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 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.

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.

scholarly journals Knot lattice homology in L-spaces

2016 ◽  
Vol 25 (01) ◽  
pp. 1650003
Author(s):  
Peter Ozsváth ◽  
András I. Stipsicz ◽  
Zoltán Szabó
Keyword(s):  
Polynomial Ring ◽  
Floer Homology ◽  
Homotopy Equivalent ◽  
Heegaard Floer Homology ◽  
Homology Sphere ◽  
Rational Homology ◽  
Knot Floer Homology ◽  
Chain Complexes ◽  
Chain Homotopy ◽  
Rational Homology Sphere

We show that the knot lattice homology of a knot in an [Formula: see text]-space is chain homotopy equivalent to the knot Floer homology of the same knot (viewed these invariants as filtered chain complexes over the polynomial ring [Formula: see text]). Suppose that [Formula: see text] is a negative definite plumbing tree which contains a vertex [Formula: see text] such that [Formula: see text] is a union of rational graphs. Using the identification of knot homologies we show that for such graphs the lattice homology [Formula: see text] is isomorphic to the Heegaard Floer homology [Formula: see text] of the corresponding rational homology sphere [Formula: see text].

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.

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 Splicing knot complements and bordered Floer homology

2016 ◽  
Vol 2016 (720) ◽  
Author(s):  
Matthew Hedden ◽  
Adam Simon Levine
Keyword(s):  
Floer Homology ◽  
Heegaard Floer Homology ◽  
Homology Sphere ◽  
Knot Complements ◽  
Heegaard Floer

AbstractWe show that the integer homology sphere obtained by splicing two nontrivial knot complements in integer homology sphere L-spaces has Heegaard Floer homology of rank strictly greater than one. In particular, splicing the complements of nontrivial knots in the 3-sphere never produces an L-space. The proof uses bordered Floer homology.

Euler classes of vector bundles over manifolds

2021 ◽  
Vol 71 (1) ◽  
pp. 199-210
Author(s):  
Aniruddha C. Naolekar
Keyword(s):  
Vector Bundle ◽  
Vector Bundles ◽  
Euler Class ◽  
Homology Sphere ◽  
Rational Homology ◽  
Rational Homology Sphere ◽  
Orientable Manifolds

Abstract Let 𝓔 k denote the set of diffeomorphism classes of closed connected smooth k-manifolds X with the property that for any oriented vector bundle α over X, the Euler class e(α) = 0. We show that if X ∈ 𝓔2n+1 is orientable, then X is a rational homology sphere and π 1(X) is perfect. We also show that 𝓔8 = ∅ and derive additional cohomlogical restrictions on orientable manifolds in 𝓔 k .

IWASAWA TYPE FORMULA FOR COVERS OF A LINK IN A RATIONAL HOMOLOGY SPHERE

2008 ◽  
Vol 17 (10) ◽  
pp. 1199-1221 ◽  
Author(s):  
TERUHISA KADOKAMI ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Number Field ◽  
Class Number ◽  
Homology Sphere ◽  
Rational Homology ◽  
Homology Groups ◽  
Cyclic Covers ◽  
Class Number Formula ◽  
Number Formula ◽  
Iwasawa Invariants ◽  
Rational Homology Sphere

Based on the analogy between links and primes, we present an analogue of the Iwasawa's class number formula in a Zp-extension for the p-homology groups of pn-fold cyclic covers of a link in a rational homology 3-sphere. We also describe the associated Iwasawa invariants precisely for some examples and discuss analogies with the number field case.

Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Yongqiang Liu ◽  
Laurenţiu Maxim ◽  
Botong Wang
Keyword(s):  
Integral Homology ◽  
Algebraic Varieties ◽  
Finiteness Property ◽  
Rational Homology ◽  
Mellin Transformation ◽  
Hopf Conjecture ◽  
Ball Quotients ◽  
Proper Map ◽  
Complex Projective ◽  
Aspherical Manifolds

Abstract In their paper from 2012, Bobadilla and Kollár studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer–Hopf conjecture in the complex projective setting.

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