Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
Other MSP Journals
GKM manifolds are not rigid

Oliver Goertsches, Panagiotis Konstantis and Leopold Zoller

Algebraic & Geometric Topology 22 (2022) 3511–3532

We construct effective GKM T3–actions with connected stabilizers on the total spaces of the two S2–bundles over S6 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.

GKM theory, Hamiltonian action, equivariant cohomology, cohomological rigidity
Mathematical Subject Classification
Primary: 57R91
Secondary: 53D20, 55N91
Received: 3 March 2021
Accepted: 30 September 2021
Published: 30 January 2023
Oliver Goertsches
Mathematik und Informatik
Philipps-Universität Marburg
Panagiotis Konstantis
Mathematik und Informatik
Philipps-Universität Marburg
Leopold Zoller
Mathematisches Institut
Ludwig-Maximilians-Universität München