[Paper Review] Topological minimal genus and $L^2$-signatures

Jae Choon|ArXiv.org|Sep 14, 2006

Geometric and Algebraic Topology23 references4 citations

TL;DR

This paper establishes new lower bounds for the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using von Neumann-Cheeger-Gromov $\rho$-invariants and $L^2$-signatures. It demonstrates that these invariants detect arbitrarily large slice genus in knots for which all prior invariants vanish, providing optimal bounds where previous methods fail.

ABSTRACT

We obtain new lower bounds of the minimal genus of a locally flat surface representing a 2-dimensional homology class in a topological 4-manifold with boundary, using the von Neumann-Cheeger-Gromov $ρ$-invariant. As an application our results are employed to investigate the slice genus of knots. We illustrate examples with arbitrarily large slice genus for which our lower bound is optimal but all previously known invariants vanish.

Motivation & Objective

• To derive new lower bounds for the minimal genus of locally flat surfaces in topological 4-manifolds with boundary.
• To address the minimal genus problem in 4-manifolds where the boundary has nontrivial homology, a case not well-covered by classical invariants.
• To apply these bounds to the slice genus problem for knots, particularly in cases where existing invariants are ineffective.
• To demonstrate that $\rho$-invariants can detect minimal genus values that are arbitrarily large and beyond the reach of classical methods.

Proposed method

• The method uses the von Neumann-Cheeger-Gromov $\rho$-invariant of the boundary 3-manifold associated to a homomorphism $\phi: \pi_1(M) \to \Gamma$, where $\Gamma$ is a PTFA group.
• It relates the $\rho$-invariant to the $L^2$-signature of a 4-manifold $W$ with boundary $M$, estimating the $L^2$-signature in terms of ordinary and $L^2$-Betti numbers.
• A key technical tool is the use of the semi-direct product $(\mathcal{K}/\mathcal{R}) \rtimes \Gamma$, where $\mathcal{K}$ is a skew-field of quotients of $\mathbb{Z}\Gamma$, to construct lifts of $\phi$ to $W$.
• The method applies the structure of the Alexander module and its homomorphisms to $\mathcal{K}/\mathcal{R}$, leveraging results from Gilmer and Cochran-Orr-Teichner on coefficient system extensions.
• It uses the Blanchfield linking form and intersection forms over $\mathcal{R}$-coefficients to analyze when a twisted homology class extends over $W$, enabling iterative lower bounds.
• The approach combines algebraic topology of group rings, $L^2$-invariants, and 4-dimensional bordism theory to derive genus bounds.

Experimental results

Research questions

• RQ1 Can $\rho$-invariants detect minimal genus in 4-manifolds with non-homology-spherical boundary, where classical invariants fail?
• RQ2 What is the minimal second Betti number of a topological null-bordism of a 3-manifold with a given group homomorphism $\phi: \pi_1(M) \to \Gamma$?
• RQ3 How can $L^2$-signatures and $\rho$-invariants be used to derive lower bounds on the slice genus of knots?
• RQ4 Are there knots with arbitrarily large slice genus that are undetected by classical invariants like the Rokhlin invariant or adjunction inequality?
• RQ5 Can the extension of twisted homomorphisms $\phi: \pi_1(M) \to \Gamma$ to $\pi_1(W)$ be used to iteratively improve genus bounds?

Key findings

• The paper establishes the inequality $|\rho(M,\phi)| \leq 2\beta_2(W)$, providing a lower bound on the second Betti number of any topological 4-manifold $W$ with boundary $M$ and fundamental group map factoring through $\pi_1(W)$.
• When the twisted homology $H_1(M;\mathcal{R})$ is $\mathcal{R}$-torsion and not generated by $\beta_2(W)$ elements, a nontrivial submodule of homomorphisms to $\mathcal{K}/\mathcal{R}$ exists, enabling a lift $\phi_1$ of $\phi$ to $W$, which allows iterative application of the bound.
• For knots constructed via connected sums and satellite operations, the method produces examples where the slice genus is exactly $g$, and the $\rho$-invariant lower bound is optimal.
• The construction yields knots with arbitrarily large slice genus $g$ for which all previously known invariants (e.g., Rokhlin, adjunction, signature) vanish, yet the $\rho$-invariant detects the genus.
• The method shows that $g_*^h(K) \geq g$ for knots $K$ obtained by tying $g$ copies of a knot and its inverse, under conditions where $\rho$-invariants of the summands are large and non-canceling.
• The paper proves that such knots are not topologically slice and can be $(1)$-solvable but not $(1.5)$-solvable, showing the invariants detect finer concordance structure.