### Abstract

Original language | English |
---|---|

Pages (from-to) | 394-507 |

Number of pages | 114 |

Journal | Advances in Mathematics |

Volume | 281 |

Early online date | 29 May 2015 |

DOIs | |

Publication status | Published - 20 Aug 2015 |

### Fingerprint

### Keywords

- string topology
- field theories
- classifying spaces

### Cite this

*Advances in Mathematics*,

*281*, 394-507. https://doi.org/10.1016/j.aim.2015.03.022

**On string topology of classifying spaces.** / Hepworth, Richard; Lahtinen, Anssi.

Research output: Contribution to journal › Article

*Advances in Mathematics*, vol. 281, pp. 394-507. https://doi.org/10.1016/j.aim.2015.03.022

}

TY - JOUR

T1 - On string topology of classifying spaces

AU - Hepworth, Richard

AU - Lahtinen, Anssi

PY - 2015/8/20

Y1 - 2015/8/20

N2 - Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.

AB - Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs. The result is a novel kind of field theory which is related to both the diffeomorphism groups of surfaces and the automorphism groups of free groups with boundaries. Our work shows that the algebraic structures in string topology of classifying spaces can be brought into line with, and in fact far exceed, those available in string topology of manifolds. For simplicity, we restrict to the characteristic 2 case. The generalization to arbitrary characteristic will be addressed in a subsequent paper.

KW - string topology

KW - field theories

KW - classifying spaces

U2 - 10.1016/j.aim.2015.03.022

DO - 10.1016/j.aim.2015.03.022

M3 - Article

VL - 281

SP - 394

EP - 507

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -