## Algebraic & Geometric Topology

### Connected Heegaard Floer homology of sums of Seifert fibrations

Irving Dai

#### Abstract

We compute the connected Heegaard Floer homology (defined by Hendricks, Hom, and Lidman) for a large class of $3$–manifolds, including all linear combinations of Seifert fibered homology spheres. We show that for such manifolds, the connected Floer homology completely determines the local equivalence class of the associated $ι$–complex. Some identities relating the rank of the connected Floer homology to the Rokhlin invariant and the Neumann–Siebenmann invariant are also derived. Our computations are based on combinatorial models inspired by the work of Némethi on lattice homology.

#### Article information

Source
Algebr. Geom. Topol., Volume 19, Number 5 (2019), 2535-2574.

Dates
Revised: 8 October 2018
Accepted: 30 October 2018
First available in Project Euclid: 26 October 2019

https://projecteuclid.org/euclid.agt/1572055267

Digital Object Identifier
doi:10.2140/agt.2019.19.2535

Mathematical Reviews number (MathSciNet)
MR4023322

Zentralblatt MATH identifier
07142612

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds 57R58: Floer homology

#### Citation

Dai, Irving. Connected Heegaard Floer homology of sums of Seifert fibrations. Algebr. Geom. Topol. 19 (2019), no. 5, 2535--2574. doi:10.2140/agt.2019.19.2535. https://projecteuclid.org/euclid.agt/1572055267

#### References

• I Dai, On the ${\rm Pin}(2)$–equivariant monopole Floer homology of plumbed $3$–manifolds, Michigan Math. J. 67 (2018) 423–447
• I Dai, C Manolescu, Involutive Heegaard Floer homology and plumbed three-manifolds, J. Inst. Math. Jussieu 18 (2019) 1115–1155
• I Dai, M Stoffregen, On homology cobordism and local equivalence between plumbed manifolds, Geom. Topol. 23 (2019) 865–924
• R Fintushel, R J Stern, Instanton homology of Seifert fibred homology three spheres, Proc. London Math. Soc. 61 (1990) 109–137
• K A Frøyshov, Equivariant aspects of Yang–Mills Floer theory, Topology 41 (2002) 525–552
• M Furuta, Homology cobordism group of homology $3$–spheres, Invent. Math. 100 (1990) 339–355
• K Hendricks, J Hom, T Lidman, Applications of involutive Heegaard Floer homology, J. Inst. Math. Jussieu (online publication April 2019)
• K Hendricks, C Manolescu, Involutive Heegaard Floer homology, Duke Math. J. 166 (2017) 1211–1299
• K Hendricks, C Manolescu, I Zemke, A connected sum formula for involutive Heegaard Floer homology, Selecta Math. 24 (2018) 1183–1245
• F Lin, ${\rm Pin}(2)$–monopole Floer homology, higher compositions and connected sums, J. Topol. 10 (2017) 921–969
• F Lin, The surgery exact triangle in ${\rm Pin}(2)$–monopole Floer homology, Algebr. Geom. Topol. 17 (2017) 2915–2960
• F Lin, A Morse–Bott approach to monopole Floer homology and the triangulation conjecture, Mem. Amer. Math. Soc. 1221, Amer. Math. Soc., Providence, RI (2018)
• C Manolescu, ${\rm Pin}(2)$–equivariant Seiberg–Witten Floer homology and the triangulation conjecture, J. Amer. Math. Soc. 29 (2016) 147–176
• A Némethi, Lattice cohomology of normal surface singularities, Publ. Res. Inst. Math. Sci. 44 (2008) 507–543
• W D Neumann, An invariant of plumbed homology spheres, from “Topology Symposium Siegen 1979” (U Koschorke, W D Neumann, editors), Lecture Notes in Math. 788, Springer (1980) 125–144
• P Ozsváth, Z Szabó, On the Floer homology of plumbed three-manifolds, Geom. Topol. 7 (2003) 185–224
• L Siebenmann, On vanishing of the Rohlin invariant and nonfinitely amphicheiral homology $3$–spheres, from “Topology Symposium Siegen 1979” (U Koschorke, W D Neumann, editors), Lecture Notes in Math. 788, Springer (1980) 172–222
• M Stoffregen, ${\rm Pin}(2)$–equivariant Seiberg–Witten Floer homology of Seifert fibrations, preprint (2015)
• M Stoffregen, Manolescu invariants of connected sums, Proc. Lond. Math. Soc. 115 (2017) 1072–1117