scholarly journals LINEAR INDEPENDENCE IN THE RATIONAL HOMOLOGY COBORDISM GROUP

2019 ◽  
pp. 1-12 ◽  
Author(s):  
Marco Golla ◽  
Kyle Larson
Keyword(s):  
Infinite Order ◽  
Double Cover ◽  
Rational Homology ◽  
Cobordism Group ◽  
Connected Sums ◽  
Knot Concordance ◽  
Linearly Independent ◽  
Rational Homology Spheres ◽  
Homology Cobordism ◽  
Correction Terms

We give simple homological conditions for a rational homology 3-sphere $Y$ to have infinite order in the rational homology cobordism group $\unicode[STIX]{x1D6E9}_{\mathbb{Q}}^{3}$ , and for a collection of rational homology spheres to be linearly independent. These translate immediately to statements about knot concordance when $Y$ is the branched double cover of a knot, recovering some results of Livingston and Naik. The statements depend only on the homology groups of the 3-manifolds, but are proven through an analysis of correction terms and their behavior under connected sums.

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 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 Brieskorn manifolds, positive Sasakian geometry, and contact topology

2016 ◽  
Vol 28 (5) ◽  
pp. 943-965
Author(s):  
Charles P. Boyer ◽  
Leonardo Macarini ◽  
Otto van Koert
Keyword(s):  
Moduli Space ◽  
Einstein Metrics ◽  
Contact Structures ◽  
Rational Homology ◽  
Contact Topology ◽  
Connected Sums ◽  
New Family ◽  
Rational Homology Spheres ◽  
Sasakian Geometry ◽  
Sasakian Structures

AbstractUsing ${S^{1}}$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn–Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of ${S^{2}\times S^{3}}$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki–Einstein metrics on certain homotopy spheres. Finally, a new family of Sasaki–Einstein metrics of real dimension 20 on ${S^{5}}$ is exhibited.

scholarly journals On independence of iterated Whitehead doubles in the knot concordance group

2018 ◽  
Vol 27 (01) ◽  
pp. 1850003
Author(s):  
Kyungbae Park
Keyword(s):  
Correction Term ◽  
Khovanov Homology ◽  
Floer Theory ◽  
Knot Concordance ◽  
Torus Knot ◽  
Linearly Independent ◽  
Heegaard Floer

Let [Formula: see text] be the positively clasped untwisted Whitehead double of a knot [Formula: see text], and [Formula: see text] be the [Formula: see text] torus knot. We show that [Formula: see text] and [Formula: see text] are linearly independent in the smooth knot concordance group [Formula: see text] for each [Formula: see text]. Further, [Formula: see text] and [Formula: see text] generate a [Formula: see text] summand in the subgroup of [Formula: see text] generated by topologically slice knots. We use the concordance invariant [Formula: see text] of Manolescu and Owens, using Heegaard Floer correction term. Interestingly, these results are not easily shown using other concordance invariants such as the [Formula: see text]-invariant of knot Floer theory and the [Formula: see text]-invariant of Khovanov homology. We also determine the infinity version of the knot Floer complex of [Formula: see text] for any [Formula: see text] generalizing a result for [Formula: see text] of Hedden, Kim and Livingston.

Strongly-cyclic branched coverings and the Alexander polynomial of knots in rational homology spheres

2007 ◽  
Vol 142 (2) ◽  
pp. 259-268 ◽  
Author(s):  
YUYA KODA
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Alexander Polynomial ◽  
Homology Sphere ◽  
Cyclic Covering ◽  
Rational Homology ◽  
Branched Coverings ◽  
Branched Covering ◽  
Rational Homology Spheres ◽  
Rational Homology Sphere

AbstractLet K be a knot in a rational homology sphere M. In this paper we correlate the Alexander polynomial of K with a g-word cyclic presentation for the fundamental group of the strongly-cyclic covering of M branched over K. We also give a formula for the order of the first homology group of the strongly-cyclic branched covering.

scholarly journals ON ALEXANDER MODULES AND BLANCHFIELD FORMS OF NULL-HOMOLOGOUS KNOTS IN RATIONAL HOMOLOGY SPHERES

2012 ◽  
Vol 21 (05) ◽  
pp. 1250042 ◽  
Author(s):  
DELPHINE MOUSSARD
Keyword(s):  
Isomorphism Class ◽  
The Other ◽  
Rational Homology ◽  
Rational Homology Spheres

In this paper, we give a classification of Alexander modules of null-homologous knots in rational homology spheres. We characterize these modules [Formula: see text] equipped with their Blanchfield forms ϕ, and the modules [Formula: see text] such that there is a unique isomorphism class of [Formula: see text], and we prove that for the other modules [Formula: see text], there are infinitely many such classes. We realize all these [Formula: see text] by explicit knots in ℚ-spheres.

Sign in / Sign up

Export Citation Format

Share Document