## Differential Forms on Singular Spaces

Differential forms are an extremely useful concept for smooth manifolds. They are the input for integration, allowing one to find volumes, centres of mass, among other measurements; they are the foundation for global invariants such as de Rham cohomology; and they can encode geometric information such as symplectic forms and connections. But what happens in the case of singular spaces? How do we even begin to define them?

First, we must carefully consider the definition of differential forms for manifolds. Interestingly, there are many different yet equivalent definitions. Here are four of them. Fix a smooth manifold $M.$

- A differential $k$ -form $\alpha $ is an antisymmetric fibre-wise $k$ -linear map $\alpha :{\displaystyle \prod ^{k}}TM\to \mathbb{R},$ taking $k$ tangent vectors and producing a real number.
- A differential $k$ -form $\alpha $ is an antisymmetric $k$ - ${C}^{\infty}\left(M\right)$ -linear map $\alpha :\mathrm{vect}\left(M\right)\to {C}^{\infty}\left(M\right),$ taking $k$ vector fields and producing a smooth function.
- A differential $k$ -form $\alpha $ is a smooth section of the tensor bundle $\stackrel{k}{\bigwedge}{T}^{*}M$ , the $k$ th wedge product of the cotangent bundle, taking a point $x$ of $M$ and producing an antisymmetric covariant $k$ -tensor ${\alpha}_{x}:{\displaystyle \prod ^{k}}{T}_{x}M\to \mathbb{R}.$
- A differential
$k$
-form
$\alpha $
is an antisymmetric covariant
$k$
-tensor field that satisfies the following transformation law in local coordinates: if locally
$\alpha $
has the form

$\sum _{{i}_{1}<\dots <{i}_{k}}}{\alpha}_{{i}_{1}{\cdots}_{}{i}_{\mathrm{k}}}d{x}^{{i}_{1}}\wedge \dots \wedge d{x}^{{i}_{k}},$

then given a coordinate transformation (a diffeomorphism) $\phi $ , the pullback ${\phi}^{*}\alpha $ (the expression of $\alpha $ in the new coordinate system) takes the form

$\sum _{{i}_{1}<\dots <{i}_{k}}}{\displaystyle \sum _{{j}_{1}<\dots <{j}_{k}}}({\alpha}_{{i}_{1}{\cdots}_{}{i}_{\mathrm{k}}}\circ \phi )\genfrac{}{}{0.1ex}{}{\partial ({\phi}_{{i}_{1}},\dots ,{\phi}_{{i}_{k}})}{\partial ({y}^{{j}_{1}},\dots ,{y}^{{j}_{k}})}d{y}^{{j}_{1}}\wedge \dots \wedge d{y}^{{j}_{k}},$

where $\genfrac{}{}{0.1ex}{}{\partial ({\phi}_{{i}_{1}},\dots ,{\phi}_{{i}_{k}})}{\partial ({y}^{{j}_{1}},\dots ,{y}^{{j}_{k}})}$ is the Jacobian of $\phi $ with respect to the coordinates ${y}^{i}$ .

In general, one should not expect any of these definitions to produce the same set of differential forms when dealing with singular spaces, and indeed, depending on various definitions of smooth structure, tangent space, and vector field, they all produce different things that act in different ways. The question is, what exactly does one want when dealing with differential forms? The answer to this question helps determine the definition that one might want to use.

I studied and compared some of these definitions in my MSc thesis [W:MSc] for arbitrary subsets of ${\mathbb{R}}^{n}$ with an induced smooth structure. In [W:PhD], [KW], and [W:gpds], it is shown that diffeological differential forms on the orbit space of a compact Lie group action, a proper Lie group action, and a proper Lie groupoid, respectively, match basic forms on the base space. In fact, [KW] generalises this to a Lie group whose identity component acts properly.

- [KW] Yael Karshon and Jordan Watts, Basic forms and orbit spaces: a diffeological approach, SIGMA Symmetry Integrability Geom. Methods Appl., Vol. 12 (2016), 026, 19 pages.
- [W:gpds] Jordan Watts, The orbit space and basic forms of a proper Lie groupoid, In: Current Trends in Analysis, its Applications and Computation: Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019; Trends in Mathematics: Research Perspectives; Birkhäuser, 2022.
- [W:PhD] Jordan Watts, Diffeologies, Differential Spaces, and Symplectic Geometry, PhD Thesis, Department of Mathematics, University of Toronto, 108 pages, 2012.
- [W:MSc] Jordan Watts, The Calculus on Subcartesian Spaces, MSc Thesis, Department of Mathematics & Statistics, University of Calgary, 116 pages, 2006.