---
abstract: 'We study families of objects in Fukaya categories, specifically ones whose deformation behaviour is prescribed by the choice of an odd degree cohomology class. This leads to invariants of symplectic manifolds, which we apply to blowups along symplectic mapping tori.'
author:
- Paul Seidel
date: August 2011
title: |
    **Abstract analogues of flux\
    as symplectic invariants**
---

Introduction {#introduction .unnumbered}
============

[**Motivation.**]{} An interesting invariant of a closed symplectic manifold $M$ is its flux group, a subgroup of $H^1(M;{\mathbb{R}})$ obtained from the topology of loops of symplectic automorphisms. This can be effectively studied using Floer cohomology, one of the notable insights being that the flux group is always discrete [@ono06]. Now consider the following question:

> *Are there flux-type subgroups in $H^{2k-1}(M;{\mathbb{R}})$, for $k>1$, which can be nontrivial for manifolds with $H^1(M;{\mathbb{R}}) = 0$?*

The last clause excludes one obvious direction, which is to consider the action of loops of symplectic automorphisms on $H^*(M;{\mathbb{R}})$ (that action vanishes if $H^1(M;{\mathbb{R}}) = 0$ by the rigidity theorem [@lalonde-mcduff-polterovich97]). Really, what the question is aiming for is a formalism in which higher degree differential forms replace the closed one-forms in their usual relation to symplectic vector fields, so anything related to 