Smooth homotopy of infinite-dimensional C8-manifolds / Hiroshi Kihara.

Format:

Book

Author/Creator:

Kihara, Hiroshi, author.

Series:

Memoirs of the American Mathematical Society ; Volume 289.

Memoirs of the American Mathematical Society Series ; Volume 289

Language:

English

Subjects (All):

Algebraic topology.

Physical Description:

1 online resource (144 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

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 and N, the smooth singular complex of the diffeological space C$C^{\infty }$(M,N) is weakly equivalent to the ordinary singular complex of the topological space C0(M,N) under the hereditary C$C^{\infty }$-paracompactness and semiclassicality conditions on M. We next generalize this result to sections of fiber bundles over a C$C^{\infty }$-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$C^{\infty }$-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$C^{\infty }$ of C$C^{\infty }$-manifolds into the category D of diffeological spaces and develop the smooth homotopy theory of diffeological spaces via a homotopical algebraic study of the model category D and the model category C0 of arc-generated spaces, also known as Δ-generated spaces. Then, the hereditary C$C^{\infty }$-paracompactness and semiclassicality conditions on M imply that M has the smooth homotopy type of a cofibrant object in 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$C^{\infty }$-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.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Fundamental problems on ^{∞}-manifolds

1.2. Main results on ^{∞}-manifolds

1.3. Smooth homotopy theory of diffeological spaces

1.4. Notation and terminology

1.5. Organization of the paper

Chapter 2. Diffeological spaces, arc-generated spaces, and ^{∞}-manifolds

2.1. Categories \dcal and \czero

2.2. Fully faithful embedding of ^{∞} into \dcal

2.3. Standard -simplices and model structure on \dcal

2.4. Quillen pairs (||_{\dcal}, ^{\dcal}) and (̃⋅, )

Chapter 3. Quillen equivalences between \scal, \dcal, and \czero

3.1. Singular homology of a diffeological space

3.2. Proof of Theorem 1.5

3.3. Proof of Corollary 1.6

Chapter 4. Smoothing of continuous maps

4.1. Enrichment of cartesian closed categories

4.2. Simplicial categories \czero and

4.3. Function complexes and homotopy function complexes for \czero and \dcal

4.4. Proof of Theorem 1.7

Chapter 5. Smoothing of continuous principal bundles

5.1. \ccal-partitions of unity

5.2. Principal bundles in \ccal

5.3. Fiber bundles in \ccal

5.4. Smoothing of principal bundles

Chapter 6. Smoothing of continuous sections

6.1. Quillen equivalences between the overcategories of \scal,\dcal, and \czero

6.2. Enrichment of overcategories

6.3. Simplicial categories \czero/ and /

6.4. Function complexes and homotopy function complexes for \czero/ and /

6.5. Proof of Theorem 1.8

Chapter 7. Dwyer-Kan equivalence between ( \dcal / )_{ } and ( \czerõ /̃ )_{ }

7.1. Enrichment of categories embedded into \mcal

7.2. Enriched groupoid \mcal{ }/

7.3. Smoothing of gauge transformations

7.4. Proof of Theorem 1.9

Chapter 8. Diffeological polyhedra

8.1. Basic properties of two kinds of diffeological polyhedra.

8.2. \dcal-homotopy equivalence between two kinds of diffeological polyhedra

Chapter 9. Homotopy cofibrancy theorem

9.1. Diffeological spaces associated to a covering

9.2. Hurewicz cofibrations in \dcal

9.3. Proof of Theorem 1.10

Chapter 10. Locally contractible diffeological spaces

Chapter 11. Applications to ^{∞}-manifolds

11.1. Proofs of Theorems 1.1-1.3

11.2. Classical atlases

11.3. Hereditary ^{∞}-paracompactness

11.4. Hereditarily ^{∞}-paracompact, semiclassical ^{∞}-manifolds

Appendix A. Pathological diffeological spaces

Appendix B. Keller's ^{∞}_{ }-theory and diffeological spaces

Appendix C. Smooth regularity and smooth paracompactness

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

Other Format:

Print version: Kihara, Hiroshi Smooth Homotopy of Infinite-Dimensional C8- Manifolds

ISBN:

1-4704-7591-X