Mayer–Vietoris Sequence for Differentiable/Diffeological Spaces

Norio Iwase; Nobuyuki Izumida

doi:10.1007/978-981-13-5742-8_8

Algebraic Topology and Related Topics pp 123-151 | Cite as

Mayer–Vietoris Sequence for Differentiable/Diffeological Spaces

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Norio Iwase
Nobuyuki Izumida

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First Online: 03 February 2019

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Abstract

The idea of a space with smooth structure is a generalization of an idea of a manifold. K. T. Chen introduced such a space as a differentiable space in his study of a loop space to employ the idea of iterated path integrals [2, 3, 4, 5]. Following the pattern established by Chen, Souriau [10] introduced his version of a space with smooth structure, which is called a diffeological space. These notions are strong enough to include all the topological spaces. However, if one tries to show de Rham theorem, he must encounter a difficulty to obtain a partition of unity and thus the Mayer–Vietoris exact sequence in general. In this paper, we introduce a new version of differential forms to obtain a partition of unity, the Mayer–Vietoris exact sequence, and a version of de Rham theorem in general. In addition, if we restrict ourselves to consider only CW complexes, we obtain de Rham theorem for a genuine de Rham complex, and hence the genuine de Rham cohomology coincides with the ordinary cohomology for a CW complex.

Keywords

Differentiable Diffeology Partition of unity Differential form De Rham theory Singular cohomology

1991 Mathematics Subject Classification

Primary 58A40 Secondary 58A03 58A10 58A12 55N10

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Notes

Acknowledgements

This research was supported by Grant-in-Aid for Scientific Research (B) #22340014, Scientific Research (A) #23244008, Exploratory Research #24654013 and Challenging Exploratory Research #18K18713 from Japan Society for the Promotion of Science.

Appendix A: Smooth CW Complex

A smooth CW complex $X=(X,\{X^{(n)}\}_{n\ge -1})$ is a differentiable or diffeological space built up from $X^{(-1)} = \emptyset $ by inductively attaching n-balls $\{B^{n}_{j}\}_{j \in J_{n}}$ by $C^{\infty }$ maps from their boundary spheres $\{S^{n-1}_{j}\}_{j \in J_{n}}$ to $n{-}1$-skeleton $X^{(n-1)}$ to obtain n-skeleton $X^{(n)}$, $n \!\ge \! 0$, where the smooth structures of balls and spheres are given by their manifold structures. Thus a plot in $X^{(n)}$ is a map $P : A \rightarrow X$ with an open covering $\{A_{\alpha }\}_{\alpha \in \Lambda }$ of A such that, for any $\alpha $, $P(A_{\alpha })$ is in $X^{(n-1)}$ or $B^{n}_{j}$ for some $j \!\in \! J_{n}$ and $P\vert _{A_{\alpha }}$ is a plot of $X^{(n-1)}$ or $B^{n}_{j}$, respectively. Then as the colimit of $\{X^{(n)}\}$, X exists in ${\mathsf {Differentiable}}$ or ${\mathsf {Diffeology}}$.

For a given CW complex, we can deform attaching maps of n-balls from their boundary spheres $\{S^{n-1}_{j}\}_{j \in J_{n}}$ to $n{-}1$-skeleton $X^{(n-1)}$ to be $C^{\infty }$ maps and obtain the following.

Theorem A.1

A CW complex is homotopy equivalent to a smooth CW complex as topological spaces. Thus we may assume that any CW complex is smooth up to homotopy.

Let $X=(X,\{X^{(n)}\})$ be a smooth CW complex in either ${\mathsf {Differentiable}}$ or ${\mathsf {Diffeology}}$ with the set of n-balls $\{B^{n}_{j}\,;\,j \!\in \!J_{n}\}$. Then for any plot $P : A \rightarrow X^{(n)}$, there is an open covering $\{A_{\alpha }\}$ of A, such that $P(A_{\alpha })$ is in either $X^{(n-1)}$ or $B^{n}_{j}$ for some $j \!\in \! J_{n}$ and $P_{\alpha }=P\vert _{A_{\alpha }}$ is a plot of $X^{(n-1)}$ or $B^{n}_{j}$, respectively. Let $U=X^{(n)} \smallsetminus X^{(n-1)}$ and $V=X^{(n)}\smallsetminus (\underset{j \in J_{n}}{\cup }\{{\varvec{0}}_{j}\})$, where ${\varvec{0}}_{j} \in B^{n}_{j}$ denotes the element corresponding to ${\varvec{0}} \in B^{n}$.

Case (${\text {Im}}{P_{\alpha }} \subset X^{(n-1)}$): $P_{\alpha }^{-1}(U) = \emptyset $, and $P_{\alpha }^{-1}(V) = A_{\alpha }$.
Case (${\text {Im}}{P_{\alpha }} \subset B^{n}_{j}$): $P_{\alpha }^{-1}(U) = P_{\alpha }^{-1}({\text {Int}}{B^{n}_{j}})$, and $P_{\alpha }^{-1}(V) = P_{\alpha }^{-1}(B^{n}_{j} \smallsetminus \{{\varvec{0}}_{j}\})$.

In each case, $P_{\alpha }^{-1}(U)$ and $P_{\alpha }^{-1}(V)$ are open in $A_{\alpha }$ and hence in A, which implies that $P^{-1}(U)$ and $P^{-1}(V)$ are open in A for any plot P. Thus U and V are open sets in $X^{(n)}$.

Similarly to the case when X is a topological CW complex, $\mathcal {U} = \{U,V\}$ is a nice open covering of $X^{(n)}$ with a normal partition of unity $\{\rho ^{U},\rho ^{V}\}$, since $\lambda $ is a smooth function. Then, similar arguments for a topological CW complex lead us to the following result.

Theorem A.2

For a smooth CW complex X, there are natural isomorphisms

$$ \begin{array}{l} H_{\mathcal {D}}^{q}(X) \cong H_{\mathcal {C}}^{q}(X) \cong H_{\underline{{\mathsf {\Box }}}}^{q}(X) \cong H^{q}(X,{\mathbb {R}}) \cong {\text {Hom}}(H_{q}(X),{\mathbb {R}}), \end{array} $$

for any $q \ge 0$, and hence we have $H_{\mathcal {D}}^{1}(X) \cong H_{\mathcal {C}}^{1}(X) \cong H_{\underline{{\mathsf {\Box }}}}^{1}(X) \overset{\rho }{\cong }{\text {Hom}}(\pi _{1}(X),{\mathbb {R}})$.

Conjecture A.3

For a smooth CW complex X, there are natural isomorphisms

$$\begin{aligned} H_{\mathcal {D}_{c}}^{q}(X) \cong H_{\mathcal {C}_{c}}^{q}(X) \cong H_{\underline{{\mathsf {\Box }}}_{c}}^{q}(X), \quad \text {for any}\, \ q \ge 0. \end{aligned}$$

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Authors and Affiliations

Norio Iwase
- 1
Email author
Nobuyuki Izumida
- 2

1.Faculty of MathematicsKyushu UniversityFukuokaJapan
2.Puropera CorporationShibuya, TokyoJapan

Mayer–Vietoris Sequence for Differentiable/Diffeological Spaces

Abstract

Keywords

1991 Mathematics Subject Classification

Notes

Acknowledgements

References

Copyright information

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