Manolescu invariants of connected sums

Stoffregen, Matthew

doi:10.1112/plms.12060

Cited by 32 publications

(47 citation statements)

References 39 publications

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“…In [24,Theorem 1.4], Stoffregen calculated the homology cobordism invariants α, β, γ (coming from Pinp2q-equivariant Seiberg-Witten Floer homology, cf. [13]) in the case of connected sums of Seifert fibered integral homology spheres of projective type.…”

Section: Introductionmentioning

confidence: 99%

Involutive Heegaard Floer Homology and Plumbed Three-Manifolds

Dai¹,

Manolescu²

2017

J. Inst. Math. Jussieu

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We compute the involutive Heegaard Floer homology of the family of threemanifolds obtained by plumbings along almost-rational graphs. (This includes all Seifert fibered homology spheres.) We also study the involutive Heegaard Floer homology of connected sums of such three-manifolds, and explicitly determine the involutive correction terms in the case that all of the summands have the same orientation. Using these calculations, we give a new proof of the existence of an infinite-rank subgroup in the three-dimensional homology cobordism group.Theorem 1.2. Let Y and s be as in Theorem 1.1. Then, the involutive Heegaard Floer correction terms are given by dpY, sq "´2μpY, sq,dpY, sq " dpY, sq, where dpY, sq is the Ozsváth-Szabó d-invariant andμpY, sq is the Neumann-Siebenmann invariant from [17], [23]. Theorems 1.1 and 1.2 should be compared with the corresponding results for Pinp2q-monopole Floer homology, obtained by the first author in [5]. For the smaller class of rational homology spheres Seifert fibered over a base orbifold with underlying space S 2 , the Pinp2q-equivariant Seiberg-Witten Floer homology was computed by Stoffregen in [25].The proof of Theorem 1.1 is in two steps. First, we identify the action of J 0 on H´pR k q (or, more precisely, H´pR k qrσ`2sq with the conjugation involution on HF´pY, sq, using an argument from [5]. Second, we prove that, in the case at hand, the action of J 0 on H´pR k q 1 In this paper, by ras we will denote a grading shift by a, so that an element in degree 0 in a module M becomes an element in degree´a in the shifted module M ras. This is the standard convention, but opposite to the one in [14].

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Section: Introductionmentioning

confidence: 99%

Involutive Heegaard Floer Homology and Plumbed Three-Manifolds

Dai¹,

Manolescu²

2017

J. Inst. Math. Jussieu

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“…Consider the Brieskorn homology spheres Σpp, 2p´1, 2p`1q for p ě 3 odd. In [27] it was shown that these are linearly independent in Θ 3 Z by using the Manolescu correction terms α, β, and γ. An analogous argument was given in [6] using the involutive Floer homology.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…Recently, work by the second author has involved understanding the Pin(2)-equivariant Seiberg-Witten Floer homology of Seifert fibered spaces (see [26], [27]). These Floer homologies have explicit algebraic models which make them amenable to computation; and, in addition, one can attempt to identify classical invariants such as the Neumann-Siebenmann invariant (defined in [19], [25]) for such spaces in terms of their Floer homology (see [15,Conjecture 4.1]).…”

Section: Overview and Motivationmentioning

confidence: 99%

“…Although this gives a good understanding of the Pin(2)-homology of individual AR manifolds, a general description of the Pin(2)-homology of connected sums is more difficult. See [27] for results in this direction. In the present paper, we will study the subgroups of Θ 3 Z generated by Seifert spaces and AR plumbed manifolds; these will be denoted by Θ SF and Θ AR , respectively.…”

Section: Overview and Motivationmentioning

confidence: 99%

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On homology cobordism and local equivalence between plumbed manifolds

Dai

Stoffregen

2019

Geom. Topol.

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We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism invariant for all linear combinations of AR plumbed homology spheres. As a corollary, we prove that if Y is a linear combination of AR plumbed homology spheres with µpY q " 1, then Y is not torsion in the homology cobordism group. A general computation of the involutive Heegaard Floer correction terms for these spaces is also included.2010 Mathematics Subject Classification. 57R58, 57M27.

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“…From this viewpoint, the spectral sequence described in Corollary 1 is then the Eilenberg-Moore spectral sequence computing the Pin(2)-equivariant homology of the smash product of two spaces from the Pin(2)-equivariant homology of the factors (see Section 1 for more details). In [25] many interesting computations for Seifert fibered spaces are provided using the fact that the homotopy type associated to a disjoint union is locally equivalent to the smash product of the homotopy types.…”

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confidence: 99%

Pin(2)‐monopole Floer homology, higher compositions and connected sums

Lin

2017

Journal of Topology

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We study the behavior of Pin(2)-monopole Floer homology under connected sums. After constructing a (partially defined) A∞-module structure on the Pin(2)-monopole Floer chain complex of a three-manifold (in the spirit of Baldwin and Bloom's monopole category), we identify up to quasi-isomorphism the Floer chain complex of a connected sum with a version of the A ∞-tensor product of the modules of the summands. There is a naturally associated spectral sequence converging to the Floer groups of the connected sum whose E 2 page is the Tor of the Floer groups of the summands. We discuss in detail a simple example, and use this computation to show that the Pin(2)-monopole Floer homology of S 3 has non-trivial Massey products.

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Manolescu invariants of connected sums

Cited by 32 publications

References 39 publications

Involutive Heegaard Floer Homology and Plumbed Three-Manifolds

Involutive Heegaard Floer Homology and Plumbed Three-Manifolds

On homology cobordism and local equivalence between plumbed manifolds

Pin(2)‐monopole Floer homology, higher compositions and connected sums

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