A Survey on Frölicher Spaces

What is it about?

This survey paper highlights a series of results in recent research on topology, geometry and categorical properties of spaces provided with a new structure in the mathematical literature, called Frölicher spaces. Without any fear of contradiction, these smooth spaces are stated to generalize the theory of smooth manifold, in the Riemannian and symplectic settings. More precisely, the study is giving the state of research on this topic which historically links a fundamental theorem of calculus on real Euclidean spaces so-called Boman's theorem (and its generalization) to abstract spaces, whether they are normable or not.

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Why is it important?

Constructing differentiable atlases on a set through normed modeling spaces in order to provide the given set with a manifold structure has been a tedious task over centuries, as it depends on the topological nature of the underlying set. The Frölicher structure is a model of smooth structure rather constructed directly on the set, the topologies being induced by both curves and functions of the structure. This structure is then canonically lifted from the smooth spaces to the sets of smooth maps between these spaces in such a way that functions sets are made smooth (without norms), a property that is called the Cartesian closedness. In this way, differential geometry extends easily to the infinite dimensional setting and is well suited to singular spaces . This study gathered all that is found to date on this research.