[Paper Review] Manolescu correction terms and knots in the three-sphere

Francesco Lin|arXiv (Cornell University)|Jul 18, 2016

Geometric and Algebraic Topology16 references3 citations

TL;DR

This paper establishes constraints on Manolescu correction terms for homology spheres obtained by Dehn surgery on knots in the three-sphere or in integral L-spaces, using monopole Floer homology and the Eilenberg-Moore spectral sequence. It shows that for even surgeries, two of the three correction terms coincide, and provides explicit formulas for the invariants in terms of the knot's genus and surgery coefficient, offering new obstructions to homology cobordism.

ABSTRACT

Manolescu correction terms are numerical invariants of homology three-spheres arising from $\mathrm{Pin}(2)$-equivariant Seiberg-Witten theory that contain information about homology cobordism. We discuss several constraints on these invariants for homology spheres obtained by Dehn surgery on a knot in the three-sphere (and, more generally, in an integral homology $L$-space) in terms of the surgery coefficient, the concordance order, and the genus.

Motivation & Objective

• To understand the structure of the homology cobordism group by analyzing correction terms of homology spheres arising from Dehn surgery on knots.
• To determine how Manolescu correction terms behave under even and odd Dehn surgeries on knots in integral L-spaces.
• To investigate whether every homology sphere is homology cobordant to a surgery on a knot in S³, a key question in low-dimensional topology.
• To extend the use of Floer-theoretic invariants to detect obstructions in homology cobordism, especially for knots like Whitehead doubles.
• To provide explicit formulas for correction terms in terms of knot invariants such as genus and concordance order, using algebraic topology tools like the Eilenberg-Moore spectral sequence.

Proposed method

• Uses monopole Floer homology to define Manolescu correction terms α(Y), β(Y), γ(Y), and δ(Y), which are invariants under homology cobordism.
• Applies the Eilenberg-Moore spectral sequence to compute the homology of the free loop space of a 3-manifold, leveraging projective resolutions of R-modules.
• Employs a two-step projective resolution of the module Mₖ* to compute Tor groups, which encode the correction terms via the spectral sequence's E²-page.
• Uses Poincaré duality and grading shifts to relate the Tor groups to the correction terms, particularly focusing on the bottom and top gradings.
• Analyzes the action of operators Q and V on the Tor groups to determine the structure of the correction terms, especially in connected sums.
• Applies the spectral sequence to connected sums of homology spheres of simple type Mₖ*, showing that correction terms depend only on the two largest indices.

Experimental results

Research questions

• RQ1 What constraints do Manolescu correction terms satisfy for homology spheres obtained by even Dehn surgery on a knot in S³?
• RQ2 How do the correction terms α, β, γ behave under odd surgeries, and what invariants of the knot determine them?
• RQ3 Can the correction terms detect whether a homology sphere is not homology cobordant to any surgery on a knot in S³?
• RQ4 Do the correction terms of Whitehead doubles carry new concordance information beyond the knot's δ-invariant?
• RQ5 To what extent can the correction terms of connected sums of homology spheres of simple type be computed algebraically using spectral sequences?

Key findings

• For even surgery on a knot in an integral L-space, two of the three Manolescu correction terms coincide: if the surgery is positive, α(Y') = β(Y'), and if negative, β(Y') = γ(Y').
• For positive even surgery, γ(Y') = δ(Y') if δ(Y') is even, and γ(Y') = δ(Y') - 1 otherwise; for negative surgery, α(Y') = δ(Y') if δ(Y') is even, and α(Y') = δ(Y') + 1 otherwise.
• The correction term δ(Y') for a surgery on a knot K is determined by the knot's δ-invariant and the sign of the surgery coefficient, with parity of the coefficient playing a crucial role.
• For Whitehead doubles Wh(K) with a positive clasp, α(Wh(K)) = β(Wh(K)) = 0, and γ(Wh(K)) = δ(K) if δ(K) is even, δ(K) - 1 otherwise, showing no new concordance information beyond δ(K).
• The Eilenberg-Moore spectral sequence collapses at the E²-page for connected sums of homology spheres of simple type, allowing explicit computation of correction terms via Tor groups.
• For any N, there exist hyperbolic homology spheres whose monopole Floer homology contains sequences of elements with complex R-module structure, showing that the homology can be arbitrarily complicated.