[Paper Review] A comparison between two de Rham complexes in diffeology

Katsuhiko Kuribayashi|arXiv (Cornell University)|Feb 17, 2020

Homotopy and Cohomology in Algebraic Topology12 references4 citations

TL;DR

This paper compares two de Rham complexes in diffeology—the original Souriau complex and the singular de Rham complex—using the Čech–de Rham spectral sequence and the factor map between them. It shows that the first singular de Rham cohomology of the irrational torus $T_{\theta}$ is isomorphic to the direct sum of the original de Rham cohomology and the group of equivalence classes of flow bundles with connection 1-forms, and that the full singular de Rham cohomology algebra is isomorphic to the tensor product of the original cohomology and the exterior algebra generated by such a flow bundle.

ABSTRACT

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech--de Rham spectral sequence by making use of the {\it factor map} which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus $T_θ$ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over $T_θ$.

Motivation & Objective

• To compare the first de Rham cohomology groups of the original Souriau de Rham complex and the singular de Rham complex in diffeology.
• To understand the geometric and cohomological difference between these two complexes using the factor map and the Čech–de Rham spectral sequence.
• To investigate the structure of the singular de Rham cohomology algebra of the irrational torus $T_{\theta}$, particularly its relation to flow bundles.
• To clarify why the singular de Rham cohomology of $T_{\theta}$ differs from that of the standard torus $\mathbb{T}^2$, despite similar homotopy types.

Proposed method

• The paper uses the Čech–de Rham spectral sequence to compare the cohomology of the two de Rham complexes.
• It employs the factor map $\alpha: \Omega(X) \to A(X)$, a morphism of differential graded algebras, to relate the original and singular de Rham complexes.
• The analysis is conducted via the spectral sequence's edge map and the induced maps on cohomology, particularly in degree 2.
• The proof relies on the injectivity of the edge map under the assumption $H^1(A(F)) = 0$, where $F$ is the fibre of a bundle.
• A generating family of plots and a monoid action on the nerve of the diffeological space are used to construct the spectral sequence.
• The construction is generalized using a multi-set of plots and a morphism of monoids $\pi'$ to relate the spectral sequences of the total space and the base space.

Experimental results

Research questions

• RQ1 How do the first de Rham cohomology groups of the original and singular de Rham complexes compare in diffeology?
• RQ2 What is the geometric origin of the difference between the cohomology of the two complexes, particularly in non-manifold or non-simply connected spaces?
• RQ3 Why does the singular de Rham cohomology of the irrational torus $T_{\theta}$ contain a non-trivial flow bundle component not present in the standard torus $\mathbb{T}^2$?
• RQ4 To what extent does the factor map $\alpha$ fail to be a quasi-isomorphism in general, and what does this imply for the cohomological structure?
• RQ5 How does the Čech–de Rham spectral sequence detect the presence of non-trivial flow bundles in the cohomology of $T_{\theta}$?

Key findings

• The first singular de Rham cohomology group $H^1(A(T_{\theta}))$ is isomorphic to the direct sum of the original de Rham cohomology $H^1(\Omega(T_{\theta})) \cong \mathbb{R}$ and the group of equivalence classes of flow bundles over $T_{\theta}$ with connection 1-forms.
• The full singular de Rham cohomology algebra $H^*(A(T_{\theta}))$ is isomorphic to the tensor product of the original de Rham cohomology algebra and the exterior algebra generated by a non-trivial flow bundle over $T_{\theta}$.
• The edge map $H^2(A(X)) \to \check{H}^2(X)$ is injective when $H^1(A(F)) = 0$, which holds for the irrational torus bundle with fibre $\mathbb{R}$.
• The spectral sequence analysis reveals that the non-triviality of $d_2$ in the $\Omega$-complex on the base space $M$ leads to a contradiction unless the cohomology group $H^1(\Omega(M))$ is non-zero, which is not the case for $T_{\theta}$, thus confirming the existence of new cohomology classes.
• The singular de Rham cohomology of $T_{\theta}$ is isomorphic to that of $\mathbb{T}^2$ as an algebra, but the presence of a non-trivial flow bundle in $H^*(A(T_{\theta}))$ distinguishes it from $H^*(A(\mathbb{T}^2))$, where such bundles are trivial due to the contractibility of the fibre.
• The factor map $\alpha$ induces a monomorphism $H^1(\Omega(X)) \to H^1(A(X))$ for any diffeological space $X$, and this monomorphism is strict for $T_{\theta}$, indicating that the singular complex detects additional cohomological structure not visible in the original complex.