Nonsplittability of the rational homology cobordism group of 3-manifolds
type: Item
instance of: scholarly article
title: Nonsplittability of the rational homology cobordism group of 3-manifolds
zbMATH Open document ID: 1312.57015
author: Se-Goo Kim
author: Charles Livingston
published in: Pacific Journal of Mathematics
publication date: 2014-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1211.3712
review text: The three dimensional homology cobordism group is very large, and only partially understood. This paper proves new results about the smooth rational homology cobordism group. In particular one may consider the relation between the smooth cobordism groups with \(\mathbb{Z}[1/p]\) coefficients and the rational homology cobordism group. It would be reasonable to conjecture that the topological rational homology cobordism group is the direct sum of the topological \(\mathbb{Z}[1/p]\) homology cobordism groups classified by the Witt groups. This is far from the case for the smooth homology cobordism groups. Indeed earlier work shows that the kernel of the map taking the sum of \(p\)-cobordism groups into the rational homology cobordism group is infinitely generated. This paper shows that the cokernel of this map contains an infinite rank free group as well as in infinite subgroup generated by elements of order two. The kernel contains an infinite subgroup generated by \(3\)-mainfolds that topologically bound rational balls. The results are obtained via Heegaard-Floer theory.
reviewed by: David Auckly
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57M25
zbMATH DE Number: 6369633
zbMATH Keywords: three-manifold
zbMATH Keywords: connected sum
zbMATH Keywords: homology cobordism
zbMATH Keywords: knot concordance
MaRDI profile type: Publication
OpenAlex ID: W3099418033
arXiv ID: 1211.3712
DOI: 10.2140/PJM.2014.271.183
description: scientific article; zbMATH DE number 6369633