Smooth Homotopy of Infinite-Dimensional $C^{\infty}$-Manifolds

Abstract

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional $C^\infty$-manifolds in convenient calculus. More precisely, we discuss the smoothing of maps, sections, principal bundles, and gauge transformations. We first introduce the notion of hereditary $C^\infty$-paracompactness along with the semiclassicality condition on a $C^\infty$-manifold, which enables us to use local convexity in local arguments. Then, we prove that for $C^\infty$-manifolds $M$ and $N$, the smooth singular complex of $C^\infty(M,N)$ is weakly equivalent to the ordinary singular complex of $\mathcal{C}^0(M,N)$ under the hereditary $C^\infty$-paracompactness and semiclassicality conditions on $M$. We next generalize this result to sections of fiber bundles over a $C^\infty$-manifold $M$ under the same conditions on $M$. Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal $G$-bundles over $M$ and that of continuous principal $G$-bundles over $M$ for a Lie group $G$ and a $C^\infty$-manifold $M$ under the same conditions on $M$, encoding the smoothing results for principal bundles and gauge transformations. For the proofs, we fully faithfully embed the category $C^\infty$ of $C^\infty$-manifolds into the category $\mathcal{D}$ of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category $\mathcal{D}$ and the model category $\mathcal{C}^0$ of arc-generated spaces. Then, the hereditary $C^\infty$-paracompactness and semiclassicality conditions on $M$ imply that $M$ has the smooth homotopy type of a cofibrant object in $\mathcal{D}$. This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey on the homotopy type of infinite-dimensional topological manifolds.