Smooth homotopy of infinite-dimensional \(C^{\infty}\)-manifolds. (English) Zbl 1542.58001
Memoirs of the American Mathematical Society 1436. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-6542-1/pbk; 978-1-4704-7591-8/ebook). vii, 129 p. (2023).
This paper develops a smooth homotopy theory for the category of diffeological spaces and applies it to the study of global analysis for infinite-dimensional \(C^\infty\)-manifolds.
A parametrization of a set \(X\) is a (set-theoretic) map from an open subset of \({\mathbb R}^n\) for some \(n\) to \(X\). A diffeological space is a set \(X\) with a specified set \({D}\) of parametrizations of \(X\) that satisfy the covering property, the locality and the smooth compatibility. A smooth map \(f : (X, {D}_X) \to (Y, {D}_Y)\) between diffeological spaces is a map \(f : X\to Y\) with \(f\circ p \in {D}_Y\) for any \(p \in {D}_X\). The category \(\mathcal{D}\) of diffeological spaces and smooth maps is cocomplete, complete and cartesian closed. These are characteristic properties of \(\mathcal{D}\).
In [H. Kihara, J. Homotopy Relat. Struct. 14, No. 1, 51–90 (2019; Zbl 1459.58001)], the author establishes a compactly generated model structure on the category \(\mathcal{D}\), which is recalled in Definition 2.13. To this end, maps \(\varphi_i : \Delta^{p-1} \times [0, 1) \to \Delta^p\) (\(i = 0,\dots, p\)) with \(\varphi_i(x, t)= (1-t)(i) + td^i(x)\) are introduced, where \(d^i\) denotes the \(i\)th coface map. Additionaly, the standard \(p\)-simplices \(\Delta^p\) (\(p\geq 0\)) are defined inductively by using \(\varphi_i\) and the final structure for the maps, where for \(p \leq 1\), \(\Delta^p\) is the canonical simplex \(\Delta_\text{sub}\) endowed with the sub-diffeology of the affine space \(\mathbb{A}^p\).
Let \(\mathcal{S}\) denote the category of simplicial spaces. The standard simplices mentioned above define the singular functor \(S^\mathcal{D} : \mathcal{D} \to \mathcal{S}\) and the realization functor \(| \ |_{\mathcal{D}} : \mathcal{S} \to \mathcal{D}\) by the usual way. Recall that a topological space \(X\) is arc-generated if its topology is final for the continuous curves from \(\mathbb{R}\) to \(X\). Let \(\mathcal{C}^0\) be the full sub-category of the category of topological spaces consisting of arc-generated spaces. By definition, the underlying topology functor \(\widetilde{\cdot} : \mathcal{D} \to \mathcal{C}^0\) assigns to a diffeological space \((X, \mathcal{D}_X)\) the underlying set \(X\) with the final topology for \(\mathcal{D}_X\). Moreover, for an arc-generated space \(X\), we define a diffeology \(\mathcal{D}_{RX}\) as the set of continuous parametrizations of \(X\). Then, the assignment \(R\) gives rise to a functor \(R : \mathcal{C}^0 \to \mathcal{D}\). These functors enable us to obtain Quillen equivalences of categories.
{Theorem 1.5} (1) \(| \ |_{\mathcal{D}} : \mathcal{S} \rightleftarrows \mathcal{D} : S^{\mathcal{D}}\) is a pair of Quillen equivalences.
(2) \(\widetilde{\cdot} : \mathcal{D} \rightleftarrows \mathcal{C}^0 : R\) is a pair of Quillen equivalences.
We observe that the underlying topology functor \(\widetilde{\cdot}\) preserves finite products ([loc. cit., Proposition 2.13]). Thus, the functor induces the natural inclusion between simplicial complexes \[ S^{\mathcal{D}}\mathcal{D}(A, X) \to S\mathcal{C}^0(\widetilde{A}, \widetilde{X}), \] where \(S\) denotes the singular simplex functor. To describe the first main theorem of this paper, we need two subclasses of \(\mathcal{D}\). One is the subclass \(\mathcal{W}_{\mathcal{D}}\) defined by \[ \mathcal{W}_{\mathcal{D}}:= \{A \in \mathcal{D} \mid \text{\(A\) has the \(\mathcal{D}\)-homotopy type of a cofibrant diffeological space}\}. \] The other is the subclass \(\mathcal{V}_{\mathcal{D}}\) defined by \[ \mathcal{V}_{\mathcal{D}}: = \{ A \in \mathcal{D} \mid \text{\(\mathrm{id} : A \to R\widetilde{A}\) is a weak equivalent in \(\mathcal{D}\)} \}. \] Corollary 1.6 asserts that there is an inclusion from \(\mathcal{W}_{\mathcal{D}}\) to \(\mathcal{V}_{\mathcal{D}}\).
The author proves the smoothing theorem for continuous maps (Theorem 1.7), continuous sections (Theorem 1.8) and principal bundles (Theorem 1.9). In particular, Theorem 1.7 asserts that if \(A\) is in \(\mathcal{W}_{\mathcal{D}}\) and \(X\) is in \(\mathcal{V}_{\mathcal{D}}\), then the natural inclusion \(S^{\mathcal{D}}\mathcal{D}(A, X) \to S\mathcal{C}^0(\widetilde{A}, \widetilde{X})\) is a weak equivalence in \(\mathcal{S}\).
As the title shows, the paper applies these diffeological results to \(C^\infty\)-manifolds in the sense of A. Kriegl and P. W. Michor [The convenient setting of global analysis. Providence, RI: American Mathematical Society (1997; Zbl 0889.58001), Section 27] with which infinite-dimensional calculus is developed. Lemma 2.5 allows us to obtain a fully faithful embedding of the category of \(C^\infty\)-manifolds into \(\mathcal{D}\). Thus, \(C^\infty\)-manifolds become important objects in \(\mathcal{D}\).
Theorem 1.10 provides a sufficient condition for a diffeological space to be in the subclass \(\mathcal{W}_D\). It is worthwhile to mention that a diffeological version of the nerve lemma (Proposition 9.5) is used to prove the theorem. Moreover, Theorem 1.11 asserts that every locally contractible diffeological space is in \(\mathcal{V}_{\mathcal{D}}\). These results yield that every hereditarily \(C^\infty\)-paracompact, semiclassical \(C^\infty\)-manifold is in \(\mathcal{W}_{\mathcal{D}}\) (Theorem 11.1) and that every \(C^\infty\)-manifold is in \(\mathcal{V}_{\mathcal{D}}\) (Theorem 11.2). In particular, every submanifold of \(M\), which is modeled on Hilbert space, nuclear Fréchet space, or nuclear Silva space, is in \(\mathcal{W}_{\mathcal{D}}\). Thus, Theorems 1.7, 1.8 and 1.9 are applicable to such manifolds.
Appendix A deals with pathological diffeological spaces. We define other subclass \(\widetilde{}\mathcal{W}_{\mathcal{C}^0}\) of \(\mathcal{D}\) by \[ \widetilde{}\mathcal{W}_{\mathcal{C}^0}:=\{A \in \mathcal{D} \mid \text{\(\widetilde{A}\) has the homotopy type of cofibrant arc-generated space}\}. \] Then, the irrational torus is in \(\widetilde{}\mathcal{W}_{\mathcal{C}^0}\backslash \mathcal{V}_{\mathcal{D}}\). Moreover, the author gives examples which are in \(\mathcal{D} \backslash (\mathcal{V}_{\mathcal{D}}\cup \widetilde{}\mathcal{W}_{\mathcal{C}^0})\), \(\widetilde{}\mathcal{W}_{\mathcal{C}^0}\backslash \mathcal{V}_{\mathcal{D}}\), \(\mathcal{V}_{\mathcal{D}}\backslash \widetilde{}\mathcal{W}_{\mathcal{C}^0}\) and \((\mathcal{V}_{\mathcal{D}}\cap \widetilde{}\mathcal{W}_{\mathcal{C}^0})\backslash \mathcal{W}_D\).
A parametrization of a set \(X\) is a (set-theoretic) map from an open subset of \({\mathbb R}^n\) for some \(n\) to \(X\). A diffeological space is a set \(X\) with a specified set \({D}\) of parametrizations of \(X\) that satisfy the covering property, the locality and the smooth compatibility. A smooth map \(f : (X, {D}_X) \to (Y, {D}_Y)\) between diffeological spaces is a map \(f : X\to Y\) with \(f\circ p \in {D}_Y\) for any \(p \in {D}_X\). The category \(\mathcal{D}\) of diffeological spaces and smooth maps is cocomplete, complete and cartesian closed. These are characteristic properties of \(\mathcal{D}\).
In [H. Kihara, J. Homotopy Relat. Struct. 14, No. 1, 51–90 (2019; Zbl 1459.58001)], the author establishes a compactly generated model structure on the category \(\mathcal{D}\), which is recalled in Definition 2.13. To this end, maps \(\varphi_i : \Delta^{p-1} \times [0, 1) \to \Delta^p\) (\(i = 0,\dots, p\)) with \(\varphi_i(x, t)= (1-t)(i) + td^i(x)\) are introduced, where \(d^i\) denotes the \(i\)th coface map. Additionaly, the standard \(p\)-simplices \(\Delta^p\) (\(p\geq 0\)) are defined inductively by using \(\varphi_i\) and the final structure for the maps, where for \(p \leq 1\), \(\Delta^p\) is the canonical simplex \(\Delta_\text{sub}\) endowed with the sub-diffeology of the affine space \(\mathbb{A}^p\).
Let \(\mathcal{S}\) denote the category of simplicial spaces. The standard simplices mentioned above define the singular functor \(S^\mathcal{D} : \mathcal{D} \to \mathcal{S}\) and the realization functor \(| \ |_{\mathcal{D}} : \mathcal{S} \to \mathcal{D}\) by the usual way. Recall that a topological space \(X\) is arc-generated if its topology is final for the continuous curves from \(\mathbb{R}\) to \(X\). Let \(\mathcal{C}^0\) be the full sub-category of the category of topological spaces consisting of arc-generated spaces. By definition, the underlying topology functor \(\widetilde{\cdot} : \mathcal{D} \to \mathcal{C}^0\) assigns to a diffeological space \((X, \mathcal{D}_X)\) the underlying set \(X\) with the final topology for \(\mathcal{D}_X\). Moreover, for an arc-generated space \(X\), we define a diffeology \(\mathcal{D}_{RX}\) as the set of continuous parametrizations of \(X\). Then, the assignment \(R\) gives rise to a functor \(R : \mathcal{C}^0 \to \mathcal{D}\). These functors enable us to obtain Quillen equivalences of categories.
{Theorem 1.5} (1) \(| \ |_{\mathcal{D}} : \mathcal{S} \rightleftarrows \mathcal{D} : S^{\mathcal{D}}\) is a pair of Quillen equivalences.
(2) \(\widetilde{\cdot} : \mathcal{D} \rightleftarrows \mathcal{C}^0 : R\) is a pair of Quillen equivalences.
We observe that the underlying topology functor \(\widetilde{\cdot}\) preserves finite products ([loc. cit., Proposition 2.13]). Thus, the functor induces the natural inclusion between simplicial complexes \[ S^{\mathcal{D}}\mathcal{D}(A, X) \to S\mathcal{C}^0(\widetilde{A}, \widetilde{X}), \] where \(S\) denotes the singular simplex functor. To describe the first main theorem of this paper, we need two subclasses of \(\mathcal{D}\). One is the subclass \(\mathcal{W}_{\mathcal{D}}\) defined by \[ \mathcal{W}_{\mathcal{D}}:= \{A \in \mathcal{D} \mid \text{\(A\) has the \(\mathcal{D}\)-homotopy type of a cofibrant diffeological space}\}. \] The other is the subclass \(\mathcal{V}_{\mathcal{D}}\) defined by \[ \mathcal{V}_{\mathcal{D}}: = \{ A \in \mathcal{D} \mid \text{\(\mathrm{id} : A \to R\widetilde{A}\) is a weak equivalent in \(\mathcal{D}\)} \}. \] Corollary 1.6 asserts that there is an inclusion from \(\mathcal{W}_{\mathcal{D}}\) to \(\mathcal{V}_{\mathcal{D}}\).
The author proves the smoothing theorem for continuous maps (Theorem 1.7), continuous sections (Theorem 1.8) and principal bundles (Theorem 1.9). In particular, Theorem 1.7 asserts that if \(A\) is in \(\mathcal{W}_{\mathcal{D}}\) and \(X\) is in \(\mathcal{V}_{\mathcal{D}}\), then the natural inclusion \(S^{\mathcal{D}}\mathcal{D}(A, X) \to S\mathcal{C}^0(\widetilde{A}, \widetilde{X})\) is a weak equivalence in \(\mathcal{S}\).
As the title shows, the paper applies these diffeological results to \(C^\infty\)-manifolds in the sense of A. Kriegl and P. W. Michor [The convenient setting of global analysis. Providence, RI: American Mathematical Society (1997; Zbl 0889.58001), Section 27] with which infinite-dimensional calculus is developed. Lemma 2.5 allows us to obtain a fully faithful embedding of the category of \(C^\infty\)-manifolds into \(\mathcal{D}\). Thus, \(C^\infty\)-manifolds become important objects in \(\mathcal{D}\).
Theorem 1.10 provides a sufficient condition for a diffeological space to be in the subclass \(\mathcal{W}_D\). It is worthwhile to mention that a diffeological version of the nerve lemma (Proposition 9.5) is used to prove the theorem. Moreover, Theorem 1.11 asserts that every locally contractible diffeological space is in \(\mathcal{V}_{\mathcal{D}}\). These results yield that every hereditarily \(C^\infty\)-paracompact, semiclassical \(C^\infty\)-manifold is in \(\mathcal{W}_{\mathcal{D}}\) (Theorem 11.1) and that every \(C^\infty\)-manifold is in \(\mathcal{V}_{\mathcal{D}}\) (Theorem 11.2). In particular, every submanifold of \(M\), which is modeled on Hilbert space, nuclear Fréchet space, or nuclear Silva space, is in \(\mathcal{W}_{\mathcal{D}}\). Thus, Theorems 1.7, 1.8 and 1.9 are applicable to such manifolds.
Appendix A deals with pathological diffeological spaces. We define other subclass \(\widetilde{}\mathcal{W}_{\mathcal{C}^0}\) of \(\mathcal{D}\) by \[ \widetilde{}\mathcal{W}_{\mathcal{C}^0}:=\{A \in \mathcal{D} \mid \text{\(\widetilde{A}\) has the homotopy type of cofibrant arc-generated space}\}. \] Then, the irrational torus is in \(\widetilde{}\mathcal{W}_{\mathcal{C}^0}\backslash \mathcal{V}_{\mathcal{D}}\). Moreover, the author gives examples which are in \(\mathcal{D} \backslash (\mathcal{V}_{\mathcal{D}}\cup \widetilde{}\mathcal{W}_{\mathcal{C}^0})\), \(\widetilde{}\mathcal{W}_{\mathcal{C}^0}\backslash \mathcal{V}_{\mathcal{D}}\), \(\mathcal{V}_{\mathcal{D}}\backslash \widetilde{}\mathcal{W}_{\mathcal{C}^0}\) and \((\mathcal{V}_{\mathcal{D}}\cap \widetilde{}\mathcal{W}_{\mathcal{C}^0})\backslash \mathcal{W}_D\).
Reviewer: Katsuhiko Kuribayashi (Nagano)
MSC:
| 58B05 | Homotopy and topological questions for infinite-dimensional manifolds |
| 58A40 | Differential spaces |
| 18N40 | Homotopical algebra, Quillen model categories, derivators |
Keywords:
model category; diffeological spaces; smooth homotopy; convenient calculus; \(C^{\infty}\)-manifoldsReferences:
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