Smooth Homotopy of Infinite-Dimensional 𝐶^{∞}-Manifolds

Abstract

In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C ∞ 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 ∞ C^\infty -paracompactness along with the semiclassicality condition on a C ∞ C^\infty -manifold, which enables us to use local convexity in local arguments. Then, we prove that for C ∞ C^\infty -manifolds M M and N N , the smooth singular complex of the diffeological space C ∞ ( M , N ) C^\infty (M,N) is weakly equivalent to the ordinary singular complex of the topological space C 0 ( M , N ) {\mathcal {C}^0}(M,N) under the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M . We next generalize this result to sections of fiber bundles over a C ∞ C^\infty -manifold M M under the same conditions on M M . Further, we establish the Dwyer-Kan equivalence between the simplicial groupoid of smooth principal G G -bundles over M M and that of continuous principal G G -bundles over M M for a Lie group G G and a C ∞ C^\infty -manifold M M under the same conditions on M M , encoding the smoothing results for principal bundles and gauge transformations. For the proofs, we fully faithfully embed the category C ∞ C^{\infty } of C ∞ C^{\infty } -manifolds into the category D {\mathcal {D}} of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category D {\mathcal {D}} and the model category C 0 {\mathcal {C}^0} of arc-generated spaces, also known as Δ \Delta -generated spaces. Then, the hereditary C ∞ C^\infty -paracompactness and semiclassicality conditions on M M imply that M M has the smooth homotopy type of a cofibrant object in D {\mathcal {D}} . This result can be regarded as a smooth refinement of the results of Milnor, Palais, and Heisey, which give sufficient conditions under which an infinite-dimensional topological manifold has the homotopy type of a C W CW -complex. We also show that most of the important C ∞ C^\infty -manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily C ∞ C^\infty -paracompact and semiclassical, and hence, results can be applied to them.