# Rigidity Theorems in Stable Homotopy Theory

Lead Research Organisation:
University of Sheffield

Department Name: Pure Mathematics

### Abstract

Algebraic topologists study the stable homotopy of spaces and the generalisation to spectra by separating out various types of periodic phenomena using Bousfield localisation to produce for each prime number localised categories in which only one type of periodicity dominates. This chromatic approach has proved extremely important in organising calculations as well as giving a conceptual framework. Detailed structure of such localised categories can be studied at the level of triangulated categories, but there are finer details available in the form of the Quillen model structures whose homotopy categories realise them. Natural questions arise concerning the uniqueness or otherwise of such model structures. For the first case, closely associated with K-theory, it is now known due to work of Roitzheim based on ideas of Schwede that at the prime 2, the model structure is essentially unique, whereas at odd primes there are exotic model structures discovered by Franke.Our goal is to investigate these questions for higher periodicities. We also intend to study possible connections with the non-existence of certain Smith-Toda complexes as proved by Nave.

## People |
## ORCID iD |

Sarah Ann Whitehouse (Principal Investigator) |

### Publications

Roitzheim C
(2007)

*Rigidity and exotic models for the -local stable homotopy category*in Geometry & Topology
Roitzheim C
(2011)

*Uniqueness of -structures and Hochschild cohomology*in Algebraic & Geometric Topology
Roitzheim C
(2010)

*Uniqueness of A-infinity algebras and Hochschild cohomology*
Roitzheim C
(2008)

*On the algebraic classification of $K$-local spectra*in Homology, Homotopy and ApplicationsDescription | Key findings relate to derived A-infinity algebra structures. A criterion for uniqueness of such structures was developed, in terms of Hochschild cohomology. Results for differential graded algebras were also obtained, in the form of a criterion for intrinsic formality. |

Exploitation Route | This research is expected to form a basis for continued work in the area of rigidity of model structures both in algebra and topology. New approaches to derived A-infinity structures should be developed, with many possible applications. |

Sectors | Other |

Description | The findings have been used as the basis of further work in pure mathematics, by Livernet. Roitzheim and Whitehouse and also by, for example J. Zhao. |

First Year Of Impact | 2013 |

Sector | Other |

Description | Baker |

Organisation | University of Glasgow |

Country | United Kingdom of Great Britain & Northern Ireland (UK) |

Sector | Academic/University |

PI Contribution | The grant was jointly held, with the postdoc spending the first half of the project in Sheffield. |

Collaborator Contribution | The postdoc spent the second half of the project working with Andy Baker in Glasgow. |

Impact | See all the publications. |

Start Year | 2007 |