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I have this paragraph from K.C. Chang Infinite dimensional Morse theory

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In comparison with degree theory, which has proved very useful in nonlinear analysis in proving existence and in estimating the number of solutions to an operator equation, Morse theory has a great advantage if the equation is variational. Relative homology groups and critical groups are series of groups that provide both a finer structure and better estimate of the number of solutions than does the degree, which is only an integer. The relationship between the Leray-Schauder index and critical groups is established.

And I don't Understand how to see that Morse theory is better then degree theory,

how to see that Relative homology groups and critical groups provide both a finer structure and better estimate of the number of solutions than does the degree ?

Thank you.

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For instance, the mountain pass lemma, says that if you have a connected "manifold" with a Morse function with two local minima then there is a path between them passing through an index one critical point. I placed manifold in parentheses because it might be infinite dimensional with appropriate conditions on the Morse function. This has been used to prove the existence of a closed geodesic in any metric on a sphere. It has also been used to construct unstable minimal surfaces, that allowed the resolution of problems that went beyond minimal surface theory.

A little looser interpretation of Morse theory is Conley index theory. Given an flow on a manifold that has isolated invariant sets you can define the index of an invariant set to be the quotient of a closed isolating neighborhood of the invariant set, by the "out ramp", the points where the flow goes outwards. From this you get a spectral sequence that converges to the homology of the manifold. This has been used to great effect to restrict the dynamics of a flow on such a manifold. For instance it was used to prove Arnold's conjecture about fixed points of Hamiltonian flows on tori.

The idea was furthered by Floer to construct a "Morse Homology Theory" describing the paths joining two Lagrangian subspaces in a symplectic manifold. Floer homology is arguably the most important tool in low dimensional topology developed in the last 25 years, and it has been used to resolve a myriad of problems that could not be solved otherwise.

Not to throw shade on degree theory, which has been important in its own light, but Morse theory gives rise to a "categorification" of degree and Euler characteristic, so it is just a more sophisticated invariant. Sometimes all you need is degree though.

What is categorification? For instance, degree gives you an integer. For example if $S$ is a compact surface in $\mathbb{R}^3$ then it bounds a solid, and the outward pointing unit normal $g:S\rightarrow S^2$ is a map to the sphere, and it's degree is one half the Euler characteristic of the surface.

In categorification you replace an integer by a vector space, or better yet a graded vector space, so that it's rank, which would be the alternating sum of the dimensions of the vector spaces recovers the integer. The alternating sum of the ranks of the homology groups of the surface $S$ is it's Euler characteristic.

So $$rnkH_0(S)-rnkH_1(S)+rnkH_2(S)$$ recovers the data from the degree of the Gauss map. Now however if you have a continuous map $f:S_1\rightarrow S_2$ from one surface to another it induces linear maps on the homology groups, which then lets you relate information about critical points on one object to the other.

In general you want the sum of integers to go over to direct sum of vector spaces and you want multiplication to go over to tensor product.

Categorification is one of the biggest movements in modern geometry and it has vastly changed the way we think about the physical and geometric worlds. The process started in the $19$th century with the introduction of the idea of homology, and went through the construction of the modern idea of algebraic variety to the modern idea of what a quantum field theory is.

Here is a sophisticated modern statement of the problem of categorification. Given a ring $R$ you want to know if there is a category of bimodules over an algebra $A$ so that the Grothendieck group (which is a ring) of that category is isomorphic to $R$.

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I know very little Morse theory but a little degree theory. The basic concept is that if a map has a non-trivial degree then there exist some singular points of it whose indices must sum to the degree (The index of an isolated singular point being the 'local degree', or the degree of the map constrained to the neighbourhood of the singular point). However, this alone tells us nothing about the number or nature of the critical points - if a map has index one we can have one singular point with index one or thirty singular points for which with index $-1$ and a singular point with index $31$. Of course, there are clever ways of gleaning more information from other considerations, but this hopefully gives some insight as to why raw degree theory can be a bit of a blunt instrument.

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