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Extending Petya's answer, if you want something like indices, handle attachments and things of that flavour the answer is no. You can prove there's no such thing in that level of generality. But that's a pretty "close read" of Morse theory that does not translate to other contexts. If you take a "broad" reading of Morse theory: classify the local behavior of "generic" maps between manifolds, and study the properties that are conserved via generic homotopies, etc, then what you have is singularity theory, the work of people like Thom and Mather applies. This kind of thing gets applied in many ways -- a nice recent example that I like would be the paper of D.Thurston and Costantino where they use the properties of generic maps from 3-manifolds into $\mathbb R^2$ to construct efficiently-triangulated 4-manifolds bounding triangulated 3-manifolds. But that's only one of many examples.

A question in return: how would you expect a Morse theory of sections of bundles? You don't have "level sets" so perhaps you're interested in properties of generic maps? Or some relations between local properties of the section and a global property of the bundle -- things like Euler characteristics?

edit: I read the "letters to nature" blog you linked to. The author says his motivation is to "classify functions"? It would be helpful if the author could specify what equivalence relation on functions he's considering. Presumably an object like the $C^1$ norm is too restrictive?
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Extending Petya's answer, if you want something like indices, handle attachments and things of that flavour the answer is no. You can prove there's no such thing in that level of generality. But that's a pretty "close read" of Morse theory that does not translate to other contexts. If you take a "broad" reading of Morse theory: classify the local behavior of "generic" maps between manifolds, and study the properties that are conserved via generic homotopies, etc, then what you have is singularity theory, the work of people like Thom and Mather applies. This kind of thing gets applied in many ways -- a nice recent example that I like would be the paper of D.Thurston and Costantino where they use the properties of generic maps from 3-manifolds into $\mathbb R^2$ to construct efficiently-triangulated 4-manifolds bounding triangulated 3-manifolds. But that's only one of many examples.

A question in return: how would you expect a Morse theory of sections of bundles? You don't have "level sets" so perhaps you're interested in properties of generic maps? Or some relations between local properties of the section and a global property of the bundle -- things like Euler characteristics?

edit: I read the "letters to nature" blog you linked to. The author says his motivation is to "classify functions"? It would be helpful if the author could specify what equivalence relation on functions he's considering.

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Extending Petya's answer, if you want something like indices, handle attachments and things of that flavour the answer is no. You can prove there's no such thing in that level of generality. But that's a pretty "close read" of Morse theory that does not translate to other contexts. If you take a "broad" reading of Morse theory: classify the local behavior of "generic" maps between manifolds, and study the properties that are conserved via generic homotopies, etc, then what you have is singularity theory, the work of people like Thom and Mather applies. This kind of thing gets applied in many ways -- a nice recent example that I like would be the paper of D.Thurston and Costantino where they use the properties of generic maps from 3-manifolds into $\mathbb R^2$ to construct efficiently-triangulated 4-manifolds bounding triangulated 3-manifolds. But that's only one of many examples.

A question in return: how would you expect a Morse theory of sections of bundles? You don't have "level sets" so perhaps you're interested in properties of generic maps? Or some relations between local properties of the section and a global property of the bundle -- things like Euler characteristics?