Homology cobordism and Seifert fibered 3-manifolds

TL;DR

This paper investigates the homology cobordism classes of 3-manifolds, showing many classes do not contain Seifert fibered spaces by analyzing their cohomology rings and invariants.

Contribution

It determines the cohomology ring types of Seifert fibered 3-manifolds without 2-torsion and constructs examples of manifolds not homology cobordant to any Seifert fibered space.

Findings

Many homology cobordism classes lack Seifert fibered representatives.

Constructed examples with fixed linking form and cohomology ring are not cobordant to Seifert fibered spaces.

Higher Massey products and Milnor's u-invariants distinguish these classes.

Abstract

It is known that every closed oriented 3-manifold is homology cobordant to a hyperbolic 3-manifold. By contrast we show that many homology cobordism classes contain no Seifert fibered 3-manifold. This is accomplished by determining the isomorphism type of the rational cohomology ring of all Seifert fibered 3-manifolds with no 2-torsion in their first homology. Then we exhibit families of examples of 3-manifolds (obtained by surgery on links), with fixed linking form and cohomology ring, that are not homology cobordant to any Seifert fibered space (as shown by their rational cohomology ring). These examples are shown to represent distinct homology cobordism classes using higher Massey products and Milnor's u-invariants for links.