FLOER HOMOLOGY AND INVARIANTS OF HOMOLOGY COBORDISM

1998 ◽  
Vol 09 (07) ◽  
pp. 885-919 ◽  
Author(s):  
NIKOLAI SAVELIEV
Keyword(s):  
Floer Homology ◽  
Integral Homology ◽  
Homology Cobordism

By using surgery techniques, we compute Floer homology for certain classes of integral homology 3-spheres homology cobordant to zero. We prove that Floer homology is two-periodic for all these manifolds. Based on this fact, we introduce a new integer valued invariant of integral homology 3-spheres. Our computations suggest its homology cobordism invariance.

scholarly journals Pin(2)-equivariant Seiberg–Witten Floer homology of Seifert fibrations

2019 ◽  
Vol 156 (2) ◽  
pp. 199-250 ◽  
Author(s):  
Matthew Stoffregen
Keyword(s):  
Isomorphism Class ◽  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Homology Sphere ◽  
Integral Homology ◽  
Fiber Space ◽  
Rational Homology ◽  
Seifert Fiber Space ◽  
Homology Cobordism ◽  
Seifert Fiber Spaces

We compute the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer homology of Seifert rational homology three-spheres in terms of their Heegaard Floer homology. As a result of this computation, we prove Manolescu’s conjecture that $\unicode[STIX]{x1D6FD}=-\bar{\unicode[STIX]{x1D707}}$ for Seifert integral homology three-spheres. We show that the Manolescu invariants $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$ give new obstructions to homology cobordisms between Seifert fiber spaces, and that many Seifert homology spheres $\unicode[STIX]{x1D6F4}(a_{1},\ldots ,a_{n})$ are not homology cobordant to any $-\unicode[STIX]{x1D6F4}(b_{1},\ldots ,b_{n})$. We then use the same invariants to give an example of an integral homology sphere not homology cobordant to any Seifert fiber space. We also show that the $\text{Pin}(2)$-equivariant Seiberg–Witten Floer spectrum provides homology cobordism obstructions distinct from $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD},$ and $\unicode[STIX]{x1D6FE}$. In particular, we identify an $\mathbb{F}[U]$-module called connected Seiberg–Witten Floer homology, whose isomorphism class is a homology cobordism invariant.

scholarly journals Rational cobordisms and integral homology

2020 ◽  
Vol 156 (9) ◽  
pp. 1825-1845
Author(s):  
Paolo Aceto ◽  
Daniele Celoria ◽  
JungHwan Park
Keyword(s):  
Homology Group ◽  
Floer Homology ◽  
Lens Spaces ◽  
Integral Homology ◽  
Rational Homology ◽  
Cobordism Class ◽  
Connected Sum ◽  
Knot Floer Homology ◽  
Infinite Rank ◽  
Homology Cobordism

We consider the question of when a rational homology $3$-sphere is rational homology cobordant to a connected sum of lens spaces. We prove that every rational homology cobordism class in the subgroup generated by lens spaces is represented by a unique connected sum of lens spaces whose first homology group injects in the first homology group of any other element in the same class. As a first consequence, we show that several natural maps to the rational homology cobordism group have infinite-rank cokernels. Further consequences include a divisibility condition between the determinants of a connected sum of $2$-bridge knots and any other knot in the same concordance class. Lastly, we use knot Floer homology combined with our main result to obstruct Dehn surgeries on knots from being rationally cobordant to lens spaces.

Casson-Lin's Invariant and Floer Homology

1997 ◽  
Vol 06 (06) ◽  
pp. 851-877 ◽  
Author(s):  
Weiping Li
Keyword(s):  
Euler Number ◽  
Floer Homology ◽  
Natural Generalization ◽  
Conjugacy Classes ◽  
Irreducible Representations ◽  
Representation Space ◽  
Integral Homology

Casson defined an invariant which can be thought of as the number of conjugacy classes of irreducible representations of π1(Y) into SU(2) counted with signs, where Y is an oriented integral homology 3-sphere. Lin defined a similar invariant (the signature of a knot) for a braid representative of a knot in S3. In this paper, we give a natural generalization of Casson-Lin's invariant. Our invariant is the symplectic Floer homology for the representation space of π1(S3 \ K) into SU(2) with trace-zero along all meridians. The symplectic Floer homology of braids is a new invariant of knots and its Euler number is the negative of Casson-Lin's invariant.

scholarly journals Lescop's invariant and gauge theory

2015 ◽  
Vol 24 (09) ◽  
pp. 1550050 ◽  
Author(s):  
Prayat Poudel
Keyword(s):  
Gauge Theory ◽  
Euler Characteristic ◽  
Betti Number ◽  
Floer Homology ◽  
Integral Homology ◽  
Similar Relationship

Taubes proved that the Casson invariant of an integral homology 3-sphere equals half the Euler characteristic of its instanton Floer homology. We extend this result to all closed oriented 3-manifolds with positive first Betti number by establishing a similar relationship between the Lescop invariant of the manifold and its instanton Floer homology. The proof uses surgery techniques.

scholarly journals INVOLUTIVE HEEGAARD FLOER HOMOLOGY AND PLUMBED THREE-MANIFOLDS

2017 ◽  
Vol 18 (06) ◽  
pp. 1115-1155 ◽  
Author(s):  
Irving Dai ◽  
Ciprian Manolescu
Keyword(s):  
Floer Homology ◽  
Three Dimensional ◽  
Heegaard Floer Homology ◽  
Cobordism Group ◽  
Connected Sums ◽  
Infinite Rank ◽  
The Family ◽  
Homology Cobordism ◽  
Heegaard Floer ◽  
Correction Terms

We compute the involutive Heegaard Floer homology of the family of three-manifolds 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.

scholarly journals INSTANTON FLOER HOMOLOGY FOR TWO-COMPONENT LINKS

2012 ◽  
Vol 21 (05) ◽  
pp. 1250054 ◽  
Author(s):  
ERIC HARPER ◽  
NIKOLAI SAVELIEV
Keyword(s):  
Euler Characteristic ◽  
Floer Homology ◽  
Integral Homology ◽  
Oriented Link ◽  
Linking Number ◽  
Two Component

For any oriented link of two components in an integral homology 3-sphere, we define an instanton Floer homology whose Euler characteristic is twice the linking number between the components of the link. We show that, for two-component links in the 3-sphere, this Floer homology does not vanish unless the link is split. We also relate our Floer homology to the Kronheimer–Mrowka instanton Floer homology for links.

scholarly journals APPLICATIONS OF INVOLUTIVE HEEGAARD FLOER HOMOLOGY

2019 ◽  
pp. 1-38 ◽  
Author(s):  
Kristen Hendricks ◽  
Jennifer Hom ◽  
Tye Lidman
Keyword(s):  
Spin Structure ◽  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Homology Sphere ◽  
Rational Homology ◽  
Cobordism Group ◽  
Homology Cobordism ◽  
Heegaard Floer ◽  
Correction Terms ◽  
Rational Homology Sphere

We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms$\bar{d}$and$\text{}\underline{d}$for certain families of three-manifolds.

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