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The cohomology of free loop spaces of homogeneous spaces

The cohomology of free loop spaces of homogeneous spaces
The cohomology of free loop spaces of homogeneous spaces
The free loops space ΛX of a space X has become an important object of study particularly in the case when X is a manifold. The study of free loop spaces is motivated in particular by two main examples. The first is their relation to geometrically distinct periodic geodesics on a manifold, originally studied by Gromoll and Meyer in 1969. More recently the study of string topology and in particular the Chas-Sullivan loop product has been an active area of research.

A complete flag manifold is the quotient of a Lie group by its maximal torus and is one of the nicer examples of a homogeneous space. Both the cohomology and Chas-Sullivan product structure are understood for spaces Sn, CPn and most simple Lie groups. Hence studying the topology of the free loops space on homogeneous space is a natural next step.

In the thesis we compute the differentials in the integral Leray-Serre spectral sequence associated to the free loops space fibrations in the cases of SU(n+1)/Tn and Sp(n)/Tn. Study in detail the structure of the third page of the spectral sequence in the case of SU(n) and give the module structure of H*(Λ(SU(3)/T2);Z) and H*(Λ(Sp(2)/T2);Z).
University of Southampton
Burfitt, Matthew Ingram
db984de8-4a63-42cf-8b9a-3d9bca458e6a
Burfitt, Matthew Ingram
db984de8-4a63-42cf-8b9a-3d9bca458e6a
Grbic, Jelena
daaea124-d4cc-4818-803a-2b0cb4362175

Burfitt, Matthew Ingram (2017) The cohomology of free loop spaces of homogeneous spaces. University of Southampton, Doctoral Thesis, 121pp.

Record type: Thesis (Doctoral)

Abstract

The free loops space ΛX of a space X has become an important object of study particularly in the case when X is a manifold. The study of free loop spaces is motivated in particular by two main examples. The first is their relation to geometrically distinct periodic geodesics on a manifold, originally studied by Gromoll and Meyer in 1969. More recently the study of string topology and in particular the Chas-Sullivan loop product has been an active area of research.

A complete flag manifold is the quotient of a Lie group by its maximal torus and is one of the nicer examples of a homogeneous space. Both the cohomology and Chas-Sullivan product structure are understood for spaces Sn, CPn and most simple Lie groups. Hence studying the topology of the free loops space on homogeneous space is a natural next step.

In the thesis we compute the differentials in the integral Leray-Serre spectral sequence associated to the free loops space fibrations in the cases of SU(n+1)/Tn and Sp(n)/Tn. Study in detail the structure of the third page of the spectral sequence in the case of SU(n) and give the module structure of H*(Λ(SU(3)/T2);Z) and H*(Λ(Sp(2)/T2);Z).

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Published date: July 2017

Identifiers

Local EPrints ID: 415383
URI: http://eprints.soton.ac.uk/id/eprint/415383
PURE UUID: f65ed695-51cd-4558-93af-97c63d0261a8
ORCID for Jelena Grbic: ORCID iD orcid.org/0000-0002-7164-540X

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Date deposited: 08 Nov 2017 17:30
Last modified: 31 Jul 2019 00:34

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