GKM manifolds are not rigid

Goertsches, Oliver; Konstantis, Panagiotis; Zoller, Leopold

doi:10.2140/agt.2022.22.3511

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Editorial Interests

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author Index

To Appear

Other MSP Journals

This article is available for purchase or by subscription. See below.

GKM manifolds are not rigid

Oliver Goertsches, Panagiotis Konstantis and Leopold Zoller

Algebraic & Geometric Topology 22 (2022) 3511–3532

DOI: 10.2140/agt.2022.22.3511

Abstract

We construct effective GKM $T^{3}$ –actions with connected stabilizers on the total spaces of the two $S^{2}$ –bundles over $S^{6}$ with identical GKM graphs. This shows that the GKM graph of a simply connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.

PDF Access Denied

We have not been able to recognize your IP address 3.137.171.121 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recommendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

Keywords

GKM theory, Hamiltonian action, equivariant cohomology, cohomological rigidity

Mathematical Subject Classification

Primary: 57R91

Secondary: 53D20, 55N91

References

Publication

Received: 3 March 2021

Accepted: 30 September 2021

Published: 30 January 2023

Authors

Oliver Goertsches
Mathematik und Informatik Philipps-Universität Marburg Marburg Germany

Panagiotis Konstantis
Mathematik und Informatik Philipps-Universität Marburg Marburg Germany

Leopold Zoller
Mathematisches Institut Ludwig-Maximilians-Universität München Munich Germany

Volume 22, issue 7 (2022)