AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

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

About this Title

Hiroshi Kihara

Publication: Memoirs of the American Mathematical Society
Publication Year: 2023; Volume 289, Number 1436
ISBNs: 978-1-4704-6542-1 (print); 978-1-4704-7591-8 (online)
DOI: https://doi.org/10.1090/memo/1436
Published electronically: August 23, 2023
Keywords: Smooth homotopy, $C^{\infty }$-manifolds, convenient calculus, diffeological spaces, model category

View full volume as PDF

Table of Contents

Chapters

• 1. Introduction
• 2. Diffeological spaces, arc-generated spaces, and $C^{\infty }$-manifolds
• 3. Quillen equivalences between ${\mathcal {S}}$, ${\mathcal {D}}$, and ${\mathcal {C}^0}$
• 4. Smoothing of continuous maps
• 5. Smoothing of continuous principal bundles
• 6. Smoothing of continuous sections
• 7. Dwyer-Kan equivalence between $(\mathsf {P}{\mathcal {D}}G / X)_{\mathrm {num}}$ and $(\mathsf {P}{\mathcal {C}^0}\widetilde {G} / \widetilde {X})_{\mathrm {num}}$
• 8. Diffeological polyhedra
• 9. Homotopy cofibrancy theorem
• 10. Locally contractible diffeological spaces
• 11. Applications to $C^{\infty }$-manifolds
• A. Pathological diffeological spaces
• B. Keller’s $C^{\infty }_{c}$-theory and diffeological spaces
• C. Smooth regularity and smooth paracompactness

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 the diffeological space $C^\infty (M,N)$ is weakly equivalent to the ordinary singular complex of the topological space ${\mathcal {C}^0}(M,N)$ 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, also known as $\Delta$-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, which give sufficient conditions under which an infinite-dimensional topological manifold has the homotopy type of a $CW$-complex. We also show that most of the important $C^\infty$-manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily $C^\infty$-paracompact and semiclassical, and hence, results can be applied to them.