Thin loop groups
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Thin loop groups Moncef Ghazel1
· Sadok Kallel2,3
In fond memory of S¸ tefan Papadima Received: 1 February 2019 / Revised: 19 September 2019 / Accepted: 23 December 2019 © Springer Nature Switzerland AG 2020
Abstract We verify that for a finite simplicial complex X and for piecewise linear loops on X , the “thin” loop space is a topological group of the same homotopy type as the space of continuous loops. This turns out not to be the case for the higher loops. Keywords Loop space · Piecewise linear map · Quasifibration · Topological group Mathematics Subject Classification 55P35 · 55U10
1 Introduction A loop on a topological space X is any continuous map of the circle S 1 ⊂ C into X . A loop is “based” if it sends 1 ∈ S 1 to a preferred basepoint x0 ∈ X . Physicists and geometers, see e.g. [4,8], introduced the “thin fundamental group” of a smooth manifold to be the space of based smooth loops, “stationary” at the basepoint, modulo “thin homotopy”. A homotopy is “thin” if its image “does not enclose area” (precise definitions in Sect. 3). If X is a smooth manifold, then its thin fundamental group is denoted by π11 (X ). This is indeed a group which is a quotient of the corresponding loop space (X ) and of which π1 (X ) is a quotient; i.e., (X ) π11 (X ) π1 (X ).
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Sadok Kallel [email protected] Moncef Ghazel [email protected]
1
Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis, Université de Tunis El Manar, 2092 El Manar Tunis, Tunisia
2
American University of Sharjah, University City, PO Box 26666, Sharjah, United Arab Emirates
3
Laboratoire Painlevé, Université de Lille 1, 59655 Villeneuve d’Ascq, France
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M. Ghazel, S. Kallel
This space plays an important role in studying holonomy, connections and transport. As shown in [4,21], and even earlier in [19,33], there is a bijective correspondence between h homomorphisms π11 (X ) − → G with G a compact Lie group, up to conjugation, and principal G-bundles with connections on X , unique up to equivalence. A technical condition is put on the topology of loops so that there is the notion of a smooth family of loops which has a smoothly varying holonomy image in G [4, Section 2]. The interesting part of this observation is that if f , g ∈ (X ) are two smooth loops related by a thin homotopy, then parallel transport along f gives the same result as parallel transport along g [8,19]. The question we ask, and then answer in this paper, is: how “close” is π11 (X ) to π1 (X ) or to (X ), the space of all continuous loops, when X is a connected finite complex? When X is a graph or a manifold of dimension 1, all homotopies are thin so the epimorphism π11 (X ) π1 (X ) becomes an isomorphism. This situation turns out to be rather exceptional. In order to answer the question in general, we need to pay a particular attention to the topology we put on the loop space [1]. By abuse of terminology, a “simplicial complex” refers to either the complex or its realization (the underlying “polyhedral space”). Let X be a finit
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