scholarly journals Nonsplittability of the rational homology cobordism group of 3-manifolds

2014 ◽  
Vol 271 (1) ◽  
pp. 183-211 ◽  
Author(s):  
Se-Goo Kim ◽  
Charles Livingston
Keyword(s):  
Rational Homology ◽  
Cobordism Group ◽  
Homology Cobordism

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 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.

Homology cobordism group of homology 3-spheres

1990 ◽  
Vol 100 (1) ◽  
pp. 339-355 ◽  
Author(s):  
Mikio Furuta
Keyword(s):  
Cobordism Group ◽  
Homology Cobordism

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 The cobordism group of homology cylinders

2010 ◽  
Vol 147 (3) ◽  
pp. 914-942 ◽  
Author(s):  
Jae Choon Cha ◽  
Stefan Friedl ◽  
Taehee Kim
Keyword(s):  
Mapping Class Group ◽  
Betti Number ◽  
Boundary Component ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Cobordism Group ◽  
Infinite Rank ◽  
Homology Cobordism

AbstractGaroufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as an enlargement of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis and Levine, and Goda and Sakasai. Furthermore, we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component. These results also hold for the homology cylinder analogue of the Torelli group.

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.

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