##
**Link cobordisms and functoriality in link Floer homology.**
*(English)*
Zbl 1455.57020

The paper under review constructs a ‘TQFT’-like invariant associated to link Floer homology. The main result is, roughly speaking, the construction, for a cobordism \((W,\Sigma)\) between pairs \((Y_1,L_1)\), \((Y_2,L_2)\) consisting of a three-manifold and a link, with \(\partial W=Y_1-Y_2\) and \(\partial \Sigma=L_1-L_2\), of well-defined cobordism maps \(\mathcal{CFL}^-(Y_1,L_1)\to \mathcal{CFL}^-(Y_2,L_2)\). To be precise, the surface \(\Sigma\) must be equipped with a decoration; a primary feature of the paper is explaining the necessity of this extra data in order to get well-defined maps. The maps constructed here are an extension of Ozsváth-Szabó’s construction of the original Heegaard Floer \((3+1)\)-dimensional TQFT (note that in the paper under review the \(3\)-manifolds involved may be disconnected), taking account of elementary moves on links inside a three-manifold, and a large portion of the paper is devoted to proving that the maps constructed by composing maps associated to elementary moves are independent of the choice of a decomposition of a complicated link cobordism into elementary moves (which is a very involved and technical problem).

Maps in link Floer homology associated to link cobordisms had been constructed in various limited contexts before (the introduction to the present paper contains a good overview of the state of the art at the time of writing), but the current paper works in much greater generality (compare also with the contemporaneous construction of A. Alishahi and E. Eftekhary [J. Topol. 13, No. 4, 1582–1657 (2020; Zbl 1511.57019)], which is at the same level of generality as the present paper).

The maps constructed in the present paper have a number of features. Principally, Zemke shows a version of functoriality in Theorem B, that the cobordism map of a composite cobordism is the composition of cobordism maps (suitably interpreting the involved \(spin^c\) structures). Using the algebraic properties of the maps constructed here, Zemke is able to extract many topological applications in later papers. Indeed, some highlights include the observation that the link cobordism maps respect the Maslov and Alexander gradings, up to some simple topological terms, and conjugation-invariance of these maps is sufficient to prove a very useful connected sum formula for involutive knot Floer homology. Another main high point is Zemke’s proof that the cobordism maps are injective under ribbon cobordisms, which, combined with basic properties of the knot Floer homology package itself, provides extremely strong constraints on the existence of ribbon cobordisms. The current paper is a fundamental ingredient in all of these applications (and others developed by other authors), and so has been a radical improvement in our ability to use Heegaard Floer homology in a wide range of situations.

Maps in link Floer homology associated to link cobordisms had been constructed in various limited contexts before (the introduction to the present paper contains a good overview of the state of the art at the time of writing), but the current paper works in much greater generality (compare also with the contemporaneous construction of A. Alishahi and E. Eftekhary [J. Topol. 13, No. 4, 1582–1657 (2020; Zbl 1511.57019)], which is at the same level of generality as the present paper).

The maps constructed in the present paper have a number of features. Principally, Zemke shows a version of functoriality in Theorem B, that the cobordism map of a composite cobordism is the composition of cobordism maps (suitably interpreting the involved \(spin^c\) structures). Using the algebraic properties of the maps constructed here, Zemke is able to extract many topological applications in later papers. Indeed, some highlights include the observation that the link cobordism maps respect the Maslov and Alexander gradings, up to some simple topological terms, and conjugation-invariance of these maps is sufficient to prove a very useful connected sum formula for involutive knot Floer homology. Another main high point is Zemke’s proof that the cobordism maps are injective under ribbon cobordisms, which, combined with basic properties of the knot Floer homology package itself, provides extremely strong constraints on the existence of ribbon cobordisms. The current paper is a fundamental ingredient in all of these applications (and others developed by other authors), and so has been a radical improvement in our ability to use Heegaard Floer homology in a wide range of situations.

Reviewer: Matthew Stoffregen (Cambridge)

### MSC:

57K18 | Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.) |

57R58 | Floer homology |

57R56 | Topological quantum field theories (aspects of differential topology) |