[Paper Review] A connected sum formula for involutive Heegaard Floer homology

Kristen Hendricks, Ciprian Manolescu+1 more|arXiv (Cornell University)|Jul 25, 2016

Geometric and Algebraic Topology20 references4 citations

TL;DR

This paper establishes a connected sum formula for involutive Heegaard Floer homology, proving that the conjugation involution on the Floer complex of a connected sum of three-manifolds is chain homotopy equivalent to the tensor product of the individual involutions. The key result is the construction of a homomorphism from the homology cobordism group to an algebraically defined Abelian group of complexes with involutions, and the first example of a three-manifold with distinct involutive correction terms: $\underline{d}(Y) \neq d(Y) \neq \bar{d}(Y)$.

ABSTRACT

We prove a connected sum formula for involutive Heegaard Floer homology, and use it to study the involutive correction terms of connected sums. In particular, we give an example of a three-manifold with $\underline{d}(Y) eq d(Y) eq \overline{d}(Y)$. We also construct a homomorphism from the three-dimensional homology cobordism group to an algebraically defined Abelian group, consisting of certain complexes (equipped with a homotopy involution) modulo a notion of local equivalence.

Motivation & Objective

• To establish a connected sum formula for involutive Heegaard Floer homology, extending the known formula for standard Heegaard Floer homology.
• To study the behavior of involutive correction terms $\underline{d}$ and $\bar{d}$ under connected sums.
• To construct a homomorphism from the three-dimensional homology cobordism group $\Theta^3_{\mathbb{Z}}$ to an algebraically defined Abelian group of complexes with homotopy involutions modulo local equivalence.
• To provide the first example of a three-manifold with $\underline{d}(Y) \neq d(Y) \neq \bar{d}(Y)$, demonstrating the refined detection power of involutive invariants.

Proposed method

• Prove that the conjugation involution $\iota$ on $\mathit{CF}^{-}(Y_1\#Y_2, \mathfrak{s}_1\#\mathfrak{s}_2)$ is chain homotopy equivalent to $\iota_1 \otimes \iota_2$ over $\mathbb{Z}_2[U]$, using graph cobordism maps and algebraic arguments.
• Use the theory of mapping cones to compute $\mathit{CFI}^{-}(Y_1\#Y_2, \mathfrak{s}_1\#\mathfrak{s}_2)$ as the cone of $Q(1 + \iota_1 \otimes \iota_2)$ on $\mathit{CF}^{-}(Y_1) \otimes \mathit{CF}^{-}(Y_2)$.
• Leverage the $\mathbb{Z}$-grading of $\mathit{CF}^{-}$ complexes to show that an extra term $U(\Phi_1\iota_1 \otimes \Phi_2\iota_2)$ is chain homotopic to zero.
• Define involutive Heegaard Floer homology $\mathit{HFI}^{-}$ as the homology of the mapping cone complex over $\mathcal{R} = \mathbb{Z}_2[Q,U]/(Q^2)$.
• Explore the possibility of an $\mathcal{A}_\infty$-module structure on $\mathit{HFI}^{-}$, suggesting a conjectural Eilenberg-Moore spectral sequence for connected sums.
• Compare the homology of the connected sum cone with the tensor product of individual $\mathit{CFI}^{-}$ complexes, showing quasi-isomorphism over $\mathbb{Z}_2[U]$ but not $\mathcal{R}$-equivariantly.

Experimental results

Research questions

• RQ1 Does the conjugation involution on the Heegaard Floer complex of a connected sum of three-manifolds decompose as the tensor product of the individual involutions?
• RQ2 Can the involutive correction terms $\underline{d}$ and $\bar{d}$ detect non-trivial elements in the homology cobordism group beyond the standard $d$-invariant?
• RQ3 Is there a homomorphism from the three-dimensional homology cobordism group $\Theta^3_{\mathbb{Z}}$ to an algebraically defined group of complexes with involutions modulo local equivalence?
• RQ4 Does the connected sum formula for involutive Heegaard Floer homology lead to a spectral sequence of Eilenberg-Moore type?
• RQ5 Can the $\mathcal{A}_\infty$-module structure on $\mathit{HFI}^{-}$ be constructed to realize a conjectural Künneth-type isomorphism?

Key findings

• The conjugation involution on $\mathit{CF}^{-}(Y_1\#Y_2, \mathfrak{s}_1\#\mathfrak{s}_2)$ is chain homotopy equivalent to $\iota_1 \otimes \iota_2$ over $\mathbb{Z}_2[U]$, proving the connected sum formula for involutive Heegaard Floer homology.
• The involutive correction terms satisfy $\underline{d}(Y_1\#Y_2) = \underline{d}(Y_1) + \underline{d}(Y_2)$ and $\bar{d}(Y_1\#Y_2) = \bar{d}(Y_1) + \bar{d}(Y_2)$, extending the additivity of the standard $d$-invariant.
• The paper constructs a homomorphism from $\Theta^3_{\mathbb{Z}}$ to an Abelian group of complexes with homotopy involutions modulo local equivalence, providing a new algebraic invariant.
• An explicit example is given where $\underline{d}(Y) \neq d(Y) \neq \bar{d}(Y)$, showing that the involutive invariants detect more structure than the standard $d$-invariant.
• The homology of the mapping cone for $Y_1\#Y_2$ is quasi-isomorphic to the tensor product of individual $\mathit{CFI}^{-}$ complexes over $\mathbb{Z}_2[U]$, though not $\mathcal{R}$-equivariantly.
• The paper conjectures that $\mathit{HFI}^{-}(Y_1\#Y_2) \cong \mathit{HFI}^{-}(Y_1) \tilde{\otimes}_{\mathfrak{R}} \mathit{HFI}^{-}(Y_2)$ as $\mathcal{A}_\infty$-modules, with a corresponding Eilenberg-Moore spectral sequence.