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On string topology of classifying spaces
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).ORCID iD: 0000-0002-3717-8609
2015 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 281, 394-507 p.Article in journal (Refereed) Published
Abstract [en]

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.

Place, publisher, year, edition, pages
2015. Vol. 281, 394-507 p.
Keyword [en]
Classifying spaces, Field theories, Primary, Secondary, String topology
National Category
URN: urn:nbn:se:kth:diva-170196DOI: 10.1016/j.aim.2015.03.022ISI: 000358460000009ScopusID: 2-s2.0-84930959598OAI: diva2:828235

QC 20150630

Available from: 2015-06-30 Created: 2015-06-29 Last updated: 2015-08-27Bibliographically approved

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Lahtinen, Anssi
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Mathematics (Dept.)
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