Morse theory Vs degree theory I asked this question on http://math.stackexchange.com but no unswers!
I have this paragraph from K.C. Chang Infinite dimensional Morse theory

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 ?
Please help me 
Thank you.
 A: To see why  Morse theory  gives more info in the variational case consider system of equations on a torus $T^n$: 
$$f_1(\theta_1,\dotsc, \theta_n)=\cdots=f_n(\theta_1,\dotsc, \theta_n)=0.$$
where $f_,\dotsc, f_n$ are smooth functions. Degree theory    cannot say much about this system. However if  
$$ (f_1,\dotsc, f_n)= \nabla g (\theta_1,\dotsc, \theta_n) $$
for some smooth function  $g$,  then Morse theory predicts that   system of equations
$$ \nabla g=0 $$
has at least  $n+1$ solutions.  If we a priori know that all the solutions are nondegenerate, then  we can conclude that  that are at least $2^n$ solutions.
Look at the simplest  case when $f(\theta)$ is a smooth $2\pi$-periodic function.  The equation $f(\theta)=0$ may have no solutions, but the equation $f'(\theta)=0$ has at least two solutions.  
The variational nature of the equation $f'(\theta)=0$  adds  some subtle   features which Morse  theory speculates to  produce more refined results.
A: Here is an example to illustrate the difference between the two approaches to find critical points. 
Consider the two-dimensional sphere $S^2$ and a function $f \in C^2(S^2 \to \mathbb{R})$ and assume that $f$ has two nondegenerate minimum points. The question is whether you can find other critical points.
You can apply degree theory to the map $F = \nabla f \in C^1 (S^2, TS^2)$. The nondegenerate critical points give you two nondegenerates zeroes of the vector field $F$, and that is consistent with the degree theory for maps in $C^1 (S^2, TS^2)$ for which the sum of the degrees should be $2$.
On the other hand, if you apply Morse theory, the Morse inequalities give you that $f$ should have two additional critical points corresponding to a maximum and a saddle point.
An explanationis that, in the two-dimensional case, degree theory does not distinguish between a maximum and a minimum point, whereas Morse theory does. In general, in dimension higher than $1$, the degree of a zero of the gradient around a critical point is a weaker information then the Morse index of the function at a critical point.
