SL}(3,\mathbb{Z})$ give rise to compactifiable $T^3$-fibrations over $S^3$ with singular fibers over a knot $\mathfrak{k}$?
• RQ2 How can the topological invariants of the total space $X$ of such a fibration be computed directly from the monodromy representation $\varrho$?
• RQ3 What are the conditions on the parameters $p,q$ for a torus knot $\mathfrak{t}(p,q)$ to support non-trivial $M$-representations into $\mathrm{SL}(3,\mathbb{Z})$?
• RQ4 Which compact six-manifolds admit $T^3$-fibrations with discriminant locus a torus knot $\mathfrak{t}(2p',3q')$?
• RQ5 Can the classification of $M$-representations of $\pi_1(S^3 \setminus \mathfrak{t}(p,q))$ be reduced to that of the $(2,3)$ or $(4,3)$ torus knots?

Key findings

• The paper constructs an infinite family of non-isomorphic $T^3$-fibrations on $S^3 \times S^3$ with discriminant locus a torus knot $\mathfrak{t}(2p',3q')$ where $p'$ is odd.
• It constructs another infinite family of $T^3$-fibrations on $(S^3 \times S^3)\#(S^3 \times S^3)\#(S^4 \times S^2)$ with discriminant locus $\mathfrak{t}(2p',3q')$ where $p'$ is even.
• For the $(4,3)$-torus knot, the paper explicitly constructs an infinite family of non-abelian $M$-representations $\varrho_k$ of the knot group into $\mathrm{SL}(3,\mathbb{Z})$, with $\varrho_{-k} = (\varrho_k^{-1})^T$.
• The $M$-representations of the $(p,q)$-torus knot group exist if and only if $p = 2^m p'$ and $q = 3^k q'$ with $\gcd(p',6) = \gcd(q',6) = \gcd(p',q') = 1$, $m,k > 0$.
• There is a bijection between $M$-representations of $G(p,q)$ and those of $G(2,3)$ when $m=1$, and of $G(4,3)$ when $m>1$, via a canonical lifting of representations using exponentiation by $\pm 1$.
• The topological invariants of the total space $X$, such as Betti numbers and Euler characteristic, are fully determined by the monodromy representation $\varrho$ without requiring geometric or topological data beyond the group representation.