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I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$. Let $\alpha'$ be a closed $k$-form on $X'$, then $f_1^*(\alpha')$ is cohomologous to $f_0^*(\alpha')$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\Omega^p$ denotes the space of differential $p$-forms, $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x \in X$ let:

$$ \phi(x) = [s \mapsto f_s(x)], \ \text{then} \ \phi \in C^\infty\big( X,\mathrm{Paths}(X')\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha'$ $$ \phi^*[K d\alpha' + d K\alpha'] = \phi^*(\hat 1^*(\alpha')) - \phi^*(\hat 0^*(\alpha')). $$ Since $d\alpha' = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha') = f_0^*(\alpha') + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha'). $$ Therefore, $f_1^*(\mathrm{class}(\alpha')) = f_0^*(\mathrm{class}(\alpha'))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.

Addendum: We maintain a Monthly Global Seminar in Diffeology, all informations on http://diffeology.net

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X'$ to $X$. Let $\alpha$ be a closed $k$-form on $X$, then $f_1^*(\alpha)$ is cohomologous to $f_0^*(\alpha)$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\Omega^p$ denotes the space of differential $p$-forms, $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x' \in X'$ let:

$$ \phi(x') = [s \mapsto f_s(x')], \ \text{then} \ \phi \in C^\infty\big( X',\mathrm{Paths}(X)\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha$ $$ \phi^*[K d\alpha + d K\alpha] = \phi^*(\hat 1^*(\alpha)) - \phi^*(\hat 0^*(\alpha)). $$ Since $d\alpha = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha) = f_0^*(\alpha) + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha). $$ Therefore, $f_1^*(\mathrm{class}(\alpha)) = f_0^*(\mathrm{class}(\alpha))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.

Addendum: We maintain a Monthly Global Seminar in Diffeology, all informations on http://diffeology.net

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$. Let $\alpha'$ be a closed $k$-form on $X'$, then $f_1^*(\alpha')$ is cohomologous to $f_0^*(\alpha')$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\Omega^p$ denotes the space of differential $p$-forms, $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x \in X$ let:

$$ \phi(x) = [s \mapsto f_s(x)], \ \text{then} \ \phi \in C^\infty\big( X,\mathrm{Paths}(X')\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha'$ $$ \phi^*[K d\alpha' + d K\alpha'] = \phi^*(\hat 1^*(\alpha')) - \phi^*(\hat 0^*(\alpha')). $$ Since $d\alpha' = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha') = f_0^*(\alpha') + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha'). $$ Therefore, $f_1^*(\mathrm{class}(\alpha')) = f_0^*(\mathrm{class}(\alpha'))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$. Let $\alpha'$ be a closed $k$-form on $X'$, then $f_1^*(\alpha')$ is cohomologous to $f_0^*(\alpha')$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\Omega^p$ denotes the space of differential $p$-forms, $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x \in X$ let:

$$ \phi(x) = [s \mapsto f_s(x)], \ \text{then} \ \phi \in C^\infty\big( X,\mathrm{Paths}(X')\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha'$ $$ \phi^*[K d\alpha' + d K\alpha'] = \phi^*(\hat 1^*(\alpha')) - \phi^*(\hat 0^*(\alpha')). $$ Since $d\alpha' = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha') = f_0^*(\alpha') + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha'). $$ Therefore, $f_1^*(\mathrm{class}(\alpha')) = f_0^*(\mathrm{class}(\alpha'))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.

Addendum: We maintain a Monthly Global Seminar in Diffeology, all informations on http://diffeology.net

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$. Let $\alpha'$ be a closed $k$-form on $X'$, then $f_1^*(\alpha')$ is cohomologous to $f_0^*(\alpha')$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x \in X$ let:

$$ \phi(x) = [s \mapsto f_s(x)], \ \text{then} \ \phi \in C^\infty\big( X,\mathrm{Paths}(X')\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha'$ $$ \phi^*[K d\alpha' + d K\alpha'] = \phi^*(\hat 1^*(\alpha')) - \phi^*(\hat 0^*(\alpha')). $$ Since $d\alpha' = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha') = f_0^*(\alpha') + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha'). $$ Therefore, $f_1^*(\mathrm{class}(\alpha')) = f_0^*(\mathrm{class}(\alpha'))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.

I would add to the previous list of "applications" of diffeology in ordinary differential geometry the theorem about Homotopic invariance of De Rham cohomology (§6.88 of Diffeology). In brief:

Theorem: Let $X$ and $X'$ be two diffeological spaces. Let $f_0$ and $f_1$ be two homotopic smooth maps from $X$ to $X'$. Let $\alpha'$ be a closed $k$-form on $X'$, then $f_1^*(\alpha')$ is cohomologous to $f_0^*(\alpha')$.

The diffeological proof uses the Chain-Homotopy operator (§6.83 op.cit)

$$ K : \Omega^p(X) \to \Omega^{p-1} (\mathrm{Paths}(X)), \ \text{that satisfies} \ K \circ d + d \circ K = \hat 1^* - \hat 0^*, $$

where $\Omega^p$ denotes the space of differential $p$-forms, $\mathrm{Paths}(X) = C^\infty({\mathbf R}, X)$ is the set of smooth paths in $X$, and for all $\gamma \in \mathrm{Paths}(X)$, $\hat t(\gamma) = \gamma(t)$.

Proof: For all $x \in X$ let:

$$ \phi(x) = [s \mapsto f_s(x)], \ \text{then} \ \phi \in C^\infty\big( X,\mathrm{Paths}(X')\big). $$

Apply $\phi^*$ to the identity above satisfied by $K$, evaluated on $\alpha'$ $$ \phi^*[K d\alpha' + d K\alpha'] = \phi^*(\hat 1^*(\alpha')) - \phi^*(\hat 0^*(\alpha')). $$ Since $d\alpha' = 0$ and $\hat t \circ \phi = f_t$ we get $$ f_1^*(\alpha') = f_0^*(\alpha') + d\beta, \ \text{with} \ \beta = \phi^*(K\alpha'). $$ Therefore, $f_1^*(\mathrm{class}(\alpha')) = f_0^*(\mathrm{class}(\alpha'))$. $\square$

This general theorem applies in particular for $X$ and $X'$ manifolds since the category of manifolds is a full subcategory of diffeological spaces. This is an example where diffeology helps in a proof of a classical differential geometry theorem, using a shortcut thru a space which is not a manifold: the space $\mathrm{Paths}(X)$. And as we see it was important to have a good definition of differential forms on every diffeological space, whatever they are.

By the way the Chain-Homotopy operator is used also in Symplectic Diffeology for the definition and the generalisation of The Moment Map in Diffeology. You will find in that Memoir a few of examples of application of diffeology in symplectic geometry.