Nonsplittability of the rational homology cobordism group of 3-manifolds

Se-Goo Kim and Charles Livingston

Vol. 271 (2014), No. 1, 183–211

DOI: 10.2140/pjm.2014.271.183

Abstract

Let $\mathbb{Z}\left[1∕p\right]$ denote the ring of integers with the prime $p$ inverted. There is a canonical homomorphism $\Psi :\oplus {\Theta }_{\mathbb{Z}\left[1∕p\right]}^3\to {\Theta }_{\mathbb{Q}}^3$, where ${\Theta }_R^3$ denotes the three-dimensional smooth $R$-homology cobordism group of $R$-homology spheres and the direct sum is over all prime integers. Gauge-theoretic methods prove the kernel is infinitely generated. Here we prove that $\Psi$ is not surjective, with cokernel infinitely generated. As a basic example we show that for $p$ and $q$ distinct primes, there is no rational homology cobordism from the lens space $L\left(pq,1\right)$ to any $M_p\#M_q$, where $H_1\left(M_p\right)={\mathbb{Z}}_p$ and $H_1\left(M_q\right)={\mathbb{Z}}_q$. More subtle examples include cases in which a cobordism to such a connected sum exists topologically but not smoothly. (Conjecturally such a splitting always exists topologically.) Further examples can be chosen to represent 2-torsion in ${\Theta }_{\mathbb{Q}}^3$. Let $\mathcal{K}$ denote the kernel of ${\Theta }_{\mathbb{Q}}^3\to {\hat{\Theta }}_{\mathbb{Q}}^3$, where ${\hat{\Theta }}_{\mathbb{Q}}^3$ denotes the topological homology cobordism group. Freedman proved that ${\Theta }_{\mathbb{Z}}^3\subset \mathcal{K}$. A corollary of results here is that $\mathcal{K}∕{\Theta }_{\mathbb{Z}}^3$ is infinitely generated. We also demonstrate the failure in dimension three of splitting theorems that apply to higher-dimensional knot concordance groups.

Keywords

three-manifold, connected sum, homology cobordism, knot concordance

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57M25

Milestones

Received: 17 June 2013

Revised: 21 March 2014

Accepted: 8 April 2014

Published: 10 September 2014