Smooth Homotopy of Infinite-Dimensional C1-Manifolds Hiroshi Kihara Author address: Center for Mathematical Sciences, University of Aizu, Tsuruga, Ikki-machi, Aizu-Wakamatsu City, Fukushima, 965-8580, Japan Email address: ([email protected]) Dedicated to My Parents Contents 1. Introduction 1 1.1. Fundamental problems on C1-manifolds 1 1.2. Main results on C1-manifolds 4 1.3. Smooth homotopy theory of diffeological spaces 5 1.4. Notation and terminology 8 1.5. Organization of the paper 9 2. Diffeological spaces, arc-generated spaces, and C1-manifolds 10 2.1. Categories D and C0 10 2.2. Fully faithful embedding of C1 into D 12 2.3. Standard p-simplices and model structure on D 14 D 2.4. Quillen pairs (j jD;S ) and ( e·;R) 17 3. Quillen equivalences between S, D, and C0 18 3.1. Singular homology of a diffeological space 18 3.2. Proof of Theorem 1.5 22 3.3. Proof of Corollary 1.6 25 4. Smoothing of continuous maps 26 4.1. Enrichment of cartesian closed categories 26 4.2. Simplicial categories C0 and D 27 4.3. Function complexes and homotopy function complexes for C0 and D 29 4.4. Proof of Theorem 1.7 35 5. Smoothing of continuous principal bundles 35 5.1. C-partitions of unity 36 5.2. Principal bundles in C 37 5.3. Fiber bundles in C 39 5.4. Smoothing of principal bundles 40 6. Smoothing of continuous sections 42 6.1. Quillen equivalences between the overcategories of S; D; and C0 42 6.2. Enrichment of overcategories 43 6.3. Simplicial categories C0=X and D=X 47 6.4. Function complexes and homotopy function complexes for C0=X and D=X 48 6.5. Proof of Theorem 1.8 51 0 7. Dwyer-Kan equivalence between (PDG=X)num and (PC G=e Xe)num 52 7.1. Enrichment of categories embedded into M 52 7.2. Enriched groupoid PMG=X 53 7.3. Smoothing of gauge transformations 54 7.4. Proof of Theorem 1.9 56 8. Diffeological polyhedra 56 8.1. Basic properties of two kinds of diffeological polyhedra 57 v vi HIROSHI KIHARA 8.2. D-homotopy equivalence between two kinds of diffeological polyhedra 60 9. Homotopy cofibrancy theorem 65 9.1. Diffeological spaces associated to a covering 65 9.2. Hurewicz cofibrations in D 71 9.3. Proof of Theorem 1.10 77 10. Locally contractible diffeological spaces 79 11. Applications to C1-manifolds 83 11.1. Proofs of Theorems 1.1-1.3 83 11.2. Classical atlases 84 11.3. Hereditary C1-paracompactness 85 11.4. Hereditarily C1-paracompact, semiclassical C1-manifolds 89 Appendix A. Pathological diffeological spaces 100 1 Appendix B. Keller's Cc -theory and diffeological spaces 104 Appendix C. Smooth regularity and smooth paracompactness 106 Bibliography 110 Abstract In this paper, we use homotopical algebra (or abstract homotopical methods) to study smooth homotopical problems of infinite-dimensional C1-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 C1-paracompactness along with the semiclassicality condition on a C1-manifold, which enables us to use local convexity in local arguments. Then, we prove that for C1-manifolds M and N, the smooth singular complex of the diffeological space C1(M; N) is weakly equivalent to the ordinary singular complex of the topological space C0(M; N) under the hereditary C1-paracompactness and semiclassicality conditions on M. We next generalize this result to sections of fiber bundles over a C1-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 C1-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 C1 of C1-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 ∆-generated spaces. Then, the hereditary C1-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 C1-manifolds introduced and studied by Kriegl, Michor, and their coauthors are hereditarily C1-paracompact and semiclassical, and hence, results can be applied to them. Received by the editor February 5, 2020. 2010 Mathematics Subject Classification. Primary 58B05; Secondary 58A40,18G55. Key words and phrases. Smooth homotopy, C1-manifolds, convenient calculus, diffeological spaces, model category. I would like to show my greatest appreciation to Prof. Toshiro Watanabe who offered con- tinuing support and constant encouragement. vii 1. INTRODUCTION 1 1. Introduction This paper aims to develop a smooth homotopy theory of diffeological spaces and apply it to global analysis on infinite-dimensional C1-manifolds by embedding C1-manifolds fully faithfully into the category of diffeological spaces. In Section 1.1, we formulate fundamental problems on C1-manifolds, and in Section 1.2, we provide answers to these problems, as the main results on C1- manifolds. In Section 1.3, we outline the results obtained herein on smooth homo- topy for diffeological spaces, to which most of this paper is devoted, and explain how they yield the results in Section 1.2. 1.1. Fundamental problems on C1-manifolds. Fr¨olicher, Kriegl, and Mi- chor [46] established the foundation of infinite-dimensional calculus, which is called convenient calculus and is regarded as a prime candidate for the final theory of infinite-dimensional calculus. However, it has no efficient approach for solving one of the most critical problems: to investigate how many smooth maps exist between the given infinite-dimensional C1-manifolds M and N. Since the study of contin- uous maps between M and N is done by topological homotopy theory (or algebraic topology), we formulate the problem as follows: (a) When do the smooth homotopy classes of smooth maps between M and N bijectively correspond to the continuous homotopy classes of continuous maps ? The following two problems are also important; Problem (b) is a generalization of Problem (a), and Problem (c) is closely related to Problems (a) and (b). (b) Let p : E −! M be a smooth fiber bundle. When do the vertical smooth homotopy classes of smooth sections of E bijectively correspond to the vertical continuous homotopy classes of continuous sections ? (c) Let G be a Lie group. When do the isomorphism classes of smooth princi- pal G-bundles over M bijectively correspond to those of continuous princi- pal G-bundles over M? Let π : P −! M be a smooth principal G-bundle. When do the isotopy classes of smooth gauge transformations of P bijec- tively correspond to those of continuous gauge transformations of P ? If all C1-manifolds involved are finite-dimensional, then the correspondences in Problems (a) and (b) are always bijective by the Steenrod approximation theorem ([66, Section 6.7]), which is one of the most basic results in differential topology. In more general settings, Problems (a), (b), and (c) were addressed in [47], [71], and [57, 70], respectively. Roughly speaking, as answers to these questions, it has been shown that the correspondences in the questions are bijective, provided that M is finite-dimensional. Precisely, M¨ullerand Wockel [57, 70, 71] did not work 1 in convenient calculus but in Keller's Cc -theory; moreover, to prove the smooth- ing result of continuous gauge transformations, Wockel imposed even compactness condition on M along with additional conditions on G [70, Proposition 1.20]. How- ever, no essential answer is known in the case where M is infinite-dimensional since the existing approaches are essentially based on the finite dimensionality (or local compactness) of M (cf. [71, Section 1] and [57, Introduction]). In the rest of this subsection, we more precisely formulate the fundamental problems mentioned above; we actually address the higher homotopical versions of Problems (a)-(c) by observing that the involved relevant categories and functors can be enriched over the category S of simplicial sets. 2 HIROSHI KIHARA Throughout this paper, C1-manifolds are ones in the sense of [46, Section 27] unless stated otherwise, and C1 denotes the category of (separated) C1-manifolds (see Section 2.2). Lie groups are defined as groups in C1 ([49, p. 75]). The under- lying topological space Mf of a C1-manifold M is defined as the set M endowed with the final topology for the smooth curves ([46, 27.4]). Then, we have the 1 0 0 underlying topological space functor e· : C −! C , where C is the category of arc-generated spaces and continuous maps (see Section 2.1). Since the category C1 is not closed under various categorical constructions, we fully faithfully embed C1 into the category D of diffeological spaces (see Section 2.2). Recall that the underlying topological space Xe of a diffeological space X is defined to be the set X endowed with the final topology for the diffeology DX . Then, we can see that the fully faithful embedding C1 ,−! D and the underlying topological space functors for C1 and D form the commutative diagram C1 D e· e· C0 consisting of functors that preserve finite products (Proposition 2.6).