Thin loop groups

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.

Appendix: Group models for loop spaces

The loop space $\Omega (X)$ is an H-space which is associative only up to homotopy, and more generally an $A_\infty$ space [30]. It is of interest in both homotopy theory and geometry to replace $\Omega (X)$ by a weakly homotopic space that is a topological group. We list below the various known ways to do this.

1.1 Combinatorial models

Milnor in [24] is perhaps the earliest to have exhibited such a group replacement for $\Omega (X)$ when X is of the homotopy type of a connected countable complex. For X a countable simplicial complex, consider $S_k$ as in (2) and define

where $(x_0,\ldots , x_{i-1}, x_i, \ldots , x_n)\sim (x_0,\ldots , x_{i-1},\widehat{x}_i,x_{i+1},\ldots , x_n)$ whenever the triple $x_{i-1}=x_i$ or $x_{i-1}=x_{i+1}$. Then $M(X)=\lim M_n$ is the group of a universal bundle, thus it is weakly homotopy equivalent to $\Omega (X)$. Our model in this paper S(X) has the advantage of being homeomorphic to the thin loop space for piecewise linear loops. Note that there is a homomorphism of topological groups $M(X)\rightarrow \omega (X)$.

Another model appears in [3] in the case X is a Riemannian manifold. The loop space model by Bahri and Cohen is obtained by considering composable small geodesics.

1.2 Simplicial (Kan) model

Let X be a based space which is of the homotopy type of a CW complex. Take the simplicial total singular complex SX of X which is a based simplical set. The Kan loop group GSX of SX is then a simplicial group and its geometric realization |GSX| is a topological group object in the category of compactly generated spaces. This is a model for the loop space. This is discussed in [18].

The generality of this model allows us to deduce for example that any strictly associative monoid (or H-space) such that $\pi _0$ is a group can be “rigidified” to a topological group; that is is weakly homotopy equivalent to a topological group. Indeed for such spaces, Dold and Lashof show the existence of a classifying space $B_X$ and a weak homotopy equivalence $X\rightarrow \Omega (B_X)$. One then uses the above construction to replace the loop space up to equivalence by a topological group.

Note that May has another model that produces this time a topological monoid weakly equivalent to any $A_\infty$-space [23, Theorem 13.5].

1.3 The “geometric model”

Lefschetz [20, Chapter V, Section 4] and Kobayashi introduce the earliest known thin loop models for piecewise smooth loops on smooth manifolds. Teleman in [33] introduces a similar model which is a quotient of $P(M,x_0)$, the space of piecewise smooth paths starting at $x_0$, by three equivalence relations and uses this model to classify bundles. Gajer introduces a similar model in [12, Section 1.3], which with the first relation ensuring that in the quotient space G(M), the product of loops is associative, $\sim _2$ ensures that the class of the constant map is the identity in G(M), and the final relation $\sim _3$ implies that $[\gamma ^{-1}]$ is a genuine inverse for $[\gamma ]$, much as in our Sect. 3. However, the proof is not quite complete, see Remark 5.6.

1.4 The Quillen model

Let $X=G$ be a simply-connected Lie group with simple Lie algebra, and let $G_{{\mathbb {C}}}$ be its complexification. An “algebraic loop” in $G_{{\mathbb {C}}}$ is a regular map ${{\mathbb {C}}}^*\!\rightarrow G_{{\mathbb {C}}}$ (i.e., a morphism of algebraic varieties). The set of all such loops is denoted by ${\widetilde{G}}_{{\mathbb {C}}}$. One then sets

This is an algebraic group. By a theorem of Quillen and Garland–Raghunathan, the restriction map to $S^1$ yields a homotopy equivalence $\Omega _{\mathrm{alg}}(G)\simeq \Omega (G)$ (one reference is [25]).