Involutive Heegaard Floer homology

Hendricks, Kristen; Manolescu, Ciprian

doi:10.1215/00127094-3793141

Cited by 109 publications

(280 citation statements)

References 91 publications

(261 reference statements)

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“…Lin constructed a

G

‐equivariant refinement of monopole Floer homology in the setting of Kronheimer–Mrowka . In the setting of Heegaard Floer homology introduced by Ozsváth–Szabó in , Hendricks and Manolescu defined an analogue,

HFI (Y, s)

, of

Z / 4

‐equivariant Seiberg–Witten Floer homology (where we regard

⟨ j ⟩ = Z / 4 \subset G

).…”

Section: Introductionmentioning

confidence: 99%

Manolescu invariants of connected sums

Stoffregen

2017

Proc. London Math. Soc.

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Abstract. We give inequalities for the Manolescu invariants α, β, γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a Z 8 subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furuta's Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of n copies of a three-manifold Y , for large n.

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“…Lin constructed a

G

HFI (Y, s)

, of

Z / 4

‐equivariant Seiberg–Witten Floer homology (where we regard

⟨ j ⟩ = Z / 4 \subset G

).…”

Section: Introductionmentioning

confidence: 99%

Manolescu invariants of connected sums

Stoffregen

2017

Proc. London Math. Soc.

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“…Hendricks and Manolescu construct a refinement of Heegaard Floer homology, called involutive Heegaard Floer homology . They define a module

{italicHFI}^{-} false(Y, frakturs false)

over

{double-struckF}_{2} false[U, Q false] / false(Q^{2} false)

as the homology of the mapping cone

{italicCFI}^{-} false(Y, frakturs false) : = Cone ({italicCF}^{-} false(Y, frakturs false) \overset{Q \cdot (id + ι)}{\to} Q \cdot {italicCF}^{-} false(Y, frakturs false)),

where

Q

is a formal variable.…”

Section: Introductionmentioning

confidence: 99%

“…The chain complex

C F {scriptL}^{\infty} false(Y, double-struckK, frakturs false)

has a filtration over

Z \oplus Z

. If

K

is null‐homologous and

frakturs

is a self‐conjugate

{Spin}^{c}

structure, Hendricks and Manolescu consider a conjugation map

ι_{K} : C F {scriptL}^{\infty} false(Y, double-struckK, frakturs false) \to C F {scriptL}^{\infty} false(Y, double-struckK, frakturs false) .

Unlike the map

ι

{italicCF}^{-} false(Y, frakturs false)

, the map

ι_{K}

is not a homotopy involution. Instead, the map

ι_{K}

satisfies

ι_{K}^{2} ≃ ρ_{*},

where

ρ_{*}

is the map induced by the diffeomorphism

ρ : (Y, K, p, q) \to (Y, K, p, q)

obtained by twisting the knot

K

in one full twist, in the direction of its orientation.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 8, we use the above formula to compute the involutive correction terms

{\bar{V}}_{0}

and

{\underset{̲}{V}}_{0}

from large surgeries from for several examples of connected sums of knots. The following is a straightforward consequence of several example computations which we perform.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of the above proposition uses the computation of the triple

({\bar{V}}_{0}, V_{0}, {\underset{̲}{V}}_{0})

for thin knots,

L

‐space knots, and mirrors of

L

‐space knots from . For such knots, the involutive correction terms have a simple pattern: either all three are non‐negative and

{\underset{̲}{V}}_{0} - {\bar{V}}_{0} ⩽ 1

, or

{\bar{V}}_{0} ⩽ 0 = V_{0} = {\underset{̲}{V}}_{0}

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Connected sums and involutive knot Floer homology

Zemke

2019

Proc. London Math. Soc.

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We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected sum formula, we perform some example computations of Hendricks and Manolescu's involutive invariants on large surgeries of connected sums of knots.

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A Slicing Obstruction from the 10/8+4 Theorem

Truong

2021

2019-20 MATRIX Annals

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Involutive Heegaard Floer homology

Cited by 109 publications

References 91 publications

Manolescu invariants of connected sums

Manolescu invariants of connected sums

Connected sums and involutive knot Floer homology

A Slicing Obstruction from the 10/8+4 Theorem

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